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[ "Fast and Flexible Temporal Point Processes with Triangular Maps", "Fast and Flexible Temporal Point Processes with Triangular Maps" ]
[ "Oleksandr Shchur [email protected] \nTechnical University of Munich\nGermany\n", "Nicholas Gao \nTechnical University of Munich\nGermany\n", "Marin Biloš [email protected] \nTechnical University of Munich\nGermany\n", "Stephan Günnemann [email protected] \nTechnical University of Munich\nGermany\n" ]
[ "Technical University of Munich\nGermany", "Technical University of Munich\nGermany", "Technical University of Munich\nGermany", "Technical University of Munich\nGermany" ]
[]
Temporal point process (TPP) models combined with recurrent neural networks provide a powerful framework for modeling continuous-time event data. While such models are flexible, they are inherently sequential and therefore cannot benefit from the parallelism of modern hardware. By exploiting the recent developments in the field of normalizing flows, we design TriTPP-a new class of non-recurrent TPP models, where both sampling and likelihood computation can be done in parallel. TriTPP matches the flexibility of RNN-based methods but permits orders of magnitude faster sampling. This enables us to use the new model for variational inference in continuous-time discrete-state systems. We demonstrate the advantages of the proposed framework on synthetic and real-world datasets.
null
[ "https://arxiv.org/pdf/2006.12631v1.pdf" ]
219,980,565
2006.12631
95c2d27517296527b8769880a60d3fe99e670e54
Fast and Flexible Temporal Point Processes with Triangular Maps Oleksandr Shchur [email protected] Technical University of Munich Germany Nicholas Gao Technical University of Munich Germany Marin Biloš [email protected] Technical University of Munich Germany Stephan Günnemann [email protected] Technical University of Munich Germany Fast and Flexible Temporal Point Processes with Triangular Maps Temporal point process (TPP) models combined with recurrent neural networks provide a powerful framework for modeling continuous-time event data. While such models are flexible, they are inherently sequential and therefore cannot benefit from the parallelism of modern hardware. By exploiting the recent developments in the field of normalizing flows, we design TriTPP-a new class of non-recurrent TPP models, where both sampling and likelihood computation can be done in parallel. TriTPP matches the flexibility of RNN-based methods but permits orders of magnitude faster sampling. This enables us to use the new model for variational inference in continuous-time discrete-state systems. We demonstrate the advantages of the proposed framework on synthetic and real-world datasets. Introduction Temporal data lies at the heart of many high-impact machine learning applications. Electronic health records, financial transaction ledgers and server logs contain valuable information. A common challenge encountered in all these settings is that both the number of events and their times are variable. The framework of temporal point processes (TPP) allows us to naturally handle data that consists of variable-number events in continuous time. Du et al. [1] have shown that the flexibility of TPPs can be improved by combining them with recurrent neural networks (RNN). While such models are expressive and can achieve good results in various prediction tasks, they are poorly suited for sampling: sequential dependencies preclude parallelization. We show that it's possible to overcome the above limitation and design flexible TPP models without relying on RNNs. For this, we use the framework of triangular maps [2] and recent developments in the field of normalizing flows [3]. Our main contributions are: (1) We propose a new parametrization for several classic TPPs. This enables efficient parallel likelihood computation and sampling, which was impossible with existing parametrizations. (2) We propose TriTPP-a new class of non-recurrent TPPs. TriTPP matches the flexibility of RNN-based methods, while allowing orders of magnitude faster sampling. (3) We derive a differentiable relaxation for non-differentiable sampling-based TPP losses. This allows us to design a new variational inference scheme for Markov jump processes. Background Temporal point processes (TPP) [4] are stochastic processes that model the distribution of discrete events on some continuous time interval [0, T ]. A realization of a TPP is a variable-length sequence of strictly increasing arrival times t = (t 1 , . . . , t N ), t i ∈ [0, T ]. We make the standard assumption and focus our discussion on regular finite TPPs [4]. One way to specify such a TPP is by using the (strictly positive) conditional intensity function λ * (t) := λ(t|H t ) that defines the rate of arrival of new events given the history H t = {t j : t j < t}. The * symbol reminds us of the dependence on the history [5]. Preprint Equivalently, we can consider the cumulative conditional intensity Λ * (t) := Λ(t|H t ) = t 0 λ * (u)du, also known as the compensator. 1 We can compute the likelihood of a realization t on [0, T ] as p(t) = N i=1 λ * (t i ) exp − T 0 λ * (u)du = N i=1 ∂ ∂t i Λ * (t i ) exp (−Λ * (T ))(1) Triangular maps [2] provide a framework that connects autoregressive models, normalizing flows and density estimation. Bogachev et al. [6] have shown that any density p(x) on R N can be equivalently represented by another densityp(z) on R N and an increasing differentiable triangular map F = (f 1 , . . . , f N ) : R N → R N that pushes forward p intop. 2 A map F is called triangular if each component function f i depends only on (x 1 , . . . , x i ) and is an increasing function of x i . Intuitively, we can think of F as converting a random variable x ∼ p into a random variable z := F (x) with a densityp. We can compute the density p(x) using the change of variables formula p(x) = |det J F (x)|p (F (x)) = N i=1 ∂ ∂x i f i (x 1 , . . . , x i ) p (F (x))(2) where det J F (x) is the Jacobian determinant of F at x. Here, we used the fact that J F (x) is a positive-definite lower-triangular matrix. To specify a complex density p(x), we can pick some simple densityp(z) and learn the triangular map F that pushes p intop. It's important that F and its Jacobian determinant can be evaluated efficiently if we are learning p(x) via maximum likelihood. We can sample from p(x) by applying the inverse map F −1 to the samples drawn fromp(z). Note that F −1 : R N → R N is also an increasing differentiable triangular map. Fast computation of F −1 is important when learning p(x) via sampling-based losses (e.g., in variational inference). Defining temporal point processes using triangular maps We can notice the similarity between the right-hand sides of Equations 1 and 2, which seems to suggest some connection between TPPs and triangular maps. Indeed, it turns out that triangular maps can also be used to specify densities of point processes. Let t = (t 1 , . . . , t N ) be a realization of a TPP on [0, T ] with compensator Λ * (i.e. with density p(t)). The random time change theorem states that in this case z = (Λ * (t 1 ), . . ∂ ∂ti Λ * (t i ) in Equation 1 corresponds to the Jacobian determinant of F . Similarly, the second term,p(z) =p(F (t)) = exp(−Λ * (T )), corresponds to the density of a HPP with unit rate on [0, Λ * (T )] for any realization z. This demonstrates the one-to-one correspondence between TPP densities (Equation 1) and increasing triangular maps (Equation 2). Our main idea is to define TPP densities p(t) by directly specifying the respective maps F . In Section 3.1, we show how maps that satisfy certain properties allow us to efficiently compute density and generate samples. We demonstrate this by designing a new parametrization for several established models in Section 3.2. Finally, we propose a new class of fast and flexible TPPs in Section 3.3. Requirements for efficient TPP models Density evaluation. The time complexity of computing the density p(t) for various TPP models can be understood by analyzing the respective map F . For a general triangular map F : R N → R N , computing F (t) takes O(N 2 ) operations. For example, this holds for Hawkes processes with arbitrary kernels [7]. If the compensator Λ * has Markov property, the complexity of evaluating F can be reduced to O(N ) sequential operations. This class of models includes Hawkes processes with exponential kernels [8,9] and RNN-based autoregressive TPPs [1,10,11]. Unfortunately, such models do not benefit from the parallelism of modern hardware. Defining an efficient TPP model will require specifying a forward map F that can be computed in O(N ) parallel operations. F (t) = (Λ * (t 1 ), ..., Λ * (t N )) is used for computing p(t). t1 t2 T t3 z1 z2 z3 Discard ti > T Λ * Figure 2: Sampling is done by applying F −1 to a sample z from a HPP with unit rate. Sampling. As a converse of the random time change theorem, we can sample from a TPP density p(t) by first drawing z from an HPP on [0, Λ * (T )] and applying the inverse map, t = F −1 (z) [4]. There are, however, several caveats to this method. Not all parametrizations of F allow computing F −1 (z) in closed form. Even if F −1 is available, its evaluation for most models is again sequential [1,9]. Lastly, the number of points N that will be generated (and thus Λ * (T ) for HPP) is not known in advance. Therefore, existing methods typically resort to generating the samples one by one [5,Algorithm 4.1]. We show that it's possible to do better than this. If the inverse map F −1 can be applied in parallel, we can produce large batches of samples t i , and then discard the points t i > T (Figure 2). Even though this method may produce samples that are later discarded, it is much more efficient than sequential generation on GPUs (Section 6.1). To summarize, defining a TPP efficient for both density computation and sampling requires specifying a triangular map F , such that both F and its inverse F −1 can be evaluated analytically in O(N ) parallel operations. We will now show that maps corresponding to several classic TPP models can be defined to satisfy these criteria. Fast temporal point process models Inhomogeneous Poisson process (IPP) [4] is a TPP whose conditional intensity doesn't depend on the history, Λ(t|H t ) = Λ(t). The corresponding map is F = Λ, where Λ simply applies the function Λ : [0, T ] → R + elementwise to the sequence (t 1 , ..., t N ). Renewal process (RP) [12] is a TPP where each inter-event time t i − t i−1 is sampled i.i.d. from the same distribution with the cumulative hazard function Φ : R + → R + . The compensator of an RP is Λ(t|H t ) = Φ(t − t i ) + i j=1 Φ(t j − t j−1 ), where t i is the last event before t. The triangular map of an RP can be represented as a composition F = C • Φ • D, where D ∈ R N ×N is the pairwise difference matrix, C ≡ D −1 ∈ R N ×N is the cumulative sum matrix, and Φ applies Φ elementwise. Modulated renewal process (MRP) [13] generalizes both inhomogeneous Poisson and renewal processes. The cumulative intensity is Λ(t|H t ) = Φ(Λ(t) − Λ(t i )) + i j=1 Φ(Λ(t j ) − Λ(t j−1 )). Again, we can represent the triangular map of an MRP as a composition, F = C • Φ • D • Λ. All three above models permit fast density evaluation and sampling. Since Φ and Λ (as well as their inverses Φ −1 and Λ −1 ) are elementwise transformations, they can obviously be applied in O(N ) parallel operations. Same holds for multiplication by the matrix D, as it is bidiagonal. Finally, the cumulative sum defined by C can also be computed in parallel in O(N ) [14]. Therefore, by reformulating IPP, RP and MRP using triangular maps, we can satisfy our efficiency requirements. Parametrization for Φ and Λ must satisfy several conditions. First, to define a valid TPP, Φ and Λ have to be positive, strictly increasing and differentiable. Next, both functions, their derivatives (for density computation) and inverses (for sampling) must be computable in closed form to meet the efficiency requirements. Lastly, we want both functions to be highly flexible. Constructing such functions is not trivial. While IPP, RP and MRP are established models, none of their existing parametrizations satisfy all the above conditions simultaneously. Luckily, the same properties are necessary when designing normalizing flows [15]. Recently, Durkan et al. [3] used rational quadratic splines (RQS) to define functions that satisfy our requirements. We propose to use RQS to define Φ and Λ for (M)RP and IPP. This parametrization is flexible, while also allowing efficient density evaluation and sampling -something that existing approaches are unable to provide (see Section 5). Defining more flexible triangular maps Even though the splines can make the functions Φ and Λ arbitrarily flexible, the overall expressiveness of MRP is still limited. Its conditional intensity λ * (t) depends only on the global time and the time since the last event. This means, MRP cannot capture, e.g., self-exciting [7] or self-correcting [16] behavior. We will now construct a model that is more flexible without sacrificing the efficiency. Figure 3: TriTPP defines an expressive map F as a composition of easy-to-invert transformations. C Φ B L B 1 D Λ . . . Jacobian of each transform The efficiency of the MRP stems from the fact that the respective triangular map F is defined as a composition of easy-to-invert transformations. More specifically, we are combining learnable element-wise nonlinear transformations Φ and Λ with fixed lower-triangular matrices D and C. We can make the map F more expressive by adding learnable lower-triangular matrices into the composition. Using full N × N lower triangular matrices would be inefficient (multiplication and inversion are O(N 2 )), and also would not work for variable-length sequences (i.e., arbitrary values of N ). Instead, we define block-diagonal matrices B l , where each block is a repeated H × H lower-triangular matrix with strictly positive diagonal entries. Computing F = C • Φ • B L • · · · • B 1 • D • Λ. The blocks in every other layer are shifted by an offset H/2 to let the model capture long-range dependencies. Figure 3 visualizes the overall sequence of maps and the Jacobians of each transformation. We name the temporal point process densities defined by the triangular map F as TriTPP. Both the forward map F and its inverse F −1 can be evaluated in parallel in linear time, making TriTPP efficient for density computation and sampling. Our insight that TPP densities can be represented by increasing triangular maps was crucial for arriving at this result. Alternative representations of TriTPP, e.g., in terms of the compensator Λ * or the conditional intensity λ * , are cumbersome and do not emphasize the parallelism of the model. TriTPP and our parametrizations of IPP, RP, MRP can be efficiently implemented on GPU to handle batches of variable-length sequences (Appendix C). Differentiable sampling-based losses for temporal point processes Fast parallel sampling allows us to efficiently answer prediction queries such as "How many events are expected to happen in the next hour given the history?". More importantly, it enables us to efficiently train TPP models using objective functions of the form E p [g(t)]. This includes using p(t) to specify the policy in reinforcement learning [17], to impute missing data during training [11] or to define an approximate posterior in variational inference (Section 4.2). In all but trivial cases the expression E p [g(t)] has no closed form, so we need to estimate its gradients w.r.t. the parameters of p(t) using Monte Carlo (MC). Recall that we can sample from p(t) by applying the map F −1 to z drawn from an HPP with unit rate. This corresponds to the so-called reparametrization trick [18]. Unfortunately, this is not enough. Sampling-based losses for TPPs are in general not differentiable. This is a property of the loss functions that is independent of the parametrization of p(t) or the sampling method. In the following, we provide a simple example and a solution to this problem. Entropy maximization Consider the problem of maximizing the entropy of a TPP. An entropy penalty can be used as a regularizer during density estimation [19] or as a part of the ELBO in variational inference. Let p λ (t) be a homogeneous Poisson process on [0, T ] with rate λ > 0. It is known that the entropy is maximized when λ = 1 [20], but for sake of example assume that we want to learn λ that maximizes the entropy −E p [log p λ (t)] with gradient ascent. We sample from p λ (t) by drawing a sequence z = (z 1 , z 2 , ...) from a HPP with unit rate and applying the inverse map t = F −1 λ (z) = 1 λ z ( Figure 2). We obtain an MC estimate of the entropy using a single such sample t = (t 1 , t 2 , ...) as −E p [log p λ (t)] ≈ λT − ∞ i=1 1(t i ≤ T ) log λ = λT − ∞ i=1 1 1 λ z i ≤ T log λ(3) Here, the indicator function 1(·) discards all the events t i > T . We can see that for any sample z the right-hand side of Equation 3 is not continuous w.r.t. λ at points λ = 1 T z i . At such points, decreasing λ by an infinitesimal amount will "push" the sample t i = 1 λ z i outside the [0, T ] interval, thus increasing log p λ (t) by a constant log λ. We plot the right-hand side of Equation 3 as a function of λ in Figure 4, estimated with 5 MC samples. Clearly, such function cannot be optimized with gradient ascent. Increasing the number of MC samples almost surely adds more points of discontinuity and does not fix the problem. In general, non-differentiability arises when estimating expectations of a function g(t) that depends on the events t i inside [0, T ]. For any TPP density p(t), the discontinuities occur at the parameter values that map the HPP realizations z i exactly to the interval boundary T . Relaxation. We obtain a differentiable approximation to Equation 3 by relaxing the indicator functions as 1(t i ≤ T ) ≈ σ γ (T − t i ), where σ γ (x) = 1/(1 + exp(−x/γ)) is the sigmoid function with a temperature parameter γ > 0. Decreasing the temperature γ makes the approximation more accurate, but complicates optimization, similarly to the Gumbel-softmax trick [21]. Figure 5 shows convergence plots for different values of γ. Our relaxation applies to MC estimation of any function g(t) that can be expressed in terms of the indicator functions. This method also enables differentiable sampling with reparametrization from a Poisson distribution, which might be of independent interest. Variational inference for Markov jump processes Combining fast sampling (Section 3) with the differentiable relaxation opens new applications for TPPs. As an example, we design a variational inference scheme for Markov jump processes. Note that N may vary for different trajectories. The prior over the trajectories p(t, s|π, A) of an MJP is governed by an initial state distribution π and a K × K generator matrix A (see Appendix B.1). Background. A Markov jump process (MJP) {s(t)} MJPs are commonly used to model the unobserved (latent) state of a system. In a latent MJP, the state s(t) influences the behavior of the system and indirectly manifests itself via some observations o. For concreteness, we consider the Markov-modulated Poisson process (MMPP) [22]. In an MMPP, each of the K states of the MJP has an associated observation intensity λ k . An MMPP is an inhomogeneous Poisson process where the intensity depends on the current MJP state as λ(t) = λ s(t) . For instance, we can model a user of an online service as switching between "active" and "inactive" states with different rates of activity using a 2-state MMPP ( Figure 6). Given the observations o, we might be interested in inferring the trajectory (t, s), the model parameters θ = {π, A, λ}, or both. Variational inference. The posterior distribution p(t, s|o, θ) of MMPP is intractable, so we approximate it with a variational distribution q(t, s) = q(t)q(s|t). Note that this is not a mean-field approximation used in other works [23]. We model the distribution over the jump times q(t) with TriTPP (Section 3.3). We find the best approximate posterior by maximizing the ELBO [24] max q(t) max q(s|t) E q(t) E q(s|t) [log p(o|t, s, θ) + log p(t, s|θ) − log q(t, s)](4) Given jump times t, the true posterior over the states p(s|t, o, θ) is just the posterior of a discrete hidden Markov model (HMM). This means that we only need to model q(t); the optimal q (s|t), i.e. q (s|t) = arg max q(s|t) E q(s|t) [log p(o|t, s, θ) + log p(t, s|θ) − log q(s|t)] = p(s|t, o, θ) (5) can be found by doing inference in an HMM -doable efficiently via the forward-backward algorithm [25]. The inner expectation w.r.t. q(s|t) in Equation 4 can be computed analytically. We approximate the expectation w.r.t. q(t) with Monte Carlo. Since all terms of Equation 4 are not differentiable, we apply our relaxation from Section 4.1. We provide a full derivation of the ELBO and the implementation details in Appendix B.2. The proposed framework is not limited to approximating the posterior over the trajectories. With small modifications (Appendix B.3), we can simultaneously learn the parameters θ, either obtaining a point estimate θ or a full approximate posterior q(θ). Our variational inference scheme can also be extended to other continuous-time discrete-state models, such as semi-Markov processes [26]. Related work Triangular maps [2] can be seen as a generalization of autoregressive normalizing flows [15,27,28]. Existing normalizing flow models are either limited to fixed-dimensional data [29,30] or are inherently sequential [31,32]. Our model proposed in Section 3.3 can handle variable-length inputs, and allows for both F and F −1 to be evaluated efficiently in parallel. Sampling from TPPs. Inverse method for sampling from inhomogeneous Poisson processes can be dated back to Çinlar [33]. However, traditional inversion methods for IPPs are different from our approach (Section 3). First, they are typically sequential. Second, existing methods either use extremely basic compensators Λ(t), such as λt or e αt , or require numerical inversion [34]. As an alternative to inversion, thinning approaches [35] became the dominant paradigm for generating IPPs, and TPPs in general. Still, sampling via thinning has a number of disadvantages. Thinning requires a piecewise-constant upper bound on λ(t), which might not always be easy to find. If the bound is not tight, a large fraction of samples will be rejected. Moreover, thinning is not differentiable, doesn't permit reparametrization, and is hard to express in terms of parallel operations on tensors [36]. Our inversion-based sampling addresses all the above limitations. It's also possible to generate an IPP by first drawing N ∼ Poisson(Λ(T )) and then sampling N points t i i.i.d. from a density p(t) = λ(t)/Λ(T ) [37]. Unlike inversion, this method is only applicable to Poisson processes. Also, the operation of sampling N is not differentiable, which limits the utility of this method. Inhomogeneous Poisson processes are commonly defined by specifying the intensity function λ(t) via a latent Gaussian process [38]. Such models are flexible, but highly intractable. It's possible to devise approximations by, e.g., bounding the intensity function [39,40]. Our spline parametrization of IPP compares favorably to the above models: it is also highly flexible, has a tractable likelihood and places no restrictions on the intensity. Importantly, it is much easier to implement and train. If uncertainty is of interest, we can perform approximate Bayesian inference on the spline coefficients [24]. Recently, Morgan et al. [41] used splines to model the intensity function of IPPs. Since Λ −1 cannot be computed analytically for their model, sampling via thinning is the only available option. Modulated renewal processes have been known for a long time [13,42], but haven't become as popular as IPPs among practitioners. This is not surprising, since inference and sampling in MRPs are even more challenging than in Cox processes [43,44]. Our proposed parametrization addresses the shortcomings of existing approaches and makes MRPs straightforward to apply in practice. Neural TPPs. Du et al. [1] proposed a TPP model based on a recurrent neural network. Follow-up works improved the flexibility of RNN-based TPPs by e.g. changing the RNN architecture [45], using more expressive conditional hazard functions [10] or modeling the inter-event time distribution with normalizing flows [11]. All the above models are inherently sequential and therefore inefficient for sampling (Section 6.1). Recently, Turkmen et al. [36] proposed to speed up RNN-based marked TPPs by discretizing the interval [0, T ] into a regular grid. Samples within each grid cell can be produced in parallel for each mark, but the cells themselves still must be processed sequentially. Latent space models. TPPs governed by latent Markov dynamics have intractable likelihoods that require approximations [46,47]. For MJPs, the state-of-the-art approach is the Gibbs sampler by Rao & Teh [48]. It allows to exactly sample from the posterior p(t, s|o, θ), but is known to converge slowly if the parameters θ are to be learned as well [49]. Existing variational inference approaches for MJPs can only learn a fixed time discretization [23] or estimate the marginal statistics of the posterior [50,51]. In contrast, our method (Section 4.2) produces a full distribution over the jump times. Table 1: Average test set NLL on synthetic and real-world datasets (lower is better). Best NLL in bold, second best underlined. Results with standard deviations can be found in Appendix F.1. the two models. We implemented TriTPP and RNN model using PyTorch [52]. The architecture of the RNN model is nearly identical to the ones used in [1,10,11], except that the cumulative conditional hazard function is parametrized with a spline [3] to enable closed-form sampling. Appendix E contains the details for this and other experiments. We measure the runtime of (a) computing the log-likelihood (and backpropagate the gradients) for a batch of 100 sequences of varying lengths and (b) sample sequences of the same sizes. We used a machine with an Intel Xeon E5-2630 v4 @ 2.20 GHz CPU, 256GB RAM and an Nvidia GTX1080Ti GPU. The results are averaged over 100 runs. Results. Figure 7 shows the runtimes for varying sequence lengths. Training is rather fast for both models, on average taking 1-10ms per iteration. RNN is slightly faster for short sequences, but is outperformed by TriTPP on sequences with more than 400 events. Note that during training we used a highly optimized RNN implementation based on custom CUDA kernels (since all the event times t i are already known). In contrast, TriTPP is implemented using generic PyTorch operations. When it comes to sampling, we notice a massive gap in performance between TriTPP and the RNN model, since the RNN must process all the preceding arrival times before being able to generate the next event in the sequence. TriTPP generates all the events in a sequence in parallel, which makes it more than 100 times faster than the recurrent model for longer sequences. Density estimation Setup. A fast TPP model is of little use if it cannot accurately learn the data distribution. The main goal of this experiment is to establish whether TriTPP can match the flexibility of RNN-based TPPs. As baselines, we use the IPP, RP and MRP models from Section 3.2 and Hawkes process [53]. Datasets. We use 6 synthetic datasets from Omi et al. [10]: Hawkes1&2 [7], self-correcting (SC) [16], inhomogeneous Poisson (IPP), renewal (RP) and modulated renewal (MRP) processes. Note that the data generators for IPP, RP and MRP by Omi et al. are not parametrized using splines, so these datasets are not guaranteed to be fitted perfectly by our models. We also consider 7 real-world datasets: PUBG (online gaming), Reddit-Comments, Reddit-Submissions (online discussions), Taxi (customer pickups), Twitter (tweets) and Yelp1&2 (check-in times). See Appendix D for more details. Metrics. The standard metric for comparing generative models, including TPPs, is negative loglikelihood (NLL) on a hold-out set [10,11,36]. We partitioned the sequences in each dataset into train/validation/test sequences (60%/20%/20%). We trained the models by minimizing the NLL of the train set using Adam [54]. We tuned the following hyperparameters: While NLL is a popular metric, it has known failure modes [55]. For this reason, we additionally computed maximum mean discrepancy (MMD) [56] between the test sets and the samples drawn from each model after training. To measure similarity between two realizations t and t , we use a Gaussian kernel k(t, t ) = exp(−d(t, t )/2σ 2 ), where d(t, t ) is the "counting measure" distance from [57,Equation 3]. For completeness, we provide the definitions in Appendix E.2. MMD quantifies the dissimilarity between the true data distribution p (t) and the learned density p(t) -lower is better. L 2 regularization {0, 10 −5 , 10 −4 , 10 −3 }, Results. Table 1 shows the test set NLLs for all models and datasets. We can see that the RNN model achieves excellent scores and outperforms the simpler baselines, which is consistent with earlier Table 2: MMD between the hold-out test set and the generated samples (lower is better). findings [1]. TriTPP is the only method that is competitive with the RNN -our method is within 0.05 nats of the best score on 11 out of 13 datasets. TriTPP consistently beats MRP, RP and IPP, which confirms that learnable block-diagonal transformations improve the flexibility of the model. The gap get larger on the datasets such as Hawkes, SC, PUBG and Twitter, where the inability of MRP to learn self-exciting and self-correcting behavior is especially detrimental. While Hawkes process is able to achieve good scores on datasets with "bursty" event occurrences (Reddit, Twitter), it is unable to adequately model other types of behavior (SC, MRP, PUBG). Table 2 reports the MMD scores. The results are consistent with the previous experiment: models with lower NLL typically obtain lower MMD. One exception is the Hawkes process that achieves low NLL but high MMD on Taxi and Twitter. TriTPP again consistently demonstrates excellent performance. Note that MMD was computed using the test sequences that were unseen during training. This means that TriTPP models the data distribution better than other methods, and does not just simply overfit the training set. In Appendix F.1, we provide additional experiments for quantifying the quality of the distributions learned by different models. Overall, we conclude that TriTPP is flexible and able to model complex densities, in addition to being significantly more efficient than RNN-based TPPs. Variational inference True MJP trajectory Setup. We apply our variational inference method (Section 4.2) for learning the posterior distribution over the latent trajectories of an MMPP. We simulate an MMPP with K = 3 latent states. As a baseline, we use the stateof-the-art MCMC sampler by Rao & Teh [48]. Results. Figure 6 shows the true latent MJP trajectory, as well as the marginal posterior probabilities learned by our method and the MCMC sampler of Rao & Teh. We can see that TriTPP accurately recovers the true posterior distribution over the trajectories. The two components that enable our new variational inference approach are our efficient parallel sampling algorithm for TriTPP (Section 3) and the differential relaxation (Section 4). Additional experiments on VI for parameter learning & segmenting real-world data are included in Appendix F.2. Future work & conclusions Future work & limitations. We parametrized the nonlinear transformations of our TPP models with splines. Making a spline more flexible requires increasing the number of knots, which increases the number of parameters and might lead to overfitting. New deep analytically invertible functions will improve both our models, as well as normalizing flows in general. Currently, TriTPP is not applicable to marked TPPs [5]. Extending our model to this setting is an important task for future work. Conclusions. We have shown that TPP densities can be represented with increasing triangular maps. By directly parametrizing the respective transformations, we are able to construct TPP models, for which both density evaluation and sampling can be done efficiently in parallel. Using the above framework, we defined TriTPP-a new class of flexible probability distributions over variable-length sequences. In addition to being highly efficient thanks to its parallelism, TriTPP shows excellent performance on density estimation, as shown by our experiments. High flexibility and efficiency of TriTPP allow it to be used as a plug-and-play component of other machine learning models. Broader impact Existing works have applied TPPs and MJPs for analyzing electronic health records [58,59], detecting anomalies in network traffic [60,61] and modeling user behavior on online platforms [62,63]. Thanks to fast sampling, our model can be used for solving new prediction tasks on such data, and the overall improved scalability allows practitioners to work with larger datasets. We do not find any of the above use cases ethically questionable, though, general precautions must be implemented when handling sensitive personal data. Since our model exploits fast parallel computations, has fewer parameters and converges in fewer iterations, it is likely to be more energy-efficient compared to RNN-based TPPs. However, we haven't performed experiments analyzing this specific aspect of our model. A Abbreviations and notation Abbreviations: • RNN -recurrent neural network • TPP -temporal point process • HPP -homogeneous Poisson process (compensator is Λ * (t) = λt for some λ > 0) • IPP -inhomogeneous Poisson process • RP -renewal process • MRP -modulated renewal process • MJP -Markov jump process • MMPP -Markov modulated Poisson process • MC -Monte Carlo • VI -variational inference Table 3: Notation used throughout the paper. Φ(τ ) Cumulative hazard function of a renewal process. C The N × N cumulative sum matrix, C ij = 1 if i ≤ j, 0 else. D ≡ C −1 The N × N difference matrix, D ij =    1 if i = j, −1 if i = j + 1, 0 else. F = (f 1 , ..., f N ) Increasing lower-triangular map that converts a realization t of an arbitrary TPP with compensator Λ * into a sample z from an HPP with unit rate. f i (t 1 , ..., t i ) = Λ(t i |t 1 , ..., t i−1 ) Component function of F . 1(x) Indicator function, 1(x) = 1 if x is True, 0 else. γ Temperature parameter for the diff. relaxation (Section 4.1). B Variational inference for Markov jump processes B.1 Generative model for MJP and MMPP Markov jump process. We represent the trajectory of an MJP as a tuple (t, s), where t = (t 1 , ..., t N ) are the (strictly increasing) jump times and s = (s 1 , ..., s N +1 ) is the sequence of visited states. For convenience, we additionally set t 0 = 0 and t N +1 = T . The distribution over the trajectories (t, s) is defined by a K × K generator matrix A and an initial state distribution π. Each entry A kl ∈ R + denotes the rate of transition from state k to state l of the the MJP. Note that we use the formulation that permits self-jumps [64] (i.e., it may happen that s i = s i+1 ). We can denote the total transition rate of state s i as A si = K k=1 A sik . We can simulate an MJP trajectory using the following procedure s 1 ∼ Categorical(π) t i − t i−1 =: τ i ∼ Exponential (A si ) s i+1 ∼ Categorical (A si: /A si )(6) Here, A si: /A si is the s i 'th row of A that is normalized to sum up to 1. The likelihood of a trajectory (t, s) for an MJP with parameters (π, A) can be computed as p(t, s|π, A) = π s1 N i=1 A si−1si exp − N +1 i=1 (t i − t i−1 ) K l=1 A sil We reformulate this expression using indicators 1(·), which will make the ELBO computation easier = K k=1 π 1(s1=k) k N i=1 K k=1 K l=1 A 1(si=k,si+1=l) kl exp − N +1 i=1 (t i − t i−1 ) K k=1 1(s i = k) K l=1 A kl By applying the logarithm to the above equation, we obtain log p(t, s|π, A) = K k=1 1(s 1 = k) log π k + N i=1 K k=1 K l=1 1(s i = k, s i+1 = l) log A kl − N +1 i=1 (t i − t i−1 ) K k=1 1(s i = k) K l=1 A kl Likelihood of the observations o given (t, s) and λ can be computed as p(o|t, s, λ) = N +1 i=1 λ M [t i−1 ,t i ) si exp − N +1 i=1 (t i − t i−1 )λ si where M [ti−1,ti) is the number of events o j in the interval [t i−1 , t i ). Again, using indicator functions, we rewrite it as =   N +1 i=1 M j=1 K k=1 λ 1(si=k) k 1(oj ∈[ti−1,ti))   exp − N +1 i=1 (t i − t i−1 ) K k=1 1(s i = k)λ k By applying the logarithm, we obtain log p(o|t, s, λ) =   N +1 i=1 M j=1 1(o j ∈ [t i−1 , t i ) K k=1 1(s i = k) log λ k   − N +1 i=1 (t i − t i−1 ) K k=1 1(s i = k)λ k (8) B.2 Derivation of the ELBO The true posterior of an MMPP p(t, s|o, π, A, λ) ∝ p(t, s, o|π, A, λ) = p(t, s|π, A)p(o|t, s, λ) is intractable. We approximate it with a variational distribution q(t, s) by maximizing the evidence lower bound (ELBO) [24] ELBO(q, θ) = E q(t,s) log p(t, s|π, A) trajectory log-likelihood + log p(o|t, s, λ) observations log-likelihood − log q(t, s) entropy Recall that we model the approximate posterior as q(t, s) = q(t)q(s|t), where q(t) is defined using TriTPP and q(s|t) is evaluated exactly for each Monte Carlo sample t. We rewrite the ELBO as ELBO(q, θ) = E q(t) E q(s|t) log p(t, s|π, A) + log p(o|t, s, λ) − log q(s|t) − log q(t) We already derived the expressions for log p(t, s|π, A) (Equation 7) and log p(o|t, s, λ) (Equation 8). The expression for log q(s|t) can be obtained similarly log q(s|t) = N +1 i=1 K k=1 1(s i = k) log q(s i = k|t)(9) Finally, to compute the log-density log q(t) of a single sample t = (t 1 , ..., t N ) we use the procedure described in Appendix C. We denote z = (z 1 , ..., z N , z N +1 ) = F (t 1 , ..., t N , T ), and z −1 is the last entry of z. log q(t) = N i=1 log ∂z i ∂t i − z −1(10) ELBO (non-differentiable version). Putting everything together, we get ELBO(q, θ) = E q(t) E q(s|t) K k=1 1(s 1 = k) log π k − N +1 i=1 (t i − t i−1 ) K k=1 1(s i = k) K l=1 A kl + N i=1 K k=1 K l=1 1(s i = k, s i+1 = l) log A kl + N +1 i=1 M j=1 1(o j ∈ [t i , t i+1 )) K k=1 1(s i = k) log λ k − N +1 i=1 (t i − t i−1 ) K k=1 1(s i = k)λ k − N +1 i=1 K k=1 1(s i = k) log q(s i = k|t) − N i=1 log ∂z i ∂t i + z −1(11) Note that N is the length of the sample t generated from q(t), so N will take different values for different samples. We can evaluate the inner expectation w.r.t. q(s|t) by using the following fact transition probabilities q(s i = k, s i+1 = l|t) using the forward-backward algorithm [25,Equations 7,8]. Therefore, the inner expectation w.r.t. q(s|t) in Equation 11 can be computed analytically. ELBO (differentiable relaxation). The ELBO, as defined above in Equation 11 , is discontinuous w.r.t. the parameters of the density q(t) for the reasons described in Section 4.1. The expression inside the expectation depends only on the events t i that happen before T . Infinitesimal change in the parameters of q(t) may "push" the point t i outside [0, T ], thus changing the function value by a fixed amount and resulting in a discontinuity. We fix this problem using the approach described in Appendix C and Section 4.1. We obtain an "extended" samplet by first simulating a sequencez = (z 1 , ...,z N ) from a HPP with unit rate and computingt = F −1 (z) (more on this in Appendix C). We get a "clipped" / "padded" sample t = (t 1 , ..., t N ) as t i = min{t i , T } ( Figure 9). Finally, we compute z = (z 1 , ..., z N ) = F (t) (this is necessary for computing the correct cumulative intensity Λ * (T ) after clipping). We can now express the ELBO in terms of the "extended" samplest and "clipped" samples t: ELBO(q, θ) = E q(t) E q(s|t) K k=1 1(s 1 = k) log π k − N i=1 (t i − t i−1 ) K k=1 1(s i = k) K l=1 A kl + N i=1 1(t i < T ) K k=1 K l=1 1(s i = k, s i+1 = l) log A kl + N i=1 M j=1 1(o j ∈ [t i , t i+1 )) K k=1 1(s i = k) log λ k − N i=1 (t i − t i−1 ) K k=1 1(s i = k)λ k − N i=1 1(t i−1 < T ) K k=1 1(s i = k) log q(s i = k|t) − N i=1 1(t i < T ) log ∂z i ∂t i + z −1(13) Changes from Equation 11 are highlighted in red. Even though the formula looks different, the result of evaluating Equation 13 will be exactly the same as for Equation 11. By using different notation we only made the process of "discarding" the events t i > T explicit. The new formulation allows us to obtain a differentiable relaxation. For this, we replace the indicator functions 1(t i < T ) with sigmoids σ γ (T − t i ). The indicator function 1(o j ∈ [t i , t i+1 )) can also be relaxed as 1(o j ∈ [t i , t i+1 )) = 1(t i+1 > o j ) − 1(t i ≥ o j ) ≈ σ γ (t i+1 − o j ) − σ γ (t i − o j )(14) By combining all these facts, we obtain a differentiable relaxation of the ELBO. Our method leads to an efficient implementation that uses batches of samples. We sample a batch of jump times {t (1) , t (2) , ...} from q(t), evaluate the posterior q(s|t) using with forward-backward for all of them in parallel, and evaluate the relaxed ELBO (Equation 13). B.3 Parameter estimation In Section 4.2, we perform approximate posterior inference over the trajectories (t, s) by maximizing the ELBO w.r.t. q(t, s) max q(t,s) E q [log p(t, s|θ) + log p(o|t, s, θ) − log q(t, s)](15) Since ELBO(q, θ) provides a lower bound on the marginal log-likelihood log p(o|θ), we can also simultaneously learn the model parameters θ = {π, A, λ} by solving the following optimization problem (subject to appropriate constraints on θ) max θ max q(t,s) E q [log p(t, s|θ) + log p(o|t, s, θ) − log q(t, s)](16) Finally, we can perform fully Bayesian treatment and approximate the posterior distribution over the parameters as well as the trajectories. For this, we can place a prior p(θ) and approximate p(θ, t, s|x) with q(θ, t, s) = q(θ)q(t)q(s|t, θ). This corresponds to the following optimization problem max q(θ,t,s) E q [log p(t, s|θ) + log p(o|t, s, θ) − log q(t, s)] − KL(q(θ) p(θ))(17) where KL denotes KL-divergence. By applying our relaxation from Section 4, it's possible to solve all of the above optimization problems (Equations 15, 16, 17) using gradient ascent. C Implementation details C.1 Batch processing By representing TPP densities with transformations, we can implement both (log-)density evaluation and sampling efficiently and in parallel. Our implementation enables parallelism not only for the events t i of a single sequence, but also for entire batches consisting of multiple sequences t of different length. First, consider a single sequence t = (1, 2.5, 4) with N = 3 events, sampled from a TPP on the interval [0, 5]. We pad this sequence with T = 5, and additionally introduce a mask m that tells us which entries of the padded vector t correspond to actual events (i.e., not padding) t = 1 2.5 4 5 m = 1 1 1 0 We implement the transformation F (corresponding to the TPP density p(t)) similarly to normalizing flow frameworks like torch.distributions [52]. We define a method forward that computes z, the result of the transformation, and j, logarithm of the diagonal entries of the Jacobian J F (t): z = F (t) = z 1 z 2 z 3 z 4 j = log ∂z1 ∂t1 log ∂z2 ∂t2 log ∂z3 ∂t3 log ∂z4 ∂t4 From the definition of F (Table 3), we can see that the last entry of z (that we denote as z −1 ) corresponds to Λ * (T ). Also, each entry j i of j corresponds to log ∂Λ * (ti) ∂ti . Therefore, we can compute the log-density log p(t) as (18) where N denotes the length with the padding. We can verify that this is equal to the logarithm of the TPP density in Equation 1. Note that if we use a longer padding, such as t = 1 2.5 4 5 5 5 5 m = 1 1 1 0 0 0 0 then Equation 18 will still correctly compute the log-likelihood for the sequence. This observation allows to process multiple sequences {t (1) , t (2) , ...} in a single batch. We simply pad all the sequences with T up to the length of the longest sequence, stack them into a matrix of shape [batch_size, max_seq_len] and process all of them in parallel. log p(t) = sum(m j) − z −1 = N i=1 m i log ∂z i ∂t i − z −1 = N i=1 log ∂Λ * (t i ) ∂t i − Λ * (T ) As described in Section 3.3, we actually define F by stacking multiple transformations. We sequentially call the forward method for each transformation in the chain to obtain the final z, and sum up the log-diagonals of the Jacobians j along the way. Each transformation and its Jacobian can be evaluated in parallel in linear time, making the whole operation efficient. C.2 Sampling Sampling is implemented similarly. We start by simulating a vectorz from a homogeneous Poisson process with unit rate. The length ofz must be "long enough" (more on this later). We define the method inverse that computest = F −1 (z). We obtain a final sample t by clipping the entries of t as t i = min{t i , T }. If we would like to compute the density of the generated sample t, we will also need the mask m that can be obtained as m i = 1(t i < T ). In some use cases, such as entropy maximization (Section 4.1) or variational inference (Appendix B), we need to use a differentiable approximation to the mask m i = σ γ (T −t i ). This recovers our relaxation from Section 4.1. By slightly abusing the notation, we use N to denote the number of events in our initial HPP samplẽ z = (z 1 , ...,z N ). N must be large enough, such that the eventt N (corresponding toz N ) happens after T . We can easily ensure this by setting N to some large number (e.g., 100 or 1000), and increasing it if for some samplet N is less than T . As we saw in Figure 7, using larger sequence length leads to no noticeable computational overhead when using GPU. C.3 Parametrizing transformations using splines (Unconstrained) rational quadratic splines used by Durkan et al. [3] define a function f : R → R that is nonlinear on the interval (0, 1) → (0, 1) and equals to the identity function everywhere else. When defining our TPP models in Section 3 by stacking transformations, we need functions that are nonlinear on all of R or R + . We accomplish this by using the following elementwise transformations: 1. Exponential of negative h(x) = exp(−x), where σ : R + → (0, 1) 2. Negative log h −1 (y) = − log(y), where h −1 : (0, 1) → R + 3. Sigmoid σ(x) = 1/(1 + exp(−x)), where σ : R → (0, 1) 4. Logit σ −1 (p) = log p − log(1 − p) where σ −1 : (0, 1) → R The details can be found in our code. Note that transformations (1) and (2) above are decreasing, so we need to use an even number of them to ensure that the overall composition of functions is still increasing. D Datasets For each synthetic TPP model from Omi Reddit-Submissions. Each sequence contains the timestamps of submissions to the /r/politics subreddit within a single day (24 hours). We consider the period from 01.01.2017 until 31.12.2019. If several events happen at the exact same time, we only keep a single event. The data is again collected using the pushshift API. Taxi 5 contains the records of taxi pick-ups in New York. We restrict our attention to the south of Manhattan, which corresponds to the points with latitude in the interval (40.700084, 40.707697) and longitude in (-74.019871, -73.999443). Twitter 6 contains the timestamps of the tweets by user 25073877, recorded over several years. (1) , t (2) , ...} min θ − 1 |D train | 1 N avg t∈Dtrain log p θ (t) We normalize the loss by N avg , the average number of events in a sequence in the training set, in order to obtain values that are at least somewhat comparable across the datasets. We perform full-batch training since the all the considered datasets easily fit into the GPU memory. For all models, we use learning rate scheduling: if the training loss does not improve for 100 iterations, the learning rate is halved. The training is stopped after 5000 epochs or if the validation loss stops improving for 300 epochs, whichever happens first. We train all models using the parameter configurations reported in Section 6.2 and pick the configuration with the best validation loss. MMD. We train the models & tune the hyperparameters using the same procedure as in the NLL experiment. Then, we compare the distribution p(t) learned by each model with the empirical distribution p (t) on the hold-out test set by estimating the maximum mean discrepancy (MMD) [56]. The MMD between distributions p and p is defined as MMD(p, p ) = E t,t ∼p [k(t, t )] − 2E t∼p,u∼p [k(t, u)] + E u,u ∼p [k(u, u )] Here, t = (t 1 , ..., t N ) and u = (u 1 , ..., u M ) denote variable-length TPP realizations from different distributions, and k(·, ·) is a positive semi-definite kernel function that quantifies the similarity between two TPP realizations. We use the Gaussian kernel where d(t, u) is the counting measure distance between two TPP realizations from [57,Equation 3], defined as k(t, u) = exp − d(t, u) 2σ 2 0 t0 t1 t2 t3 u0 u1 u2 u3 T Timed(t, u) = N i=1 |t i − u i | + M i=N +1 (T − u i ) Here, we assume w.l.o.g. that N ≤ M . Following Section 8 of Gretton et al. [56], the parameter σ is estimated as the median of d(t, u) with t, u ∼ p ∪ p . E.3 Variational inference We simulate an MMPP with K = 3 states and the following parameters We use the following configuration for TriTPP in this experiment: L = 2 blocks of size H = 4, learning rate 0.01, no weight decay. We estimate the ELBO using 512 Monte Carlo samples from q(t) and use the temperature γ = 0.1 for the relaxation. We implemented the MCMC sampler by Rao & Teh [48] in Pytorch. We discard the first 100 samples (burn-in stage), and use 1000 samples to compute the marginal distribution of the posterior. Visualization: effect block-diagonal matrices. A completely arbitrary compensator Λ * leads to a completely arbitrary increasing triangular map F . However, by picking a parametric class of models, such as MRP or TriTPP, we restrict the set of possible maps F that our model represent. One way to visualize the dependencies captured by the map F is by looking at its Jacobian J F . Figures 11 and 12 show the Jacobians of the component transformations for the modulated renewal process and TriTPP. We can obtain the overall (accumulated) Jacobian of the entire transformation by multiplying the component Jacobians from right to left. We can see that thanks to the block-diagonal layers TriTPP is able to capture more complex transformations, and thus richer densities, than MRP. Distribution of sequence lengths. In this experiment, we additionally quantify how well each model captures the true data distribution. Like before, we train all models on the training set. We then generate sequences t from a trained model and compare the distribution of their lengths to the distribution of the lengths of the true data using Wasserstein distance. We use the whole dataset since the test sets is too small in some cases. Using Python pseudocode, this procedure can be expressed as lengths_sampled = [len(t) for t in model_samples] lengths_true = [len(t) for t in dataset] wd = wasserstein_distance(lengths_sampled, lengths_true) Figure 13 shows the distributions for the Twitter dataset together with the respective Wasserstein distances. Note that the histograms are used only for visualization purposes, the Wasserstein distance is computed on the raw distributions. Quantitative results are reported in Table 7. We observe the same trend as before: the RNN-based model and TriTPP consistently outperform the other methods. Recall that Hawkes process achieves a good NLL on the Twitter data (Table 1). However, when we sample sequences from the trained Hawkes model, the distribution of their lengths doesn't actually match the true data, as can be seen in Figure 13c. Figure 13: Histograms of sequence lengths (true and generated) for Twitter. The difference between the two is quantified using Wasserstein distance (WD) -lower is better. F.2 Variational inference Random initliazations. In order to show that ours results are not cherry-picked, we provide the plots of marginal posterior trajectories (similar to Figure 8) obtained with 3 different random seeds. Figure 14 shows that our results are consistent across the random seeds. Variational inference on real-world data. We apply our model to the server log data 8 . More specifically, we perform segmentation on the interval that contains the first 200 events. We estimate the posterior over the trajectories (t, s) and learn the model parameters θ = {π, A, λ} using the procedure described in Equation 16. Like before, we compare our approach to the MCMC sampler of Rao closed-form parameter updates for θ and simulating the posterior trajectories. Figure 15 shows the obtained posterior trajectories for the two approaches. Both model learn to segment the sequence into a high-event-rate and a low-event-rate states. Figure 15: Segmentation of server data obtained using our VI approach and MCMC. In both cases, we estimate the posterior p(t, s|o, θ) as well as the MMPP parameters θ. F.3 Miscellaneous Convergence plots for density estimation. Convergence plots for variational inference. . , Λ * (t N )) is a realization of a homogeneous Poisson process (HPP) with unit rate on the interval [0, Λ * (T )] [4, Theorem 7.4.I][5, Proposition 4.1] (Figure 1). The transformation F = (f 1 , . . . , f N ) : t → z is an increasing triangular map. Each component function f i (t) = Λ(t i |t 1 , . . . , t i−1 ) only depends on (t 1 , . . . , t i ) and is increasing in t i since∂ ∂ti Λ * (t i ) = λ * (t i ) > 0. The number N of the component functions f i depends on the length of the specific realization t. Notice that the term N i=1 Figure 1 : 1Triangular map O(H 2 ), and multiplication by B l or B −1 l can be done in O(N H 2 ) in parallel. We stack L such matrices B l and define the triangular map Figure 5 :Figure 6 : 56Maximizing Markov modulated Poisson process with 2 states. t≥0 is a piecewise-constant stochastic process on R + . At any time t, the process occupies a discrete state s(t) ∈ {1, ..., K}. The times when the state changes are called jumps. A trajectory of an MJP on an interval [0, T ] with N jumps can be represented by a tuple (t, s) of jump times t = (t 1 , ..., t N ) and the visited states s = (s 1 , ..., s N +1 ). Figure 7 : 7Scalability analysis. Standard devs. are below 1ms. Figure 8 : 8Posterior distributions over the latent trajectory of an MMPP learned using our VI approach & MCMC. t = (t 1 , ..., t N ) Variable-length realization of a TPP.p(t)Density of a point process, also called likelihood (Equation 1).λ * (t) = λ(t|t 1 , ..., t i−1 ) Conditional intensity at time t, where t i−1 is the last event before t. Λ * (t) = Λ(t|t 1 , ..., t i−1 ) = T 0 λ * (u)duCumulative conditional intensity at time t, a.k.a. compensator.Λ(t)(Unconditional) cumulative intensity of a Poisson process. ( 7 ) 7Markov modulated Poisson process. Distribution of the observations o = (o 1 , ..., o M ) of an MMPP depends on the latent MJP trajectory (t, s) and the rates of each state λ ∈ R K + . The observations o are sampled from an inhomogeneous Poisson process with piecewise-constant intensity that depends on the current state: λ(t) = λ s(t) . EFigure 9 : 9q(s|t) [1(s i = k)] = q(s i = k|t) E q(s|t) [1(s i = k, s i+1 = l)] = q(s i = k, s i+1 = l|t) (12) Recall that we set q(s|t) to the true posterior p(s|t, o, θ) over states given the jumps (Equation 5). This allows us to exactly compute both the posterior marginals q(s i = k|t) Relaxed indicator σ γ (T − t) (1.00, 0.97, 0.01, 0.00) Examples of values involved in the ELBO computation. the user check-in times for the McCarran International Airport and for all businesses in the city of Mississauga in 2018, respectively. Figure 10 : 10The blue area represents the counting measure distance (figure adapted from[57]). Figure 11 :Figure 12 : 1112Jacobians of the component transformations of the modulated renewal process. We obtain the Jacobian of the combined transformation F = C • Φ • D • Λ by multiplying the Jacobians of each transform (right to left). Jacobians of the component transformations of TriTPP. We obtain the Jacobian of the combined transformation F = C • Φ • B 4 • B 3 • B 2 • B 1 • D • Λ by multiplying the Jacobians of each transform (right to left). Figure 14 : 14Marginal posterior trajectories obtained when using different random seeds. Figure 16 : 16Training loss convergence for TriTPP and RNN model with different random seeds. Figure 17 : 17Convergence of our variational inference procedure when using 5 different random seeds. . Under review. arXiv:2006.12631v1 [cs.LG] 22 Jun 2020 Setup.The key feature of TriTPP is its ability to compute likelihood and generate samples in parallel, which is impossible for RNN-based models. We quantify this difference by measuring the runtime of6 Experiments 6.1 Scalability Hawkes1 Hawkes2 SC IPP MRP RP PUBG Reddit-C Reddit-S Taxi Twitter Yelp1 Yelp2 IPP 1.06 1.03 1.00 0.71 0.70 0.89 -0.06 -1.59 -4.08 -0.68 1.60 0.62 -0.05 RP 0.65 0.08 0.94 0.85 0.68 0.24 0.12 -2.08 -4.00 -0.58 1.20 0.67 -0.02 MRP 0.65 0.07 0.93 0.71 0.36 0.25 -0.83 -2.13 -4.38 -0.68 1.23 0.61 -0.10 Hawkes 0.51 0.06 1.00 0.86 0.98 0.39 0.11 -2.40 -4.19 -0.64 1.04 0.69 0.01 RNN 0.52 -0.03 0.79 0.73 0.37 0.24 -1.96 -2.40 -4.89 -0.66 1.08 0.67 -0.08 TriTPP 0.56 0.00 0.83 0.71 0.35 0.24 -2.41 -2.36 -4.49 -0.67 1.06 0.64 -0.09 number of spline knots {10, 20, 50}, learning rate {10 −3 , 10 −2 }, hidden size {32, 64} for RNN, number of blocks {2, 4} and block size {8, 16} for TriTPP. We used the validaiton set for hyperparameter tuning, early stopping and model development. We computed the results for the test set only once before including them in the paper. All results are averaged over 5 runs. Table 4 4shows the number of sequences, average sequence length and the duration of the [0, T ] interval for all the datasets. Dataset name Number of sequences Average sequence length Interval duration Hawkes 1 1000 95.4 100 Hawkes 2 1000 97.2 100 SC 1000 100.2 100 IPP 1000 100.3 100 MRP 1000 98.0 100 RP 1000 109.2 100 PUBG 3001 76.5 40 Reddit-C 1356 295.7 24 Reddit-S 1094 1129.0 24 Taxi 182 98.4 24 Twitter 2019 14.9 24 Yelp 1 319 30.5 24 Yelp 2 319 55.2 24 Table 4 : 4Statistics for the synthetic & real-world datasetsE Experimental setup E.1 Scalability For both the RNN-based model and TriTPP we used 20 spline knots. We ran TriTPP with blocks of size H = 16 and a total of L = 4 block-diagonal layers. This is the configuration of TriTPP with the largest number of parameters that we used across our experiments. For the RNN model, we used the hidden size of 32. This is the configuration of the RNN model with the smallest number of parameters that we used across our experiments. We did not use JIT compilation for either the RNN model or TriTPP, even though enabling JIT would make TriTPP even faster. When measuring the sampling time, we disabled the gradient computation with torch.no_grad(). To remove outliers for the RNN model, we removed 10 longest runtimes for both models. tablewith standard deviations. For most models & datasets the results are nearly independent of the random initialization and the standard deviations are very close to zero. In the following table, we show the standard deviations of the NLL computed over 5 random initializations for all datasets where at least one of the models has the standard deviation above 0.005.PUBG Reddit-S Twitter Yelp2 mean std mean std mean std mean std TriTPP -2.41 0.34 -4.49 0.06 1.06 0.01 -0.09 0.01 RNN -1.97 0.16 -4.89 0.29 1.08 0.01 -0.07 - MRP -0.83 0.08 -4.38 0.05 1.23 - -0.1 - RP 0.12 - -4.01 0.03 1.2 --0.02 - IPP -0.06 - -4.08 - 1.61 --0.05 - Table 5 : 5Average test set NLL with standard deviations. Datasets where all models have a standard deviation below 0.005 are excluded.Effect of the block size and number on TriTPP performance. In this experiment, we show that TriTPP works well with different numbers L and sizes H of block-diagonal layers. We use the same setup as in the density estimation experiment.Table 6shows the test set NLL scores for different configurations. Smaller block are helpful for datasets with a clear global trend (e.g., Reddit-S, Taxi, Yelp), and larger blocks help for datasets with bursty behavior (Reddit-C, Twitter). In all cases, TriTPP is still better than simpler baselines(Table 1).Configuration Hawkes1 Hawkes2 SC IPP MRP RP PUBG Reddit-C Reddit-S Taxi Twitter Yelp1 Yelp2 TriTPP (L = 2, H = 4) 0.58 0.01 0.86 0.71 0.35 0.24 -0.95 -2.26 -4.69 -0.68 1.11 0.62 -0.1 TriTPP (L = 4, H = 4) 0.57 0.01 0.85 0.71 0.35 0.24 -2.04 -2.28 -4.57 -0.68 1.06 0.63 -0.1 TriTPP (L = 2, H = 8) 0.56 0.01 0.84 0.71 0.35 0.24 -1.93 -2.3 -4.42 -0.66 1.06 0.64 -0.09 TriTPP (L = 4, H = 8) 0.56 0.0 0.83 0.71 0.35 0.24 -2.41 -2.33 -4.46 -0.67 1.06 0.64 -0.09 TriTPP (L = 2, H = 16) 0.56 0.0 0.84 0.71 0.36 0.25 -1.78 -2.35 -4.45 -0.64 1.06 0.67 -0.06 TriTPP (L = 4, H = 16) 0.56 0.0 0.84 0.72 0.36 0.25 -1.83 -2.36 -4.49 -0.64 1.07 0.67 -0.06 Table 6 : 6Test set NLL for different configurations of TriTPP. & Teh. For the MCMC sampler, we adopt an EM-like approach, where we alternate betweenHawkes1 Hawkes2 SC IPP MRP RP PUBG Reddit-C Reddit-S Taxi Twitter Yelp1 Yelp2 IPP 0.11 0.11 0.03 0.00 0.03 0.07 0.01 0.76 0.27 0.10 0.52 0.07 0.12 RP 0.07 0.09 0.19 0.14 0.38 0.02 0.67 0.67 0.28 0.85 0.28 0.31 0.20 MRP 0.08 0.07 0.02 0.00 0.01 0.01 0.05 0.66 0.27 0.09 0.28 0.06 0.11 Hawkes 0.01 0.05 0.03 0.15 0.15 0.03 0.12 0.25 0.65 0.09 0.50 0.10 0.15 RNN 0.00 0.01 0.00 0.04 0.03 0.00 0.08 0.40 0.07 0.08 0.10 0.05 0.12 TriTPP 0.05 0.03 0.00 0.01 0.01 0.00 0.05 0.53 0.24 0.08 0.17 0.05 0.09 Table 7 : 7Wasserstein distance between the distributions of lengths of true and sampled sequences. 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[]
[ "Descent-to-Delete: Gradient-Based Methods for Machine Unlearning", "Descent-to-Delete: Gradient-Based Methods for Machine Unlearning" ]
[ "Seth Neel \nUniversity of Pennsylvania\n\n", "Aaron Roth \nUniversity of Pennsylvania\n\n", "Saeed Sharifi-Malvajerdi \nUniversity of Pennsylvania\n\n" ]
[ "University of Pennsylvania\n", "University of Pennsylvania\n", "University of Pennsylvania\n" ]
[]
We study the data deletion problem for convex models. By leveraging techniques from convex optimization and reservoir sampling, we give the first data deletion algorithms that are able to handle an arbitrarily long sequence of adversarial updates while promising both per-deletion run-time and steadystate error that do not grow with the length of the update sequence. We also introduce several new conceptual distinctions: for example, we can ask that after a deletion, the entire state maintained by the optimization algorithm is statistically indistinguishable from the state that would have resulted had we retrained, or we can ask for the weaker condition that only the observable output is statistically indistinguishable from the observable output that would have resulted from retraining. We are able to give more efficient deletion algorithms under this weaker deletion criterion.
null
[ "https://arxiv.org/pdf/2007.02923v1.pdf" ]
220,364,296
2007.02923
617dad3ccbbf9e93c4faf54e19e449649433a10d
Descent-to-Delete: Gradient-Based Methods for Machine Unlearning 6 Jul 2020 July 7, 2020 Seth Neel University of Pennsylvania Aaron Roth University of Pennsylvania Saeed Sharifi-Malvajerdi University of Pennsylvania Descent-to-Delete: Gradient-Based Methods for Machine Unlearning 6 Jul 2020 July 7, 2020 We study the data deletion problem for convex models. By leveraging techniques from convex optimization and reservoir sampling, we give the first data deletion algorithms that are able to handle an arbitrarily long sequence of adversarial updates while promising both per-deletion run-time and steadystate error that do not grow with the length of the update sequence. We also introduce several new conceptual distinctions: for example, we can ask that after a deletion, the entire state maintained by the optimization algorithm is statistically indistinguishable from the state that would have resulted had we retrained, or we can ask for the weaker condition that only the observable output is statistically indistinguishable from the observable output that would have resulted from retraining. We are able to give more efficient deletion algorithms under this weaker deletion criterion. Introduction Users voluntarily provide huge amounts of personal data to online services, such as Facebook, Google, and Amazon, in exchange for useful services. But a basic principle of data autonomy asserts that users should be able to revoke access to their data if they no longer find the exchange of data for services worthwhile. Indeed, each of these organizations provides a way for users to request that their data be deleted. This is related to, although distinct from the "Right to be Forgotten" from the European Union's General Data Protection Act (GPDR). The Right to be Forgotten entails the right for users, in certain circumstances, to request that negative information concerning them to be removed. Like basic data autonomy, it sometimes obligates companies to delete data. But what does it mean to delete data? Typically, user data does not sit siloed in a database, but rather is used to produce derivatives such as predictive models. Deleting a user's data from a database may prevent it from influencing the training of future models 1 , but does not remove the influence of a user's data on existing models -and that influence may be significant. For example, it is possible to extract information about specific data points used for training from models that have been trained in standard ways [SSSS17]. So deleting a user's data naively, by simply removing it from a database, may not accomplish much: what we really want is to remove (or at least rigorously limit) the influence that an individual's data has on the behavior of any part of the system. How should we accomplish this? We could retrain all predictive models from scratch every time a user requests that their data be removed, but this would entail an enormous computational cost. Ginart et al. [ GGVZ19] propose a compelling alternative: full retraining is unnecessary if we can design a deletion operation that produces a (distribution of) model output(s) that is statistically indistinguishable from the (distribution of) model output(s) that would have arisen from full retraining. Ginart et al. [GGVZ19] also propose an approximate notion of deletion that uses a differential-privacy like measure of "approximate" statistical indistinguishability that we adopt in this work. Our Results and Techniques In this paper, we consider convex models that are trained to some specified accuracy, and then are deployed while a sequence of requests arrive to delete (or add) additional data points. The deletion or addition must happen immediately, before the next point comes in, using only a fixed running time (which we measure in terms of gradient computations) per update. We require that the distribution on output models be (ǫ, δ)-indistinguishable from the distribution on output models that would result from full retraining (see Section 2 for the precise definition: this is a notion of approximate statistical indistinguishability from the differential privacy literature). In a departure from prior work, we make the distinction between whether the entire internal state of the algorithm must be indistinguishable from full re-training, or whether we only require statistical indistinguishability with respect to the observable outputs of the algorithms. If we require indistinguishability with respect to the full internal state, we call these update or unlearning algorithms perfect. This is similar to the distinction made in the differential privacy literature, which typically only requires indistinguishability for the outputs of private algorithms, but which has a strengthening (called pan privacy [DNPR10,AJM19]) which also requires that the internal state satisfy statistical indistinguishability. We remark that while unlearning algorithms that are allowed to maintain a "secret state" that need not satisfy the data deletion notion require additional trust in the security of the training system, this is orthogonal to privacy. Indeed, [CZW + 20] show that even without secret state, algorithms satisfying standard deletion guarantees can exacerbate membership inference attacks if the attacker can observe the model both before and after a deletion (because standard deletion guarantees promise nothing about what can be learned about an individual from two model outputs). In contrast, although some of our unlearning algorithms maintain a secret state that does not satisfy the statistical indistinguishability property, our model outputs themselves satisfy (ǫ, δ)-differential privacy. This in particular prevents membership inference attacks from observers who can observe a small number of output models, so long as they cannot observe the secret state. All prior work has focused on perfect unlearning. We introduce another novel distinction between strong unlearning algorithms and weak unlearning algorithms. For an unlearning algorithm to be strong, we require that for a fixed accuracy target, the run-time of the update operation be constant (or at most logarithmic) in the length of the update sequence. A weak unlearning algorithm may have run-time per update (or equivalently, error) that grows polynomially with the length of the update sequence. All prior work has given weak unlearning algorithms. We give two sets of results. The first, which operates under the most permissive set of assumptions, is a simple family of gradient descent algorithms. After each addition or deletion request, the update algorithm starts from the previous model, and performs a small number of gradient descent updatessufficient to guarantee that the model parameter is boundedly close to the optimal model parameter in Euclidean distance. It then perturbs the model parameter with Gaussian noise of sufficient magnitude to guarantee (ǫ, δ)-indistinguishability with respect to anything within a small neighborhood of the optimal model. We prove that this simple approach yields a strong, perfect unlearning algorithm for loss functions that are strongly convex and smooth. Without the strong convexity assumption, we can still derive strong unlearning algorithms, but ones which must maintain secret state. We can further improve our accuracy guarantees if we are willing to settle for weak unlearning algorithms. The per-round computation budget and the achievable steady state accuracy can be smoothly traded off against one another. Our second algorithm improves over the straightforward approach above (under slightly stronger regularity assumptions) when the data dimension is sufficiently large. It first takes a bootstrap sample from the underlying dataset, and then randomly partitions it into K parts. The initial training algorithm separately and independently optimizes the loss function on each part, and then averages the parameter vector from each part, before finally releasing the perturbed average. Zhang et al [ZDW12] analyzed this algorithm (absent the final perturbation) and proved accuracy bounds with respect to the underlying distribution (which for us is the dataset from which we draw the bootstrap sample). Our update operation involves first using a variant of reservoir-sampling that maintains the property that the union of the partitions continue to be distributed as independent samples drawn with replacement from our current dataset. We then use the simple gradient based update algorithms from our first set of results to update the parameters only from the partitions that have been modified by the addition or deletion. Because each of these partitions contains only a fraction of the dataset, we can use our fixed gradient computation budget to perform more updates. Because we have maintained the marginal distributions on partition elements via our reservoir sampling step, the overall accuracy analysis of [ZDW12] carries over even after an arbitrary sequence of updates. This is also crucial for our statistical indistinguishability guarantee. The result is a strong unlearning algorithm that yields an improved tradeoff between per-round run-time and steady state accuracy for sufficiently high dimensional data. Related Work At a high level, our work differs from prior work in several ways. We call deletion algorithms that do not maintain secret state perfect. All prior work focuses on perfect deletion algorithms, but we give improved bounds for several problems by allowing our algorithms to maintain secret state. Second, we allow arbitrary sequences of updates, which can include additions and deletions (rather than just deletions). Finally, we distinguish between weak and strong unlearning algorithms, and give the first strong unlearning algorithms. Cao and Yang [CY15] first considered the problem of efficiently deleting data from a trained model under a deterministic notion of deletion, and coined the term "machine unlearning". They gave efficient deletion methods for certain statistical query algorithms -but in general, their methods (or indeed, any deterministic notion of deletion) can apply to only very structured problems. Ginart et al. [GGVZ19] gave the first definition of data deletion that can apply to randomized algorithms, in terms of statistical indistinguishability. We adopt the approximate deletion notion they introduced, which is itself based on differential privacy [DMNS06,DR14]. Ginart Guo et al. [GGHvdM19] give deletion algorithms for linear and logistic regression, using the same notion of approximate statistical indistinguishability that we use. Their algorithm is similar to our first algorithm: it performs a convex optimization step, followed by a Gaussian perturbation. They use a second order update (a Newton step) rather than first order updates as we do, and their algorithm yields error that grows linearly with the number of updates, and so is a weak deletion algorithm. Izzo et al [ISCZ20] focus on linear regression and show how to improve the run-time per deletion of the algorithm given in [GGHvdM19] from quadratic to linear in the dimension. Our main result leverages a distributed optimization algorithm that partitions the data, separately optimizes on each partition, and then averages the parameters, analyzed by Zhang et al [ZDW12]. Optimizing separately on different partitions of the data, and then aggregating the results is also a well known general technique in differential privacy known as "Subsample and Aggregate" [NRS07] which has found applications in private learning [PAE + 16]. Bourtoule et al. [BCCC + 19] use a similar technique in the context of machine unlearning that they call "SISA" (Sharded, Isolated, Sliced, Aggregated). Their goal is more ambitious (to perform deletion for non-convex models), but they have a weaker deletion criterion (that it simply be possible that the model arrived at after deletion could have arisen from the retraining process), and they give no error guarantees. Their algorithm involves full retraining on the affected partitions, a different aggregation function, no randomization, and does not include the reservoir sampling step that is crucial to our stronger indistinguishability guarantees. This distributed optimization algorithm also bears similarity to the well-known FederatedAveraging algorithm of [MMRyA16] used for deep learning in the federated setting. Chen et al. [CZW + 20] observe that deterministic deletion procedures such as SISA [BCCC + 19] can exacerbate privacy problems when an attacker can observe both the model before and after the deletion of a particular user's data point, and show how to perform membership inference attacks against SISA in this setting. Our method leverages techniques from differential privacy, and so in addition to being an (ǫ, δ)-deletion algorithm, a view of the two outputs of our algorithm before and after a deletion is (2ǫ, 2δ)differentially private, which precludes non-trivial membership inference for reasonable values of ǫ and δ. This follows because our deletion algorithm is randomized: procedures such as the one from [GGHvdM19] which have randomized training procedure but deterministic deletion procedure do not share this property. Summary of Results In Table 1, we state bounds for all our unlearning algorithms, and (in the 2nd column) the assumptions that they require (convexity, strong convexity, smoothness, etc.) The 3rd column of the table states whether our algorithms are strong or weak update algorithms (i.e. whether or not their runtimes grow polynomially with the length of the update sequence or not). The 4th column states the steady-state accuracy of the algorithm as a function of the desired run time I of the first update (each algorithm has a budget of nI gradient computations per update). The 5th column lists the run-time of the i'th update. The 6th column measures the run-time of the baseline approach that would fully retrain the model after each update, to the accuracy achieved by our algorithms in the 4th column. Most of these guarantees are for algorithms that maintain a secret state. But for strongly convex and smooth functions we can obtain a perfect unlearning algorithm (i.e. one that satisfies the indistinguishability guarantee not just with respect to observable outputs, but with respect to the entire saved state) with the same asymptotic accuracy/runtime tradeoff, so long as the per-update run-time is at least logarithmic in the dimension. See Theorem 3.2 for details. For non strongly convex functions, our techniques do not appear to be able to give perfect unlearning algorithms for non-trivial parameters; this is an intriguing direction for future work. Our "Distributed PGD" algorithm is somewhat more complex (see Section 4), but has the advantage that it obtains improved accuracy/run-time tradeoffs for sufficiently high dimensional data. It divides the same gradient computation budget nI between different numbers of iterations on different parts of the dataset. See Remark 4.6 for the exact conditions on when it yields an improvement over our simpler algorithms. Model and Preliminaries We write Z to denote the data domain. A dataset D is a multi-set of elements from Z. Datasets can be modified by updates which are requests to either add or remove one element from the dataset. Definition 2.1 (Update). An update u is a pair (z, •) where z ∈ Z is a data point and • ∈ T = { ′ add ′ , ′ delete ′ } determines the type of the update. An update sequence U is a sequence (u 1 , u 2 , . . .) where u i ∈ Z × T for all i. Given a dataset D and an update u = (z, •), the update operation is defined as follows. D • u D ∪ {z} if • = ′ add ′ D \ {z} if • = ′ delete ′ We use Θ to denote the space of models. In our setting, a learning or training algorithm is a mapping A : Z * → Θ that maps datasets to models. An unlearning or update algorithm for A is a mapping R A : Z * × (Z × T ) × Θ → Θ that takes as input a dataset accompanied by a single update, and a model, and outputs an updated model. Some of our update algorithms will also take as input auxiliary information, that we elide here but will be clear from context. The output of the unlearning algorithm itself will not be made public: before any model is made public, it must pass through a publishing function. A publishing function is a mapping f publish : Θ → Θ that maps a (secret) model to the model that will be made publicly available. Our unlearning guarantee will informally require that there should be no way to distinguish whether the published model resulted from full retraining, or an arbitrary sequence of updates via the unlearning algorithm. Depending on whether we demand perfect unlearning or not (to be defined shortly), we may save either the (secret) output of the unlearning algorithm as persistent state, or save only the (public) output of the publishing function. Definition 2.2 (D i , θ i ,θ i ,θ i ). Fix any pair (A, R A ) of learning and unlearning algorithms, any publishing function f publish , any dataset D, and any update sequence U = (u 1 , u 2 , . . .). We write D 0 = D and for any i ≥ 1, D i = D i−1 • u i . For any i ≥ 1, we write θ i for the model input to the unlearning algorithm R A on round i. We writeθ 0 = A (D 0 ), and for any i ≥ 1,θ i = R A (D i−1 , u i , θ i ). For any i ≥ 0, we defineθ i = f publish (θ i ). In other words, whenever A, R A , f publish , D, and U are clear from context, we write {D i } i≥0 to represent the sequence of updated datasets, {θ i } i≥1 for the sequence of input models to R A , {θ i } i≥0 to denote the (secret) output models of A and R A , and {θ i } i≥0 to denote their corresponding sequence of published models. Our (ǫ, δ)-unlearning notion is similar to the deletion notion proposed in [GGVZ19] but generalizes it to an update sequence consisting of both additions and deletions. Definition 2.3 ((ǫ, δ)-indistinguishability). Let X and Y be random variables over some domain Ω. We say X and Y are (ǫ, δ)-indistinguishable and write X ǫ,δ ≈ Y , if for all S ⊆ Ω, Pr [X ∈ S] ≤ e ǫ Pr [Y ∈ S] + δ, Pr [Y ∈ S] ≤ e ǫ Pr [X ∈ S] + δ Definition 2.4 ((ǫ, δ)-unlearning). We say that R A is an (ǫ, δ)-unlearning algorithm for A with respect to a publishing function f publish , if for all data sets D and all update sequences U = (u i ) i , the following condition holds. For every update step i ≥ 1, for θ i =θ i−1 f publish (R A (D i−1 , u i , θ i )) ǫ,δ ≈ f publish (A (D i )) If the above condition holds for θ i =θ i−1 , R A is an (ǫ, δ)-perfect unlearning algorithm for A. Remark 2.5. Observe that an unlearning algorithm takes as input the model output by the previous round's unlearning algorithm, whereas a perfect unlearning algorithm takes as input the model output by the previous round's publishing algorithm. Since we require that the published outputs satisfy (ǫ, δ)-indistinguishability, this means that unlearning algorithms may need to maintain secret state that does not satisfy the indistinguishability guarantee, but that perfect unlearning algorithms do not need to. Assumption 2.6. For notational simplicity (so that we can state asymptotic bounds in terms of n) We assume throughout that over the course of an update sequence, the size of the updated datasets never drops below n/2 where n is the size of the original training dataset: ∀ i, n i ≥ n/2 where n i is the size of D i . Note that this is consistent with update sequences being of arbitrary length, since we allow additions as well as deletions. This assumption is not necessary, but otherwise bounds would have to be stated in terms of n i . Learning Framework: ERM We consider an Empirical Risk Minimization (ERM) setting in this paper where models are (parameter) vectors in d-dimensional space R d equipped with the (Euclidean) ℓ 2 -norm which will be denoted by · 2 . Let Θ ⊆ R d be a convex and closed subset of R d , and let D = sup θ,θ ′ ∈Θ θ − θ ′ 2 be the diameter of Θ. We denote a loss function by a mapping f : Θ × Z → R that takes as input a parameter θ ∈ Θ and a data point z ∈ Z, and outputs the loss of θ on z, f (θ, z) -which we may also denote by f z (θ). Given a dataset D = {z i } n i=1 ∈ Z n , with slight abuse of notation, let f D (θ) denote the empirical loss of θ on the dataset D. In other words, f D (θ) 1 n n i=1 f zi (θ) (1) Definition 2.7 ((α, β)-accuracy). We say a pair (A, R A ) of learning and unlearning algorithms is (α, β)accurate with respect to a publishing function f publish , if for every dataset D and every update sequence U, the following condition holds. For every i ≥ 0, Pr f Di (θ i ) − min θ∈Θ f Di (θ) > α < β Definition 2.8 (strong vs. weak unlearning). Fix any pair (A, R A ) of learning and unlearning algorithms that satisfy (α, β)-accuracy with respect to some publishing function f publish . Let C i represent the overall computational cost of the unlearning algorithm at step i of the update. We say R A is a "strong" unlearning algorithm for A if 1. α and β are independent of the length of the update sequence, and 2. For every i ≥ 1, C i /C 1 = O (log(i)), i.e., the computation cost of the unlearning algorithm must grow at most logarithmically with i. If (1) holds and ∀i ≥ 1, C i /C 1 = Ω (poly(i)), we say R A is a "weak" unlearning algorithm for A. We remark that we have defined update sequences as if they are non-adaptively chosen, but that our basic algorithms in Section 3 have guarantees also for adaptively chosen update sequences. Loss Function Properties Definition 2.9 (Strong Convexity). A function h : Θ → R is said to be m-strongly convex for some m ≥ 0, if for any θ 1 , θ 2 ∈ Θ, and any t ∈ (0, 1), h (tθ 1 + (1 − t)θ 2 ) ≤ th(θ 1 ) + (1 − t)h(θ 2 ) − m 2 t(1 − t) θ 1 − θ 2 2 2 if the above condition holds for m = 0, we say h is convex. Definition 2.10 (Lipschitzness). A function h : Θ → R is said to be L-Lipschitz if for all θ 1 , θ 2 ∈ Θ, |h(θ 1 ) − h(θ 2 )| ≤ L θ 1 − θ 2 2 Definition 2.11 (Smoothness). A function h : Θ → R is said to be M -smooth, if it is differentiable and for all θ 1 , θ 2 ∈ Θ, ∇h(θ 1 ) − ∇h(θ 2 ) 2 ≤ M θ 1 − θ 2 2 Strong Convexity and Sensitivity Throughout the paper we will leverage the fact that the optimizers of strongly convex functions have low sensitivity to individual data points. We will formally state this fact in Lemma 2.12 and defer its proof to Appendix B. Lemma 2.12 (Sensitivity). Suppose for any z ∈ Z, f z is L-Lipschitz and m-strongly convex. For any dataset D, let θ * D argmin θ∈Θ f D (θ). We have that for any integer n, any data set D of size n, and any update u, θ * D − θ * D•u 2 ≤ 2L mn . Convergence Results for Gradient Descent We make use of projected gradient descent extensively throughout this paper. Here, we state two convergence results for gradient descent that we will use. A crucial feature of these bounds (and one not shared by all bounds for gradient descent and its variants) is that they improve as a function of how close our initial parameter is to the optimal parameter. Let h : Θ → R where Θ ⊆ R d is convex, closed, and bounded. Our goal is to approximate min θ∈Θ h(θ). The Gradient Descent (GD) algorithm starts with an initial point θ 0 ∈ Θ and proceeds as follows: ∀ t ≥ 1 : θ t = Proj Θ (θ t−1 − η t ∇h(θ t−1 )) Proj Θ (θ) = argmin θ ′ ∈Θ θ − θ ′ 2 is a projection onto Θ, and η t is the step size used in round t. Theorem 2.13 (Strongly Convex and Smooth [Che19]). Let h be m-strongly convex and M -smooth, and let θ * = argmin θ∈Θ h(θ). We have that after T steps of GD with step size η t = 2 m+M , θ T − θ * 2 ≤ M − m M + m T θ 0 − θ * 2 Theorem 2.14 (Convex and Smooth [ABD17]). Let h be convex and M -smooth, and let θ * ∈ argmin θ∈Θ h(θ). We have that after T steps of GD with step size η t = 1 M , h(θ T ) − min θ∈Θ h(θ) ≤ M θ 0 − θ * 2 2 2T Basic Perturbed Gradient Descent A key building block for our main result (and a simple and effective deletion scheme in its own right, that requires fewer assumptions than our main result) is perturbed gradient descent. The basic idea is as follows, for both the training algorithm and the deletion algorithm: we will perform gradient descent updates until we are guaranteed that we have found aθ t which is within Euclidean distance α of the optimizer, for some small α. Our publishing algorithm f publish adds Gaussian noise scaled as a function of α to every coordinate. This guarantees (ǫ, δ)-indistinguishability with respect to any other parameter that is within distance α of the optimizer -and hence between the outcomes of full retraining and updating. Depending on whether we want a perfect deletion algorithm or not, we save either the perturbed or unperturbed parameter as our initialization point for the next update. Our update algorithm will be the same as our training algorithm -except that it will be initialized at the learned parameter from the previous round, which will guarantee faster convergence. This is becauseif we allow secret state -the initialization parameter will be within α of the optimizer before the update, and if f is strongly convex, within O(α + 1 mn ) of the optimal parameter after the update by the sensitivity Lemma 2.12. If we require a perfect deletion algorithm, we will necessarily need to start further from the optimizer, because our saved state will have been additionally perturbed with Gaussian noise. Here we leverage the fact that gradient descent converges quickly when its initialization point is near the optimal solution. This algorithm relies crucially on leveraging strong convexity, which guarantees us that updates only change the empirical risk minimizer by a small amounts in parameter space. In Section 3.2 we solve the non-strongly-convex case by adding a strongly convex regularizer. θ ′ t = Proj Θ θ ′ t−1 − η t ∇f D (θ ′ t−1 ) 5: Output:θ 0 = θ ′ T ⊲ Secret output We parameterize our results by the computational cost of the update operations, and we can trade off run-time for accuracy. We measure computational cost by gradient computations. In this section, we parameterize our strong unlearning algorithms by the number of iterations I that they run for, which corresponds to a budget of ≈ nI gradient computations per update. For weak unlearning algorithms, this is the number of iterations they run for at their first update. Algorithm 2 R A : ith Unlearning for Perturbed Gradient Descent 1: Input: dataset D i−1 , update u i , model θ i 2: Update dataset D i = D i−1 • u i 3: Initialize θ ′ 0 = θ i 4: for t = 1, 2, . . . T i do 5: θ ′ t = Proj Θ θ ′ t−1 − η t ∇f Di (θ ′ t−1 ) 6: Outputθ i = θ ′ Ti ⊲ Secret output 1: Input:θ ∈ R d 2: Draw Z ∼ N 0, σ 2 I d 3: Output:θ =θ + Z ⊲ Public output Perturbed GD Analysis: Strongly Convex Loss In this section we analyze Algorithms 1 and 2 in the case when f is m-strongly convex. Let the learning algorithm A (Algorithm 1) run with η t = η and T ≥ I + log( Dmn 2L )/ log (1/γ) where n is the size of the input dataset, and let the unlearning algorithm R A (Algorithm 2) run with input models θ i ≡θ i−1 and η t = η and T i = I iterations, for all i ≥ 1. Let the unlearning parameters ǫ and δ be such that ǫ = O (log (1/δ)), and let σ = 4 √ 2Lγ I mn (1 − γ I ) log (1/δ) + ǫ − log (1/δ) in f publish (Algorithm 3). We have that 1. Unlearning: R A is a strong (ǫ, δ)-unlearning algorithm for A with respect to f publish . 2. Accuracy: for any β, (A, R A ) is (α, β)-accurate with respect to f publish where α = O M L 2 γ 2I d log (1/δ) log 2 (d/β) (1 − γ I ) 2 m 2 ǫ 2 n 2 Proof of Theorem 3.1. We first prove the unlearning guarantee. Fix a training dataset D of size n and an update sequence U = (u i ) i . Recall from Definition 2.2 the notation we use: {D i } i≥0 for the sequence of updated datasets according to the update sequence U, {θ i } i≥0 for the sequence of secret non-noisy parameters, and {θ i } i≥0 for the sequence of published noisy parameters. We also use n i to denote the size of D i . Note that n 0 = n and that by Assumption 2.6, n i ≥ n/2 for all i. Let θ * i argmin θ f Di (θ) denote the optimizer of f Di , for any i ≥ 0. We have that for any i ≥ 0, f publish (A (D i )) ∼ N µ i , σ 2 I d , where it follows by the convergence guarantee of Theorem 2.13 that µ i − θ * i 2 ≤ γ T θ ′ 0 − θ * i 2 = 2Lγ I θ ′ 0 − θ * i 2 Dmn i ≤ 4L mn · γ I(2) We also have that for any i ≥ 1, f publish (R A (D i−1 , u i , θ i )) ∼ N µ ′ i , σ 2 I d where µ ′ i − θ * i 2 ≤ 4L mn · γ I 1 − γ I(3) We use induction on i to prove this claim. Let's focus on the base case i = 1. We have that µ ′ 1 − θ * 1 2 ≤ γ I θ 0 − θ * 1 2 ≤ γ I θ 0 − θ * 0 2 + θ * 0 − θ * 1 2 ≤ γ I 4L mn · γ I 1 − γ I + 4L mn = 4L mn · γ I 1 − γ I The first inequality follows from Theorem 2.13 and the fact that when running Algorithm 2 for the first update i = 1, the initial point θ ′ 0 = θ 1 ≡θ 0 saved by the training algorithm. The second inequality is a simple triangle inequality, and the third follows from Equation (2) (noting thatθ 0 ≡ µ 0 ) and the sensitivity Lemma 2.12. Let's move on to the induction step of the argument. Suppose Equation (3) holds for some i ≥ 1. We will show that it holds for (i + 1) as well. We have that µ ′ i+1 − θ * i+1 2 ≤ γ I θ i − θ * i+1 2 ≤ γ I θ i − θ * i 2 + θ * i − θ * i+1 2 ≤ γ I 4L mn · γ I 1 − γ I + 4L mn = 4L mn · γ I 1 − γ I The first inequality follows from Theorem 2.13 and the fact that when running Algorithm 2 for the (i + 1)th update, the initial point θ ′ 0 = θ i+1 =θ i saved by the previous run of the unlearning algorithm. The second inequality is a simple triangle inequality, and the third follows from the induction assumption for i (noting thatθ i ≡ µ ′ i ), the sensitivity Lemma 2.12, and the assumption that n i ≥ n/2. We therefore have shown that for any i ≥ 1, for θ i ≡θ i−1 f publish (A (D i )) ∼ N µ i , σ 2 I d , f publish (R A (D i−1 , u i , θ i )) ∼ N µ ′ i , σ 2 I d where Equations (2) and (3) imply µ i − µ ′ i 2 ≤ ∆ 8L mn · γ I 1 − γ I It follows from Lemma A.3 that R A is a ( ∆ 2 2σ 2 + ∆ σ 2 log (1/δ), δ)-unlearning algorithm for A, where, with σ specified in the theorem statement, we get (ǫ, δ)-unlearning guarantee. Now let's prove the accuracy statement of the theorem. We will make use of Equations (2) and (3) and a Gaussian tail bound (see Lemma A.2). Recall that for any i ≥ 0, the published outputθ i =θ i + Z, and thatθ 0 ≡ µ 0 andθ i ≡ µ ′ i for i ≥ 1. We therefore have that, for any β, and for any update step i ≥ 0, Pr θ i − θ * i 2 ≥ 4L mn · γ I 1 − γ I + σ √ 2d log (2d/β) ≤ β The choice of σ in the theorem and the fact that for ǫ = O (log (1/δ)), we have log (1/δ) + ǫ− log (1/δ) = Ω(ǫ/ log (1/δ)), imply that for any i ≥ 0, with probability at least 1 − β, θ i − θ * i 2 = O Lγ I d log (1/δ) log (d/β) (1 − γ I ) ǫmn(4) Finally, since f z is M -smooth for all z, we get that for any update step i ≥ 0, with probability at least 1 − β, f Di (θ i ) − f Di (θ * i ) ≤ M 2 θ i − θ * i 2 2 = O M L 2 γ 2I d log (1/δ) log 2 (d/β) (1 − γ I ) 2 m 2 ǫ 2 n 2 The same algorithm can be analyzed as a perfect unlearning algorithm (i.e. without maintaining secret state). It obtains the same asymptotic tradeoff between running time and accuracy, under the condition that the per-update run-time is at least logarithmic in the relevant parameters. Intuitively, this run-time lower bound is required so that the update algorithm can "recover" from the effect of the added noise. (1/δ)). Let the learning algorithm A (Algorithm 1) run with η t = η and T ≥ I + log( Dmn 2L )/ log (1/γ) where n is the size of the input dataset, and let the unlearning algorithm R A (Algorithm 2) run with input models θ i ≡θ i−1 and η t = η and T i = I + log (log (4di/δ)) / log (1/γ) iterations for all i ≥ 1 where I ≥ log √ 2d (1−γ) −1 √ 2 log(2/δ)+ǫ− √ 2 log(2/δ) log (1/γ) , and σ = 8Lγ I 1 − γ I −1 mn 2 log (2/δ) + 3ǫ − 2 log (2/δ) + 2ǫ in f publish (Algorithm 3). We have that 1. Unlearning: R A is a strong (ǫ, δ)-perfect unlearning for A with respect to f publish . 2. Accuracy: for any β, (A, R A ) is (α, β + δ)-accurate with respect to f publish where α = O M L 2 γ 2I d log (1/δ) log 2 (d/β) (1 − γ I ) 2 m 2 ǫ 2 n 2 The proof of this theorem can be found in Appendix D. Convex Loss: Regularized Perturbed GD If our loss function is not strongly convex, we can regularize it to enforce strong convexity, and apply our algorithms to the regularized loss function. When we do this, we must manage a basic tradeoff: the more aggressively we regularize the loss function, the less sensitive it will be, and so the less noise we will need to add in our f publish routine. This reduced noise will increase accuracy. On the other hand, the more aggressively we regularize, the less well the optimizer of the regularized loss function will optimize the original loss function of interest, which will decrease accuracy. More aggressive regularization will also degrade the Lipschitz and smoothness guarantees of the loss function. We choose our regularization parameter carefully to trade off these various sources of error. Suppose in this section, without loss of generality, that Θ contains the origin: 0 ∈ Θ. This will imply that sup θ∈Θ θ 2 ≤ D where D is the diameter of Θ, as before. Our strategy is to regularize f so as to make it strongly convex, and have our learning and unlearning algorithms run on the regularized version of f . Let, for any z ∈ Z and any θ ∈ Θ, for some m > 0, where σ is the noise level in f publish . We have that 1. Unlearning: R A is a strong (ǫ, δ)-unlearning algorithm for A with respect to f publish . g z (θ) f z (θ) + m 2 θ 2 2 (5) Accuracy: for any β, (A, R A ) is (α, β)-accurate with respect to f publish where α = O   M 3 2 LD 4 d log (1/δ) ǫnI 2 5 log 2 (d/β)   + O n − 4 5 + O n − 6 5 Proof of Theorem 3.4. The unlearning guarantee of the theorem holds for any m > 0, and follows from Theorem 3.1 by the choice of σ in the theorem statement. Let's prove the accuracy statement. Let θ * r i = argmin θ∈Θ g Di (θ) denote the optimizer of the regularized g Di , for all i ≥ 0. It follows from the proof of Theorem 3.1 (see Equation (4)) that for any update step i ≥ 0, with probability 1 − β, θ i − θ * r i 2 = O (L + mD) γ I d log (1/δ) log (d/β) (1 − γ I ) ǫmn(6) Also note that γ I 1 − γ I = 1 (1 + 2 (m/M )) I − 1 ≤ M mI (7) Now let θ * i ∈ argmin θ∈Θ f Di (θ) denote an optimizer of the original loss function f Di , for any i ≥ 0. We have that, for any i ≥ 0, f Di (θ i ) − f Di (θ * i ) = f Di (θ i ) − f Di (θ * r i ) + f Di (θ * r i ) − f Di (θ * i ) = O M 3 (L + mD) 2 d log (1/δ) log 2 (d/β) m 4 ǫ 2 n 2 I 2 + mD 2(8) where inequality (1) follows from f Di being M -smooth. (2) follows from the fact that for all θ, ∇f Di (θ) = ∇g Di (θ) − mθ and that by optimality of θ * r i for g Di , we have ∇g Di (θ * r i ) = 0. (3) follows from a simple application of Cauchy-Schwarz: for all θ 1 , θ 2 ∈ Θ, we have θ ⊤ 1 θ 2 ≤ θ 1 2 θ 2 2 ≤ D 2 . (4) follows from the optimality of θ * r i for g Di , and (5) is implied by Equations (6) and (7), and it holds with probability 1 − β. Now for the choice of m in the theorem, we conclude that for any i ≥ 0, with probability 1 − β, f Di (θ i ) − f Di (θ * i ) = O   M 3 2 LD 4 d log (1/δ) ǫnI 2 5 log 2 (d/β)   + O n − 4 5 + O n − 6 5 If our goal is to satisfy only weak unlearning (i.e. to allow our run-time to grow with the length of the update sequence), we can obtain error bounds that have a better dependence on n. Theorem 3.5 (Accuracy, Unlearning, and Computation Tradeoffs). Suppose for all z ∈ Z, the loss function f z is convex, L-Lipschitz, and M -smooth, and let g z be defined as in Equation (5) for some m specified later. Define γ M/(M + 2m) and η 2/(M + 2m). Let the learning algorithm A (Algorithm 1) run on the regularized g with η t = η and T ≥ I + log( Dmn 2L )/ log (1/γ) where n is the size of the input dataset, and let the unlearning algorithm R A (Algorithm 2) run on the regularized g with input model θ i ≡θ i−1 and η t = η and T i = i 2 · I iterations, for the ith update. Let the unlearning parameters ǫ and δ be such that ǫ = O (log (1/δ)), and let σ = 2 √ 2M (L + mD) m √ mIn log (1/δ) + ǫ − log (1/δ) , m = LM d log (1/δ) Dǫn √ I where σ is the noise level in f publish . We have that 1. Unlearning: R A is a weak (ǫ, δ)-unlearning algorithm for A with respect to f publish . 2. Accuracy: for any β, (A, R A ) is (α, β)-accurate with respect to f publish where α = O   M LD 3 d log (1/δ) ǫn √ I log 2 (d/β)   + O n −1 + O n − 3 2 Remark 3.6. We remark that we can further explore the tradeoff between each update's runtime T i and dependence on sample size n. Let ξ ≥ 1 be any constant (Theorem 3.5 corresponds to ξ = 1). We have that under the setting of Theorem 3.5, with T i = i 2ξ · I iterations, and σ = 2 √ 2M 1 2ξ (L + mD) m(mI) 1 2ξ n log (1/δ) + ǫ − log (1/δ) , m = L 2 M 1+ξ ξ d log (1/δ) D 2 ǫ 2 n 2 I 1 ξ ξ 3ξ+1 1. Unlearning: R A is a weak (ǫ, δ)-unlearning algorithm for A with respect to f publish . Accuracy: for any β, (A, R A ) is (α, β)-accurate with respect to f publish where α = O   M 1+ξ ξ L 2 D 2+4ξ ξ d log (1/δ) ǫ 2 n 2 I 1 ξ ξ 3ξ+1 log 2 (d/β)   + O n − 4ξ 3ξ+1 + O n − 6ξ 3ξ+1 The proof of Theorem 3.5 can be found in Appendix E. Perturbed Distributed Descent Our next algorithm obtains additional running time improvements for sufficiently high dimensional data. The basic idea is as follows: we randomly partition the dataset into K parts, separately optimize to find a model that approximates the empirical risk minimizer on each part, and then take the average of each of the K models. Zhang et al [ZDW12] analyze this algorithm and show that its out of sample guarantees match the out of sample guarantees of non-distributed gradient descent, whenever K ≤ √ n. For us, this algorithm has a key advantage: the element involved in an update will only appear in a small number of the partitions, and we only need to update the parameters corresponding to those partitions. Our algorithm will improve over basic gradient descent because those partitions are smaller in size than the entire dataset by a factor of K, and hence our run-time budget of nI gradient computations will allow us to perform more than I gradient descent operations per modified partition. We provide deletion guarantees by using a publishing function that adds noise to the average of the K parameters. There are several difficulties that we must overcome. Primary among these is that the analysis of [ZDW12] provides out of sample guarantees for a dataset that is drawn i.i.d. from some fixed distribution. In our case (because our dataset results from an arbitrary and possibly adversarial sequence of additions and deletions), there is no distribution from which the dataset is drawn. To deal with this, our initial training algorithm does not directly partition the dataset, but instead draws a bootstrap sample (i.e. a sample with replacement) from the empirical distribution defined by the dataset, so that the "out of sample" guarantees of [ZDW12] correspond to empirical risk bounds in our case. Because the accuracy analysis depends on this distributional property, as updates come in, before we use gradient descent to update the models corresponding to the appropriate partitions, we must apply a form of reservoir sampling to guarantee that each partition continues to be distributed as a set of samples drawn i.i.d. from the empirical distribution defined by the current dataset (i.e. after the update). This is also crucial to our unlearning guarantee. Finally, the basic instantiation of this algorithm only gives guarantees on the expected error of the learned model [ZDW12], and we want high probability guarantees. To achieve these, we run C = O(log(1/β)) copies of the algorithm in parallel, and at every round, only publish a noisy version of the parameter achieving the lowest loss among all C candidates. We now go into more detail. To facilitate the technical development in this section, we introduce some notation: Definition 4.1. Fix any update round i ≥ 0. In this section we use S i = (S ij ) K j=1 for the partitioned dataset at round i. We use S i (unbold) to denote the union of partitions in S i and D i for the unique data points in S i (i..e D i removes the duplicates in S i which results from our sampling scheme). We useθ i = (θ ij ) K j=1 for the learned parameters in each partition.θ i = f publish (θ i ) represents the published model of round i. In this section, the unlearning algorithm for update i takes as input the partitioned dataset of previous round S i−1 , an update u i , and the learned models of previous roundθ i−1 , and outputs the updated modelsθ i and the updated datasets S i for use in the next update. Draw S ∼ P B (D). ⊲ Bootstrap B data points. 4: Partition S randomly into K equally-sized datasets: S 0,l = (S j ) K j=1 . 5: for j = 1, 2, . . . , K do 6: Initialize θ ′ 0 ∈ Θ. 7: for t = 1, 2, . . . T do 8: θ ′ t = Proj Θ θ ′ t−1 − η t ∇f Sj (θ ′ t−1 ) 9:θ j = θ ′ T 10:θ 0,l = (θ j ) K j=1 ⊲ l'th set of models. 11: Call f publish (θ 0,l * ) where l * = argmin l f D (avg(θ 0,l )). ⊲ Publish the best model. Throughout we denote the distribution on datasets of size B sampled with replacement from D by P B (D). We need to maintain the condition that the marginal distribution of the sampled dataset S i at round i is P B (D i ). To do this, at each update, we iteratively update each partition using a technique called reservoir sampling with replacement (that we need to extend to handle both additions and deletions). The algorithm S B rep is detailed below. Lemma 4.2. Fix any training dataset D and any non-adaptively chosen update sequence U. Let S 0 ∼ P B (D) (as in the learning algorithm) and for every i ≥ 1, S i ∼ S B rep (S i−1 , u i ) (as in the unlearning algorithm). We have that for all i ≥ 0: S i d = P B (D i ). Algorithm 5 S B rep : Reservoir Sampling with Replacement for ith update Input: Subsample S i−1 , update u i = (z i , • i ). S i = S i−1 . if • i = ′ add ′ then Draw N ∼ Binomial(B, n −1 i ). ⊲ n i : size of D i . Pick distinct indices i 1 , . . . , i N at random from [B]. for k = 1, 2, . . . , N do Replace z i k with z i in S i . else for z k ∈ S i : z k = z i do Replace z k with z ∼ P(D i ) in S i ⊲ P(D i ): empirical distribution of D i . Output: S i . 5: Let (S i,j ) K j=1 ≡ S i,l , (S i−1,j ) K j=1 ≡ S i−1,l , (θ i−1,j ) K j=1 ≡θ i−1,l . 6: Let ind = {j : S i−1,j = S i,j } ⊲ Modified partitions. 7: for j = 1, 2, . . . , K do 8: if j ∈ ind then 9: Initialize θ ′ 0 =θ i−1,j 10: for t = 1, 2, . . . , T = KnTi B|ind| do 11: θ ′ t = Proj Θ θ ′ t−1 − η t ∇f Si,j (θ ′ t−1 ) 12:θ i,j = θ ′ T 13: else 14:θ i,j =θ i−1,j 15:θ i,l = (θ i,j ) K j=1 ⊲ l'th set of models. 16: Call f publish (θ i,l * ) where l * = argmin l f Di (avg(θ i,l )) ⊲ Publish the best model. 17: Output:θ i = (θ i,l ) C l=1 , S i = (S i,l ) C l=1 . ⊲ For use in next update. 1: Input:θ = (θ j ) K j=1 2: Draw Z ∼ N 0, σ 2 I d 3: Output:θ = avg(θ) + Z ⊲ avg(·) averages input models. We now state the accuracy and strong unlearning bounds for perturbed distributed gradient descent. The convergence analysis on each partition is similar to the analysis in the proof of Theorem 3.1, with the added complexity of handling the number of partitions updated at each round, and the number of duplicated points (that could possibly be removed) in each partition. In order to obtain accuracy bounds we need to leverage an accuracy bound for the averaged parameter in a distributed setting, which we quote below from [ZDW12]. They remark that the required assumptions hold in most common settings, including in linear and logistic regression as long as the data distribution satisfies standard regularity conditions. Theorem 4.4 (Corollary 2 of [ZDW12]). Let θ * avg = K −1 K j=1 θ * j , where θ * j are the empirical risk minimizers on partition j of a dataset of size B sampled i.i.d. from some distribution P. Let θ * = argmin θ∈Θ E z∼P [f z (θ)]. Then under the assumption that f z is m-strongly convex for all z, and satisfies the following smoothness conditions for all θ ∈ Θ: E z∼P ∇f z (θ) 8 2 ≤ L 8 , E z∼P ∇ 2 f z (θ) − ∇ 2 E z∼P [f z (θ)] 8 2 ≤ H 8 , and the Hessian matrix ∇ 2 f z (·) is G-Lipschitz continuous for all z, then, for some constant c: log (2/δ) + ǫ − log (2/δ) in f publish (Algorithm 7). We have that 1. Unlearning: R A is a strong (ǫ, δ)-unlearning algorithm for A with respect to f publish . E θ * avg − θ * 2 2 ≤ 2L 2 m 2 B + cK 2 L 2 m 4 B 2 H 2 log d + L 2 G 2 m 2 + O K B 2 + O K 3 B 3 2. Accuracy: for any β, letting C = log (2/β) / log 2, we get that (A, R A ) is (α, β)-accurate with respect to f publish where α = O   M L 2 γ 2In 4−3ξ 2 d log (1/δ) log 2 (d/β) m 2 (1 − γ) 2 ǫ 2 n 2   + O log d n ξ + O 1 n 3ξ 2 Remark 4.6. For any 1 ≤ ξ ≤ 4/3: α =Õ   γ In 4−3ξ 2 d ǫ 2 n 2 + 1 n ξ   This improves over the bound of Theorem 3.1 (Basic Perturbed GD), whenever d =Ω ǫ 2 n 2−ξ γ I − γ In 4−3ξ 2 The proof of Theorem 4.5 can be found in Appendix F. Lemma A.4. Let X ≥ 0 be any random variable drawn from a distribution P, with finite expectation µ = E X∼P [X]. Let X 1 , . . . X N iid ∼ P. Then if X min min j X j , for N ≥ log(1/δ) log 2 , with probability at least 1 − δ: X min < 2µ. Proof. By Markov's inequality, for any X j , Pr [X j ≥ 2µ] ≤ 1 2 . Hence, Pr [X min ≥ 2µ] = N j=1 Pr [X j ≥ 2µ] ≤ 1 2 N ≤ 1 2 log(1/δ) log 2 = δ as desired. Lemma A.5 (Chernoff Bound). Let X ∼ Binomial (m, p), and let µ = mp. Then for any δ ′ ≥ 0, Pr [X ≥ (1 + δ ′ )µ] ≤ e − µδ ′2 2+δ ′ B Proof of Sensitivity Lemma 2.12 To prove Lemma 2.12, we will need the following claim. Claim B.1. Suppose h : Θ → R is m-strongly convex and let θ * = argmin θ∈Θ h(θ). We have that for any θ ∈ Θ, h(θ) ≥ h(θ * ) + m 2 θ − θ * 2 2 . Proof. First, recall the definition of m-strong convexity: for any θ 1 , θ 2 ∈ Θ, and any t ∈ (0, 1), h(tθ 1 + (1 − t)θ 2 ) ≤ th(θ 1 ) + (1 − t)h(θ 2 ) − m 2 t(1 − t) θ 1 − θ 2 2 2 Now fix some θ ∈ Θ. We have that for any t ∈ (0, 1), h(θ * ) ≤ h(tθ + (1 − t)θ * ) ≤ th(θ) + (1 − t)h(θ * ) − m 2 t(1 − t) θ − θ * 2 2 where the first inequality follows because θ * is the minimizer of h, and the second is due to m-strong convexity of h. Rearranging the above inequality and dividing both sides by t, we get that for any t ∈ (0, 1), h(θ) ≥ h(θ * ) + m 2 (1 − t) θ − θ * 2 2 We therefore have that h(θ) ≥ h(θ * ) + m 2 sup t∈(0,1) (1 − t) θ − θ * 2 2 = h(θ * ) + m 2 θ − θ * 2 2 Proof of Lemma 2.12. Fix n, a data set D = {z i } n i=1 , and an update u = (z, •), and let D ′ = D • u. Assume • = ′ delete ′ . If z / ∈ D, then the claim immediately follows; so suppose z ∈ D. We have that f D (θ * D ′ ) = n − 1 n f D ′ (θ * D ′ ) + 1 n f z (θ * D ′ ) ≤ n − 1 n f D ′ (θ * D ) + 1 n f z (θ * D ′ ) = f D (θ * D ) + 1 n f z (θ * D ′ ) − 1 n f z (θ * D ) ≤ f D (θ * D ) + L n θ * D ′ − θ * D 2(9) where the first inequality follows by optimality of θ * D ′ for D ′ , and the second follows by L-Lipschitzness of f z . Note that since f D is m-strongly convex, Claim B.1 implies f D (θ * D ′ ) ≥ f D (θ * D ) + m 2 θ * D ′ − θ * D 2 2(10) Combining Equations (9) and (10) completes the proof for the case when • = ′ delete ′ . Note that when • = ′ add ′ , one can take u ′ (z, ′ delete ′ ), and use the bound for deletion to conclude that θ * D − θ * D•u 2 = θ * D•u − θ * (D•u)•u ′ 2 ≤ 2L mn C Proofs of Lemmas in Section 4 Proof of Lemma 4.2. We prove the claim by induction on i. For i = 0, S 0 is explicitly drawn from P B (D 0 ) and so the claim holds. Now assume the claim holds for i − 1. In the case of addition, where u i = (z i , ′ add ′ ) this is exactly what is known as "Reservoir Sampling with Replacement" and we refer the reader to [Vit85]. So we need only establish the claim for deletion updates. Let us perform an update u i = (z i , ′ delete ′ ). We show that after conditioning on u i , after the deletion update, each element of S i is independent and has marginal distribution P(D i−1 • u i ) = P(D i ), which will establish the claim. Conditioning on u i , let h ui : Z → Z be the randomized function: h ui (z) = z z = z i z ′ ∼ P(D i ) z = z i Then for any data point z l ∈ S i−1 , the corresponding element in S i is h ui (z l ). Since by assumption the {z l } = S i−1 are independent, since h ui is a fixed randomized function conditioned on u i , the {h ui (z l )} = S i are conditionally independent given u i . It remains to show that the marginal distribution of any z ′ l = h ui (z l ) is P(D i−1 • u i ) ≡ P(D i ). If z l = z i , then z ′ l ∼ P(D i ) by design. If z l = z i , then z ′ l = z l , and the distribution of z ′ l is z l |z l = z i , u i . Since U is a non-adaptive sequence of updates, z l |z l = z i , u i ∼ z l |z l = z i . Then by inductive assumption z l ∼ P(D i−1 ), and so the distribution of z l |z l = z i for z i ∈ D i−1 is uniform over D i−1 \ {z i } = D i−1 • u i = D i , which is exactly P(D i ), as desired. This establishes the induction. Proof of Lemma 4.3. At any round i of update, by Lemma 4.2, we know S i ∼ P B (D i ). By Assumption 2.6, n i ≥ n/2 where n i is the size of dataset D i . Hence for any data point z, the number of copies of z subsampled in S i is distributed as Binomial(B, p), where p ≤ 2/n. Let µ = (2B)/n and note that µ ≥ 1. Now by a Chernoff bound (see Lemma A.5) for a Bernoulli random variable, we get that for any i, the number of repeated points of any one type in S i (including the ones subject to update) satisfies, with probability 1 − δ ′ : s i ≤ µ + log 2 (1/δ ′ ) + 8µ log (1/δ ′ ) = µ   1 + log 2 (1/δ ′ ) µ 2 + 8 log (1/δ ′ ) µ   ≤ µ 1 + log 2 (1/δ ′ ) + 8 log (1/δ ′ ) ≤ 5µ log (1/δ ′ ) as desired. Note the last inequality follows because log (1/δ ′ ) ≥ 1 by assumption. D Proof of Theorem 3.2 Proof of Theorem 3.2. We first prove the unlearning guarantee. Fix a training dataset D of size n and an update sequence U = (u i ) i . Similar to the proof of Theorem 3.1, we first recall a few notations from Definition 2.2: {D i } i≥0 for the sequence of updated datasets according to the update sequence U, {θ i } i≥0 for the sequence of secret non-noisy parameters, and {θ i } i≥0 for the sequence of published noisy parameters. Let Z i denote the Gaussian noise added by f publish on round i of update, and recall thatθ i =θ i + Z i . We use n i (≥ n/2) to denote the size of D i . Let θ * i argmin θ f Di (θ) denote the optimizer of f Di . We have that for any i ≥ 0, f publish (A (D i )) ∼ N µ i , σ 2 I d , where it follows by the convergence guarantee of Theorem 2.13 that, µ i − θ * i 2 ≤ γ T θ ′ 0 − θ * i 2 = 2Lγ I θ ′ 0 − θ * i 2 Dmn i ≤ 4L mn · γ I(11) We also have that for any update step i ≥ 1, conditioned on the noise of previous rounds {Z 0 , . . . , Z i−1 }, f publish (R A (D i−1 , u i , θ i )) ∼ N µ ′ i , σ 2 I d , where for any β ′ > 0, Pr Z0,...,Zi−1 µ ′ i − θ * i 2 ≥ γ Ti 1 − γ I 4L mn + σ √ 2d log (2d/β ′ ) ≤ iβ ′(12) We use induction on i to prove this claim. Fix any β ′ . Let's focus on the base case i = 1. We have that µ ′ 1 − θ * 1 2 ≤ γ T1 θ 0 − θ * 1 2 ≤ γ T1 Z 0 2 + θ 0 − θ * 0 2 + θ * 0 − θ * 1 2 ≤ γ T1 4L mn γ I + 4L mn + σ √ 2d log (2d/β ′ ) ≤ γ T1 γ I 1 − γ I 4L mn + σ √ 2d log (2d/β ′ ) + 4L mn + σ √ 2d log (2d/β ′ ) = γ T1 1 − γ I 4L mn + σ √ 2d log (2d/β ′ ) The first inequality follows from Theorem 2.13 and the fact that when running Algorithm 2 for the first update i = 1, the initial point of the algorithm θ ′ 0 = θ 1 ≡θ 0 . The second inequality is a simple triangle inequality, and the third holds with probability at least 1 − β ′ and follows from Equation (11) (noting that θ 0 ≡ µ 0 ), the sensitivity Lemma 2.12, and a Gaussian tail bound for Z 0 (Lemma A.2). Let's move on to the induction step of the argument. Suppose Equation (12) holds for some i ≥ 1. We will show that it holds for (i + 1) as well. We have that µ ′ i+1 − θ * i+1 2 ≤ γ Ti+1 θ i − θ * i+1 2 ≤ γ Ti+1 Z i 2 + θ i − θ * i 2 + θ * i − θ * i+1 2 ≤ γ Ti+1 γ I 1 − γ I 4L mn + σ √ 2d log (2d/β ′ ) + 4L mn + σ √ 2d log (2d/β ′ ) = γ Ti+1 1 − γ I 4L mn + σ √ 2d log (2d/β ′ ) The first inequality follows from Theorem 2.13 and the fact that when running Algorithm 2 for the (i + 1)th update, the initial point of the algorithm θ ′ 0 = θ i+1 ≡θ i . The second inequality is a simple triangle inequality, and the third holds with probability at least 1 − (i + 1)β ′ and follows from the induction assumption for i (noteθ i ≡ µ ′ i and T i ≥ I), the sensitivity Lemma 2.12 (note n i ≥ n/2), and a Gaussian tail bound for Z i (Lemma A.2). Now with the choice of β ′ = δ/(2i), Equation (12) implies with probability at least 1 − δ/2 over the Gaussian noise draws {Z 0 , . . . , Z i−1 }, µ ′ i − θ * i 2 ≤ γ I 1 − γ I 4L mn + σ √ 2d (13) because γ Ti ≤ (log (4di/δ)) −1 γ I . We therefore have shown that for any i ≥ 1, conditioned on {Z 0 , . . . , Z i−1 } f publish (A (D i )) ∼ N µ i , σ 2 I d , f publish (R A (D i−1 , u i , θ i )) ∼ N µ ′ i , σ 2 I d where Equations (11) and (13) imply, with probability 1 − δ/2 over {Z 0 , . . . , Z i−1 }, µ i − µ ′ i 2 ≤ γ I 1 − γ I 4L mn + σ √ 2d + 4L mn · γ I ≤ 2γ I 1 − γ I 4L mn + σ √ 2d ∆ It then follows from Lemma A.3, as well as the choice of σ and the assumption on I in the theorem statement, that for any i ≥ 1, with probability 1 − δ/2 over {Z 0 , . . . , Z i−1 }, f publish (A (D i )) ǫ,δ/2 ≈ f publish (R A (D i−1 , u i , θ i )) Now we can apply Lemma A.1 to conclude that for any i ≥ 1, f publish (A (D i )) ǫ,δ ≈ f publish (R A (D i−1 , u i , θ i )) And this shows R A is an (ǫ, δ)-unlearning algorithm for A, as desired. Now let's prove the accuracy statement of the theorem. We will make use of Equations (11) and (13) and a Gaussian tail bound (see Lemma A.2). Recall that for any i ≥ 0, the published outputθ i =θ i + Z, and thatθ 0 ≡ µ 0 andθ i ≡ µ ′ i for i ≥ 1. We therefore have that, for any β, and for any update step i ≥ 0, Pr Z0,...,Zi θ i − θ * i 2 ≥ γ I 1 − γ I 4L mn + σ √ 2d + σ √ 2d log (2d/β) ≤ β + δ 2 The choice of σ in the theorem and the fact that for ǫ = O (log (1/δ)), we have log (1/δ) + ǫ− log (1/δ) = Ω(ǫ/ log (1/δ)), imply for any update step i ≥ 0, with probability at least 1 − β − δ/2, θ i − θ * i 2 = O Lγ I d log (1/δ) log (d/β) (1 − γ I ) ǫmn(14) Finally, since f z is M -smooth for all z, we get that for any update step i ≥ 0, with probability at least 1 − β − δ/2, f Di (θ i ) − f Di (θ * i ) ≤ M 2 θ i − θ * i 2 2 = O M L 2 γ 2I d log (1/δ) log 2 (d/β) (1 − γ I ) 2 m 2 ǫ 2 n 2 E Proof of Theorem 3.5 Proof of Theorem 3.5. We first prove the unlearning guarantee. Fix a training dataset D of size n and an update sequence U = (u i ) i . Recall from Definition 2.2 the notation we use: {D i } i≥0 for the sequence of updated datasets according to the update sequence U, {θ i } i≥0 for the sequence of secret non-noisy parameters, and {θ i } i≥0 for the sequence of published noisy parameters. We also use n i to denote the size of D i . Note that n 0 = n and that by Assumption 2.6, n i ≥ n/2 for all i. Let θ * i ∈ argmin θ f Di (θ) denote an optimizer of f Di . Let θ * r i = argmin θ∈Θ g Di (θ) denote the optimizer of the regularized loss g Di . Fact E.1. Note that for any positive integer T ′ , γ T ′ = 1 1 + 2 (m/M ) T ′ ≤ 1 1 + 2 (m/M ) T ′ ≤ M mT ′ (15) where the last inequality follows because for all x ≥ 0, 1 + x ≥ 2 √ x. Fact E.2 (Generalizing Fact E.1). In general, for any constant ξ ≥ 1 and any integer T ′ , we have γ T ′ = γ ξT ′ 1 ξ ≤ M mT ′ 1 2ξ(16) We will use Fact E.1 later on in the proof and we note that Remark 3.6 follows by using the more general Fact E.2. of Let L ′ L + mD which is the Lipschitz constant of the regularized loss function g. We have that for any i ≥ 0, f publish (A (D i )) ∼ N µ i , σ 2 I d , where it follows by the convergence guarantee of Theorem 2.13 that µ i − θ * r i 2 ≤ γ T θ ′ 0 − θ * r i 2 ≤ 2L ′ γ I θ ′ 0 − θ * r i 2 Dmn i ≤ 2L ′ mn i · γ I(17) We also have that for any i ≥ 1, f publish (R A (D i−1 , u i , θ i )) ∼ N µ ′ i , σ 2 I d where µ ′ i − θ * r i 2 ≤ 4L ′ mn · i · γ i 2 I(18) We use induction on i to prove this claim. Let's focus on the base case i = 1. We have that µ ′ 1 − θ * r 1 2 ≤ γ I θ 0 − θ * r 1 2 ≤ γ I θ 0 − θ * r 0 2 + θ * r 0 − θ * r 1 2 ≤ γ I 2L ′ mn · γ I + 2L ′ mn ≤ 4L ′ mn · γ I The first inequality follows from Theorem 2.13 and the fact that when running Algorithm 2 for the first update i = 1, the initial point θ ′ 0 = θ 1 ≡θ 0 saved by the training algorithm. The second inequality is a simple triangle inequality, and the third follows from Equation (17) (noting thatθ 0 ≡ µ 0 ) and the sensitivity Lemma 2.12. Let's move on to the induction step of the argument. Suppose Equation (3) holds for some i ≥ 1. We will show that it holds for (i + 1) as well. We have that µ ′ i+1 − θ * r i+1 2 ≤ γ (i+1) 2 I θ i − θ * r i+1 2 ≤ γ (i+1) 2 I θ i − θ * r i 2 + θ * r i − θ * r i+1 2 ≤ γ (i+1) 2 I 4L ′ mn · i · γ i 2 I + 4L ′ mn ≤ 4L ′ mn · (i + 1) · γ (i+1) 2 I The first inequality follows from Theorem 2.13 and the fact that when running Algorithm 2 for the (i + 1)th update, the initial point θ ′ 0 = θ i+1 ≡θ i saved by the previous run of the unlearning algorithm. The second inequality is a simple triangle inequality, and the third follows from the induction assumption for i (noting thatθ i ≡ µ ′ i ), the sensitivity Lemma 2.12, and the assumption that n i ≥ n/2. ∀ i ≥ 0, µ i − θ * r i 2 ≤ 4L ′ √ M m √ mIn , ∀ i ≥ 1, µ ′ i − θ * r i 2 ≤ 4L ′ √ M m √ mIn (19) We therefore have shown that for any i ≥ 1, f publish (A (D i )) ∼ N µ i , σ 2 I d , f publish (R A (D i−1 , u i , θ i )) ∼ N µ ′ i , σ 2 I d where Equation (19) implies µ i − µ ′ i 2 ≤ 8L ′ √ M m √ mIn ∆ It then follows from Lemma A.3 that R A is a ( ∆ 2 2σ 2 + ∆ σ 2 log (1/δ), δ)-unlearning algorithm for A, where, with σ specified in the theorem statement, we get (ǫ, δ)-unlearning guarantee. Now let's focus on the accuracy statement of the theorem. Note, similar to the proof of Theorem 3.1, the convergence bounds in Equation (19), the choice of σ in the theorem statement, as well as a Gaussian tail bound (Lemma A.2), imply that for any update step i ≥ 0, with probability at least 1 − β, θ i − θ * r i 2 = O √ M (L + mD) d log (1/δ) log (d/β) ǫm √ mIn(20) We therefore have that, using a similar analysis as in the proof of Theorem 3.4 (see Equation (8)), for any update step i ≥ 0, with probability 1 − β, f Di (θ i ) − f Di (θ * i ) = O M 2 (L + mD) 2 d log (1/δ) log 2 (d/β) m 3 ǫ 2 n 2 I + mD 2 Finally, with the choice of m in the theorem, f Di (θ i ) − f Di (θ * i ) = O   M LD 3 d log (1/δ) ǫn √ I log 2 (d/β)   + O n −1 + O n − 3 2 F Proof of Theorem 4.5 Proof of Theorem 4.5. We first prove the unlearning guarantee. We note that the boosting of our algorithms (running multiple copies of algorithms and picking the best model for publishing) won't matter in our unlearning bounds. In fact, the unlearning guarantee holds for any set l of models learned by the algorithms because they have all sufficiently come close to their respective optimizers in each chunk. Hence, until we get to the proof of accuracy statement, we imagine the algorithms are run once. We will see how this boosting will be helpful to recover high probability accuracy guarantees from the accuracy bounds of [ZDW12] which are in expectation. Fix a training dataset D of size n and a non-adaptively chosen update sequence U = (u i ) i . Similar to our previous proofs, we first recall a few notations (from Definition 4.1), as well as some new notations for our proof: • {D i } i≥0 for the sequence of updated datasets. We use n i (≥ n/2) to denote the size of D i . • {S i = (S ij ) K j=1 } i≥0 for the sequence of partitioned subsampled datasets. • {θ i = (θ ij ) K j=1 } i≥0 for the sequence of learned parameters in each partition. • {θ i,avg } i≥0 for the sequence of averaged learned parameters:θ i,avg = K −1 K j=1θ ij . • {θ i = f publish (θ i ) =θ i,avg + Z i } i≥0 for the sequence of published parameters. • {θ * i } i≥0 is the sequence of target optimizers: θ * i argmin θ f Di (θ). • {θ * i = (θ * ij ) K j=1 } i≥0 is the sequence of optimizers for partitions: θ * ij argmin θ f Sij (θ). • {θ * i,avg } i≥0 is the average of optimizers for partitions: θ * i,avg = K −1 K j=1 θ * ij . • {s i } i≥1 for the sequence of number of affected data points in the whole dataset, i.e., s i shows how many points differ between S i and S i−1 . We will also make use of notation s ij which shows how many points differ between S ij and S i−1,j . Note that s i = K j=1 s ij . Fact F.1. Lets i max l≤i s l . We have by Lemma 4.3 that for any i, with probability at least 1 − δ/2 over the sampling randomness up to round i,s i ≤ 10B n log (2i/δ). We condition on this high probability event throughout the proof. Fact F.2. We also work with general K and B for now and eventually we use the ones stated in the theorem. We note that for general K and B, we can write T ≥ Kn 2 I B 2 + log DmL −1 B (1 + 10 log (2/δ)) log (1/γ) and T i = 10 log (2i/δ) I + B 2 Kn 2 · log (1 + 10i log (2i/δ)) log (1/γ) Let T ′ i be the number of iterations in affected partitions on round i. We have that with probability at least 1 − δ/2, by Fact F.1, T ′ i ≥ Kn Bs i T i ≥ Kn 2 10B 2 log (2i/δ) T i ≥ log (1 + 10i log (2i/δ)) log (1/γ) + Kn 2 I B 2 (21) Fact F.3. We have that B ≥ n, and Kn 2 ≥ B 2 (note these are justified by the setting of these parameters in theorem statement). We will use these later on in the proof. For every i ≥ 1, let S ′ i be the partitioned dataset we would have had we retrained (using our learning algorithm A) on dataset D i , and note that by Lemma 4.2, S ′ i and S i are distributed identically. To apply Lemma 4.2 we have used the fact that U is a non-adaptive sequence of updates selected independently of any internal randomness of R A . Now let C i be a coupling of the pair (S ′ i , S i ) such that S ′ i = S i with probability one. Throughout the proof when we condition on any of S ′ i or S i being drawn from their distribution, we will think of these datasets being drawn from the coupling C i so that we are always guaranteed S ′ i = S i . Let's start proving the unlearning guarantees. For any i ≥ 0, conditioned on the draw of S ′ i , we have that f publish (A (D i )) ∼ N µ i , σ 2 I d , where µ i = K −1 K j=1 µ ij and that it follows by the convergence guarantee of Theorem 2.13 that, for all partitions j, µ ij − θ * ij 2 ≤ γ T θ ′ 0 − θ * ij 2 ≤ 4Lγ Kn 2 I B 2 θ ′ 0 − θ * ij 2 DmB (1 + 10 log (2/δ)) ≤ 4L mB (1 + 10 log (2/δ)) · γ Kn 2 I B 2(22) We also have that for any update step i ≥ 1, with probability at least 1 − δ/2 over the randomness up to step i (draws of all S l for all l ≤ i), f publish (R A (S i−1 , u i , θ i )) ∼ N µ ′ i , σ 2 I d , where we first observe that µ ′ i =θ i,avg = K −1 K j=1θ ij , and furthermore, (23) m 2 B + cL 2 K 2 m 4 B 2 H 2 log d + L 2 G 2 m 2 + O K B 2 + O K 3 B 3 = 2L 2 m 2 B + cL 2 m 4 B H 2 log d + L 2 G 2 m 2 + O B − 3 2 = 1 B 2L 2 m 2 + cL 2 m 4 H 2 log d + L 2 G 2 m 2 + O B − 3 2(29) Putting together Equations (27) Now by Lemma A.4, we have that by running the algorithm for C = log (2/β) / log 2 times and picking the best model with smallest loss (note by strong convexity, the smaller the loss of a model is, the closer the model parameter is to the optimizer. Also for notational convenience, we still useθ i,avg for the best model), with probability at least 1 − β/2, θ i,avg − θ * i 2 2 = O      γ n 2 I B √ B n 2 1 − γ n 2 I B √ B 2      + O log d B + O 1 B 3 2(30) Recall that at any given round i ≥ 0, the published modelθ i =θ i,avg + Z i . We therefore have that by Equation (30), a Gaussian tail bound (Lemma A.2), choice of σ in the theorem statement, and the fact that for ǫ = O (log (1/δ)), we have log (1/δ) + ǫ− log (1/δ) = Ω(ǫ/ log (1/δ)), with probability at least 1−β, θ i − θ * Note that (1 − γ a ) −1 ≤ (1 − γ) −1 for any a ≥ 1 (in our case a = Kn 2 I B 2 ≥ 1). The proof is complete by the choice of B = n ξ and M -smoothness of f : f Di (θ i ) − f Di (θ * i ) ≤ M 2 θ i − θ * i 2 2 Algorithm 1 A: Learning for Perturbed Gradient Descent 1: Input: dataset D 2: Initialize θ ′ 0 ∈ Θ 3: for t = 1, 2, . . . T do 4: Theorem 3. 1 ( 1Accuracy, Unlearning, and Computation Tradeoffs). Suppose for all z ∈ Z, the loss function f z is m-strongly convex, L-Lipschitz, and M -smooth. Define γ (M − m)/(M + m) and η 2/(M + m). Theorem 3. 2 ( 2Perfect Unlearning). Suppose for all z ∈ Z, the loss function f z is m-strongly convex, L-Lipschitz, and M -smooth. Define γ (M − m)/(M + m) and η 2/(M + m). Let the unlearning parameters ǫ and δ be such that ǫ = O (log Claim 3. 3 . 3If f z is convex, L-Lipschitz, and M -smooth, then g z is m-strongly convex, (L + mD)-Lipschitz, and (M + m)-smooth. Theorem 3. 4 ( 4Accuracy, Unlearning, and Computation Tradeoffs). Suppose for all z ∈ Z, the loss function f z is convex, L-Lipschitz, and M -smooth, and let g z be defined as in Equation(5)for some m specified later. Define γ M/(M + 2m) and η 2/(M + 2m). Let the learning algorithm A (Algorithm 1) run on the regularized g with η t = η and T ≥ I + log( Dmn 2L )/ log (1/γ) where n is the size of the input dataset, and let the unlearning algorithm R A (Algorithm 2) run on the regularized g with input models θ i ≡θ i−1 and η t = η and T i = I iterations for all i ≥ 1. Let the unlearning parameters ǫ and δ be such that ǫ = O (log (mD) γ I mn (1 − γ I ) log (1/δ) + ǫ − log (1/δ) Algorithm 4 A: Learning for Perturbed Distributed Gradient Descent 1: Input: dataset D 2: for l = 1, 2, . . . , C do 3: 12 : l=1 ⊲ 12l=1Output:θ 0 = (θ 0,l ) C l=1 , S 0 = (S 0,l ) C For use in first update. Lemma 4.3 shows that the reservoir sampling operation (Algorithm 5) modifies at most s i =Õ(B/n) data points, and hence, at most s i partitions containing a modified data point. Thus we can divide our budget of nT i gradient computations at round i, into (KnT i )/(Bs i ) gradient computations per modified partition. Lemma 4 . 3 . 43Fix any training dataset D and any update sequence U, and suppose B ≥ n. Let s i denote the number of data points modified by the update of round i, namely, u i . In other words, s i = |{z l : z l ∈ S i , z l / ∈ S i−1 }|. We have that for any update step i and any δ ′ ≤ e −1 , with probability at least 1 − δ ′ ,s i ≤ 10B n log (1/δ ′ )Algorithm 6 R A : ith Unlearning for Perturbed Distributed Gradient Descent 1: Input: datasets S i−1 = (S i−1,l ) C l=1 , update u i , modelsθ i−1 = (θ i−1,l ) C l=1 . 2: Update D i = D i−1 • u i . 3: for l = 1, 2, . . . , C do 4: Draw S i,l ∼ S B rep (S i−1,l , u i ) ⊲ Reservoir update + similar partition. Theorem 4. 5 ( 5Accuracy, Unlearning, and Computation Tradeoffs). Suppose for all z ∈ Z, the loss function f z is m-strongly convex, L-Lipschitz, M -smooth, and that its Hessian is G-Lipschitz and bounded by H (with respect to ℓ 2 -operator norm of matrices). Define γ (M − m)/(M + m) and η 2/(M + m). Fix any 1 ≤ ξ ≤ 4/3, and let B = n ξ and K = √ B. Let the learning algorithm A (Algorithm 4) run with η t = η and T iterations on every partition, and for any update i ≥ 1, let the unlearning algorithm R A (Algorithm 6) run with η t = η and total T i iterations per copy (i.e. total nT i gradient computations per copy), where for any I, Let the unlearning parameters ǫ and δ be such that ǫ = O (log (1/δ)) and δ = O(B −1 ), ∀ j; θ ij − θ * ij 2 ≤ 4LK K −1 + l≤i s lj mB (1 + 10i log (2i/δ)) and (28) (with K = √ B) and (29), and noting that for δ = O(B −1 ) and B ≥ n we have E θ i,avg − θ * O(log d/B), and hiding all constants under the O notation, we have E θ i,avg − θ et al. gave a deletion algorithm for the k-means problem. Their algorithm is a weak deletion algorithm, because their (amortized) running time per update scales linearly with the number of updates.tradeoffs for (ǫ, δ)-unlearning method loss function properties unlearning accuracy iterations for ith update baseline iterations PGD SC, smooth strong (Thm. 3.1) de −I ǫ 2 n 2 I I + log ǫn √ d SC, smooth strong, perfect (Thm. 3.2) de −I ǫ 2 n 2 log i · I I ≥ log (d/ǫ) I + log ǫn √ d Regularized PGD C, smooth strong (Thm. 3.4) √ d ǫnI 2 5 I ǫnI √ d 2 5 C, smooth weak (Thm. 3.5) √ d ǫn √ I i 2 · I ǫn √ I √ d Distributed PGD SC, smooth, Lipschitz and bounded Hessian strong (Thm. 4.5) de −In 4−3ξ 2 ǫ 2 n 2 + 1 n ξ log i · I min log n, In 4−3ξ 2 + log ǫn √ d Table 1 : 1Summary of Results. In this table, SC: strongly convex, C: convex, n is training dataset size, d is dimension, ξ ∈ [1, 4/3] is a parameter. Now that we can apply Equation (15) to Equations (17) and (18) to conclude Or perhaps not, if previously trained models (trained before a user's data deletion) are used as inputs to the subsequent models. Appendix A Probabilistic ToolsLemma A.1. Suppose X, Y are random variables over the same domain Ω, and let Z be any random variable. If with probability at least 1 − δ over Z, we have X|ZProof. Define for any z, the following (good) event:and note that Pr z∼Z [z / ∈ G(Z)] ≤ δ. We have that for any S ⊆ Ω,where 1(A) is the indicator function of event A, for any A. This completes the proof because we can similarly show,Lemma A.2 (Gaussian Tail Bound). Let Z ∼ N (0, σ 2 I d ). We have that for any β > 0,We have that for any δ > 0, X ǫ,δ ≈ Y , where ǫ = ∆ 2 2σ 2 + ∆ σ 2 log (1/δ) Convex analysis and nonsmooth optimization. Aleksandr Aravkin, James Burke, Dmitriy Drusvyatskiy, Aleksandr Aravkin, James Burke, and Dmitriy Drusvyatskiy. Convex analysis and nonsmooth optimization, 2017. . Kareem Amin, Matthew Joseph, Jieming Mao, arXiv:1911.01452Pan-private uniformity testing. arXiv preprintKareem Amin, Matthew Joseph, and Jieming Mao. Pan-private uniformity testing. arXiv preprint arXiv:1911.01452, 2019. Varun Bccc + 19] Lucas Bourtoule, Christopher Chandrasekaran, Hengrui Choquette-Choo, Adelin Jia, Travers, arXiv:1912.03817Baiwu Zhang, David Lie, and Nicolas Papernot. Machine unlearning. arXiv preprintBCCC + 19] Lucas Bourtoule, Varun Chandrasekaran, Christopher Choquette-Choo, Hengrui Jia, Adelin Travers, Baiwu Zhang, David Lie, and Nicolas Papernot. Machine unlearning. arXiv preprint arXiv:1912.03817, 2019. Concentrated differential privacy: Simplifications, extensions, and lower bounds. Mark Bun, Thomas Steinke, Theory of Cryptography Conference. SpringerMark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. In Theory of Cryptography Conference, pages 635-658. Springer, 2016. Ele 522 lecture notes: Gradient methods for unconstrained problems. Yuxin Chen, Yuxin Chen. Ele 522 lecture notes: Gradient methods for unconstrained problems, 2019. Towards making systems forget with machine unlearning. Yinzhi Cao, Junfeng Yang, 2015 IEEE Symposium on Security and Privacy. IEEEYinzhi Cao and Junfeng Yang. Towards making systems forget with machine unlearning. In 2015 IEEE Symposium on Security and Privacy, pages 463-480. IEEE, 2015. Min Chen, Zhikun Zhang, Tianhao Wang, Michael Backes, Mathias Humbert, Yang Zhang, arXiv:2005.02205When machine unlearning jeopardizes privacy. arXiv preprintMin Chen, Zhikun Zhang, Tianhao Wang, Michael Backes, Mathias Humbert, and Yang Zhang. When machine unlearning jeopardizes privacy. arXiv preprint arXiv:2005.02205, 2020. Calibrating noise to sensitivity in private data analysis. Cynthia Dwork, Frank Mcsherry, Kobbi Nissim, Adam Smith, Theory of cryptography conference. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Theory of cryptography conference, pages 265-284. . Springer, Springer, 2006. Differential privacy under continual observation. Cynthia Dwork, Moni Naor, Toniann Pitassi, Guy N Rothblum, Proceedings of the forty-second ACM symposium on Theory of computing. the forty-second ACM symposium on Theory of computingCynthia Dwork, Moni Naor, Toniann Pitassi, and Guy N Rothblum. Differential privacy under continual observation. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 715-724, 2010. The algorithmic foundations of differential privacy. Cynthia Dwork, Aaron Roth, Foundations and Trends R in Theoretical Computer Science. 93-4Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Foun- dations and Trends R in Theoretical Computer Science, 9(3-4):211-407, 2014. Certified data removal from machine learning models. 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H , Brendan Mcmahan, Eider Moore, Daniel Ramage, Blaise Agüera Y Arcas, abs/1602.05629CoRRH. Brendan McMahan, Eider Moore, Daniel Ramage, and Blaise Agüera y Arcas. Federated learning of deep networks using model averaging. CoRR, abs/1602.05629, 2016. Smooth sensitivity and sampling in private data analysis. Kobbi Nissim, Sofya Raskhodnikova, Adam Smith, Proceedings of the thirty-ninth annual ACM symposium on Theory of computing. the thirty-ninth annual ACM symposium on Theory of computingKobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Smooth sensitivity and sampling in private data analysis. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 75-84, 2007. Semisupervised knowledge transfer for deep learning from private training data. Nicolas Papernot, Martin Abadi, Ulfar Erlingsson, Ian Goodfellow, Kunal Talwar, Nicolas Papernot, Martin Abadi, Ulfar Erlingsson, Ian Goodfellow, and Kunal Talwar. Semi- supervised knowledge transfer for deep learning from private training data, 2016. Membership inference attacks against machine learning models. Reza Shokri, Marco Stronati, Congzheng Song, Vitaly Shmatikov, 2017 IEEE Symposium on Security and Privacy (SP). IEEEReza Shokri, Marco Stronati, Congzheng Song, and Vitaly Shmatikov. Membership inference attacks against machine learning models. In 2017 IEEE Symposium on Security and Privacy (SP), pages 3-18. IEEE, 2017. Random sampling with a reservoir. Jeffrey S Vitter, ACM Trans. Math. Softw. 111Jeffrey S. Vitter. Random sampling with a reservoir. ACM Trans. Math. Softw., 11(1):37?57, March 1985. Comunication-efficient algorithms for statistical optimization. Yuchen Zhang, John C Duchi, Martin Wainwright, Yuchen Zhang, John C. Duchi, and Martin Wainwright. Comunication-efficient algorithms for statistical optimization, 2012. Let's focus on the base case i = 1. For any partition j such that s 1j = 0, because the update algorithm doesn't make any updates. We use induction on i to prove this claim. we haveWe use induction on i to prove this claim. Let's focus on the base case i = 1. For any partition j such that s 1j = 0, because the update algorithm doesn't make any updates, we have
[]
[ "How benign is benign overfitting?", "How benign is benign overfitting?" ]
[ "Amartya Sanyal \nDepartment of Computer Science\nUniversity of Oxford\n\n\nThe Alan Turing Institute\n\n", "Puneet K Dokania \nDepartment of Engineering Science\nUniversity of Oxford\n\n\nFive AI Ltd\nUK\n", "Varun Kanade \nDepartment of Computer Science\nUniversity of Oxford\n\n\nThe Alan Turing Institute\n\n", "Philip H S Torr \nDepartment of Engineering Science\nUniversity of Oxford\n\n" ]
[ "Department of Computer Science\nUniversity of Oxford\n", "The Alan Turing Institute\n", "Department of Engineering Science\nUniversity of Oxford\n", "Five AI Ltd\nUK", "Department of Computer Science\nUniversity of Oxford\n", "The Alan Turing Institute\n", "Department of Engineering Science\nUniversity of Oxford\n" ]
[]
We investigate two causes for adversarial vulnerability in deep neural networks: bad data and (poorly) trained models. When trained with SGD, deep neural networks essentially achieve zero training error, even in the presence of label noise, while also exhibiting good generalization on natural test data, something referred to as benign overfitting [2, 10]. However, these models are vulnerable to adversarial attacks. We identify label noise as one of the causes for adversarial vulnerability, and provide theoretical and empirical evidence in support of this. Surprisingly, we find several instances of label noise in datasets such as MNIST and CIFAR, and that robustly trained models incur training error on some of these, i.e. they don't fit the noise. However, removing noisy labels alone does not suffice to achieve adversarial robustness. Standard training procedures bias neural networks towards learning "simple" classification boundaries, which may be less robust than more complex ones. We observe that adversarial training does produce more complex decision boundaries. We conjecture that in part the need for complex decision boundaries arises from sub-optimal representation learning. By means of simple toy examples, we show theoretically how the choice of representation can drastically affect adversarial robustness.
null
[ "https://arxiv.org/pdf/2007.04028v1.pdf" ]
220,403,547
2007.04028
dfb6250ae1c8f4d0ec3e28ed84596f77704485ab
How benign is benign overfitting? Amartya Sanyal Department of Computer Science University of Oxford The Alan Turing Institute Puneet K Dokania Department of Engineering Science University of Oxford Five AI Ltd UK Varun Kanade Department of Computer Science University of Oxford The Alan Turing Institute Philip H S Torr Department of Engineering Science University of Oxford How benign is benign overfitting? We investigate two causes for adversarial vulnerability in deep neural networks: bad data and (poorly) trained models. When trained with SGD, deep neural networks essentially achieve zero training error, even in the presence of label noise, while also exhibiting good generalization on natural test data, something referred to as benign overfitting [2, 10]. However, these models are vulnerable to adversarial attacks. We identify label noise as one of the causes for adversarial vulnerability, and provide theoretical and empirical evidence in support of this. Surprisingly, we find several instances of label noise in datasets such as MNIST and CIFAR, and that robustly trained models incur training error on some of these, i.e. they don't fit the noise. However, removing noisy labels alone does not suffice to achieve adversarial robustness. Standard training procedures bias neural networks towards learning "simple" classification boundaries, which may be less robust than more complex ones. We observe that adversarial training does produce more complex decision boundaries. We conjecture that in part the need for complex decision boundaries arises from sub-optimal representation learning. By means of simple toy examples, we show theoretically how the choice of representation can drastically affect adversarial robustness. Introduction Modern machine learning methods achieve a very high accuracy on wide range of tasks, e.g. in computer vision, natural language processing, etc. [28,17,20,60,55,45], but especially in vision tasks, they have been shown to be highly vulnerable to small adversarial perturbations that are imperceptible to the human eye [12,7,51,16,8,42,38]. This vulnerability poses serious security concerns when these models are deployed in real-world tasks (cf. [30,29,43,49,24,32]). A large body of research has been devoted to crafting defences to protect neural networks from adversarial attacks (e.g. [16,41,11,57,22,9,53,36,63]). However, such defences have usually been broken by future attacks [1,52]. This arms race between attacks and defences suggests that to create a truly robust model would require a deeper understanding of the source of this vulnerability. Our goal in this paper is not to propose new defences, but to provide better answers to the question: what causes adversarial vulnerability? In doing so, we also seek to understand how existing methods designed to achieve adversarial robustness overcome some of the hurdles pointed out by our work. We identify two sources of vulnerability that, to the best of our knowledge, have not been properly studied before: a) memorization of label noise, and b) the implicit bias in the decision boundaries of neural networks trained with stochastic gradient descent (SGD). First, in the case of label noise, starting with the celebrated work of Zhang et al. [62] it has been observed that neural networks trained with SGD are capable of memorizing large amounts of label noise. Recent theoretical work (e.g. [34,4,3,19,5,6,2,39,10]) has also sought to explain why fitting training data perfectly (also referred to as memorization or interpolation) does not lead to a large drop in test accuracy, as the classical notion of overfitting might suggest. We show through simple theoretical models, as well as experiments, that there are scenarios where label noise does cause significant adversarial vulnerability, even when high natural (test) accuracy can be achieved. Surprisingly, we find that label noise is not at all uncommon in datasets such as MNIST and CIFAR-10 (see Figure 1). Our experiments show that robust training methods like Adversarial training (AT) [36] and TRADES [63] produce models that incur training error on at least some of the noisy examples, 1 but also on atypical examples from the classes. Viewed differently, robust training methods are unable to differentiate between atypical correctly labelled examples (rare dog) and a mislabelled example (cat labelled as dog) and end up not memorizing either; interestingly, the lack of memorizing these atypical examples has been pointed out as an explanation for slight drops in test accuracy, as the test set often contains similarly atypical (or even identical) examples in some cases [14,61]. Second, the fact that adversarial learning may require more "complex" decision boundaries, and as a result may require more data has been pointed out in some prior work [48,59,40,36]. However, the question of decision boundaries in neural networks is subtle as the network learns a feature representation as well as a decision boundary on top of it. We develop theoretical examples that establish that choosing one feature representation over another may lead to visually more complex decision boundaries on the input space, though these are not necessarily more complex in terms of statistical learning theoretic concepts such as VC dimension. One way to evaluate whether more meaningful representations lead to better robust accuracy is to use training data with more fine-grained labels (e.g. subclasses of a class); for example, one would expect that if different breeds of dogs are labelled differently the network will learn features that are relevant to that extra information. We show both using synthetic data and CIFAR100 that training on fine-grained labels does increase robust accuracy. Tsipras et al. [54] and Zhang et al. [63] have argued that the trade-off between robustness and accuracy might be unavoidable. However, their setting involves a distribution that is not robustly separable by any classifier. In such a situation there is indeed a trade-off between robustness and accuracy. In this paper, we focus on settings where robust classifiers exist, which is a more realistic scenario for real-world data. At least for vision, one may well argue that "humans" are robust classifiers, and as a result we would expect that classes are well-separated at least in some representation space. In fact, Yang et al. [58] show that classes are already well-separated in the input space. In such situations, there is no need for robustness to be at odds with accuracy. A more plausible scenario which we posit, and provide theoretical examples in support of, is that the trained models may not be using the "right" representations. Recent empirical work has also established that modifying the training objective to favour certain properties in the learned representations can automatically lead to improved robustness [46]. Summary of Theoretical Contributions 1. We provide simple sufficient conditions on the data distribution under which any classifier that fits the training data with label noise perfectly is adversarially vulnerable. 2. The choice of the representation (and hence the shape of the decision boundary) can be important for adversarial accuracy even when it doesn't affect natural test accuracy. 3. There exists data distributions and training algorithms, which when trained with (some fraction of) random label noise have the following property: (i) using one representation, it is possible to have high natural and robust test accuracies but at the cost of having training error; (ii) using another representation, it is possible to have no training error (including fitting noise) and high test accuracy, but low robust accuracy. Furthermore, any classifier that has no training error must have low robust accuracy. The last example shows that the choice of representation matters significantly when it comes to adversarial accuracy, and that memorizing label noise directly leads to loss of robust accuracy. The proofs of the results are not technically complicated and are included in the supplementary material. We have focused on making conceptually clear statements rather than optimize the parameters to get the best possible bounds. We also perform experiments on synthetic data (motivated by the theory), as well as MNIST, CIFAR10/100 to test these hypotheses. Summary of Experimental Contributions 1. As predicted theoretically, neural nets trained to convergence with label noise have greater adversarial vulnerability. 2. Robust training methods, such as AT and TRADES that have higher robust accuracy, avoid overfitting (some) label noise. This behaviour is also partly responsible for their decrease in natural test accuracy. 3. Even in the absence of any label noise, methods like AT and TRADES have higher robust accuracy due to more complex decision boundaries. 4. When trained with more fine-grained labels, subclasses within each class, leads to higher robust accuracy. Theoretical Setting We develop a simple theoretical framework to demonstrate how overfitting, even very minimal, label noise causes significant adversarial vulnerability. We also show how the choice of representation can significantly affect robust accuracy. Although we state the results for binary classification, they can easily be generalized to multi-class problems. We formally define the notions of natural (test) error and adversarial error. Definition 1 (Natural and Adversarial Error). For any distribution D defined over (x, y) ∈ R d × {0, 1} and any binary classifier f : R d → {0, 1}, • the natural error is R(f ; D) = P (x,y)∼D [f (x) = y] ,(1)• if B γ (x) is a ball of radius γ ≥ 0 around x under some norm 2 , the γ-adversarial error is R Adv,γ (f ; D) = P (x,y)∼D [∃z ∈ B γ (x) ; f (z) = y] ,(2) In the rest of the section, we provide theoretical results to show the effect of overfitting label noise and choice of representations (and hence simplicity of decision boundaries) on the robustness of classifiers. Overfitting Label Noise The following result provides a sufficient condition under which even a small amount of label noise causes any classifier that fits the training data perfectly to have significant adversarial error. Informally, Theorem 1 states that if the data distribution has significant probability mass in a union of (a relatively small number of, and possibly overlapping) balls, each of which has roughly the same probability mass (cf. Eq. (3)), then even a small amount of label noise renders this entire region vulnerable to adversarial attacks to classifiers that fit the training data perfectly. Theorem 1. Let c be the target classifier, and let D be a distribution over (x, y), such that y = c (x) in its support. Using the notation P D [A] to denote P (x,y)∼D [x ∈ A] for any measurable subset A ⊆ R d , suppose that there exist c 1 ≥ c 2 > 0, ρ > 0, and a finite set ζ ⊂ R d satisfying P D   s∈ζ B p ρ (s)   ≥ c 1 and ∀s ∈ ζ, P D B p ρ (s) ≥ c 2 |ζ| (3) where B p ρ (s) represents a p -ball of radius ρ around s. Further, suppose that each of these balls contain points from a single class i.e. for all s ∈ ζ, for all x, z ∈ B p ρ (s) : c (x) = c (z). Let S m be a dataset of m i.i.d. samples drawn from D, which subsequently has each label flipped independently with probability η. For any classifier f that perfectly fits the training data S m i.e. ∀ x, y ∈ S m , f (x) = y, ∀δ > 0 and m ≥ |ζ| ηc2 log |ζ| δ , with probability at least 1 − δ, R Adv,2ρ (f ; D) ≥ c 1 . The goal is to find a relatively small set ζ that satisfies the condition as this will mean that even for modest sample sizes, the trained models have significant adversarial error. We remark that it is easy to construct concrete instantiations of problems that satisfy the conditions of the theorem, e.g. each class represented by a spherical (truncated) Gaussian with radius ρ, with the classes being well-separated satisfies Eq. (3). The main idea of the proof is that there is sufficient probability mass for points which are within distance 2ρ of a training datum that was mislabelled. We note that the generality of the result, namely that any classifier (including neural networks) that fits the training data must be vulnerable irrespective of its structure, requires a result like Theorem 1. There are a few things to note about Theorem 1. First, the lower bound on adversarial error applies to any classifier f that fits the training data S m perfectly and is agnostic to the type of model f is. Second, for a given c 1 , there maybe multiple ζs that satisfy the bounds in (3) and the adversarial risk holds for all of them. Thus, smaller the value of |ζ| the smaller the size of the training data it needs to fit and it can be done by simpler classifiers. Third, if the distribution of the data is such that it is concentrated around some points then for a fixed c 1 , c 2 , a smaller value of ρ would be required to satisfy (3) and thus a weaker adversary (smaller perturbation budget 2ρ) can cause a much larger adversarial error. In practice, classifiers exhibit much greater vulnerability than purely arising from the presence of memorized noisy data. Experiments in Section 3.1 show how label noise causes vulnerability in a toy MNIST model, as well as the full MNIST. Bias towards simpler decision boundaries Label noise by itself is not the sole cause for adversarial vulnerability especially in deep learning models trained with standard optimization procedures like SGD. A second cause is the choice of representation of the data, which in turn affects the shape of the decision boundary. The choice of model affects representations and introduces desirable and possibly even undesirable (cf. training image in the training set are permuted with a fixed permutation [62]. This invariance is worrying as it means that such a network can effectively classify a matrix (or tensor) that is visually nothing like a real image into an image category. While CNNs don't have this particular invariance, as Liu et al. [35] shows, location invariance in CNNs mean that they are unable to predict where in the image a particular object is. In particular, it may be that the decision boundary for robust classifiers needs to be "visually" more complex as pointed out in prior work [40], but we emphasize that this may be because of the choice of representation, and in particular in standard measures of statistical complexity, such as VC dimension, this may not be the case. We demonstrate this phenomenon by a simple (artificial) example even when there is no label noise. Our example in Section 2.3 combines the two causes and shows how classifiers that are translation invariant may be worse for adversarial robustness. Theorem 2. For some universal constant c, and any 0 < γ 0 < 1/ √ 2, there exists a family of distributions D defined on X × {0, 1} where X ⊆ R 2 such that for all distributions P ∈ D, and denoting by S m = {(x 1 , y 1 ) , · · · , (x m , y m )} a sample of size m drawn i.i.d. from P, (i) For any m ≥ 0, S m is linearly separable i.e., ∀(x i , y i ) ∈ S m , there exist w ∈ R 2 , w 0 ∈ R s.t. y i w x i + w 0 ≥ 0. Furthermore, for every γ > γ 0 , any linear separator f that perfectly fits the training data S m has R Adv,γ (f ; P) ≥ 0.0005, even though R(f ; P) → 0 as m → ∞. (ii) There exists a function class H such that for some m ∈ O(log(δ −1 )), any h ∈ H that perfectly fits the S m , satisfies with probability at least 1 − δ, R(h; P) = 0 and R Adv,γ (h; P) = 0, for any γ ∈ [0, γ 0 + 1/8]. A complete proof of this result appears in Appendix A.2, but first, we provide a sketch of the key idea here.The distributions in family D will be supported on balls of radius at most 1/ √ 2 on the integer lattice in R 2 . The true class label for any point x is provided by the parity of a + b, where (a, b) is the lattice point closest to x. However, the distributions in D are chosen to be such that there is also a linear classifier that can separate these classes, e.g. a distribution only supported on balls centered at the points (a, a) and (a, a + 1) for some integer a (See Figure 2b). Visually learning the classification problem using the parity of a + b results in a seemingly more complex decision boundary, a point that has been made earlier regarding the need for more complex boundaries to achieve adversarial robustness [40,13]. However, it is worth noting that this complexity is not rooted in any statistical theory, e.g. the VC dimension of the classes considered in Theorem 2 is essentially the same (even lower for H by 1). This visual complexity arises purely due to the fact that the linear classifier looks at a geometric representation of the data whereas the parity classifier looks at the binary representation of the sum of the nearest integer of the coordinates. In the case of neural networks, recent works [26] have indeed provided empirical results to support that excessive invariance (eg. rotation invariance) increases adversarial error. Representation Learning in the presence of label noise In this section, we show how both causes of vulnerability can interact. Informally, we show that if the correct representation is used, then in the presence of label noise, it will be impossible to fit the training data perfectly, but the classifier that best fits the training data, 3 will have good test accuracy and adversarial accuracy. However, using an "incorrect" representation, we show that it is possible to find a classifier that has no training error, has good test accuracy, but has high adversarial error. We posit this as an (partial) explanation of why classifiers trained on real data (with label noise, or at least atypical examples) have good test accuracy, while still being vulnerable to adversarial attacks. Theorem 3. [Formal version of Theorem 3] For any n ∈ Z + , there exists a family of distributions D n over R × {0, 1} and function classes C, H, such that for any P from D n , and for any 0 < γ < 1/4, and η ∈ (0, 1/2) if S m = {(x i , y i )} m i=1 denotes a sample of size m where m = O max n log n δ (1 − η) (1 − 2η) 2 + 1 , n ηγ 2 log n γδ drawn from P, and if S m,η denotes the sample where each label is flipped independently with probability η. (i) the classifier c ∈ C that minimizes the training error on S m,η , has R(c; P) = 0 and R Adv,γ (c; P) = 0 for 0 ≤ γ < 1/4. (ii) there exist h ∈ H, h has zero training error on S m,η , and R(h; P) = 0. However, for any γ > 0, and for any h ∈ H with zero training error on S m,η , R Adv,γ (h; P) ≥ 0.1. Furthermore, the required c ∈ C and h ∈ H above can be computed in O poly (n) , poly 1 1 2 −η , poly 1 δ time. We sketch the proof here and present the complete the proof in Appendix B; as in Section 2.2 we will make use of parity functions, though the key point is the representations used. Let X = [0, N ], where N = 2 n , we consider distributions that are supported on intervals (i − 1/4, i + 1/4) for i ∈ {1, . . . , N − 1} (See Figure 2a), but any such distribution will only have a small number, O(n), intervals on which it is supported. The true class label is given by a function that depends on the parity of some hidden subsets S of bits in the bit-representation of the closest integer i, e.g. as in Figure 2a if S = {0, 2}, then only the least significant and the third least significant bit of i are examined and the class label is 1 if an odd number of them are 1 and 0 otherwise. Despite the noise, the correct label on any interval can be guessed by using the majority vote and as a result, the correct parity learnt using Gaussian elimination. (This corresponds to the class C in Theorem 3.) On the other hand it is also possible to learn the function as a union of intervals, i.e. find intervals, I 1 , I 2 , . . . , I k such that any point that lies in one of these intervals is given the label 1 and any other point is given the label 0. By choosing intervals carefully, it is possible to fit all the training data, including noisy examples, but yet not compromise on test accuracy (Fig. 2a). Such a classifier, however, will be vulnerable to adversarial examples by applying Theorem 1. A classifier such as union of intervals (H in There is negligible variance between runs and thus the shaded region showing the confidence interval is invisible. Theorem 3) is translation-invariant, whereas the parity classifier is not. This suggests that using classifiers, such as neural networks, that are designed to have too many built-in invariances might hurt its robustness accuracy. Experimental results In Section 2, we provided three theoretical settings to highlight how fitting label noise and sub-optimal representation learning (leading to seemingly simpler decision boundaries) hurts adversarial robustness. In this section, we provide empirical evidence on synthetic data inspired by the theory and on the standard datasets: MNIST [31], CIFAR10, and CIFAR100 [27] to support the theory. Overfitting label noise decreases adversarial accuracy We design a simple binary classification problem, toy-MNIST, and show that when fitting a complex classifier on a training dataset with label noise, adversarial vulnerability increases with the amount of label noise, and that this vulnerability is caused by the label noise. The problem is constructed by selecting two random images from MNIST: one "0" and one "1". Each training/test example is generated by selecting one of these images and adding i.i.d. Gaussian noise sampled from N 0, σ 2 . We create a training dataset of 4000 samples by sampling uniformly from either class. Finally, η fraction of the training data is chosen randomly and its labels are flipped. We train a neural network with four fully connected layers followed by a softmax layer and minimize the cross-entropy loss using an SGD optimizer until the training error becomes zero. Then, we attack this network with a strong ∞ PGD adversary [36] with = 64 255 for 400 steps with a step size of 0.01. In Figure 3, we plot the adversarial error, test error and training error as the amount of label noise (η) varies, for three different values of sample variance (σ 2 ). For low values of σ 2 , the training data from each class is all concentrated around the same point; as a result these models are unable to memorize the label noise and the training error is high. In this case, over-fitting label noise is impossible and the test error, as well as the adversarial error, is low. However, as σ 2 increases, the neural network is flexible enough to use the "noise component" to extract features that allow it to memorize label noise and fit the training data perfectly. This brings the training error down to zero, while causing the test error to increase, and the adversarial error even more so. This is in line with Theorem 1. The case when σ 2 = 1 is particularly striking as it exhibits a range of values of η for which test error remains very close to 0 even as the adversarial error jumps considerably. This confirms the hypothesis that benign overfitting may not be so benign when it comes to adversarial error. We perform a similar experiment on the full MNIST dataset trained on a ReLU network with 4 convolutional layers, followed by two fully connected layers. The first four convolutional layers have 32, 64, 128, 256 output filters and 3, 4, 3, 3 sized kernels respectively. This is followed by a fully connected layers with a hidden dimension of 1024. For varying values of η, for a uniformly randomly chosen η fraction of the training data we assigned the class label randomly. The network is optimized with SGD with a batch size of 128, learning rate of 0.1 for 60 epochs and the learning rate is decreased to 0.01 after 50 epochs. We compute the natural test accuracy and the adversarial test accuracy for when the network is attacked with a ∞ bounded PGD adversary for varying perturbation budget , with a step size of 0.01 and for 20 steps. Figure 4 shows that the effect of over-fitting label noise is even more clearly visible here; for the same PGD adversary the adversarial error jumps sharply with increasing label noise, while the growth of natural test error is much slower. Visualizing through low-dimensional projections: For the toy-MNIST problem, we plot a 2-d projection (using PCA) of the learned representations (activations before the last layer) at various stages of training in Figure 5. (We remark that the simplicity of the data model ensures that even a 1-d PCA projection suffices to perfectly separate the classes when there is no label noise; however, the representations learned by a neural network in the presence of noise maybe very different!) We highlight two key observations: (i) The bulk of adversarial examples ("+"-es) are concentrated around the mis-labelled training data ("•"-es) of the opposite class. For example, the purple +-es (Adversarially perturbed: True: 0, Pred:1 ) are very close to the green •-es (Mislabelled: True:0, Pred: 1). This provides empirical validation for the hypothesis that if there is a mis-labelled data-point in the vicinity that has been fit by the model, an adversarial example can be created by moving towards that data point as predicted by Theorem 1. (ii) The mis-labelled training data take longer to be fit by the classifier. For example by iteration 20, the network actually learns a fairly good representation and classification boundary that correctly fits the clean training data (but not the noisy training data). At this stage, the number of adversarial examples are much lower as compared to Iteration 160, by which point the network has completely fit the noisy training data. Thus early stopping helps in avoiding memorizing the label noise, but consequently also reduces adversarial vulnerability. Early stopping has indeed been used as a defence in quite a few recent papers in context of adversarial robustness [56,23], as well as learning in the presence of label-noise [33]. Our work provides an explanation regarding why early stopping may reduce adversarial vulnerability by avoiding fitting noisy training data. Robust training avoids memorization of (some) label noise Robust training methods like AT [36] and TRADES [63] are commonly used techniques to increase adversarial robustness of deep neural networks. However, it has been pointed out that this comes at a cost to clean accuracy [44,54]. When trained with these methods, both the training and test accuracy (on clean data) for commonly used deep learning models drops with increasing strength of the PGD adversary used (see Table 1). In this section, we provide evidence to show that robust training avoids memorization of label noise and this also results in the drop of clean train and test accuracy. Table 1: Train and test accuracies on clean dataset for ResNet-50 models trained using 2 adversaries of perturbation . The = 0 setting represents the natural training. CIFAR10 MNIST Figure 6: Each pair is a training (left) and test (right) image mis-classified by the adversarially trained model. They were both correctly classified by the naturally-trained model. Figure 1 shows that label noise is not uncommon in standard datasets like MNIST and CIFAR10. In fact, upon closely monitoring the mis-classified training set examples for both AT and TRADES, we found that that neither predicts correctly on the training set labels for any of the examples identified in Figure 1, all examples that have a wrong label in the training set, whereas natural training does. Thus, in line with Theorem 1, robust training methods ignore fitting noisy labels. We also observe this in a synthetic experiment on the full MNIST dataset where we assigned random labels to 15% of the dataset. A naturally trained CNN model achieved 100% accuracy on this dataset whereas an adversarially trained model (standard setting with = 0.3 for 30 steps) mis-classified 997 examples in the training set after the same training regime. Out of these 997 samples, 994 belonged to the set of examples whose labels were randomized. Robust training ignores label noise Robust Training ignores rare examples Next, we show that though ignoring these rare samples helps in adversarial robustness, it hurts the natural test accuracy. Our hypothesis is that one of the effects of robust training is to not memorize rare examples, which would otherwise be memorized by a naturally trained model. The underlying intuition is that certain examples in the training set belong to rare sub-populations (eg. a special kind of cat) and this sub-population is sufficiently distinct from the rest of the examples of that class in the training dataset (other cats in the dataset). As Feldman [14] points out, if these sub-populations are very infrequent in the training dataset, they are indistinguishable from data-points with label noise with the difference being that examples from that sub-population are also present in the test-set. Natural training by memorizing those rare training examples reduces the test error on the corresponding test examples. Robust training, by not memorizing these rare samples (and label noise), achieves better robustness but sacrifices the test accuracy on the test examples corresponding to those training points. Experiments on MNIST and CIFAR10 We demonstrate this effect in Figure 6 with examples from CIFAR10 and MNIST. Each pair of images contains a mis-classified (by robustly trained models) test image and the mis-classified training image "responsible" for it (We describe below how they were identified.). Importantly both of these images were correctly classified by a naturally trained model. Visually, it is evident that the training images are extremely similar to the corresponding test image. Inspecting the rest of the training set, they are also very different from other images in the training set. We can thus refer to these as rare sub-populations. The notion that certain test examples were not classified correctly due to a particular training examples not being classified correctly is measured by the influence a training image has on the test image (c.f. defn 3 in Zhang and Feldman [61]). Intuitively, it measures the probability that a certain test example would be classified correctly if the model were learned using a training set that did not contain the training point compared to if the training set did contain that particular training point. We obtained the influence of each training image on each test image for that class from Zhang and Feldman [61]. We found the images in Figure 6 by manually searching for each test image, the training image that is misclassified and is visually close to it. Our search space was shortened with the help of the influence scores each training image has on the classification of a test image. We searched in the set of top-10 most influential mis-classified train images for each mis-classified test image. The model used for Figure 6 is a AT model for CIFAR10 with 2 -adversary with an = 0.25 and a model trained with TRADES for MNIST with λ = 1 6 and = 0.3. A precise notion of measuring if a sample is rare is through the concept of self-influence. Self influence of an example with respect to an algorithm (model, optimizer etc) can be defined as how unlikely it is for the model learnt by that algorithm to be correct on an example if it had not seen that example during training compared to if it had seen the example during training. For a precise mathematical definition please refer to Eq (1) in Zhang and Feldman [61]. Self-influence for a rare example, that is unlike other examples of that class, will be high as the rest of the dataset will not provide relevant information that will help the model in correctly predicting on that particular example. In Figure 7, we show that the self-influence of training samples that were mis-classified by adversarially trained models but correctly classified by a naturally trained model is higher compared to the distribution of self-influence on the entire train dataset. In other words, it means that the self-influence of the training examples mis-classified by the robustly trained models is larger than the average self-influence of (all) examples belonging to that class. This supports our hypothesis that adversarial training excludes fitting these rare (or ones that need to be memorized) samples. Experiments on a synthetic setting This phenomenon is demonstrated more clearly in a simpler distribution for different NN configurations in Figure 8. We create a binary classification problem on R 2 . The data is uniformly supported on non-overlapping circles of varying radiuses. All points in one circle have the same label i.e. it is either blue or red depending on the color of the circle. We train a shallow network with 2 Figure 8 shows that the adversarially trained (AT) models ignore the smaller circles (i.e. rare sub-populations) and tries to get a larger margin around the circles it does classify correctly whereas the naturally trained (NAT) models correctly predicts every circle but ends up with very small margin around a lot of circles. Complexity of decision boundaries When neural networks are trained they create classifiers whose decisions boundaries are much simpler than they need to be for being adversarially robust. A few recent papers [40,48] have discussed that robustness might require more complex classifiers. In Theorem 2 and 3 we discussed this theoretically and also why this might not violate the traditional wisdom of Occam's Razor. In particular, complex decision boundaries does not necessarily mean more complex classifiers in statistical notions of complexity like VC dimension. In this section, we show through a simple experiment how the decision boundaries of neural networks are not "complex" enough to provide large enough margins and are thus adversarially much more vulnerable than is possible. We train three different neural networks with ReLU activations, a shallow network (Shallow NN) with 2 layers and 100 neurons in each layer, a shallow network with 2 layers and 1000 neurons in each layer (Shallow-Wide NN), and a deep network with 4 layers and 100 neurons in each layer. We train them for 200 epochs on a binary classification problem as constructed in Figure 9. The distribution is supported on blobs and the color of each blob represent its label. On the right side, we have the decision boundary of a large margin classifier, which is simulated using a 1-nearest neighbour. From Figure 9, it is evident that the decision boundaries of neural networks trained with standard optimizers have far simpler decision boundaries than is needed to be robust (eg. the 1-nearest neighbour is much more robust than the neural networks.) Accounting for fine grained sub-populations leads to better robustness We hypothesize that learning more meaningful representations by accounting for fine-grained sub-populations within each class may lead to better robustness. We use the theoretical setup presented in Section 2.2 and Figure 2b. However, if each of the circles belonged to a separate class then the decision boundary would have to be necessarily more complex as it needs to, now, separate the balls that were previously within the same class. We test this hypothesis with two experiments. First, we test it on the the distribution defined in Theorem 2 where for each ball with label 1, we assign it a different label (say α 1 , · · · , α k ) and similarly for balls with label 0, we assign it a different label (β 1 , · · · , β k ). Now, we solve a multi-class classification problem for 2k classes with a deep neural network and then later aggregate the results by reporting all α i s as 1 and all β i s as 0.The resulting decision boundary is drawn in Figure 10a along with the decision boundary for natural training and AT. Clearly, the decision boundary for AT is the most complex and has the highest margin (and robustness) followed by the multi-class model and then the naturally trained model. Second, we also repeat the experiment with CIFAR-100. We train a ResNet50 [21] on the fine labels of CIFAR100 and then aggregate the fine labels corresponding to a coarse label by summing up the logits. We call this model the Fine2Coarse model and compare the adversarial risk of this network to a ResNet-50 trained directly on the coarse labels. Note that the model is end-to-end differentiable as the only addition is a layer to aggregate the logits corresponding to the fine classes pertaining to each coarse class. Thus PGD adversarial attacks can be applied out of the box. Figure 10b shows that for all perturbation budgets, Fine2Coarse has smaller adversarial risk than the naturally trained model. [37] established that there are concept classes with finite VC dimensions i.e. are properly PAC-learnable but are only improperly robustly PAC learnable. This implies that to learn the problem with small adversarial error, a different class of models (or representations) needs to be used whereas for small natural test risk, the original model class (or representation) can be used. Recent empirical works have also shown evidence towards this (eg. [46]). Related Work Hanin and Rolnick [18] have shown that though the number of possible linear regions that can be created by a deep ReLU network is exponential in depth, in practice for networks trained with SGD this tends to grow only linearly thus creating much simpler decision boundaries than is possible due to sheer expresssivity of deep networks. Experiments on the data models from our theoretical settings indeed show that adversarial training indeed produces more "complex" decision boundaries Jacobsen et al. [25] have discussed that excesssive invariance in neural networks might increase adversarial error. However, their argument is that excessive invariance can allow sufficient changes in the semantically important features without changing the network's prediction. They describe this as Invariance-based adversarial examples as opposed to perturbation based adversarial examples. We show that excessive (incorrect) invariance might also result in perturbation based adversarial examples. Another contemporary work [15] discusses a phenomenon they refer to as Shortcut Learning where deep learning models perform very well on standard tasks like reducing classification error but fail to perform in more difficult real world situations. We discuss this in the context of models that have small test error but large adversarial error and provide and theoretical and empirical to discuss why one of the reasons for this is sub-optimal representation learning. Conclusion Recent research has largely shone a positive light on interpolation (zero training error) by highly overparameterized models even in the presence of label noise. While overfitting noisy data may not harm generalisation, we have shown that this can be severely detrimental to robustness. This raises a new security threat where label noise can be inserted into datasets to make the models learnt from them vulnerable to adversarial attacks without hurting their test accuracy. As a result, further research into learning without memorization is ever more important [47,50]. Further, we underscore the importance of proper representation learning in regards to adversarial robustness. Representations learnt by deep networks often encode a lot of different invariances, e.g., location, permutation, rotation, etc. While some of them are useful for the particular task at hand, we highlight that certain invariances can increase adversarial vulnerability. Thus we believe that making significant progress towards training robust models with good test error requires us to rethink representation learning and closely examine the data on which we are training these models. Acknowledgement We thank Vitaly Feldman and Chiyuan Zhang for providing us with data that helped to significantly speed up some parts of this work. We also thank Nicholas Lord for feedback on the draft. AS acknowledges support from The Alan Turing Institute under the Turing Doctoral Studentship grant TU/C/000023. VK is supported in part by the Alan Turing Institute under the EPSRC grant EP/N510129/1. PHS and PD are supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1 and EPSRC/MURI grant EP/N019474/1. PHS and PD also acknowledges the Royal Academy of Engineering and FiveAI. A Proofs for Section 2 In this section, we present the formal proofs to the theorems stated in Section 2. A.1 Proof of Theorem 1 Theorem 1. Let c be the target classifier, and let D be a distribution over (x, y), such that y = c (x) in its support. Using the notation P D [A] to denote P (x,y)∼D [x ∈ A] for any measurable subset A ⊆ R d , suppose that there exist c 1 ≥ c 2 > 0, ρ > 0, and a finite set ζ ⊂ R d satisfying P D   s∈ζ B p ρ (s)   ≥ c 1 and ∀s ∈ ζ, P D B p ρ (s) ≥ c 2 |ζ| (3) where B p ρ (s) represents a p -ball of radius ρ around s. Further, suppose that each of these balls contain points from a single class i.e. for all s ∈ ζ, for all x, z ∈ B p ρ (s) : c (x) = c (z). Let S m be a dataset of m i.i.d. samples drawn from D, which subsequently has each label flipped independently with probability η. For any classifier f that perfectly fits the training data S m i.e. ∀ x, y ∈ S m , f (x) = y, ∀δ > 0 and m ≥ |ζ| ηc2 log |ζ| δ , with probability at least 1 − δ, R Adv,2ρ (f ; D) ≥ c 1 . Proof of Theorem 1. From (3), for any ζ and s ∈ ζ, P (x,y)∼D [x ∈ B ρ (s)] ≥ c 2 |ζ| As the sampling of the point and the injection of label noise are independent events, P (x,y)∼D [x ∈ B ρ (s) ∧ x gets mislabelled] ≥ c 2 η |ζ| Thus, P Sm∼D m [∃ (x, y) ∈ S m : x ∈ B ρ (s) ∧ x is mislabelled] ≥ 1 − 1 − c 2 η |ζ| m ≥ 1 − exp −c 2 ηm |ζ| Substituting m ≥ |ζ| ηc2 log |ζ| δ and applying the union bound over all s ∈ ζ, we get P Sm∼D m [∀s ∈ ζ, ∃ (x, y) ∈ S m : x ∈ B ρ (s) ∧ x is mislabelled] ≥ 1 − δ(4) As for all s ∈ R d and ∀x, z, ∈ B p ρ (s) , x − z p ≤ 2ρ, we have that R Adv,2ρ (f ; D) = P Sm∼D m P (x,y)∼D [∃z ∈ B 2ρ (x) ∧ y = f (z)] = P Sm∼D m P (x,y)∼D [∃z ∈ B 2ρ (x) ∧ c (z) = f (z)] ≥ P Sm∼D n   P (x,y)∼D   x ∈ s∈ζ B p ρ (s) ∧ {∃z ∈ B 2ρ (x) : c (z) = f (z)}     = P Sm∼D m P (x,y)∼D ∃s ∈ ζ : x ∈ B p ρ (s) ∧ {∃z ∈ B ρ (s) : c (z) = f (z)} = P (x,y)∼D   x ∈ s∈ζ B p ρ (s)   w.p. atleast 1 − δ ≥ c 1 w.p. 1 − δ where c is the true concept for the distribution D. The second equality follows from the assumptions that each of the balls around s ∈ ζ are pure in their labels. The second last equality follows from (4) by using the x that is guaranteed to exist in the ball around s and be mis-labelled with probability atleast 1 − δ. The last equality from Assumption (4). A.2 Proofs of Section 2.2 Theorem 2. For some universal constant c, and any 0 < γ 0 < 1/ √ 2, there exists a family of distributions D defined on X × {0, 1} where X ⊆ R 2 such that for all distributions P ∈ D, and denoting by S m = {(x 1 , y 1 ) , · · · , (x m , y m )} a sample of size m drawn i.i.d. from P, (i) For any m ≥ 0, S m is linearly separable i.e., ∀(x i , y i ) ∈ S m , there exist w ∈ R 2 , w 0 ∈ R s.t. y i w x i + w 0 ≥ 0. Furthermore, for every γ > γ 0 , any linear separator f that perfectly fits the training data S m has R Adv,γ (f ; P) ≥ 0.0005, even though R(f ; P) → 0 as m → ∞. (ii) There exists a function class H such that for some m ∈ O(log(δ −1 )), any h ∈ H that perfectly fits the S m , satisfies with probability at least 1 − δ, R(h; P) = 0 and R Adv,γ (h; P) = 0, for any γ ∈ [0, γ 0 + 1/8]. Proof of Theorem 2. We define a family of distribution D, such that each distribution in D is supported on balls of radius r around (i, i) and (i + 1, i) for positive integers i. Either all the balls around (i, i) have the labels 1 and the balls around (i + 1, i) have the label 0 or vice versa. Figure 2b shows an example where the colors indicate the label. Formally, for r > 0, k ∈ Z + , the (r, k)-1 bit parity class conditional model is defined over (x, y) ∈ R 2 ×{0, 1} as follows. First, a label y is sampled uniformly from {0, 1}, then and integer i is sampled uniformly from the set {1, · · · , k} and finally x is generated by sampling uniformly from the 2 ball of radius r around (i + y, i). In Lemma 1 we first show that a set of m points sampled iid from any distribution as defined above for r < 1 2 √ 2 is with probability 1 linear separable for any m. In addition, standard VC bounds show that any linear classifier that separates S m for large enough m will have small test error. Lemma 1 also proves that there exists a range of γ, r such that for any distribution defined with r in that range, though it is possible to obtain a linear classifier with 0 training and test error, the minimum adversarial risk will be bounded from 0. However while it is possible to obtain a linear classifier with 0 test error, all such linear classifiers has a large adversarial vulnerability. In Lemma 2, we show that there exists a different representation for this problem, which also achieves zero training and test error and in addition has zero adversarial risk for a range of r, γ where the linear classifier's adversarial error was atleast a constant. Lemma 1 (Linear Classifier). There exists universal constants γ 0 , ρ, such that for any perturbation γ > γ 0 , radius r ≥ ρ, and k ∈ Z + , the following holds. Let D be the family of (r, k)-1-bit parity class conditional model, P ∈ D and S n = {(x 1 , y 1 ) , · · · , (x n , y 1 )} be a set of n points sampled i.i.d. from P. 1) For any n > 0, S n is linearly separable with probability 1 i.e. there exists a h : (w, w 0 ), w ∈ R 2 , w 0 ∈ R such that the linear hyperplane x → w x + w 0 separates S n with probability 1: ∀ (x, y) ∈ S n z w x + w 0 > 0 where z = 2y − 1 2) Further there exists an universal constant c such that for any , δ > 0 with probability 1 − δ for any S n with n = c 1 2 log 1 δ , any linear classifierh that separates S n has R(h; P) ≤ . 3) Let h : (w, w 0 ) be any linear classifier that has R(h; P P ) = 0. Then, R Adv,γ (h; P) > 0.0005. We will prove the first part for any r < 1 2 √ 2 by constructing a w, w 0 such that it satisfies the constraints of linear separability. Let w = (1, −1) , w 0 = −0.5. Consider any point (x, y) ∈ S n and z = 2y − 1. Converting to the polar coordinate system there exists a θ ∈ [0, 2π] , j ∈ [0, · · · , k] such that x = j + z+1 2 + rcos (θ) , j + rsin (θ) z w x + w 0 = z j + z + 1 2 + rcos (θ) − j − rsin (θ) − 0.5 w = (1, −1) = z z 2 + 0.5 + rcos (θ) − rsin (θ) − 0.5 = 1 2 + zr (cos (θ) − sin (θ)) |cos (θ) − sin (θ)| < √ 2, z ∈ {−1, 1} > 1 2 − r √ 2 > 0 r < 1 2 √ 2 Part 2 follows with simple VC bounds of linear classifiers. Let the universal constants γ 0 , ρ be 0.02 and 1 2 √ 2 − 0.008 respectively. Note that there is nothing special about this constants except that some constant is required to bound the adversarial risk away from 0. Now, consider a distribution P 1-bit parity model such that the radius of each ball is atleast ρ. This is smaller than 1 2 √ 2 and thus satisfies the linear separability criterion. Consider h to be a hyper-plane that has 0 test error. Let the 2 radius of adversarial perturbation be γ > γ 0 . The region of each circle that will be vulnerable to the attack will be a circular segment with the chord of the segment parallel to the hyper-plane. Let the minimum height of all such circular segments be r 0 . Thus, R Adv,γ (h; P) is greater than the mass of the circular segment of radius r 0 . Let the radius of each ball in the support of P be r. Using the fact that h has zero test error; and thus classifies the balls in the support of P correctly and simple geometry 1 √ 2 ≥ r + (γ − r 0 ) + r r 0 ≥ 2r + γ − 1 √ 2(5) To compute R Adv,γ (h; P) we need to compute the ratio of the area of a circular segment of height r 0 of a circle of radius r to the area of the circle. The ratio can be written A r 0 r = cos −1 1 − r0 r − 1 − r0 r 2 r0 r − r 2 0 r 2 π(6) As (6) is increasing with r0 r , we can evaluate r 0 r ≥ 2r − 1 √ 2 + γ r Using (5) ≥ 2 − 1 √ 2 − 0.02 r γ > γ 0 = 0.02 ≥ 2 − 1 √ 2 − 0.02 1 √ 2 − 0.008 > 0.01 r > ρ = 1 2 √ 2 − 0.008 Substituting r0 r > 0.01 into Eq. (6), we get that A r0 r > 0.0005. Thus, for all γ > 0.02, we have R Adv,γ (h; P) > 0.0005. Lemma 2 (Robustness of parity classifier). There exists a concept class H such that for any γ ∈ γ 0 , γ 0 + 1 8 , k ∈ Z + , P being the corresponding (ρ, k) 1-bit parity class distribution where ρ, γ 0 are the same as in Lemma 1 there exists g ∈ H such that R(g; P) = 0 R Adv,γ (g; P) = 0 Proof of Lemma 2. We will again provide a proof by construction. Consider the following class of concepts H such that g b ∈ H is defined as g (x 1 , x 2 ) = 1 if [x 1 ] + [x 2 ] = b (mod 2) 1 − b o.w.(7) where [x] rounds x to the nearest integer and b ∈ {0, 1}. In Figure 2b, the green staircase-like classifier belongs to this class. Consider the classifier g 1 . Note that by construction R(g 1 ; P) = 0. The decision boundary of g 1 that are closest to a ball in the support of P centered at (a, b) are the lines x = a ± 0.5 and y = b ± 0.5. As γ < γ 0 + 1 8 , the adversarial perturbation is upper bounded by 1 50 + 1 8 . The radius of the ball is upper bounded by 1 2 √ 2 , and as we noted the center of the ball is at a distance of 0.5 from the decision boundary. If the sum of the maximum adversarial perturbation and the maximum radius of the ball is less than the minimum distance of the center of the ball from the decision boundary, then the adversarial error is 0. Substituting the values, 1 50 + 1 8 + 1 2 √ 2 < 0.499 < 1 2 This completes the proof. (i) the classifier c ∈ C that minimizes the training error on S m,η , has R(c; P) = 0 and R Adv,γ (c; P) = 0 for 0 ≤ γ < 1/4. B Proof of Section 2.3 (ii) there exist h ∈ H, h has zero training error on S m,η , and R(h; P) = 0. However, for any γ > 0, and for any h ∈ H with zero training error on S m,η , R Adv,γ (h; P) ≥ 0.1. Furthermore, the required c ∈ C and h ∈ H above can be computed in O poly (n) , poly 1 1 2 −η , poly 1 δ time. Proof of Theorem 3. We will provide a constructive proof to this theorem by constructing a distribution D, two concept classes C and H and provide the ERM algorithms to learn the concepts and then use Lemma 3 and 4 to complete the proof. Distribution: Consider the family of distribution D n such that D S,ζ ∈ D n is defined on X ζ × {0, 1} for S ⊆ {1, · · · , n} , ζ ⊆ {1, · · · , 2 n − 1} such that the support of X ζ is a union of intervals. supp (X ) ζ = j∈ζ I j where I j := j − 1 4 , j + 1 4(8) We consider distributions with a relatively small support i.e. where |ζ| = O (n). Each sample (x, y) ∼ D S,ζ is created by sampling x uniformly from X ζ and assigning y = c S (x) where c S ∈ C is defined below (9). We define the family of distributions D = n∈Z+ D n . Finally, we create D η S,ζ -a noisy version of D S,ζ , by flipping y in each sample (x, y) with probability η < 1 2 . Samples from D S,ζ can be obtained using the example oracle EX (D S,ζ ) and samples from the noisy distribution can be obtained through the noisy oracle EX η (D S,ζ ) Concept Class C: We define the concept class C n of concepts c S : [0, 2 n ] → {0, 1} such that c S (x) = 1, if ( [x] b XOR S) is odd. 0 o.w.(9) where [·] : R → Z rounds a decimal to its nearest integer, · b : {0, · · · , 2 n } → {0, 1} n returns the binary encoding of the integer, and ( [x] b XOR S) = j∈S [x] b [j] mod 2. [x] b [j] ish I (x) = 1 if x ∈ j I j 0 o.w.(10) Now, we look at the algorithms to learn the concepts from C and H that minimize the train error. Both of the algorithms will use a majority vote to determine the correct (de-noised) label for each interval, which will be necessary to minimize the test error. The intuition is that if we draw a sufficiently large number of samples, then the majority of samples on each interval will have the correct label with a high probability. As m ≥ 2 |ζ| 2 log 2|ζ| δ 8 (1−η) (1−2η) 2 + 1 , using δ 2 = δ 2 and k = 8(1−η) (1−2η) 2 log 2|ζ| δ in Lemma 5 guarantees that with probability 1 − δ 2 , each interval will have atleast 8(1−η) (1−2η) 2 log 2|ζ| δ samples. Then for any specific interval, using δ 1 = 2|ζ| δ in Lemma 6 guarantees that with probability atleast 1 − 2|ζ| δ , the majority vote for the label in that interval will succeed in returning the de-noised label. Applying a union bound over all |ζ| intervals, will guarantee that with probability atleast 1 − δ, the majority label of every interval will be the denoised label. Now, the problem reduces to solving a parity problem on this reduced dataset of |ζ| points (after denoising, all points in that interval can be reduced to the integer in the interval and the denoised label). We know that there exists a polynomial algorithm using Gaussian Elimination that finds a consistent hypothesis for this problem. We have already guaranteed that there is a point in S m from every interval in the support of D S,ζ . Further, f is consistent on S m and f is constant in each of these intervals by design. Thus, with probability atleast 1 − δ we have that R(f ; D S,ζ ) = 0. By construction, f makes a constant prediction on each interval j − 1 2 , j + 1 2 for all j ∈ ζ. Thus, for any perturbation radius γ < 1 4 the adversarial risk R Adv,D S, ζ (f ) = 0. Combining everything, we have shown that there is an algorithm that makes 2 |ζ| 2 log 2|ζ| δ 8 (1−η) (1−2η) 2 + 1 calls to the EX D η S,ζ oracle, runs in time polynomial in |ζ| , 1 1−2η , 1 δ to return f ∈ C such that R(f ; D S,ζ ) = 0 and R Adv,γ (f ; D S,ζ ) = 0 for γ < 1 4 . 4. Add all (x, y) ∈ N z such that y =ỹ to M z 5. Ifỹ = 1, then add the interval (z − 0.5, z + 0.5) to I. 6. Remove all elements of N z from S m i.e. S m := S m \ N z . Lemma 4 (Union of Interval Concept Class). There exists a learning algorithm For reasons similar to Lemma 3, as m ≥ 2 |ζ| 2 log 2|ζ| δ 8 (1−η) (1−2η) 2 + 1 , Lemma 5 guarantees that with probability 1 − δ 2 , each interval will have atleast 8(1−η) (1−2η) 2 log 2|ζ| δ samples. Then for any specific interval, Lemma 6 guarantees that with probability atleast 1 − 2|ζ| δ , the majority vote for the label in that interval will succeed in returning the de-noised label. Applying a union bound over all intervals, will guarantee that with probability atleast 1 − δ, the majority label of every interval will be the denoised label. As each interval inζ has atleast one point, all the intervals in ζ with label 1 will be included in I with probability 1 − δ. Thus, R(h; D S,ζ ) = 0. Now, for all (x, y) ∈ M z , add the interval [x] to I if y = 1. If y = 0 then x must lie a interval (a, b) ∈ I. Replace that interval as follows I := I \ (a, b) ∪ {(a, x), (x, b)}. As only a finite number of sets with lebesgue measure of 0 were added or deleted from I, the net test error of h doesn't change and is still 0 i.e. R(h; D S,ζ ) = 0 For the second part, we will invoke Theorem 1. To avoid confusion in notation, we will use Γ instead of ζ to refer to the sets in Theorem 1 and reserve ζ for the support of interval of D S,ζ . Let Γ be any set of disjoint intervals of width γ 2 such that |Γ| = 0.1|ζ| γ . This is always possible as the total width of all intervals in Γ is 0.1|ζ| γ γ 2 = 0.1 |ζ| 2 which is less than the total width of the support |ζ| 2 . c 1 , c 2 from Eq. (3) is c 1 = P D S,ζ [Γ] = 2 * 0.1 |ζ| 2 |ζ| = 0.1, c 2 = 2γ 2 |ζ| |ζ| = γ Proof of Lemma 6. Without loss of generality let y = 1. For the majority to be 1 we need to show that there are more than m 2 "1"s in S i.e. we need to show that the following probability is less than δ 1 . P s i < m 1 2 = P s i < m 1 2µ * µ + µ − µ µ = E s i = P s i < 1 − 1 − m 1 2µ µ ≤ exp − (1 − 2η) 2 8 (1 − η) 2 µ By Chernoff's Inequality = exp − (1 − 2η) 2 8 (1 − η) m ∵ µ = (1 − η) m ≤ δ 1 ∵ m > 8 (1 − η) (1 − 2η) 2 log 1 δ 1 Figure 1 : 1Label Noise in CIFAR10 and MNIST. Text above the image indicates the training set label. For instance, one could construct the classifier h, where h(x) = c(x), if (x, b) ∈ S m for b = 0, 1, and h(x) = y if (x, y) ∈ S m . Note that the classifier h agrees with the target c on every point of R d except the mislabelled training examples, and as a result these examples are the only source of vulnerability. The complete proof is presented in Appendix A.1. [35]) invariances; for example, training convolutional networks are invariant to (some) translations, while training fully connected networks are invariant to permutations of input features. This means that fully connected networks can learn even if the pixels of Parity and Union of Interval classifier predicts red if inside any green interval and blue if outside all intervals. The ×-es are correctly labelled and the •-es are mis-labelled points. Reference integer points on the line labelled in binary. Figure 2 : 2Visualization of the distribution and classifiers used in the Proof of Theorem 2 and 3. The Red and Blue indicate the two classes. Figure 3 : 3Adversarial error increases with label noise (η) if training error is 0. Shaded region shows 95% confidence interval. Figure 4 : 4Shows the adversarial for the full MNIST dataset for varying levels of adversarial perturbation. Figure 5 : 5Two dimensional PCA projections of the original correctly labelled (blue and orange), original mis-labelled (green and red), and adversarial examples (purple and brown) at different stages of training. The correct label for True 0 (blue), Noisy 0 (green), Adv 0 (purple +) are the same i.e. 0 and similar for the other class. Figure 7 : 7Fraction of train points that have a self-influence greater than s is plotted versus s. The blue line represents the points mis-classified by an adversarially trained model on CIFAR10. The orange lines shows the distribution of self-influence for all points in the CIFAR10 dataset (of the concerned class). Figure 8 : 8Adversarial training (AT) leads to larger margin, and thus adversarial robustness around high density regions (larger circles) but causes training error on low density sub-populations (smaller circles) whereas naturally trained models (NAT) minimizes the training error but leads to regions with very small margins.layers and 1000 neurons in each layer (Shallow-Wide NN) and a deep network with 4 layers and 100 neurons in each layer using cross entropy loss and SGD. The background color shows the decision region of the learnt neural network. Figure 9 : 9Decision boundaries of neural networks are much simpler than they should be. Region of neural networks are more complex for adversarially trained models. Treating it as a multi-class classification problem, with natural training (MULTICLASS), also increases robustness by increasing the margin.(b) Adversarial error on coarse labels of CIFAR-100. Figure 10 : 10Assigning a separate class to each sub-population within the original class during training increases robustness by learning more meaningful representations. Theorem 3 . 3[Formal version of Theorem 3] For any n ∈ Z + , there exists a family of distributions D n over R × {0, 1} and function classes C, H, such that for any P from D n , and for any 0 < γ < 1/4, and η∈ (0, 1/2) if S m = {(x i , y i )} m i=1denotes a sample of size m where m = O max n log from P, and if S m,η denotes the sample where each label is flipped independently with probability η. the j th least significant bit in the binary encoding of the nearest integer to x. It is essentially the class of parity functions defined on the bits corresponding to the indices in S for the binary encoding of the nearest integer to x. For example, as in Figure 2a if S = {0, 2}, then only the least significant and the third least significant bit of i are examined and the class label is 1 if an odd number of them are 1 and 0 otherwise. Concept Class H: Finally, we define the concept class H = ∞ k=1 H k where H k is the class of union of k intervals on the real line H k . Each concept h I ∈ H k can be written as a set of k disjoint intervals I = {I 1 , · · · , I k } on the real line i.e. for 1 ≤ j ≤ k, I j = [a, b] where 0 ≤ a ≤ b and . Lemma 3 proves that there exists an algorithm A such that A draws m = O |ζ| noisy oracle EX η (D s,ζ ) and with probability 1 − δ where the probability is over the randomization in the oracle, returns f ∈ C such that R(f ; D S,ζ ) = 0 and R Adv,γ (f ; D S,ζ ) = 0 for all γ < 1 4 . As Lemma 3 states, the algorithm involves gaussian elimination over |ζ| variables and |ζ| majority votes (one in each interval) involving a total of m samples. Thus the algorithm runs in O (poly (m) + poly (|ζ|)) time. Replacing the complexity of m and the fact that |ζ| = O (n), the complexity of the algorithm is O poly n, that there exists an algorithm A such that A draws samples and returns h ∈ H such that h has 0 training error, 0 test error and an adversarial test error of atleast 0.1. We can replace |ζ| = O (n) to get the required bound on m in the theorem. The algorithm to construct h visits every point atmost twice -once during the construction of the intervals using majority voting, and once while accommodating for the mislabelled points. Replacing the complexity of m, the complexity of the algorithm is O poly n, This completes the proof. Lemma 3 ( 3Parity Concept Class). There exists a learning algorithm A such that given access to the noisy example oracle EX η (D S,ζ ), A makes m = O |ζ| the oracle and returns a hypothesis f ∈ C such that with probability 1 − δ, we have that R(f ; D S,ζ ) = 0 and R Adv,γ (f ; D S,ζ ) = 0 for all γ <1 4 . Proof. The algorithm A works as follows. It makes m calls to the oracle EX (D m s ) to obtain a set of points {(x 1 , y 1 ) , · · · , (x m , y m )} where m ≥ 2 |ζ| 2 log 2|ζ| δ 8 (1−η) (1−2η) 2 + 1 . Then, it replaces each x i with [x i ] ([·] rounds a decimal to the nearest integer) and then removes duplicate x i s by preserving the most frequent label y i associated with each x i . For example, if S 5 = { the oracle and returns a hypothesis h ∈ H such that training error is 0 and with probability 1 − δ, R(f ; D S,ζ ) = 0.Further for any h ∈ H that has zero training error on m samples drawn fromEX η (D S,ζ ) for m > |ζ| 10ηγ 2 log |ζ| 10γδ and η ∈ 0, 1 2 then R Adv,γ (f ; D S,ζ ) ≥ 0.1 for all γ > 0.Proof of Lemma 4. The first part of the algorithm works similarly to Lemma 3. The algorithm A makes m calls to the oracle EX (D m s ) to obtain a set of points S m = {(x 1 , y 1 ) , · · · , (x m , y m )} where m ≥ 2 |ζ| 2 log 2|ζ| δ 8 (1−η) (1−2η) 2 + 1 . A computes h ∈ H as follows. To begin, let the list of intervals in h be I and M z = {} Then do the following for every (x, y) ∈ S m . 1. let z := [x], 2. Let N z ⊆ S m be the set of all (x, y) ∈ S m such that |x − z| < 0.5. 3. Compute the majority labelỹ of N z . We manually inspected all training set errors of these models. Throughout, we will mostly use the (most commonly used) ∞ norm, but the results hold for other norms. This is referred to as the Empirical Risk Minimization (ERM) in the statistical learning theory literature. it is possible to obtain h ∈ H such that h has zero training error, R(D S,ζ ; h) = 0 and R Adv,γ (h; D S,ζ ) > 0.1 for any γ > 0.Lemma 5. Given k ∈ Z + and a distribution D S,ζ , for any δ 2 > 0 if m > 2 |ζ| 2 k + 2 |ζ| 2 log |ζ| δ2 samples are drawn from EX (D S,ζ ) then with probability atleast 1 − δ 2 there are atleast k samples in each interval j − 1 4 , j + 1 4 for all j ∈ ζ. Proof of Lemma 5. We will repeat the following procedure |ζ| times once for each interval in ζ and show that with probability δ |ζ| the j th run will result in atleast k samples in the j th interval. Corresponding to each interval in ζ, we will sample atleast m samples where m = 2 |ζ| k + 2 |ζ| log |ζ| δ2 . If z j i is the random variable that is 1 when the i th sample belongs to the j th interval, then j th interval has atleast k points out of the m points sampled for that interval with probability less than δ2 |ζ| .where the last step follows from m > 2 |ζ| k + 2 |ζ| log |ζ| δ2 . With probability atleast δ |ζ| , every interval will have atleast k samples. Finally, an union bound over each interval gives the desired result. As we repeat the process for all |ζ| intervals, the total number of samples drawn will be atleast |ζ| m = 2 |ζ| 2 k + 2 |ζ| 2 log |ζ| δ2 . Lemma 6 (Majority Vote). For a given y ∈ {0, 1}, let S = {s 1 , · · · , s m } be a set of size m where each element is y with probability 1 − η and 1 − y otherwise. If m > 8(1−η)(1−2η) 2 log 1 δ1 then with probability atleast 1 − δ 1 the majority of S is y. 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[]
[ "Event Prediction in the Big Data Era: A Systematic Survey", "Event Prediction in the Big Data Era: A Systematic Survey" ]
[ "Liang Zhao \nEmory University\n\n" ]
[ "Emory University\n" ]
[]
Events are occurrences in specific locations, time, and semantics that nontrivially impact either our society or the nature, such as earthquakes, civil unrest, system failures, pandemics, and crimes. It is highly desirable to be able to anticipate the occurrence of such events in advance in order to reduce the potential social upheaval and damage caused. Event prediction, which has traditionally been prohibitively challenging, is now becoming a viable option in the big data era and is thus experiencing rapid growth, also thanks to advances in high performance computers and new Artificial Intelligence techniques. There is a large amount of existing work that focuses on addressing the challenges involved, including heterogeneous multi-faceted outputs, complex (e.g., spatial, temporal, and semantic) dependencies, and streaming data feeds. Due to the strong interdisciplinary nature of event prediction problems, most existing event prediction methods were initially designed to deal with specific application domains, though the techniques and evaluation procedures utilized are usually generalizable across different domains. However, it is imperative yet difficult to cross-reference the techniques across different domains, given the absence of a comprehensive literature survey for event prediction. This paper aims to provide a systematic and comprehensive survey of the technologies, applications, and evaluations of event prediction in the big data era. First, systematic categorization and summary of existing techniques are presented, which facilitate domain experts' searches for suitable techniques and help model developers consolidate their research at the frontiers. Then, comprehensive categorization and summary of major application domains are provided to introduce wider applications to model developers to help them expand the impacts of their research. Evaluation metrics and procedures are summarized and standardized to unify the understanding of model performance among stakeholders, model developers, and domain experts in various application domains. Finally, open problems and future directions for this promising and important domain are elucidated and discussed.
10.36227/techrxiv.12733049.v1
[ "https://arxiv.org/pdf/2007.09815v3.pdf" ]
220,647,363
2007.09815
3ca50610e34cc8273bc946514afcfc99a0752a1c
Event Prediction in the Big Data Era: A Systematic Survey Liang Zhao Emory University Event Prediction in the Big Data Era: A Systematic Survey 10.1145/nnnnnnn.nnnnnnnAdditional Key Words and Phrases: Event PredictionBig DataArtificial Intelligence Events are occurrences in specific locations, time, and semantics that nontrivially impact either our society or the nature, such as earthquakes, civil unrest, system failures, pandemics, and crimes. It is highly desirable to be able to anticipate the occurrence of such events in advance in order to reduce the potential social upheaval and damage caused. Event prediction, which has traditionally been prohibitively challenging, is now becoming a viable option in the big data era and is thus experiencing rapid growth, also thanks to advances in high performance computers and new Artificial Intelligence techniques. There is a large amount of existing work that focuses on addressing the challenges involved, including heterogeneous multi-faceted outputs, complex (e.g., spatial, temporal, and semantic) dependencies, and streaming data feeds. Due to the strong interdisciplinary nature of event prediction problems, most existing event prediction methods were initially designed to deal with specific application domains, though the techniques and evaluation procedures utilized are usually generalizable across different domains. However, it is imperative yet difficult to cross-reference the techniques across different domains, given the absence of a comprehensive literature survey for event prediction. This paper aims to provide a systematic and comprehensive survey of the technologies, applications, and evaluations of event prediction in the big data era. First, systematic categorization and summary of existing techniques are presented, which facilitate domain experts' searches for suitable techniques and help model developers consolidate their research at the frontiers. Then, comprehensive categorization and summary of major application domains are provided to introduce wider applications to model developers to help them expand the impacts of their research. Evaluation metrics and procedures are summarized and standardized to unify the understanding of model performance among stakeholders, model developers, and domain experts in various application domains. Finally, open problems and future directions for this promising and important domain are elucidated and discussed. and is the focus of this survey. Accurate anticipation of future events enables one to maximize the benefits and minimize the losses associated with some event in the future, bringing huge benefits for both society as a whole and individual members of society in key domains such as disease prevention [167], disaster management [140], business intelligence [226], and economics stability [24]. "Prediction is very difficult, especially if it's about the future. " -Niels Bohr, 1970 Event prediction has traditionally been prohibitively challenging across different domains, due to the lack or incompleteness of our knowledge regarding the true causes and mechanisms driving event occurrences in most domains. With the advent of the big data era, however, we now enjoy unprecedented opportunities that open up many alternative approaches for dealing with event prediction problems, sidestepping the need to develop a complete understanding of the underlying mechanisms of event occurrence. Based on large amounts of data on historical events and their potential precursors, event prediction methods typically strive to apply predictive mapping to build on these observations to predict future events, utilizing predictive analysis techniques from domains such as machine learning, data mining, pattern recognition, statistics, and other computational models [16,26,92]. Event prediction is currently experiencing extremely rapid growth, thanks to advances in sensing techniques (physical sensors and social sensors), prediction techniques (Artificial Intelligence, especially Machine Learning), and high performance computing hardware [78]. Event prediction in big data is a difficult problem that requires the invention and integration of related techniques to address the serious challenges caused by its unique characteristics, including: 1) Heterogeneous multi-output predictions. Event prediction methods usually need to predict multiple facets of events including their time, location, topic, intensity, and duration, each of which may utilize a different data structure [171]. This creates unique challenges, including how to jointly predict these heterogeneous yet correlated facets of outputs. Due to the rich information in the outputs, label preparation is usually a highly labor-intensive task performed by human annotators, with automatic methods introducing numerous errors in items such as event coding. So, how can we improve the label quality as well as the model robustness under corrupted labels? The multi-faceted nature of events make event prediction a multi-objective problem, which raises the question of how to properly unify the prediction performance on different facets. It is also challenging to verify whether a predicted event "matches" a real event, given that the various facets are seldom, if ever, 100% accurately predicted. So, how can we set up the criteria needed to discriminate between a correct prediction ("true positive") and a wrong one ("false positive")? 2) Complex dependencies among the prediction outputs. Beyond conventional isolated tasks in machine learning and predictive analysis, in event prediction the predicted events can correlate to and influence each other [142]. For example, an ongoing traffic incident event could cause congestion on the current road segment in the first 5 minutes but then lead to congestion on other contiguous road segments 10 minutes later. Global climate data might indicate a drought in one location, which could then cause famine in the area and lead to a mass exodus of refugees moving to another location. So, how should we consider the correlations among future events? 3) Real-time stream of prediction tasks. Event prediction usually requires continuous monitoring of the observed input data in order to trigger timely alerts of future potential events [182]. However, during this process the trained prediction model gradually becomes outdated, as real world events continually change dynamically, concepts are fluid and distribution drifts are inevitable. For example, in September 2008 21% of the United States population were social media users, including 2% of those over 65. However, by May 2018, 72% of the United States population were social media users, including 40% of those over [37]. Not only the data distribution, but also the number of features and input data sources can also vary in real time. Hence, it is imperative to periodically upgrade the models, which raises further questions concerning how to train models based on non-stationary distributions, while balancing the cost (such as computation cost and data annotation cost) and timeliness? In addition, event prediction involves many other common yet open challenges, such as imbalanced data (for example data that lacks positive labels in rare event prediction) [206], data corruption in inputs [248], the uncertainty of predictions [25], longer-term predictions (including how to trade-off prediction accuracy and lead time) [27], trade-offs between precision and recall [171], and how to deal with high-dimensionality [247] and sparse data involving many unrelated features [208]. Event prediction problems provide unique testbeds for jointly handling such challenges. In recent years, a considerable amount of research has been devoted to event prediction technique development and applications, in order to address the aforementioned challenges [157]. Recently, there has been a surge of research that both proposes and applies new approaches in numerous domains, though event prediction techniques are generally still in their infancy. Most existing event prediction methods have been designed for a specific application domains, but their approaches are usually general enough to handle problems in other application domains. Unfortunately, it is difficult to cross-reference these techniques across different application domains serving totally different communities. Moreover, the quality of event prediction results require sophisticated and specially-designed evaluation strategies due to the subject matter's unique characteristics, for example its multi-objective nature (e.g., accuracy, resolution, efficiency, and lead time) and heterogeneous prediction results (e.g., heterogeneity and multi-output). As yet, however, we lack a systematic standardization and comprehensive summarization approaches with which to evaluate the various event prediction methodologies that have been proposed. This absence of a systematic summary and taxonomy of existing techniques and applications in event prediction causes major problems for those working in the field who lacks clear information on the existing bottlenecks, traps, open problems, and potentially fruitful future research directions. To overcome these hurdles and facilitate the development of better event prediction methodologies and applications, this survey paper aims to provide a comprehensive and systematic review of the current state of the art for event prediction in the big data era. The paper's major contributions include: • A systematic categorization and summarization of existing techniques. Existing event prediction methods are categorized according to their event aspects (time, location, and semantics), problem formulation, and corresponding techniques to create the taxonomy of a generic framework. Relationships, advantages, and disadvantages among different subcategories are discussed, along with details of the techniques under each subcategory. The proposed taxonomy is designed to help domain experts locate the most useful techniques for their targeted problem settings. • A comprehensive categorization and summarization of major application domains. The first taxonomy of event prediction application domains is provided. The practical significance and problem formulation are elucidated for each application domain or subdomain, enabling it to be easily mapped to the proposed technique taxonomy. This will help data scientists and model developers to search for additional application domains and datasets that they can use to evaluate their newly proposed methods, and at the same time expand their advanced techniques to encompass new application domains. • Standardized evaluation metrics and procedures. Due to the nontrivial structure of event prediction outputs, which can contain multiple fields such as time, location, intensity, duration, and topic, this paper proposes a set of standard metrics with which to standardize existing ways to pair predicted events with true events. Then additional metrics are introduced and standardized to evaluate the overall accuracy and quality of the predictions to assess how close the predicted events are to the real ones. • An insightful discussion of the current status of research in this area and future trends. Based on the comprehensive and systematic survey and investigation of existing event prediction techniques and applications presented here, an overall picture and the shape of the current research frontiers are outlined. The paper concludes by presenting fresh insights into the bottleneck, traps, and open problems, as well as a discussion of possible future directions. Related Surveys This section briefly outlines previous surveys in various domains that have some relevance to event prediction in big data in three categories, namely: 1. event detection, 2. predictive analytics, and 3. domain-specific event prediction. Event detection has been an extensively explored domain with over many years. Its main purpose is to detect historical or ongoing events rather than to predict as yet unseen events in the future [181,222]. Event detection typically focuses on pattern recognition [26], anomaly detection [92], and clustering [92], which are very different from those in event prediction. There have been several surveys of research in this domain in the last decade [9,15,63,146]. For example, Deng et al. [63] and Atefeh and Khreich [15] provided overviews of event extraction techniques in social media, while Michelioudakis et al. [146] presented a survey of event recognition with uncerntainty. Alevizos et al. [9] provided a comprehensive literature review of event recognition methods using probabilistic methods. Predictive analysis covers the prediction of target variables given a set of dependent variables. These target variables are typically homogeneous scalar or vector data for describing items such as economic indices, housing prices, or sentiments. The target variables may not necessarily be values in the future. Larose [120] provides a good tutorial and survey for this domain. Predictive analysis can be broken down into subdomains such as structured prediction [26], spatial prediction [105], and sequence prediction [88], enabling users to handle different types of structure for the target variable. Fülöp et al. [81] provided a survey and categorization of applications that utilize predictive analytics techniques to perform event processing and detection, while Jiang [105] focused on spatial prediction methods that predict the indices that have spatial dependency. Baklr et al. [17] summarized the literature on predicting structural data such as geometric objects and networks, and Arias et al. [12] Phillips et al. [163], and Yu and Kak [232] all proposed the techniques for predictive analysis using social data. As event prediction methods are typically motivated by specific application domains, there are a number of surveys event predictions for domains such as flood events [47], social unrest [29], wind power ramp forecasting [76], tornado events [68], temporal events without location information [87], online failures [182], and business failures [6]. However, in spite of its promise and its rapid growth in recent years, the domain of event prediction in big data still suffers from the lack of a comprehensive and systematic literature survey covering all its various aspects, including relevant techniques, applications, evaluations, and open problems. Outline The remainder of this article is organized as follows. Section 2 presents generic problem formulations for event prediction and the evaluation of event prediction results. Section 3 then presents a taxonomy and comprehensive description of event prediction techniques, after which Section 4 categorizes and summarizes the various applications of event prediction. Section 5 lists the open problems and suggests future research directions and this survey concludes with a brief summary in Section 6. PROBLEM FORMULATION AND PERFORMANCE EVALUATIONS This section begins by examining the generic denotation and formulation of the event prediction problem (Section 2.1) and then considers way to standardize event prediction evaluations (Section 2.2). Problem Formulation An event refers to a real-world occurrence that happens at some specific time and location with specific semantic topic [222]. We can use y = (t, l, s) to denote an event where its time t ∈ T , its location l ∈ L, and its semantic meaning s ∈ S. Here, T , L, and S represent the time domain, location domain, and semantic domain, respectively. Notice that these domains need to have very general meanings that cover a wide range of types of entities. For example, the location L can include any features that can be used to locate the place of an event in terms of a point or a neighborhood in either Euclidean space (e.g., coordinate and geospatial region) or non-Euclidean space (e.g., a vertex or subgraph in a network). Similarly, the semantic domain S can contain any type of semantic features that are useful when elaborating the semantics of an event's various aspects, including its actors, objects, actions, magnitude, textual descriptions, and other profiling information. For example, ("11am, June 1, 2019", "Hermosillo, Sonora, Mexico", "Student Protests") and ("June 1, 2010", âĂIJBerlin, Germany", "Red Cross helps pandemics control") denote the time, location, and semantics, for two events respectively. An event prediction system requires inputs that could indicate future events, called event indicators, and these could contain both critical information on events that precede the future event, known as precursors, as well as irrelevant information [86,171]. Event indicator data can be denoted as X ⊆ T × L × F , where F is the domain of the features other than location and time. If we denote the current time as t now and define the past time and future time as T − ≡ {t |t ≤ t now , t ∈ T } and T + ≡ {t |t > t now , t ∈ T }, respectively, the event prediction problem can now be formulated as follows: Definition 2.1 (Event Prediction). Given the event indicator data X ⊆ T − × L × F and historical event data Y 0 ⊆ T − × L × S, event prediction is a process that outputs a set of predicted future eventsŶ ⊆ T + × L × S, such that for each predicted future eventŷ = (t, l, s) ∈Ŷ where t > t now . Not every event prediction method necessarily focuses on predicting all three domains of time, location, and semantics simultaneously, but may instead predict any part of them. For example, when predicting a clinical event such as the recurrence of disease in a patient, the event location might not always be meaningful [167], but when predicting outbreaks of seasonal flu, the semantic meaning is already known and the focus is the location and time [27] and when predicting political events, sometimes the location, time, and semantics (e.g., event type, participant population type, and event scale) are all necessary [171]. Moreover, due to the intrinsic nature of time, location, and semantic data, the prediction techniques and evaluation metrics of them are necessarily different, as described in the following. Event Prediction Evaluation Event Prediction Evaluation essentially investigates the goodness of fit for a set of predicted eventŝ Y against real events Y . Unlike the outputs of conventional machine learning models such as the simple scalar values used to indicate class types in classification or numerical values in regression, the outputs of event prediction are entities with rich information. Before we evaluate the quality of prediction, we need to first determine the pairs of predictions and the labels that will be used for the comparison. Hence, we must first optimize the process of matching predictions and real events (Section 2.2.1) before evaluating the prediction error and accuracy (Section 2.2.2). 2.2.1 Matching predicted events and real events. The following two types of matching are typically used: • Prefixed matching: The predicted events will be matched with the corresponding groundtrue real events if they share some key attributes. For example, for event prediction at a particular time and location point, we can evaluate the prediction against the ground truth for that time and location. This type of matching is most common when each of the prediction results can be uniquely distinguished along the predefined attributes (for example, location and time) that have a limited number of possible values, so that one-on-one matching between the predicted and real events are easily achieved [1,244]. For example, to evaluate the quality of a predicted event on June 1, 2019 in San Francisco, USA, the true event occurrence on that date in San Francisco can be used for the evaluation. • Optimized matching: In situations where one-on-one matching is not easily achieved for any event attribute, then the set of predicted events might need to assess the quality of the match achieved with the set of real events, via an optimized matching strategy [168,171]. For example, consider two predictions, Prediction 1: ("9am, June 4, 2019", "Nogales, Sonora, Mexico", "Worker Strike"), and Prediction 2: ("11am, June 1, 2019", "Hermosillo, Sonora, Mexico", "Student Protests"). The two ground truth events that these can usefully be compared with are Real Event 1: ("9am, June 1, 2019", "Hermosillo, Sonora, Mexico", "Teacher Protests"), and Real Event 2: ("June 4, 2019", "Navojoa, Sonora, Mexico", "General-population Protest"). None of the predictions are an exact match for any of the attributes of the real events, so we will need to find a "best" matching among them, which in this case is between Prediction 1 and Real Event 2 and Prediction 2 and Real Event 1. This type of matching allows some degree of inaccuracy in the matching process by quantifying the distance between the predicted and real events among all the attribute dimensions. The distance metrics are typically either Euclidean distance [105] or some other distance metric [92]. Some researchers have hired referees to manually check the similarity of semantic meanings [168], but another way is to use event coding to code the events into an event type taxonomy and then consider a match to have been achieved if the event type matches [49]. Based on the distance between each pair of predicted and real events, the optimal matching will be the one that results in the smallest average distance [157]. However, suppose there are m predicted events and n real events, then there can be as many as 2 m ·n possible ways of matching, making it prohibitively difficult to find the optimal solution. Moreover, there could be different rules for matching. For example, the "multiple-to-multiple" rule shown in Figure 2(a) allows one predicted (real) event to match multiple real (predicted) events [170], while "Bipartite matching" only allows one-to-one matching between predicted and real events (Figure 2(b)). "Non-crossing matching" requires that the real events matched by the predicted events follow the same chronological order (Figure 2(c)). In order to utilize any of these types of matching, researchers have suggested using event matching optimization to learn the optimal set of "(real event, predicted event)" pairs [151]. Metrics of Effectiveness. The effectiveness of the event predictions is evaluated in terms of two indicators: 1) Goodness of Matching, which evaluates performance metrics such as the number and percentage of matched events [26], and 2) Quality of Matched Predictions, which evaluates how close the predicted event is to the real event for each pair of matched events [171]. • Goodness of Matching. A true positive means a real event has been successfully matched by a predicted event; if a real event has not been matched by any predicted event, then it is called a false negative and a false positive means a predicted event has failed to match any real event, which is referred to as a false alarm. Assume the total number of predictions is N , the number of real events isN , the number of true positives is N T P , the number of false negatives is N F N and the number of false positives is N F P . Then, the following key evaluation metrics can be calculated: Prediction=N T P /(N T P + N F P ), Recall=N T P /(N T P + N F N ), F-measure = 2 · Precision · Recall/(Precision + Recall). Other measurements such as the area under the ROC curves are also commonly used [26]. This approach can be extended to include other items such as multi-class precision/recall, and Precision/Recall at Top K [2,103,123,252]. • Quality of Matched Predictions. If a predicted event matches a real one, it is common to go on to evaluate how close they are. This reflects the quality of the matched predictions, in terms of different aspects of the events. Event time is typically a numerical values and hence can be easily measured in terms of metrics such as mean squared error, root mean squared error, and mean absolute error [26]. This is also the case for location in Euclidean space, which can be measured in terms of the Euclidean distance between the predicted point (or region) and the real point (or region). Some researchers consider the administrative unit resolution. For example, a predicted location ("New York City", "New York State", "USA") has a distance of 2 from the real location ("Los Angeles", "California", "USA") [240]. Others prefer to measure multi-resolution location prediction quality as follows: (1/3)(l count ry + l count ry · l st at e + l count r y · l st at e · l city ), where l city , l st at e , and l count r y can only be either 0 (i.e., no match to the truth) or 1 (i.e., completely matches the truth) [171]. For a location in non-Euclidean space such as a network [187], the quality can be measured in terms of the shortest path length between the predicted node (or subgraph) and the real node (or subgraph), or by the F-measure between the detected subsets of nodes against the real ones, which is similar to the approach for evaluating community detection [92]. For event topics, in addition to conventional ways of evaluating continuous values such as population size, ordinal values such as event scale, and categorical values such as event type, actors, and actions, as well as more complex semantic values such as texts, can be evaluated using Natural Language Process measurements such as edit distance, BLEU score, Top-K precision, and ROUGE [10]. EVENT PREDICTION TECHNIQUES This section focuses on the taxonomy and representative techniques utilized for each category and subcategory. Due to the heterogeneity of the prediction output, the technique types depend on the type of output to be predicted, such as time, location, and semantics. As shown in Figure 3, all the event prediction methods are classified in terms of their goals, including time, location, semantics, and the various combinations of these three. These are then further categorized in terms of the output forms of the goals of each and the corresponding techniques normally used, as elaborated in the following. Time Prediction Location Prediction Semantic Prediction Joint Prediction Time and Semantic Time and Location Time, Location, and Semantics Semantic Sequence Causality-Based Raster-Based Discrete Time Continuous Time Occurrence Association-Based Point-Based Event Prediction Research Problems 1. Supervised: geo-featured classification; spatial multi-task learning; spatial autoregressive 2. Unsupervised: Spatial scan, network scan Step 1: Event representation Step 2: Event causality inference. Step 3: Future event inference. Fig. 3. Taxonomy of event prediction problems and techniques. Techniques Time Prediction Event time prediction focuses on predicting when future events will occur. Based on their time granularity, time prediction methods can be categorized into three types: 1) event Occurrence: Binary-valued prediction on whether an event does or does not occur in a future time period; 2) discrete-time prediction: in which future time slot will the event occur; and 3) continuous-time prediction: at which precise time point will the future event occur. Occurrence Prediction. Occurrence prediction is arguably the most extensive, classical, and generally simplest type of event time prediction task [12]. It focuses on identifying whether there will be event occurrence (positive class) or not (negative class) in a future time period [244]. This problem is usually formulated as a binary classification problem, although a handful of other methods instead leverage anomaly detection or regression-based techniques. 1. Binary classification. Binary classification methods have been extensively explored for event occurrence prediction. The goal here is essentially to estimate and compare the values of f (y = "Yes ′′ |X ) and f (y = "No ′′ |X ), where the former denotes the score or likelihood of event occurrence given observation X while the latter corresponds to no event occurrence. If the value of the former is larger than the latter, then a future event occurrence is predicted, but if not, there is no event predicted. To implement f , the methods typically used rely on discriminative models, where dedicated feature engineering is leveraged to manually extract potential event precursor features to feed into the models. Over the years, researchers have leveraged various binary classification techniques ranging from the simplest threshold-based methods [121,251], to more sophisticated methods such as logistic regression [18,248], Support Vector Machines [102], (Convolutional) Neural Networks [35,135], and decision trees [58,185]. In addition to discrminative models, generative models [11,239] have also been used to embed human knowledge for classifying event occurrences using Bayesian decision techniques. Specifically, instead of assuming that the input features are independent, prior knowledge can also be directly leveraged to establish Bayesian networks among the observed features and variables based on graphical models such as (semi-)hidden Markov models [54,183,239] and autoregresive logit models [199]. The joint probabilities p(y = "Yes ′′ , X ) of p(y = "No ′′ , X ) can thus be estimated using graphical models, and then utilized to estimate f (y = "Yes ′′ |X ) = p(y = "Yes ′′ |X ) and f (y = "No ′′ |X ) = p(y = "No ′′ |X ) using Bayesian rules [26]. 2. Anomaly detection. Alternatively, anomaly detection can also be utilized to learn a "prototype" of normal samples (typical values corresponding to the situation of no event occurrence), and then identify if any newly-arriving sample is close to or distant from the normal samples, with distant ones being identified as future event occurrences. Such methods are typically utilized to handle "rare event" occurrences, especially when the training data is highly imbalanced with little to no data for "positive" samples. Anomaly detection techniques such as one-classification [189] and hypotheses testing [100,187] are often utilized here. 3. Regression. In addition to simply predicting the occurrence or not, some researchers have sought to extend the binary prediction problem to deal with ordinal and numerical prediction problems, including event count prediction based on (auto)regression [73], event size prediction using linear regression [229], and event scale prediction using ordinal regression [84]. 3.1.2 Discrete-time Prediction. In many applications, practitioners want to know the approximate time (i.e. the date, week, or month) of future events in addition to just their occurrence. To do this, the time is typically first partitioned into different time slots and the various methods focus on identifying which time slot future events are likely to occur in. Existing research on this problem can be classified into either direct or indirect approaches. 1. Direct Approaches. These of methods discretize the future time into discrete values, which can take the form of some number of time windows or time scales such as near future, medium future, or distant future. These are then used to directly predict the integer-valued index of future time windows of the event occurrence using (auto)regression methods [147,154], or to predict the ordinal values of future time scales using ordinal regression or classification [201]. 2. Indirect Approaches. These methods adopt a two-step approach, with the first step being to place the data into a series of time bins and then perform time series forecasting using techniques such as autoregressive [26] based on the historical time series x = {x 1 , · · · , x T } to obtain the future time seriesx = {x T +1 , · · · , xT }. The second step is to identify events in the predicted future time seriesx using either unsupervised methods such as burstness detection [31] and change detection [109], or supervised techniques based on learning event characterization function. For example, existing works [165,173] first represent the predicted future time seriesx ∈ RT ×D using time-delayed embedding, intox ∈ RT ×D ′ where each observation at time t can be represented asx t = {x t −(D ′ −1)τ , · · · , x t −2τ , x t −τ , x t } and t = T ,T + 1, · · ·T . Then an event characterization function f c (x t ) is established to mapx t to the likelihood of an event, which can be fitted based on the event labels provided in the training set intuitively. Overall, the unsupervised method requires users to assume the type of patterns (e.g., burstiness and change) of future events based on prior knowledge but do not require event label data. However, in cases where the event time series pattern is difficult to assume but the label data is available, supervised learning-based methods are usually used. Continuous-time Prediction. Discrete-time prediction methods, although usually simple to establish, also suffer from several issues. First, their time-resolution is limited to the discretization granularity; increasing this granularity significantly increases the computations al resources required, which means the resolution cannot be arbitrarily high. Moreover, this trade-off is itself a hyperparameter that is sensitive to the prediction accuracy, rendering it difficult and time-consuming to tune during training. To address these issues, a number of techniques work around it by directly predicting the continuous-valued event time [191], usually by leveraging one of three techniques. 1. Simple Regression. The simplest methods directly formalize continuous-event-time prediction as a regression problem [26], where the output is the numerical-value future event time [212] and/or their duration [80,129]. Common regressors such as linear regression and recurrent neural networks have been utilized for this. Despite their apparent simplicity, this is not straightforward as simple regression typically assumes Gaussian distribution [129], which does not usually reflect the true distribution of event times. For example, the future event time needs to be left-bounded (i.e., larger than the current time), as well asbeing typically non-symmetric and usually periodic, with recurrent events having multiple peaks in the probability density function along the time dimension. 2. Point Processes. As they allow more flexibility in fitting true event time distributions, point process methods [167,219] are widely leveraged and have demonstrated their effectiveness for continuous time event prediction tasks. They require a conditional intensity function, defined as follows: λ(t |X ) = E[N (t, t + dt)/dt |X ] = д(t |X )/(1 − G(t |X ))(1) where д(t |X ) is the conditional density function of the event occurrence probability at time t given an observation X , and whose corresponding cumulative distribution function, G(t |X )), N (t, t + dt), denotes the count of events during the time period between t and t + dt, where dt is an infinitelysmall time period. Hence, by leveraging the relation between density and accumulative functions and then rearranging Equation (1), the following conditional density function is obtained: д(t |X ) = λ(t |X ) · exp · − ∫ t t 0 λ(u |X )du(2) Once the above model has been trained using a technique such as maximal likelihood [26], the time of the next event in the future is predicted as: t = ∫ ∞ t 0 t · д(t |X )dt(3) Although existing methods typically share the same workflow as that shown above, they vary in the way they define the conditional intensity function λ(t |X ). Traditional models typically utilize prescribed distributions such as the Poisson distribution [191], Gamma distribution [53], Hawks [69], Weibull process [56], and other distributions [219]. For example, Damaschke et al. [56] utilized a Weibull distribution to model volcano eruption events, while Ertekin et al. [72] instead proposed the use of a non-homogeneous Poisson process to fit the conditional intensity function for power system failure events. However, in many other situations where there is no information regarding appropriate prescribed distributions, researchers must start by leveraging nonparametric approaches to learn sophisticated distributions from the data using expressive models such as neural networks. For example, Simma and Jordan [191] utilized of RNN to learn a highly nonlinear function of λ(t |X ). 3. Survival Analysis. Survival analysis [62,204] is related to point processes in that it also defines an event intensity or hazard function, but in this case based on survival probability considerations, as follows: H (t |X ) = (ξ (t − dt |X ) − ξ (t |X )) /ξ (t |X ), where ξ (t |X ) = p(t > t)(4) where H (t |X ) is the so-called Hazard function denoting the hazard of event occurrence between time (t −dt) for a t for a given observation X . Either H (t |X ) or ξ (t |X ) could be utilized for predicting the time of future events. For example, the event occurrence time can be estimated when ξ (t |X ) is lower than a specific value. Also, one can obtain ξ (t |X ) = exp − ∫ t 0 H(u|X)du according to Equation (4) [132]. Here H (t |X ) can adopt any one of several prescribed models, such as the wellknown Cox hazard model [61,132]. To learn the model directly from the data, some researchers have recommended enhancing it using deep neural networks [119]. Vahedian et al. [204] suggest learning the survival probability ξ (t |X ) and then applying the function H (·|X ) to indicate an event at time t if H (t |X ) is larger than a predefined threshold value. A classifier can also be utilized. Instead of using the raw sequence data, the conditional intensity function can also be projected onto additional continuous-time latent state layers that eventually map to the observations [62,205]. These latent states can then be extracted using techniques such as hidden semi-Markov models [26], which ensure the elicitation of the continuous time patterns. Location Prediction Event location prediction focuses on predicting the location of future events. Location information can be formulated as one of two types: 1. Raster-based. Here, a continuous space is partitioned into a grid of cells, each of which represents a spatial region, as shown in Figure 4(a). This type of representation is suitable for situations where the spatial size of the event is non-negligible. 2. Point-based. In this case, each location is represented by an abstract point with infinitely-small size, as shown in Figure 4(b). This type of representation is most suitable for the situations where the spatial size of the event can be neglected, or the location regions of the events can only be in discrete spaces such as network nodes. Raster-based Location Prediction. There are three types of techniques used for raster-based event location prediction, namely spatial clustering, spatial embedding, and spatial convolution. 1. Spatial clustering. In raster-based representations, each location unit is usually a regular grid cell with the same size and shape. However, regions with similar spatial characteristics typically have irregular shapes and sizes, which could be approximated as composite representations of a number of grids [105]. The purpose of spatial clustering here is to group the contiguous regions who collectively exhibit significant patterns. The methods are typically agglomerative style. They typically start from the original finest-grained spatial raster units and proceed by merging the spatial neighborhood of a specific unit in each iteration. But different research works define different criteria for instantiating the merging operation. For example, Wang and Ding [211] merge neighborhoods if the unified region after merging can maintain the spatially frequent patterns. Xiong et al. [220] chose an alternative approach by merging spatial neighbor locations into the current locations sequentially until the merged region possesses event data that is sufficiently statistically significant. These methods usually run in a greedy style to ensure their time complexity remains smaller than quadratic. After the spatial clustering is completed, each spatial cluster will be input into the classifier to determine whether or not there is an event corresponding to it. 2. Spatial interpolation. Unlike spatial clustering-based methods, spatial interpolation based methods maintain the original fine granularity of the event location information. The estimation of event occurrence probability can be further interpolated for locations with no historical events and hence achieve spatial smoothness. This can be accomplished using commonly-used methods such as kernel density estimation [5,93] and spatial Kriging [105,114]. Kernel density estimation is a popular way to model the geo-statistics in numerous types of events such as crimes [5] and terrorism [93]: k(s) = 1/(n · γ ) n i=1 K ((s − s i )/γ )(5) where k(s) denotes the kernel estimation for the location point s, n is the number of historical event locations, each s i is a historical event location, γ is a tunable bandwidth parameter, and K(·) is a kernel function such as Gaussian kernel [85]. More recently, Ristea et al. [176] further extended KDE-based techniques by leveraging Localized KDE and then applying spatial interpolation techniques to estimate spatial feature values for the cells in the grid. Since each cell is an area rather than a point, the center of each cell is usually leveraged as the representative of this cell. Finally, a classifier will take this as its input to predict the event occurrence for each grid [5,176]. 3. Spatial convolution. In the last few years, Convolutional Neural Networks (CNNs) have demonstrated significant success in learning and representing sophisticated spatial patterns from image and spatial data [88]. A CNN contains multiple convolutional layers that extract the hierarchical spatial semantics of images. In each convolutional layer, a convolution operation is executed by scanning a feature map with a filter, which results in another smaller feature map with a higher level semantic. Since raster-based spatial data and images share a similar mathematical form, it is natural to leverage CNNs to process it. Existing methods [19,150,164,209] in this category typically formulate a spatial map as input to predict another spatial map that denotes future event hotspots. Such a formulation is analogous to the "image translation" problem popular in recent years in the computer vision domain [46]. Specifically, researchers typically leverage an encoder-decoder architecture, where the input images (or spatial map) are processed by multiple convolutional layers into a higher-level representation, which is then decoded back into an output image with the same size, through a reverse convolutional operations process known as transposed convolution [88]. 4. Trajectory destination prediction. This type of method typically focuses on populationbased events whose patterns can be interpreted as the collective behaviors of individuals, such as "gathering events" and "dispersal events". These methods share a unified procedure that typically consists of two steps: 1) predict future locations based on the observed trajectories of individuals, and 2) detect the occurrence of the "future" events based on the future spatial patterns obtained in Step 1. The specific methodologies for each step are as follows: • Step 1: Here, the aim is to predict each location an individual will visit in the future, given a historical sequence of locations visited. This can be formulated as a sequence prediction problem. For example, Wang and Gerber [214] sought to predict the probability of the next time point t + 1's location s t +1 based on all the preceding time points: p(s t +1 |s ≤t ) = p(s t +1 |s t , s t −1 , · · · , s 0 ), based on various strategies including a historical volume-based prior model, Markov models, and multi-class classification models. Vahedian et al. [203] adopted Bayesian theory p(s t +1 |s ≤t ) = p(s ≤t |s t +1 ) · p(s t +1 )/p(s ≤t ) which requires the conditional probability p(s ≤t |s t +1 ) to be stored. However, in many situations, there is huge number of possible trajectories for each destination. For example, with a 128 × 64 grid, one needs to store (128 × 64) 3 ≈ 5.5 × 10 11 options. To improve the memory efficiency, this can be limited to a consideration of just the source and current locations, leveraging a quad-tree style architecture to store the historical information. To achieve more efficient storage and speed up p(s ≤t |s t +1 ) queries, Vahedian et al. [203] further extended the quad-tree into a new technique called VIGO, which removes duplicate destination locations in different leaves. • Step 2: The aim in this step is to forecast future event locations based on the future visiting patterns predicted in Step 1. The most basic strategy here is to consider each grid cell independently. For example, Wang and Gerber [214] adopted supervised learning strategies to build predictive mapping between the visiting patterns and the event occurrence. A more sophisticated approach is to consider the spatial outbreaks composited by multiple grids. Scalable algorithms have also been proposed to identify regions containing statistically significant hotspots [110], such as spatial scan statistics [116]. Khezerlou et al. [110] proposed a greedy-based heuristic tailored for the grid-based data formulation, which extends the original "seed" grid containing statistically-large future event densities to four directions until the extended region is no longer a statistically-significant outbreak. Point-based Prediction. . Unlike the raster-based formulation, which covers the prediction of a contiguous spatial region, point-based prediction focuses specifically on locations of interest, which can be distributed sparsely in a Euclidean (e.g., spatial region) or non-Euclidean space (e.g., graph topology). These methods can be categorized into supervised and unsupervised approaches. 1. Supervised approaches. In supervised methods, each location will be classified as either "positive" or "negative" with regard to a future event occurrence. The simplest setting is based on the independent and identically distributed (i.i.d.) assumption among the locations, where each location is predicted by a classifier independently using their respective input features. However, given that different locations usually have strong spatial heterogeneity and dependency, further research has been proposed to tackle them based on different locations' predictors and outputs, resulting in two research directions: 1) Spatial multi-task learning, and 2) Spatial auto-regressive methods. • Spatial multi-task learning. Multi-task learning is a popular learning strategy that can jointly learn the models for different tasks such that the learned model can not only share their knowledge but also preserve some exclusive characteristics of the individual tasks [244]. This notion coincides very well with spatial event prediction tasks, where combining the outputs of models from different locations needs to consider both their spatial dependency and heterogeneity. Zhao et al. [244] proposed a spatial multi-task learning framework as follows: min W m i L(Y i , f (W i , X i )) + α · R({W i } k i , M), s.t ., C({W i } m i , D) ∈ C(6) where m is the total number of locations (i.e., tasks), W i and Y i are the model parameters and true labels (event occurrence for all time points), respectively, of task i. L(·) is the empirical loss, f (W i , X i ) is the predictor for task i, and R(·) is the spatial regularization term based on the spatial dependency information M ∈ R m×m , where M i, j records the spatial dependency between location i and j. C(·) represents the spatial constraints imposed over the corresponding models to enforce them to remain within the valid space C. Over recent years, there have been multiple studies proposing different strategies for R(·) and C(·). For example, Zhao et al. [245] assumed that all the locations would be evenly correlated and enforced their similar sparsity patterns for feature selection, while Gao et al. [85] further extended this to differentiate the strength of the correlation between different locations' tasks according to the spatial distance between them. This research has been further extended this approach to tree-structured multitask learning to handle the hierarchical relationship among locations at different administrative levels (e.g., cities, states, and countries) [246] in a model that also considers the logical constraints over the predictions from different locations who have hierachical relationships. Instead of evenly similar, Zhao, et al. [243] further estimated spatial dependency D utilizing inverse distance using Gaussian kernels, while Ning et al. [156] proposed estimating the spatial dependency D based on the event co-occurrence frequency between each pair of locations. • Spatial auto-regressive methods. Spatial auto-regressive models have been extensively explored in domains such as geography and econometrics, where they are applied to perform predictions where the i.i.d. assumption is violated due to the strong dependencies among neighbor locations. Its generic framework is as follows: Y t +1 = ρMŶ t +1 + X t · W + ε(7) where X t ∈ R m×D andŶ t +1 ∈ R m×m are the observations at time t and event predictions at time t + 1 over all the m locations, and M ∈ R m×m is the spatial dependency matrix with zero-valued diagonals. This means the prediction of each locationŶ t +1,i ∈Ŷ t +1 is jointly determined by its input X t,i and neighbors {j |M i, j 0} and ρ is a positive value to balance these two factors. Since event occurrence requires discrete predictions, simple threshold-based strategies can be used to discretizeŶ i intoŶ ′ i = {0, 1} [32]. Moreover, due to the complexity of event prediction tasks and the large number of locations, sometimes it is difficult to define the whole M manually. Zhao et al. [243] proposed jointly learning the prediction model and spatial dependency from the data using graphical LASSO techniques. Yi et al. [228] took a different approach, leveraging conditional random fields to instantiate the spatial autoregression, where the spatial dependency is measured by Gaussian kernel-based metrics. Yi et al. [227] then went on to propose leveraging the neural network model to learn the locations' dependency. 2. Unsupervised approaches. Without supervision from labels, unsupervised-based methods must first identify potential precursors and determinant features in different locations. They can then detect anomalies that are characterized by specific feature selection and location combinatorial patterns (e.g., spatial outbreaks and connected subgraphs) as the future event indicators [41]. The generic formulation is as follows: (F , R) = arg max F, R q(F , R) s.t ., supp(F i ) ∈ M(G, β), supp(R i ) ∈ C(8) where q(·) denotes scan statistics which score the significance of each candidate pattern, represented by both a candidate location combinatorial pattern R and feature selection pattern F . Specifically, F ∈ {0, 1} D ′ ×n denotes the feature selection results (where "1" means selected; "0", otherwise) and R ∈ {0, 1} m×n denotes the m involved locations for the n events. M(G, β) and C are the set of all the feasible solutions of F and R, respectively. q(·) can be instantiated by scan statistics such as Kulldorff's scan statistics [116] and the Berk-Jones statistic [41], which can be applied to detect and forecast events such as epidemic outbreaks and civil unrest events [171]. Depending on whether the embedding space is an Euclidean region (e.g., a geographical region) or a non-Euclidean region (e.g., a network topology), the pattern constraint C can be either constrained to predefined geometric shapes such as a circle, rectangle, or an irregular shape or subgraphs such as connected, cliques, and k-cliques. The problem in Equation (8) is nonconvex and sometimes even discrete, and hence difficult to solve. A generic way is to optimize F using sparse feature selection; there is a useful survey provided in [127] and R can be defined using the two-step graph-structured matching method detailed in [42]. More recently, new techniques have been developed that are capable of jointly learning both feature and location selection [42,187]. Semantic Prediction Event semantics prediction addresses the problem of forecasting topics, descriptions, or other metaattributes in addition to future events' times and locations. Unlike time and location prediction, the data in event semantics prediction usually involves symbols and natural languages in addition to numerical quantities, which means different types of techniques may be utilized. The data are categorized into three types based on how the historical data are organized and utilized to infer future events. The first of these categories covers rule-based methods, where future event precursors are extracted by mining association or logical patterns in historical data. The second type is sequence-based, considering event occurrence to be a consequence of temporal event chains. The third type further generalizes event chains into event graphs, where additional cross-chain contexts need to be modeled. These are discussed in turn below. Association-based Prediction. . Association rule-based methods are amongst the most classic approaches in data mining domain for event prediction, typically consisting of two steps: 1) learn the associations between precursors and target events, and then 2) utilize the learned associations to predict future events. For the first step, for example, an association could be x = {"election", "fraud"} → y ="protest event", which indicates that serious fraud occurring in an election process could lead to future protest events. To discover all the significant associations from the ocean of candidate rules efficiently, frequent set mining [92] can be leveraged. Each discovered rule needs to come with both sufficient support and confidence. Here, support is defined as the number of cases where both "x" and "y" co-occur, while confidence means the ratio indicating that "y" occurs once "x" happens. To better estimate these discrimination rules, further temporal constraints can be added that require the occurrence time of "x" and "y" to be sufficiently close to be considered "co-occurrences". Once the frequent set rules have been discovered, pruning strategies may be applied to retain the most accurate and specific ones, with various strategies for generating final predictions [92]. Specifically, given each new observation x ′ , one of the simplest strategies is to output the events that are triggered by any of the association rules starting from event x ′ [206]. Other strategies first rank the predicted results based on their confidence and then predict just the top r events [252]. More sophisticated and rigorous strategies tend to build a decision list where each element in the list is an association rule mapping, so once a generative model has been built for the decision process, the maximal likelihood can be leveraged to optimize the order of the decision list [124]. Causality-based prediction. This type of research leverages the causality inferred among the historical events to achieve future event predictions. The data here typically shares a generic framework consisting of the following procedures: 1) event representation, 2) event graph construction, and 3) future event inference. Step 1: Event semantic representation. This approach typically begins by extracting the events from the target texts using natural language processing techniques such as sanitization, tokenization, POS tag analysis, and name entity recognition. Several types of objects can be extracted to represent the events: i) Noun Phrase-based [39,94,111], where the noun-phrase corresponds to each event (for example, "2008 Sichuan Earthquake"); ii) Verbs and Nouns [112,168], where an event is represented as a set of noun-verb pairs extracted from news headlines (for example, "<capture, people>", "<escape, prison>", or "<send, prison>"); and iii) Tuple-based [249], where each event is represented by a tuple consisting of objects (such as actors, instruments, or receptors), a relationship (or property), and time. An RDF-based format has also been leveraged in some works [57]. Step 2: Event causality inference. The goal here is to infer the cause-effect pairs among historical events. Due to its combinatorial nature, narrowing down the number of candidate pairs is crucial. Existing works usually begin by clustering the events into event chains, each of which consist of a sequence of time-ordered events under the relevant semantics, typically the same topics, actors, and/or objects [2]. The causal relations among the event pairs can then be inferred in various ways. The simplest approach is just to consider the likelihood that y occurs after x has occurred throughout the training data. Other methods utilize NLP techniques to identify causal mentions such as causal connectives, prepositions, and verbs [168]. Some formulate causal-effect relationship identification as a classification task where the inputs are the cause and effect candidate events, often incorporating contextual information including related background knowledge from web texts. Here, the classifier is built on a multi-column CNN that outputs either "1" or "0" to indicate whether the candidate has an effect or not [115]. In many situations, the cause-effect rules learned directly using the above methods can be too specific and sparse, with low generalizability, so a typical next step is to generalize the learned rules. For example, "Earthquake hits China" → "Red Cross help sent to Beijing" is a specific rule that can be generalized to "Earthquake hits [A country]" → "Red Cross help sent to [The capital of this country]". To achieve this, some external ontology or a knowledge base is typically needed in order to establish the underlying relationships among items or provide necessary information on their properties, such as Wikipedia (https://www.wikipedia.org/), YAGO [196], WordNet [75], or ConceptNet [137]. Based on these resources, the similarity between two cause-effect pairs (c i , ε i ) and (c j , ε j ) can be computed by jointly considering the respective similarity of the putative cause and effect: σ ((c i , ε i ), (c j , ε j )) = (σ (c i , c j )+σ (ε i , ε j ))/2. An appropriate algorithm can then be utilized to apply hierarchical agglomerative clustering to group them and hence generate a data structure that can efficiently manage the task of storing and querying them to identify any cause-effect pairs. For example, [168,169,190] leverage an abstraction tree, where each leaf is an original specific cause-effect pair and each intermediate node is the centroid of a cluster. Instead of using hierarchical clustering, [249] directly uses the word ontology to simultaneously generalize cause and effect (e.g., the noun "violet" is generalized to "purple", the verb "kill" is generalized to "murder-42.1 1 ") and then leverage a hierarchical causal network to organize the generalized rules. Step 3: Future event Inference. Given an arbitrary query event, two steps are needed to infer the future events caused by it based on the causality of events learned above. First, we need to retrieve similar events that match the query event from historical event pool. This requires the similarity between the query event and all the historical events to be calculated. To achieve this, Lei et al. [123] utilized context information, including event time, location, and other environmental and descriptive information. For methods requiring event generalization, the first step is to traverse the abstraction tree starting from the root that corresponds to the most general event rule. The search frontier then moves across the tree if the child node is more similar, culminating in the nodes which are the least general but still similar to the new event being retrieved [168]. Similarly, [45] proposed another tree structure referred to as a "circular binary search tree" to manage the event occurrence pattern. We can now apply the learned predicate rules starting from the retrieved event to obtain the prediction results. Since each cause event can lead to multiple events, a convenient way to determine the final prediction is to calculate the support [168], or conditional probability [226] of the rules. Radinsky et al. [168] took a different approach, instead ranking the potential future events by their similarity defined by the length of their minimal generalization path. For example, the minimal generalization path for "London" and "Paris" is "London" capital-of − −−−−−− →"Great Brain" in-continent − −−−−−−−− →"Europe" in-continent ← −−−−−−−− −"France" capital-of ← −−−−−− −"Paris". Alternatively, Zhao et al. [249] proposed embedding the event causality network into a continuous vector space and then applying an energy function designed to rank potential events, where true cause-effect pairs are assumed to have low energies. Semantic Sequence. These methods share a very straightforward problem formulation. Given a temporal sequence for a historical event chain, the goal is to predict the semantics of the next event using sequence prediction [26]. The existing methods can be classified into four major categories: 1) classical sequence prediction; 2) recurrent neural networks; 3) Markov chains; and 4) time series predictions. Sequence classification-based methods. These methods formulate event semantic prediction as a multi-class classification problem, where a finite number of candidate events are ranked and the top-ranked event is treated as the future event semantic. The objective isĈ = arg max C i u(s T +1 = C i |s 1 , · · · , s T ), where s T +1 denotes the event semantic in time slot T +1 andĈ is the optimal semantic among all the semantic candidates C i (i = 1, · · · ). Multi-class classification problems can be split into events with different topics/semantic meaning. Three types of sequence classification methods have been utilized for this purpose, namely feature-based methods, prototype-based methods, and model-based methods such as Markov models. • Feature-based. One of the simplest methods is to ignore the temporal relationships among the events in the chain, by either aggregating the inputs or the outputs. Tama and Comuzzi [198] formulated historical event sequences with multiple attributes for event prediction, testing multiple conventional classifiers. Another type of approach based on this notion utilizes compositional based-methods [89] that typically leverage the assumption of independency among the historical input events to simplify the original problem u(s T +1 |s 1 , s 2 , · · · , s T ) = u(s T +1 |s ≤T ) into v(u(s T +1 |s 1 ), u(s T +1 |s 2 ), · · · , u(s T +1 |s T )) where v(·) is simply an aggregation function that represents a summation operation over all the components. Each component function u(s T +1 |s i ) can then be calculated by estimating how likely it is that event semantic s T +1 and s i (i ≤ T ) co-occur in the same event chain. Granroth-Wilding and Clark [89] investigated various models ranging from straightforward similarity scoring functions through bigram models and word embedding combined with similarity scoring functions to newly developed composition neural networks that jointly learn the representation of s T +1 and s i and then calculate their coherence. Some other researchers have gone further to consider the dependency among the historical events. For example, Letham et al. [125] proposed to optimizing the correct ordering among the candidate events, based on the following equation: i ∈I, j ∈ J 1 [u(s T +1 =C i |s ≤T )>u(s T +1 =C j |s ≤T )] =⇒ i ∈I, j ∈ J e u(s T +1 =C i |s ≤T )−u(s T +1 =C j |s ≤T ) + ρ ∥W ∥ 2 2 where the semantic candidate in the set I should be ranked strictly to be lower than those in J , with the goal being to penalize the "incorrect ordering". Here, 1 [·] is an indicator function which is discrete such that 1 [b ≥a] ≤ e b−a and can thus be utilized as the upperbound for minimization, as can be seen in the right-hand-side of the above equation. W is the set of parameters of the function u(·). This can now be relaxed to an exponentialbased approximation for effective optimization using gradient-based algorithms [88]. Other methods focus on first transferring the sequential data into sequence embeddings that can encode the latent sequential context. For example, Fronza et al. [79] apply random indexing to represent the words in terms of their its vector representations by embedding the information from neighboring words into each word before utilizing conventional classifiers such as Support Vector Machines (SVM) to identify the future events. • Model-based. Markov-based models have also been leveraged to characterize temporal patterns [224]. These typically use E i to denote each event under a specific type and E denotes the set of event types. The goal here is to predict the event type of the next event to occur in the future. In [7], the event types are modeled using the Markov model so given the current event type, the next event type can be inferred simply by looking up the state with the highest probability in the transition matrix. A tool called Wayeb [8] has been developed based on this method. Laxman et al. [121] developed a more complicated model, based on a mixture of Hidden Markov models and introducing new assumptions and the concept of episodes composed of a subsequence of event types. They assumed different event episodes should have different transition patterns so started by discovering the frequent episodes for events, each of which they modeled by a specific hidden Markov model over various event types. This made it possible to establish the generative process for each future event type s based on the mixture of the above episode Markov models. When predicting, the likelihood of a current observed event sequence over each possible generative process, p(X |Λ Y ) is evaluated, after which a future event type can be considered as either being larger than some threshold (as in [121]) or the largest among all the different Y values (in [239,241]). • Prototype-based. Adhikari et al. [3] took a different approach, utilizing a prototype-based strategy that first clusters the event sequences into different clusters in terms of their temporal patterns. When a new event sequence is observed, its closest cluster's centroid will then be leveraged as a "reference event sequence" whose sub-sequential events will be referred to when predicting future events for this new event sequence. Recurrent neural network (RNN)-based methods. Approaches in this category can be classified into two types: 1. Attribute-based models; and 2. Descriptive-based models. The attribute-based models, ingest feature representation of events as input, while the descriptive-based models typically ingest unstructured information such as texts to directly predict future events. • Attributed-based Methods. Here, each event y = (t, l, s) at time t is recast and represented as e t = (e t,1 , e t,2 , · · · , e t,k ), where e t,i is the i-th feature of the event at time t. The feature here can include location and other information such as event topic and semantics. Each sequence e = (e 1 , · · · , e t ) is then input into the standard RNN architecture for predicting next event e t +1 in the sequence at time point t + 1 [134]. Various types of RNN components and architecture have been utilized for this purpose [33,34], but a vanilla RNN [70,88] for sequence-based event prediction can be written in the following form: h i = tahn(a t ), a i = b + W · h i−1 + U · e i , o i = c + V · h i , ψ (i + 1) = softmax(o i ), i ≤ t where h i , o i , and a i are the latent state, output, and activation for the i-th event, respectively, and W , U , and V are the model parameters for fitting the corresponding mappings. The prediction e t +1 := ψ (t + 1) can then be calculated in a feedforward way from the first event and the model training can be done by back-propagating the error from the layer of ψ (t). Existing work typically utilizes the variants of vanilla RNN to handle the gradient vanishing problem, especially when the event chain is not short. The most commonly used methods for event prediction are LSTM and GRU [88]. For example, the architecture and equation for LSTM are as follows: a i = σ (W j · [h i−1 , e i ] + b j ),C i = tanh(W C · [h i−1 , h i ] + b C ), C i = ζ i C i−1 + a iCi ,(9)ζ i = σ (W ζ · [h i−1 , e i ] + b ζ ), o i = σ (W o [h i−1 ], e i ) + b o ), h i = o i * tahn(C i ) where the additional components C i−1 and ζ i are introduced to keep tracking the previous "history" and gating the information for forgetting in order to handle longer sequences. For example, some researchers opt to leverage a simple type LSTM architecture to extend the RNN-based sequential event prediction [33,97], while others leverage variants of LSTM, such as bi-directional LSTM instead [113,155]and yet others prefer to leverage gated-recurrent units (GRU) [70]. Moving beyond considering just the chain relationships among events, Li et al. [131] generalized this into graph-structured relationships to better incorporate the event contextual information via the Narrative Event Evolutionary Graph (NEEG). An NEEG is a knowledge graph where each node is an event and each edge denotes the association between a pair of events, enabling the NEEG to be represented by a weighted adjacency matrix A. The basic architecture can be denoted by the following, as detailed in the paper [131]: a i = A T h i−1 + b, z i = σ (W z a i + U z h i−1 ), r i = σ (W r a i + U r h i−1 ),(10)c i = tanh(W a i + U (r i h i−1 )), h i = (1 − z i )h i−1 + z i c i Here, the current activation a i is not only dependent on the previous time point but also influenced by its neighbor nodes in NEEG. • Descriptive-based Methods. Attribute-based methods require extra effort during preprocessing in order to convert the unstructured raw data into feature vectors, a process which is not only computationally labor intensive but also not always feasible. Therefore, multiple architectures have been proposed to directly process the raw (textual) event descriptions to enable them to be used to predict future event semantics or descriptions. These models share a similar generic framework [96,97,139,195,221,231], which begins by encoding each sequence of words into event representations, utilizing an RNN architecture, as shown in Figure 5. The sequence of events must then be characterized by another higher-level RNN to predict future events. Under this framework, some works begin by decoding the predicted future candidate events into event embedding, after which they are compared with each other and the one with the largest confidence score is selected as the predicted event. These methods are usually constrained by the known list of event types, but sometimes we are interested in open set predictions where the predicted event type can be a new appearance of a type that has not previously been seen in the training set. To achieve this, other methods focus on directly generating future events' descriptions that characterize event semantics that may or may not have appeared before by designing an additional sequence decoder that decodes the latent representation of future events into word sequences. More recent research has enhanced the utility and interpretability of the relationship between words and relevant events, and all the previous events for the relevant future event, by adding a hierarchical attention mechanisms. For example, Yu et al. [231] and Su and Jiang [195] both proposed word-level attention and event-level attention, while Hu [96] leveraged word-level attention in the event encoder as well as in the event decoder. Multi-faceted Prediction This section discusses the research into ways to jointly predict the time, location, and semantics of future events. Existing work in this area can be categorized into three types: 1) joint time and For example, Vilalta and Ma [206] defined LHS as a tuple (E L , τ ), where τ is the time window before the target in RHS predefined by the user. Only the events occurring within a time window before the event in RHS will satisfy the LHS. Similar techniques have also been leveraged by other researchers [45,194]. However, τ is difficult to define beforehand and it is preferable to be flexible to suit different target events. To handle this challenge, Yang et al. [225] proposed a way to automatically identify information on a continuous time interval from the data. Here, each transaction is composed of not only items but also continuous time duration information. LHS is a set of items (e.g., previous events) while RHS is a tuple (E R , [t 1 , t 2 ]) consisting of a future event semantic representation and its time interval of occurrence. To automatically learn the time interval in RHS, [225] proposed the use of two different methods . The first is called the confidence-intervalbased method, which leverages a statistical distribution (e.g., Gaussian and student-t [26]) to fit all the observed occurrence times of events in RHS, and then treats the statistical confidence interval as the time interval. The second method is known as minimal temporal region selection, which aims to find the temporal region with the smallest interval and covers all historical occurrences of the event in RHS. Time expression extraction. In contrast to the above statistical-based methods, another way to achieve event time and semantics joint prediction comes from the pattern recognition domain, aiming to directly discover time expressions that mention the (planned) future events. As this type of technique can simultaneously identify time, semantics, and other information such as locations, it is widely used and will be discussed in more details later as part of the discussion of "Planned future event detection methods" in Section 3.4.3. Time series forecasting-based methods. The methods based on time series forecasting can be separated into direct methods and indirect methods. Direct methods typically formulate the event semantic prediction problem as a multi-variate time series forecasting problem, where each variable corresponds to an event type C i (i = 1, · · · ) and hence the predicted event type at future timet is calculated asŝt = arg max C i f (st = C i |X ). For example, in [128], a longitudinal support vector regressor is utilized to predict multi-attribute events, where n support vector regressors, each of which corresponds to an attribute, is built to achieve the goal of predicting the next time point's attribute value. Weiss and Page [219] took a different approach, leveraging multiple point process models to predict multiple event types. To further estimate the confidence of their predictions, Biloš et al. [25] first leveraged RNN to learn the historical event representation and then input the result into a Gaussian process model to predict future event types. To better capture the joint dynamics across the multiple variables in the time series, Brandt et al. [30] extended this to Bayesian vector autoregression. Utilizing indirect-style methods, they focused on learning a mapping from the observed event semantics down to low-dimensional latent-topic space using tensor decompositionbased techniques. Similarly, Matsubara et al. [142] proposed a 3-way topic analysis of the original observed event tensor Y 0 ∈ R D o ×D a ×D c consisting of three factors, namely actors, objects, and time. They then went on to decompose this tensor into latent variables via three corresponding low-rank matrices P o ∈ R D k ×D o , P a ∈ R D k ×D a , and P c ∈ R D k ×D c respectively, as shown in Figure 6. Here D k is the number of latent topics. For the prediction, the time matrices P c are predicted into the futureP c via multi-variate time series forecasting, after which a future event tensor are estimated by recovering a "future event tensor"Ŷ by the multiplication among the predicted time matrixP c as well as the known actor matrix P a and object matrix P o . Raster-based. These methods usually formulate data into temporal sequences consisting of spatial snapshots. Over the last few years, various techniques have been proposed to characterize the spatial and temporal information for event prediction. The simplest way to consider spatial information is to directly treat location information as one of the input features, and then feed it into predictive models, such as linear regression [250], LSTM [174] and Gaussian processes [118]. During model training, Zhao and Tang [250] leveraged the spatiotemporal dependency to regularize their model parameters. Most of the methods in this domain aim to jointly consider the spatial and temporal dependency for predictions [64]. At present, the most popular framework is the CNN+RNN architecture, which implements sequenceto-sequence learning problems such as the one illustrated in Figure 7. Here, the multi-attributed spatial information for each time point can be organized as a series of multi-channel images, which can be encoded using convolution-based operations. For example, Huang et al. [99] proposed the addition of convolutional layers to process the input into vector representations. Other researchers have leveraged variational autoencoders [215] and CNN autoencoders [104] to learn the lowdimensional embedding of the raw spatial input data. This allows the learned representation of the input to be input into the temporal sequence learning architecture. Different recurring units have been investigated, including RNN, LSTM, convLSTM, and stacked-convLSTM [88]. The resulting representation of the input sequence is then sent to the output sequence as input. Here, another recurrent architecture is established. The output of the unit for each time point will be input into a spatial decoder component which can be implemented using transposed convolutional layers [233], transposed convLSTM [104], or a spatial decoder in a variational autoencoder [215]. A conditional random field is another popular technique often used to model the spatial dependency [105]. Point-based. The spatiotemporal point process is an important technique for spatiotemporal event prediction as it models the rate of event occurrence in terms of both spatial and time points. It is defined as: λ(t, l |X ) = lim |dt |→0, |dl |→0 E[N (dt × dl)|X ]/(|dt ||dl |)(11) Various models have been proposed to instantiate the model of the framework illustrated in Equation (11). For example, Liu and Brown et al. [136] began by assuming there to be a conditional independence among spatial and temporal factors and hence achieved the following decomposition: λ(t, l |X ) = λ(t, l |L,T , F ) = λ 1 (l |L,T , F , t) · λ 2 (t |T )(12) where X , L,T , and F denotes the whole input indicator data as well as its different facets, including location, time, and other semantic features, respectively. Then the term λ 1 (·) can be modeled based on the Markov spatial point process while λ 2 (·) can be characterized using temporal autoregressive models. To handle situations where explicit assumptions for model distributions are difficult, several methods have been proposed to involve the deep architecture during the point process. Most recently, Okawa et al. [159] have proposed the following: λ(t, l |X ) = ∫ д θ (t ′ , l ′ , F (t ′ , l ′ )) · K((t, l), (t ′ , l ′ )) dt ′ dl ′(13) where K(·, ·) is a kernel function such as a Gaussian kernel [26] that measures the similarity in time and location dimensions. F (t ′ , l ′ ) ⊆ F denotes the feature values (e.g., event semantics) for the data at location l ′ and time t ′ . д θ (·) can be a deep neural network that is parameterized by θ and returns an nonnegative scalar. The model selection of д θ (·) depends on the specific data types. For example, these authors constructed an image attention network by combining a CNN with the spatial attention model proposed by Lu et al. [138]. Time, location, and Semantics. In this section, we introduce the strategies that jointly predict the time, location, and semantics of future events, which can be grouped into either system-based or model-based strategies. System-based. The first type of the system-based methods considered here is the model-fusion system. The most intuitive approach is to leverage and integrate the aforementioned techniques for time, location, and semantics prediction into an event prediction system. For example, a system named EMBERS [171] is an online warning system for future events that can jointly predict the time, location, and semantics including the type and population of future events. This system also provides information on the confidence of the predictions obtained. Using an ensemble of predictive models for time [160], location, and semantic prediction, this system achieves a significant performance boost in terms of both precision and recall. The trick here is to first prioritize the precision of each individual prediction model by suppressing their recall. Then, due to the diversity and complementary nature of the different models, the fusion of the predictions from different models will eventually result in a high recall. A Bayesian fusion-based strategy has also been investigated [95]. Another system named Carbon [108] also leverages a similar strategy. The second type of model involves crowd-sourced systems that implement fusion strategies to generate the event predictions made by the human predictors. For example, in order to handle the heterogeneity and diversity of the human predictors' skill sets and background knowledge under limited human resources, Rostami et al. [177] proposed a recommender system for matching event forecasting tasks to human predictors with suitable skills in order to maximize the accuracy of their fused predictions. Li et al. [126] took a different approach, designing a prediction market system that operates like a futures market, integrating information from different human predictors to forecast future events. In this system, the predictors can decide whether to buy or sell the "tokens" (using virtual dollars, for example) for each specific prediction they have made according to their confidence in it. They typically make careful decisions as they will obtain corresponding awards (for correct predictions) or penalties (for erroneous predictions). Planned future event detection methods. These methods focus on detecting the planned future events, usually from various media such sources as social media and news and typically relying on NLP techniques and linguistic principles. Existing methods typically follow a workflow similar to the one shown in Figure 8, consisting of four main steps: 1) Content filtering. Methods for content filtering are typically leveraged to retain only the texts that are relevant to the topic of interest. Existing works utilize either supervised methods (e.g., textual classifiers [117] or unsupervised methods (e.g., querying techniques [152,238]); 2) Time expression identification is then utilized to identify future reference expressions and determine the time to event. These methods either leverage existing tools such as the Rosetta text analyzer [55] or propose dedicated strategies based on linguistic rules [101]; 3) Future reference sentence extraction is the core of planned event detection, and is implemented either by designing regular expression-based rules [153] or by textual classification [117]; and 4) Location identification. The expression of locations is typically highly heterogeneous and noisy. Existing works have relied heavily on geocoding techniques that can resolve the event location accurately. In order to infer the event locations, various types of locations are considered by different researchers, such as article locations [152], authors' profile locations [49], locations mentioned in the articles [22], and authors' neighbors' locations [107]. Multiple locations have been selected using a geometric median [49] or fused using logical rules such as probabilistic soft logic [152]. Tensor-based methods. Some methods formulate the data into tensor-form, with dimensions including location, time, and semantics. Tensor decomposition is then applied to approximate the original tensors as the product of multiple low-rank matrices, each of which is a mapping from latent topics to each dimension. Finally, the tensor is extrapolated towards future time periods by various strategies. For example, Mirtaheri [148] extrapolated the time dimension-matrix only, which they then multiplied with the other dimensions' matrices to recover the estimated extrapolated tensor into the future. Zhou et al. [253] took a different approach, choosing instead to add "empty values" for the entries in future time to the original tensor, and then use tensor completion techniques to infer the missing values corresponding to future events. APPLICATIONS OF EVENT PREDICTIONS Healthcare This category generally consists of two types of event prediction: 1) population level, which includes disease epidemics and outbreaks, and 2) individual level, which relates to clinical longitudinal events. Population-level. There has been extensive research on disease outbreaks for many different types of diseases and epidemics, including seasonal flu [3], Zika [200], H1N1 [158], Ebola [13], and COVID-19 [162]. These predictions target both the location and time of future events, while the disease type is usually fixed to a specific type for each model. Compartmental models such as SIR models are among the classical mathematical tools used to analyze, model, and simulate the epidemic dynamics [186,237]. More recently, individual-based computational models have begun to be used to perform network-based epidemiology based on network science and graphtheoretical models, where an epidemic is modeled as a stochastic propagation over an explicit interaction network among people [52]. Thanks to the availability of high-performance computing resources, another option is to construct a "digital twin" of the real world, by considering a realistic representation of a population, including membersâĂŹ demographic, geographic, behavioral, and social contextual information, and then using individual-based simulations to study the spread of epidemics within each network [27]. The above techniques heavily rely on the model assumptions regarding how the disease progresses individually and is transmitted from person to person [27]. The rapid growth of large surveillance data and social media data sets such as Twitter and Google flu trends in recent years has led to a massive increase of interest in using data-driven approaches to directly learn the predictive mapping [3]. These methods are usually both more time-efficient and less dependent on assumptions, while the aforementioned computational models are more powerful for longer-term prediction due to their ability to take into account the specific disease mechanisms [242]. Finally, there have also been reports of synergistic research that combines both techniques to benefit from their complementary strengths [98,242]. Individual-level. This research thread focuses on the longitudinal predictive analysis of individual health-related events, including death occurrence [62], adverse drug events [185], sudden illnesses such as strokes [124] and cardiovascular events [23], as well as other clinical events [62] and life events [59] for different groups of people, including elders and people with mental disease. The goal here is usually to predict the time before an event occurs, although some researchers have attempted to predict the type of event. The data sources are essentially the electronic health records of individual patients [172,185]. Recently, social media, forum, and mobile data has also been utilized for predicting drug adverse events [185] and events that arise during chronic disease (e.g., chemical radiation and surgery) [62]. Media This category focuses on predicting events based on information held in various types of media including: video-based, audio-based, and text-based formats. The core issue is to retrieve key information related to future events utilizing semantic pattern recognition from the data. 4.2.1 Video-and Audio-based. . While event detection has been extensively researched for video data [129] and audio mining [192], event prediction is more challenging and has been attracting increasing attention in recent years. The goal here is usually to predict the future status of the objects in the video, such as the next action of soccer players [60] or basketball players [154], or the movement of vehicles [235]. 4.2.2 Text-and script-based. A huge amount of news data has accumulated in recent decades, much of which can be used for big data predictive analytics among news events. A number of researchers have focused on predicting the location, time, and semantics of various events. To achieve this, they usually leverage the immense historical news and knowledge base in order to learn the association and causality among events, which is then applied to forecast events when given current events. Some studies have even directly generated textual descriptions of future events by leveraging NLP techniques such as sequence to sequence learning [57,97,123,153,168,170,190,195,221]. Transportation . This category can be classified into: 1) population based events, including dispersal events, gathering events, and congestion; and 2) individual-level events, which focus on fine-grained patterns such as human mobility behavior prediction. 4.3.1 Group transportation pattern. . Here, researchers typically focus on transportation events such as congestion [43,104], large gatherings [203], and dispersal events [204]. The goal is thus to forecast the future time period [80] and location [203] of such events. Data from traffic meters, GPS, and mobile devices are usually used to sense real-time human mobility patterns. Transportation and geographical theories are usually considered to determine the spatial and temporal dependencies for predicting these events. Individual transportation behavior. Another research thread focuses on individual-level prediction, such as predicting an individual's next location [130,223] or the likelihood or time duration of car accidents [19,174,233]. Sequential and trajectory analyses are usually used to process trajectory and traffic flow data. Engineering Systems Different types of engineering systems have begun to routinely apply event forecasting methods, including: 1) civil engineering, 2) electrical engineering, 3) energy engineering, and 4) other engineering domains. Despite the variety of systems in these widely different domains, the goal is essentially to predict future abnormal or failure events in order to support the system's sustainability and robustness. Both the location and time of future events are key factors for these predictions. The input features usually consist of sensing data relevant to specific engineering systems. • Civil engineering. This covers various a wide range of problems in diverse urban systems, such as smart building fault adverse event prediction [21], emergency management equipment failure prediction [66], manhole event prediction [179], and other events [99]. • Electrical engineering. This includes teleservice systems failures [61] and unexpected events in wire electrical discharge machining operations [184]. • Energy engineering. Event prediction is also a hot topic in energy engineering, as such systems usually require strong robustness to handle the disturbance from the nature environments. Active research domains here include wind power ramp prediction [83], solar power ramp prediction [1], and adverse events in low carbon energy production [50]. • Other engineering domains. There is also active research on event prediction in other domains, such as irrigation event prediction in agricultural engineering [161] and mechanical fault prediction in mechanical engineering [197]. Cyber Systems Here, the prediction models proposed generally focus on either network-level events or devicelevel events. For both types, the general goal is essentially to predict the likelihood of future system failure or attacks based on various indicators of system vulnerability. So far these two categories have essentially differed only in their inputs: the former relies on network features, including system specifications, web access logs and search queries, mismanagement symptoms, spam, phishing, and scamming activity, although some researchers are investigating the use of social media text streams to identify semantics indicating future potential attack targets of DDoS [142,217]. For device-level events, the features of interest are usually the binary file appearance logs of machines [160,210]. Some work has been done on micro-architectural attacks [90] by observing and proactively analyzing the observations on speculative branches, out-of-order executions and shared last level caches [188]. Political events Political event prediction has become a very active research area in recent years, largely thanks to the popularity of social media. The most common research topics can be categorized as: 1) offline events, and 2) online activism. 4.6.1 Offline events. This includes civil unrest [171], conflicts [218], violence [28], and riots [67]. This type of research usually targets the future events' geo-location, time, and topics by leveraging the social sensors that indicate public opinions and intentions. Utilization of social media has become a popular approach for these endeavors, as social media is a source of vital information during the event development stage [171]. Specifically, many aspects are clearly visible in social media, including complaints from the public (e.g., toward the government), discussions about their intentions regarding specific political events and targets, as well as advertisements for the planned events. Due to the richness of this information, further information on future events such as the type of event [85], the anticipated participants population [171], and the event scale [84] can also be discovered in advance. 4.6.2 Online events. Due to the major impact of online media such as online forums and social media, many events such as online activism, petitions, and hoaxes in such online platform also involve strong motivations for achieving some political purpose [213]. Beyond simple detection, the prediction of various types of events have been studied in order to enable proactive intervention to sidetrack the events such as hoaxes and rumor propagation [106]. Other researchers have sought to foresee the results of future political events in order to benefit a particular group of practitioners, for example by predicting the outcome of online petitions or presidential elections [213]. Natural disasters Different types of natural disasters have been the focus of a great deal of research. Typically, these are rare events, but mechanistic models, a long historical records (often extending back dozens or hundreds of years), and domain knowledge are usually available. The input data are typically collected by sensors or sensor networks and the output is the risk or hazard of future potential events. Since these event occurrence are typically rare but very high-stakes, many researchers strive to cover all event occurrences and hence aim to ensure high recalls. 4.7.1 Geophysics-related. Earthquakes. Predictions here typically focus on whether there will be an earthquake with a magnitude larger than a specified threshold in a certain area during a future period of time. To achieve this, the original sensor data is usually proccessed using geophysical models such as GutenbergâĂŞRichterâĂŹs inverse law, the distribution of characteristic earthquake magnitudes, and seismic quiescence [14,175]. The processed data are then input into machine learning models that treat them as input features for predicting the output, which can be either binary values of event occurrence or time-to-event values. Some studies are devoted to identifying the time of future earthquakes and their precursors, based on an ensemble of regressors and feature selection techniques [178], while others focus on aftershock prediction and the consequences of the earthquake, such as fire prevention [140]. It worth noting that social media data has also been used for such tasks, as this often supports early detection of the first-wave earthquake, which can then be used to predict the afterstocks or earthquakes in other locations [181]. Fire events. Research in this category can be grouped into urban fires and wildfires. This type of research often focuses on the time at which a fire will affect a specific location, such as a building. The goal here is to predict the risk of future fire events. To achieve this, both the external environment and the intrinsic properties of the location of interests are important. Therefore, both static input data (e.g., natural conditions and demographics) and time-varying data (e.g., weather, climate, and crowd flow) are usually involved. Shin and Kim [189] focus on building fire risk prediction, where the input is the building's profile. Others have studied wildfires, where weather data and satellite data are important inputs. This type of research focuses primarily on predicting both the time and location of future fires [216,234]. Other researchers have focused on rarer events such as volcanic eruptions. For example, some leverage chemical prior knowledge to build a Bayesian network for prediction [44], while others adopt point processes to predict the hazard of future events [56]. 4.7.2 Atmospheric science-related. Flood events. Floods may be caused by many different reasons, including atmospheric (e.g., snow and rain), hydrological (e.g., ice melting, wind-generated waves, and river flow), and geophysical (e.g., terrain) conditions. This makes the forecasting of floods highly complicated task that requires multiple diverse predictors [212]. Flood event prediction has a long history, with the latest research focusing especially on computational and simulation models based on domain knowledge. This usually involves using ensemble prediction systems as inputs for hydrological and/or hydraulic models to produce river discharge predictions. For a detailed survey on flood computational models please refer to [47]. However, it is prohibitively difficult to comprehensively consider and model all the factors correctly while avoiding all the accumulated errors from upstream predictions (e.g., precipitation prediction). Another direction, based on data-driven models such as statistical and machine learning models for flood prediction, is deemed promising and is expected to be complementary to existing computational models. These newly developed machine learning models are often based solely on historical data, requiring no knowledge of the underlying physical processes. Representative models are SVM, random forests, and neural networks and their variants and hybrids. A detailed recent survey is provided in [149]. Tornado Forecasting. Tornadoes usually develop within thunderstorms and hence most tornado warning systems are based on the prediction of thunderstorms. For a comprehensive survey, please refer to [68]. Machine learning models, when applied to tornado forecasting tasks, usually suffer from high-dimensionality issues, which are very common in meteorological data. Some methods have leveraged dimensional reduction strategies to preprocess the data [230] before prediction. Research on other atmosphere-related events such as droughts and ozone events has also been conducted [77]. Astrophysics-related. There is a large body of prediction research focusing on events outside the Earth, especially those affecting the star closest to us, the sun. Methods have been proposed to predict various solar events that could impact life on Earth, including solar flares [20], solar eruptions [4], and high energy particle storms [141]. The goal here is typically to use satellite imagery data of the sun to predict the time and location of future solar events and the activity strength [74]. Business Business intelligence can be grouped into company-based events and customer-based events. 4.8.1 Customer activity prediction. The most important customer activities in business is whether a customer will continue doing business with a company and how long a costumer will be willing to wait before receiving the service? A great deal of research has been devoted to these topics, which can be categorized based on the type of business entities namely enterprises, social media, and education, who are primarily interested in churn prediction, site migration, and student dropout, respectively. The first of these focuses on predicting whether and when a customer is likely to stop doing business with a profitable enterprise [71]. The second aims to predict whether a social media user will move from one site, such as Flickr, to another, such as Instagram, a movement known as site migration [236]. While site migration is not popular, attention migration might actually be much more common, as a user may "move" their major activities from one social media site to another. The third type, student dropout, is a critical domain for education data mining, where the goal is to predict the occurrence of absenteeism from school for no good reason for a continuous number of days; a comprehensive survey is available in [143]. For all three types, the procedure is first to collect features of a customer's profile and activities over a period of time and then conventional or sequential classifiers or regressors are generally used to predict the occurrence or time-to-event of the future targeted activity. Business process events. Financial event prediction has been attracting a huge amount of attention for risk management, marketing, investment prediction and fraud prevention. Multiple information resources, including news, company announcements, and social media data could be utilized as the input, often taking the form of time series or temporal sequences. These sequential inputs are used for the prediction of the time and occurrence of future high-stack events such as company distress, suspension, mergers, dividends, layoffs, bankruptcy, and market trends (rises and falls in the company's stock price) [36,40,65,91,122,144,226]. Crime It is difficult to deduce the precise location and time for individual crime incidences. Therefore, the focus is instead estimating the risk and probability of the location, time, and types of future crimes. This field can be naturally categorized based on the various crime types: 4.9.1 Political crimes and terrorism. This type of crime is typically highly destructive, and hence attracts huge attention in its anticipation and prevention. Terrorist activities are usually aimed at religious, political, iconic, economic or social targets. The attacker typically targets larger numbers of people and the evidences related to such attacks is retained in the long run. Though it is extremely challenging to predict the precise location and time of individual terrorism incidents, numerous studies have shown the potential utility for predicting the regional risks of terrorism attacks based on information gathered from many data sources such as geopolitical data, weather data, and economics data. The Global Terrorism Database is the most widely recognized dataset that records the descriptions of world-wide terrorism events of recent decades. In addition to terrorism events, other similar events such as mass killings [202] and armed-conflict events [193] have also been studied using similar problem formulations. Crime incidents. Most studies on this topic focus on predicting the types, intensity, count, and probability of crime events across defined geo-spatial regions. Until now, urban crimes are most commonly the topic of research due to data availability. The geospatial characteristics of the urban areas, their demographics, and temporal data such as news, weather, economics, and social media data are usually used as inputs. The geospatial dependency and correlation of the crime patterns are usually leveraged during the prediction process using techniques originally developed for spatial predictions, such as kernel density estimation and conditional random fields. Some works simplify the tasks by only focusing on specific types of crimes such as theft [180], robbery, and burglary [51]. 4.9.3 Organized and serial crimes. Unlike the above research on regional crime risks, some recent studies strive to predict the next incidents of criminal individuals or groups. This is because different offenders may demonstrate different behavioral patterns, such as targeting specific regions (e.g., wealthy neighborhoods), victims (e.g., women), for specific benefits (e,g, money). The goal here is thus is to predict the next crime site and/or time, based on the historical crime event sequence of the targeted criminal individual or group. Models such as point processes [130] or Bayesian networks [133] are usually used to address such problems. OPEN CHALLENGES AND FUTURE DIRECTIONS Despite the major advances in event prediction in recent years, there are still a number of open problems and potentially fruitful directions for future research, as follows: Model Transparency, Interpretability, and Accountability Increasingly sophisticated forecasting models have been proposed to improve the prediction accuracy, including those utilizing approaches such as ensemble models, neural networks, and the other complex systems mentioned above. However, although the accuracy can be improved, the event prediction models are rapidly becoming too complex to be interpreted by human operators. The need for better model accountability and interpretability is becoming an important issue; as big data and Artificial Intelligence techniques are applied to ever more domains this can lead to serious consequences for applications such as healthcare and disaster management. Models that are not interpretable by humans will find it hard to build the trust needed if they are to be fully integrated into the workflow of practitioners. A closely related key feature is the accountability of the event prediction system. For example, disaster managers need to thoroughly understand a model's recommendations if they are to be able to explain the reason for a decision to displace people in a court of law. Moreover, an ever increasing number of laws in countries around the world are beginning to require adequate explanations of decisions reached based on model recommendations. For example, Articles 13-15 in the European Union's General Data Protection Regulation (GDPR) [207] require algorithms that make decisions that âĂIJsignificantly affect" individuals to provide explanations ("right to explanation") by May 28, 2018. Similar laws have also been established in countries such as the United States [48] and China [166]. Vulnerability to Noise and Adversarial Attacks The massive popularity of the proposal, development, and deployment of event prediction is stimulating a surge interest in developing ways to counter-attack these systems. It will therefore not be a surprise when we begin to see the introduction of techniques to obfuscate these event prediction methods in the near future. As with many state-of-the-art AI techniques applied in other domains such as object recognition, event prediction methods can also be very vulnerable to noise and adversarial attacks. The famous failure of Google Flu trends, which missed the peak of the 2013 flu season by 140 percent due to low relevance and high disturbance affecting the input signal, is a vivid memory for practitioners in the field [82]. Many predictions relying on social media data can also be easily influenced or flipped by injecting scam messages. Event prediction models also tend to over-rely on low-quality input data that can be easily disturbed or manipulated, lacking sufficient robustness to survive noisy signals and adversarial attacks. Similar problems threaten to other application domains such as business intelligence, crime, and cyber systems. Integration of mechanistic knowledge and data driven-models Over the years, many domains have accumulated a significant amount of knowledge and experience about event development occurrence mechanisms, which can thus provide important clues for anticipating future events, such as epidiomiology models, socio-political models, and earthquake models. All of these models focus on simplifying real-world phenomena into concise principles in order to grasp the core mechanism, discarding many details in the process. In contrast, data-driven models strive to ensure the accurate fitting of large historical data sets, based on sufficient model expressiveness but cannot guarantee that the true underlying principle and causality of event occurrence modeled accurately. There is thus a clear motivation to combine their complementary strengths, and although this has already attracted great deal of interest [98,242], most of the models proposed so far are merely ensemble learning-based and simply merge the final predictions from each model. A more thorough integration is needed that can directly embed the core principles to regularize and instruct the training of data-driven event prediction methods. Moreover, existing attempts are typically specific to particular domains and are thus difficult to develop further as they require in-depth collaborations between data scientists and domain experts. A generic framework developed to encompass multiple different domains is imperative and would be highly beneficial for the various domain experts. Prescriptive and counterfactual analysis The ultimate purpose of event prediction is usually not just to anticipate the future, but to change it, for example by avoiding a system failure and flattening the curve of a disease outbreak. However, it is difficult for practitioners to determine how to act appropriately and implement effective policies in order to achieve the desired results in the future. This requires a capability that goes beyond simply predicting future events based on the current situation, requiring them instead to also take into account the new actions being taken in real time and then predict how they might influence the future. One promising direction is the use of counterfactual event [145] prediction that models what would have happened if different circumstances had occurred. Another related direction is prescriptive analysis where different actions can be merged into the prediction system and future results anticipated or optimized. Related works have been developed in few domains such as epidemiology. However, as yet these lack sufficient research in many other domains that will be needed if we are to develop generic frameworks that can benefit different domains. Multi-objective training Existing event prediction methods mostly focus primarily on accuracy. However, decision makers who utilize these predicted event results usually need much more, including key factors such as event resolution (e.g., time resolution, location resolution, description details), confidence (e.g., the probability a predicted event will occur), efficiency (whether the model can predict per day or per seccond), lead time (how many days the prediction can be made prior to the event occurring), and event intensity (how serious it is). multi-objective optimization (e.g., accuracy, confidence, resolution). There are typically trade-offs among all the above metrics and accuracy, so merely optimizing accuracy during training will inevitably mean the results drift away from the overall optimal event-prediction-based decision. A system that can flexibly balance the trade-off between these metrics based on decision makers' needs and achieve a multi-objective optimization is the ultimate objective for these models. CONCLUSION This survey has presented a comprehensive survey of existing methodologies developed for event prediction methods in the big data era. It provides an extensive overview of the event prediction challenges, techniques, applications, evaluation procedures, and future outlook, summarizing the research presented in over 200 publications, most of which were published in the last five years. Event prediction challenges, opportunities, and formulations have been discussed in terms of the event element to be predicted, including the event location, time, and semantics, after which we went on to propose a systematic taxonomy of the existing event prediction techniques according to the formulated problems and types of methodologies designed for the corresponding problems. We have also analyzed the relationships, differences, advantages, and disadvantages of these techniques from various domains, including machine learning, data mining, pattern recognition, natural language processing, information retrieval, statistics, and other computational models. In addition, a comprehensive and hierarchical categorization of popular event prediction applications has been provided that covers domains ranging from natural science to the social sciences. Based upon the numerous historical and state-of-the-art works discussed in this survey, the paper concludes by discussing open problems and future trends in this fast-growing domain. Fig. 1 . 1Various types of times in event prediction. Fig. 2 . 2Generic framework for hierarchical RNN-based event forecasting. Fig. 4 . 4Raster-based and Point-based Event Location Predictions. Fig. 5 . 5Generic framework for hierarchical RNN-based event forecasting. semantics prediction; 2) joint time and location prediction; and 3) joint time, location, and semantic prediction.3.4.1 Time and Semantics. . For joint time and semantic prediction, there are three popular types of methods, discussed in turn below Temporal association rule. A temporal association rule can be developed from the vanilla association rule LHS → RHS by embedding additional temporal information into either LHS, RHS or both, thus redefining the meaning of co-occurrence and association with temporal constraints. Fig. 6 . 6Tensor Decomposition and Forecasting for Complex Time-stamped events.3.4.2 Time and Location Prediction. This category of methods focuses on jointly predicting the location and time of future events. These methods can be classified into two subtypes: 1) rasterbased: which focus on predictions for individual time slots and location regions, and 2) point-based: which predicts continuous time and location points. Fig. 7 . 7Generic framework for spatiotemporal event prediction using CNN+RNN-based deep learning framework. Fig. 8 . 8Generic framework for planned event detection. , Vol. 1, No. 1, Article . Publication date: August 2020. , Vol. 1, No. 1, Article . Publication date: August 2020.Event Prediction in the Big Data Era: A Systematic Survey the form of verb class in VerbNet [249]. , Vol. 1, No. 1, Article . Publication date: August 2020.Event Prediction in the Big Data Era: A Systematic Survey Forecasting of solar power ramp events: A post-processing approach. 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[]
[ "Time Evolution of the Neel State", "Time Evolution of the Neel State" ]
[ "A A Soluyanov [email protected] \nDepartment of Theoretical Physics\nV. A. Fock Institute of Physics\nSt.Petersburg State University\nStariy Peterhof\nUlianovskaya 3198504St.PetersburgRussia\n", "S N Zagoulaev \nDepartment of Theoretical Physics\nV. A. Fock Institute of Physics\nSt.Petersburg State University\nStariy Peterhof\nUlianovskaya 3198504St.PetersburgRussia\n", "I V Abarenkov \nDepartment of Theoretical Physics\nV. A. Fock Institute of Physics\nSt.Petersburg State University\nStariy Peterhof\nUlianovskaya 3198504St.PetersburgRussia\n" ]
[ "Department of Theoretical Physics\nV. A. Fock Institute of Physics\nSt.Petersburg State University\nStariy Peterhof\nUlianovskaya 3198504St.PetersburgRussia", "Department of Theoretical Physics\nV. A. Fock Institute of Physics\nSt.Petersburg State University\nStariy Peterhof\nUlianovskaya 3198504St.PetersburgRussia", "Department of Theoretical Physics\nV. A. Fock Institute of Physics\nSt.Petersburg State University\nStariy Peterhof\nUlianovskaya 3198504St.PetersburgRussia" ]
[]
A quasionedimentional spin chain (s = 1 2 ) is considered as a lattice consisting of two sublattices. The attention is paid to the states which are pure spin states of the whole lattice and both sublattices, the value of the sublattices' spins being maximum. It is shown that the Neel state can be considered as a superposition of such states. The exact equation for this superposition coefficients is developed.The possibility of the Neel state to be the eigenstate of a Hamiltonian is discussed. Several model Hamiltonians are examined, the well known ones and few novel Hamiltonians being considered. The time evolution of the Neel state in different models is studied with the help of Fock-Krylov method.
10.1002/qua.21349
[ "https://export.arxiv.org/pdf/cond-mat/0701067v2.pdf" ]
93,188,000
cond-mat/0701067
f30664ff9650a70ad4ea1aaeb657ed80bc6d33a1
Time Evolution of the Neel State 2 Feb 2007 A A Soluyanov [email protected] Department of Theoretical Physics V. A. Fock Institute of Physics St.Petersburg State University Stariy Peterhof Ulianovskaya 3198504St.PetersburgRussia S N Zagoulaev Department of Theoretical Physics V. A. Fock Institute of Physics St.Petersburg State University Stariy Peterhof Ulianovskaya 3198504St.PetersburgRussia I V Abarenkov Department of Theoretical Physics V. A. Fock Institute of Physics St.Petersburg State University Stariy Peterhof Ulianovskaya 3198504St.PetersburgRussia Time Evolution of the Neel State 2 Feb 2007 A quasionedimentional spin chain (s = 1 2 ) is considered as a lattice consisting of two sublattices. The attention is paid to the states which are pure spin states of the whole lattice and both sublattices, the value of the sublattices' spins being maximum. It is shown that the Neel state can be considered as a superposition of such states. The exact equation for this superposition coefficients is developed.The possibility of the Neel state to be the eigenstate of a Hamiltonian is discussed. Several model Hamiltonians are examined, the well known ones and few novel Hamiltonians being considered. The time evolution of the Neel state in different models is studied with the help of Fock-Krylov method. I. INTRODUCTION When considering the antiferromagnetic state, one usually refers to a model of two interpenetrating equivalent sublattices. The total lattice is in a state with zero z-projection of the total spin, whereas each sublattice is in a state with maximum spin, the z-projection of one sublattice being maximum and the other being minimum. This state is known as the Neel state. The Neel state attracts with it's simplicity and clearness. However, its quantum-mechanical realization and generation of the corresponding Hamiltonians encounter serious problems. For one, the Neel state wave function is not an eigenfunction of the system's total spin 1 . Hence, the non-degenerate eigenstate of any Hamiltonian, which describes a system with definite total spin, is not the Neel state. Therefore, considering various antiferromagnetic models one usually uses Hamiltonians with indefinite total spin 2,3,4,5,6,7,8,9,10 . At the same time, if a Hamiltonian with definite total spin has degenerate spectrum the Neel or Neel-like states could be hidden in the degenerate level functional subspace. However, the structure of feasible Hamiltonians with degenerate spectrum is not at all evident and it is a problem not only to extract the given state from the functional subspace but even to recognize whether the state is a member of the functional manifold. The approach adopted in the present paper is not to consider the Hamiltonian eigenstates and compare them to the Neel state, but to investigate governed by the Hamiltonian time evolution of the Neel state. The Hamiltonians where the Neel or Neel-like state is either stationary or has a long lifetime can be considered as describing the antiferromagnetic systems. The quasionedimentional spin chain is considered in the paper. First, the pure spin states of the chain are discussed, the main attention being paid to the states with the maximum spin of each sublattice. It is shown in the paper that the Neel state can be expressed as the particular linear combination of such states and the explicit equations for the linear combination coefficients are presented. In addition to the pure Neel state various Neel-like states are also considered in this section. In subsequent sections the Neel state's time evolution is investigated with the help of Fock-Krylov method 11 . Various spin-model Hamiltonians are employed here. Among them are the well-known ones (Heisenberg model, Ising model, XXZ-model), and several new models with the degenerate ground state. The influence of the degeneracy on the Neel state's time evolution is examined. Finally the separable potential 13 was employed to obtain models with the Neel state as the ground non-degenerate state. II. STATES WITH MAXIMUM SPIN OF EACH SUBLATTICE The spin chain's states with definite total spin and maximum spins of each sublattice are considered. It is shown, that particular linear combinations of these states possess some of the Neel state's properties and, therefore, can be considered as a Neel-like state. One such combination is the Neel state itself. Consider a quasionedimentional spin chain consisting of even number N = 2n of 1/2 spins s k , k = 1, · · · , N (atomic units are used throughout the paper). The chain is divided into two sublattices, designated by the index ℓ. The first sublattice (ℓ = 1) comprises spins with odd numbers, the second sublattice (ℓ = 2) comprises spins with even numbers. The spin operators for both sublattices are S ℓ = N/2−ℓ i=0 s 2i+ℓ .(1) Because operators S 2 , S 2 1 , S 2 2 , S z commute with each other, they have a common set of orthonormalized eigenfunctions ϕ(S, S 1 , S 2 , M ). Consider particular eigenfunctions ϕ S = ϕ(S, S 0 , S 0 , 0), where S is in the range [0, N/2], M = 0, and sublattices spins have maximum value S 1 = S 2 = N/4 = S 0 . There exists only one (apart from the phase factor) function ϕ S for each S. It can be seen from the momenta coupling rule ϕ S = M ℓ G(S 0 , M ℓ , S 0 , −M ℓ |S, 0)ϕ (1) S0,M ℓ ϕ (2) S0,−M ℓ , (2) where G is a Clebsch-Gordan coefficient, M ℓ is ℓ-th sublattice spin z-projection and ϕ (1) S0,M ℓ , ϕ (2) S0,−M ℓ are normalized to unity functions of the first and the second sublattices correspondingly (the conventional phase factor 12 is used in the present paper). For any M ℓ there exist only one function ϕ (ℓ) S0,M ℓ . Therefore, the only possibility to obtain another function is to couple S 1 and S 2 in reverse order, but, according to the properties of the Clebsch-Gordan coefficients, this can change only the phase of the function ϕ S . Thus, there are only N/2 + 1 functions with a definite total spin and maximum values of sublattices' spins. In what follows we will consider a normalized to unity linear combinations of functions ϕ S Φ = N/2 S=0 C S ϕ S ,(3) It will be shown, that the particular set of the coefficients C S yields the wave function corresponding to the Neel state. Besides, the set of coefficients can be found to produce a Neel-like state, i.e. the state which is not exactly the Neel state but has properties similar to those of the Neel state. These properties are: (i) a total chain's magnetization is equal to zero, while sublattices' magnetizations are non-zero, (ii) a mean value of the scalar product of spins from the same sublattice is equal to 1/4, (iii) a mean value of the scalar product of spins from the different sublattices is close to -1/4. A. SUBLATTICES MAGNETIZATION In this subsection it will be shown that the sublattices magnetization in state (3) may be non-zero. Magnetization of a system is usually defined as a mean value of a system's total spin z-projection operator. Hence, the ℓ-th sublattice magnetization in the state Φ is Φ| S ℓz |Φ = N/2 S=0 N/2 S ′ =0 C * S C S ′ ϕ S | S ℓz |ϕ S ′ .(4) Here S ℓz is the operator of the ℓ-th sublattice spin zprojection. One has ϕ S | S ℓz |ϕ S ′ = S0 M ℓ =−S0 G(S 0 , M ℓ , S 0 , −M ℓ |S, 0)G(S 0 , M ℓ , S 0 , −M ℓ |S ′ , 0) ϕ (ℓ) S0,M ℓ | S ℓz |ϕ (ℓ) S0,M ℓ = = S0 M ℓ =0 M ℓ G(S 0 , M ℓ , S 0 , −M ℓ |S ′ , 0)G(S 0 , M ℓ , S 0 , −M ℓ |S, 0) − G(S 0 , −M ℓ , S 0 , M ℓ |S ′ , 0)G(S 0 , −M ℓ , S 0 , M ℓ |S, 0) = = 1 − (−1) 4S0−S−S ′ S0 M ℓ =0 M ℓ G(S 0 , M ℓ , S 0 , −M ℓ |S ′ , 0)G(S 0 , M ℓ , S 0 , −M ℓ |S, 0),(5) because according to 12 , G(S 0 , M ℓ , S 0 , −M ℓ |S, 0) = = (−1) 2S0−S G(S 0 , −M ℓ , S 0 , M ℓ |S, 0). From the equation (5) it follows that the sublattice magnetization will be zero if the state Φ contains spin states ϕ S of the same parity only. To have non-zero sublattices magnetization the state Φ must contain spin states ϕ S with different parity. The particular state Φ, where all coefficients C S are equal to zero except two C S and C S ′ with odd S − S ′ , is an example of the state with non-zero sublattices magnetization. B. SCALAR PRODUCTS OF SPINS In this subsection it will be shown that in the state Φ from (3) the mean value of the scalar product of the same sublattice spins has the largest possible value, while a mean value of the scalar product of the different sublattices spins can be close to -1/4. First of all, note that ( s i , s j ) = 1 4 (2 P ij − 1),(6) where P ij is a transposition operator. The mean value of operator (6) is in the range [−3/4; 1/4], because the lower and upper bounds of P ij are −1 and +1. Now consider s j and s k belonging to the same sublattice number ℓ. Then Φ| ( s j , s k ) |Φ = N/2 S=0 N/2 S ′ =0 C * S C S ′ ϕ S |( s j , s k )|ϕ S ′(7) where For the matrix element in the right hand side of (8) one has ϕ S |( s j , s k )|ϕ S ′ = S0 M ℓ =−S0 G(S 0 , M ℓ , S 0 , −M ℓ |S ′ , 0)× ×G(S 0 , M ℓ , S 0 , −M ℓ |S, 0) ϕ (ℓ) S0,M ℓ | ( s j , s k ) |ϕ (ℓ) S0,M ℓϕ (ℓ) S0,M ℓ |( s j , s k )|ϕ (ℓ) S0,M ℓ = 1 4 (9) for every s j , s k from the ℓ-th sublattice. It is evident when M ℓ is maximum, i.e. M ℓ = N/4. However, the matrix element ϕ (ℓ) S0,M ℓ |( s j , s k )|ϕ (ℓ) S0,M ℓ does not depend on M ℓ as can be seen from the following. One has S ℓ+ ϕ (ℓ) S0,M ℓ = ( S ℓx + i S ℓy )ϕ (ℓ) S0,M ℓ = B(S 0 , −M ℓ )ϕ (ℓ) S0,M ℓ +1 ,(10)S ℓ− ϕ (ℓ) S0,M ℓ = ( S ℓx − i S ℓy )ϕ (ℓ) S0,M ℓ = B(S 0 , M ℓ )ϕ (ℓ) S0,M ℓ −1 ,(11)where B(S 0 , M ℓ ) = (S 0 + M ℓ )(S 0 − M ℓ + 1).(12) and the square root is positive. At the same time the equation ( s j , s k ), S ℓ− = = ( s j , s k ), ( s jx + s kx ) − i( s jy + s ky ) = 0,(13) is valid for spins from the same sublattice because the scalar product ( s j , s k ) commutes with any component of the vector-operator s j + s k . Therefore, ϕ (ℓ) S0,M ℓ −1 |( s j , s k )|ϕ (ℓ) S0,M ℓ −1 = 1 (B(S 0 , M ℓ )) 2 S ℓ− ϕ (ℓ) S0,M ℓ |( s j , s k )| S ℓ− ϕ (ℓ) S0,M ℓ = = 1 (B(S 0 , M ℓ )) 2 ϕ (ℓ) S0,M ℓ | S ℓ+ S ℓ− ( s j , s k )|ϕ (ℓ) S0,M ℓ = 1 (B(S 0 , M ℓ )) 2 ϕ (ℓ) S0,M ℓ |( s j , s k )( S 2 ℓ − S 2 ℓz + S z )|ϕ (ℓ) S0,M ℓ = = S 0 (S 0 + 1) − M 2 ℓ + M ℓ (B(S 0 , M ℓ )) 2 ϕ (ℓ) S0,M ℓ |( s j , s k )|ϕ (ℓ) S0,M ℓ = ϕ (ℓ) S0,M ℓ |( s j , s k )|ϕ (ℓ) S0,M ℓ .(14) It can be inferred from this equality, that the mean value of the scalar product of spins from the same sublattice does not depend on M ℓ in the state ϕ (ℓ) S0,M ℓ . According to 12 , m G(s, m, s, −m|S, 0)G(s, m, s, −m|S ′ , 0) = δ SS ′ , and one can write ϕ S |( s j , s k )|ϕ S ′ = M ℓ ,M ′ ℓ G(S 0 , M ℓ , S 0 , −M ℓ |S, 0)G(S 0 , M ℓ , S 0 , −M ℓ |S ′ , 0) ϕ (ℓ) S0,M ℓ |( s j , s k )|ϕ (ℓ) S0,M ℓ = = 1 4 M ℓ G(S 0 , M ℓ , S 0 , −M ℓ |S, 0)G(S 0 , M ℓ , S 0 , −M ℓ |S ′ , 0) = δ SS ′ 1 4 .(15) Hence, for the state Φ from (3) one has Φ| ( s j , s k ) |Φ = 1 4 S |C S | 2 = 1 4 .(16) Now, suppose, that a spin j is from the first sublattice and a spin k is from the second one. In order to calculate the mean value of (ŝ j ,ŝ k ) in this case, note that from (6), (9) and (14) it follows, that ϕ (ℓ) S0,M ℓ |P jk |ϕ (ℓ) S0,M ℓ = 1 for every M ℓ . Hence, function ϕ (ℓ) S0,M ℓ is symmetric under any transposition of it's variables, and, therefore, function ϕ S is symmetric under any transposition of spins from the same sublattice. One has ϕ S |( s j , s k )|ϕ S ′ = = 1 (N/2) 2 N/2−1 p=0 N/2 q=1 ϕ S |( s 2p+1 , s 2q )|ϕ S ′ = = 1 (N/2) 2 ϕ S |( S 1 , S 2 )|ϕ S ′ .(17) Obviously, ( S 1 , S 2 ) = 1 2 ( S 2 − S 2 1 − S 2 2 ),(18) and, therefore, ϕ S |( s j , s k )|ϕ S ′ = = δ SS ′ S(S + 1) 2(N/2) 2 − N (N/4 + 1) 4(N/2) 2 = = δ SS ′ − 1 4 − 1 N + 2S(S + 1) N 2 .(19) We should note, that this result does not depend on total spin projection's value and is also true for functions, corresponding to the maximum sublattices' spins and a non-zero total spin projection. Thus, when spins j and k belong to different sublattices one can write Φ|( s j , s k )|Φ = − 1 4 − 1 N + S 2S(S + 1) N 2 |C S | 2 . (20) Equation (16) shows, that mean values of the scalar products of spins from the same sublattice are always positive and have the largest possible value independently on the particular choice of coefficients C S in the state Φ. Contrary to that, the equation (20) indicates that mean values of the scalar products of spins from different sublattices essentially depend on coefficients C S values. It is easy to present two sets of coefficients resulting correspondingly in positive and negative mean values. The simplest sets are the following, all C S are equal to zero except C N/2 = 1 in one set and C 0 = 1 in the other set. However, as it was shown above, in these states the sublattices magnetization is equal to zero. Still, there exist many states where the mean values of the scalar products of spins from different sublattices are close to −1/4 and the sublattices' magnetization is non-zero. These states can be referred to as the Neel-like states. C. NEEL STATE In this section the coefficients C S in (3) are found to make function (3) exactly coincide with the Neel state wave function. In the case of a quasionedimentional spin chain there are two Neel states with the following spin functions ψ (1) N eel = α(σ 1 )β(σ 2 )....α(σ N −1 )β(σ N ),(21)ψ (2) N eel = β(σ 1 )α(σ 2 )....β(σ N −1 )α(σ N )(22) where σ i is a spin variable of an i-th spin. Let us show, that the Neel state can be represented in the form of a linear combination (3) with the following coefficients: ψ (1) N eel = N/2 S=0 g S (N ) K ϕ S ; (23) ψ (2) N eel = N/2 S=0 (−1) (N/2−S) g S (N ) K ϕ S .(24) where g S (N ) are Wigner coefficients g S (N ) = (2S + 1)N ! (N/2 + S + 1)!(N/2 − S)! ,(25) and K = N/2 S=0 g S (N ) = N ! (N/2)!(N/2)! .(26) This statement is easy to prove. According to the momenta coupling rule (2) one has ϕ S = S0 M ℓ =−S0 (2S + 1) (2S 0 − S)!(2S 0 + S + 1)! × × z (−1) z (S 0 − M ℓ )!(S 0 + M ℓ )!(2S 0 − S)!S!S! z!(2S 0 − S − z)![(S 0 − M ℓ − z)!] 2 [(S − S 0 + M ℓ + z)!] 2 ϕ (1) S0,M ℓ ϕ (2) S0,−M ℓ ,(27) where z runs over all integers, for which the factorial argument is positive, and ϕ (ℓ) S0,M ℓ is wave function of the ℓ−th sublattice. Consider a scalar product ψ N eel |ϕ S . Calculating it with the help of equation (27), one can see that because of the relation ψ (1) N eel = ϕ (1) S0,S0 ϕ (2) S0,−S0 and functions ϕ N eel |ϕ S = (2S + 1) (N/2 + S + 1)!(N/2 − S)! N 2 ! = = g S (N ) K . Thus, the equation (23) is proved (one can easily see that the right hand side of (23) is normalized to unity). The equation (24) can be proved similarly. Note, that the only possible value of z in this case is z = N/2 − S. III. FOCK-KRYLOV METHOD IMPLICATION In this section the evolution of the Neel state in different model systems is examined. Consider the system with Hamiltonian H having the discrete and continuous parts of the spectrum H(x)ψ n (x) = E k ψ n (x), H(x)ψ(E, x) = Eψ(E, x). According to the Fock-Krylov theorem 12 , the time evolution of the system is determined by the initial state's Φ(0) energy distribution W (E) w(E) = n v n δ(E − E k ) + v(E), v n = < ψ n | Φ(0) > 2 , v(E) = < ψ(E)| Φ(0) > 2 . The initial state will decay if the distribution W (E) has only continuous part v(E). In this case the probability to find the system at a time t in the initial state L(t) = < Φ(0)| Φ(t) > 2 tends to zero when t tends to infinity. Here Φ(t) is the solution of the Schrodinger equation with the initial state Φ(0). The systems considered in the present paper have only discrete energy spectrum and therefore any initial state does not decay. Taking a particular state for the initial state Φ(0) one can see that L(t) oscillates (periodically or aperiodically) between two values L min and L max . If L min > 0 then at any time there is the non zero probability to find the system in the state Φ(0). Moreover, in any case the averaged in time probability to find the system in the state Φ(0) is not equal to zero. However, considering a property F (t) =< Φ(t)| F | Φ(t) > one can see that the deviation of F (t) from its initial value F (0) is only partially determined by L(t) and, therefore, the F (t) behavior must not be similar to that of L(t). Hence, the time evolution of the property of interest F (t) should be examined separately. In the present paper the initial state of a system is taken in the form Φ(0) = n P n ψ n(28) where P n are some numerical coefficients. The particular case of (28) is the state (3). In calculations only one of two Neel states (21 and 22), namely ψ N eel , was taken for the initial state because the results for the other Neel state will be the same. However, instead one probability L(t), two probabilities were considered: the probability L 1 (t) = Φ (t)|ψ (1) N eel 2 ,(29) to find the system in the initial state and the probability L 2 (t) = Φ (t)|ψ (2) N eel 2 .(30) to find the system in the other Neel state. Moreover, as it was discussed before, we are interested in states which have non-zero magnetization of both sublattices. Therefore, we also calculated the sublattices magnetizations in the stateΦ(t) S ℓz (t) = Φ (t)| S ℓz |Φ(t) ,(31) where ℓ is the sublattice's number. In the present paper only the systems with conserved z-projection of the total spin are considered and in the initial states the zprojection of the total spin is zero. Therefore, only the results for one sublattice magnetization are sufficient, the magnetization of the other sublattice being of the opposite sign. IV. NUMERICAL RESULTS AND DISCUSSION Let us consider, first, the Ising model H = J N −1 j=1 s j,z s j+1,z .(32) which is anisotropic because its Hamiltonian does not commute withŜ 2 operator. We consider the case J > 0. The ground state in this case is doubly degenerate and both Neel states (21) and (22) belong to it. In this case the Neel state ψ N eel energy distribution w(E) is the δfunction, the Neel state is stationary, and L 1 (t) ≡ 1; L 2 (t) ≡ 0; S 1z (t) ≡ N 4 . However, due to the degeneracy, the linear combination ψ = 1 √ 2 ψ (1) N eel + ψ (2) N eel also can be considered as the ground state, but in this state, contrary to the Neel one, the magnetization of each sublattice is equal to zero. Second, we consider the Heisenberg model H = J 1 N −1 j=1 ( s j , s j+1 ) + J 2 N −2 j=1 ( s j , s j+2 ).(33) which is isotropic as its Hamiltonian commutes withŜ 2 operator. The Neel state is not the eigenfunction of this Hamiltonian and the energy distribution w(E) has some width. The width depends on the particular values of J 1 , J 2 parameters and in case |J 1 | > |J 2 | > 0 it could be rather large, its maximum being shifted off the ground state energy. The Neel state in the Heisenberg model is not stationary and the probabilities L 1 , L 2 , and magnetization S 1z depend on time. Their time dependences are shown in figures 1 and 2 for two variants of the Heisenberg model. In figure 1 the Heisenberg model with only the nearest neighbors interaction is shown, whereas the figure 2 corresponds to the case with the second neighbors interaction included, the latter being somewhat weaker than the nearest neighbors interaction. Although the Neel state in this model does not exactly decay (the spectrum is pure discrete), one can see from figures 1 and 2 that the probability to find the system in the Neel state quickly drops down and afterwards its time average remains small. Besides, the time average of the sublattice magnetization is almost zero. Next, we consider the well-known XXZ-model with Hamiltonian H = A   N −1 j=1 s j,x s j+1,x + N −1 j=1 s j,y s j+1,y   + +W N −1 j=1 s j,z s j+1,z(34) This model has two parameters A and W and the considered above Ising and Heisenberg models Hamiltonians are particular cases of (34) with A = 0 and A = W respectively. Therefore, to consider the XXZ-model we selected the particular anisotropic case A = 1, W = 5 which is far from both Ising and Heisenberg cases discussed above. In the selected case the Neel state energy distribution has two main weights v 1 and v 2 , which correspond to the ground and first excited states. Weights v 1 and v 2 are b H = J1 P N−1 j=1 (b sj, b sj+1) + J2 P N−2 j=1 (b sj, b sj+2) with J1 = 1, J2 = 0.25. approximately equal to each other and they are at least one order of magnitude greater than the weight of any other state. Such energy distribution results in particular time dependences of L 1 , L 2 and S 1z shown in figure 3. The main feature is a harmonic oscillation of L 1 (t) and L 2 (t) superimposed by a small amplitude and high frequency oscillations. The L max of the harmonic oscillation is close to 1, and L min is close to 0. The L 1 (t) would experience exactly harmonic oscillation if all weights other than v 1 and v 2 would be exactly zero. In such oscillation the frequency is defined by the energy gap between the ground and excited states, the L max value is equal to 1, and L min value is defined by the weights difference, being zero when weights are exactly equal. In our case weights v 1 and v 2 are only close to each other and therefore, L min is small but not equal to zero. The similar oscillation (with opposite phase) of L 2 (t) indicates that the energy distribution of another Neel state ψ (2) N eel is similar to that of ψ (1) N eel but with interchanged v 1 and v 2 . From this we can conclude that both ground and first excited states are linear combinations of two Neel states with approximately equal weights and neither of them is close to a single Neel state. This conclusion is confirmed by the results of numerical calculations and is in accord with the time behavior of the sublattice magnetization shown in figure 3. From this figure it is seen that the time average of the sublattice magnetization is close to zero. The result of numerical calculations is −0.02. Besides, the sublattice magnetizations in the ground and the first excited states in the case considered are exactly zero. Hence, although the time average of the probability to find the system in the initial Neel state is essential (0.39 according to the numerical calculations), the magnetic properties of the system do not correspond to the Neel state. The next model was constructed to satisfy two conditions. First, the system was chosen to be isotropic. Hence, the system Hamiltonian must commute with the total spin operator. Second, the Neel state energy distribution in the system must have the smallest possible number of components. For this the system Hamiltonian must commute with sublattices square spin operators. Indeed, in this case equations (23) and (24) show that only states with maximum spins of each sublattices contribute to the Neel state energy distribution. Based on these considerations, the system Hamiltonian was chosen in the form H = A N −2 j=1 ( s j , s j+2 ) + W S 2 + V S 2 1 + S 2 2 .(35) with three parameters A, V , and W . Considering the Neel state we see, that all functions ϕ S from (2) onto which the Neel state is exactly expanded are eigenfunctions of this Hamiltonian. At the same time, from the equation (15) it follows that the matrix of the first term in (35), calculated on ϕ S , is proportional to the unit matrix. Besides, all ϕ S are the eigenfunctions of S 2 1 and S 2 2 with the same eigenvalue S 0 (S 0 + 1). Hence, the matrix of the third term in (35), calculated on ϕ S , is also proportional to the unit matrix. Consequently, parameters A and V define only the position of the Neel state distribution w(E) in the operator H spectrum, while the parameter W defines the energy distribution width ∆E = W N (N + 2)/4. The Neel state evolution in the considered model is periodic (however not harmonic), because energy differences between all ϕ S energy levels are commensurable (∆E jk = W (S j (S j + 1) − S k (S k + 1))). b H = A P N−2 j=1 (b sj , b sj+2) + W c S 2 + (V − W ) " b S 2 1 + b S 2 2 " . In figure 4 the time evolution of the Neel state is shown for the particular case of Hamiltonian (35) with A = −5, W = 1 V = −7. The parameters were chosen to place the energy levels of all eigenstates ϕ S , which contribute to the Neel state, below the rest of the spectrum. There are four main weights v k in the energy distribution in this case. Therefore, the oscillation shape is more complicated than in the XXZ model with two main weights, and the maximum value of L 2 (t) is well below 1. The time average of the probability to find system in the initial Neel state is about 0.29. However, the time average of the sublattice magnetization is small (about 0.001). Besides, the sublattice magnetization in the ground state is zero. Therefore, similarly to the XXZ model, the magnetic properties of the system do not correspond to the Neel state. Next model was made to have the degenerate ground state with all and only functions ϕ S belonging to it. In this case the Hamiltonian, similarly to (35), must commute with the total spin operator and with the sublattices spin square operators. In addition, the ground state must correspond to the maximum spins of sublattices and to the zero z-projection of the total spin. Besides, all other states must have higher energy than the ground state. The simplest Hamiltonian satisfying these conditions is H = −( S 2 1 + S 2 2 ) + S 2 z .(36) Similarly to the Ising model, the Neel state ψ N eel is the stationary state and in this state L 1 (t) ≡ 1; L 2 (t) ≡ 0; S 1z (t) ≡ N 4 . In this model, contrary to the Ising model, any Neel-like state is also the stationary state. However, for any Neel or Neel-like state the partner state with the opposite sublattices magnetization also belongs to the ground state. Therefore, their linear combination with zero sublattices magnetization is also the ground stationary state. Finally, the simple model with Hamiltonian H = S 2 + S k S=0 |ϕ S (A − S(S + 1)) ϕ S |, S k < N 2 , A < 0.(37) was considered where functions ϕ S are defined by (2). This model shows, that the time evolution of the Neel state helps us to recognize the existence of the stationary Neel-like state in the system where the Neel state itself is not stationary. The peculiarity of the model is that the ground state is degenerate and the corresponding functional subspace is spanned by some (not all) functions ϕ S . Therefore, the Neel-like states are eigenstates of operator (37) and the Neel state itself is not. The time evolution of the Neel state in this model is shown in figure 5. From this picture one can see that the probability L 1 (t) to find the system in the initial Neel state oscillates around some value close to 1 (L 1 time average is equal to 0.97 according to the numerical calculations) and the sublattice magnetization oscillates around a value close to the maximum (time average is equal to 1.94), the mentioned values being corresponded to the Neel-like state. Hence, the behavior of the Neel state time evolution, similar to that shown in figure 5, indicates that in the system there exists stationary Neel-like state. Note, that in this system, similarly to all systems with degenerate ground state discussed above, besides the particular Neellike state, there exists the stationary Neel-like state with the same energy and op In all discussed above models the Neel state cannot be considered as stable. Indeed, in all models, except Ising model and model with Hamiltonian (36), the Neel state is not the eigenstate. In the Ising model and the model (36) the Neel state is the eigenstate but it belongs to the degenerate level with the other Neel state also belonging to this level. In this case even if a certain Neel state was generated at a time t = 0 it will be destroyed by small external perturbations which are always present in real circumstances. To be stable the Neel state must be the ground non-degenerate state. However, it occur Hamiltonian can be transformed to have the particular Neel state as the ground non-degenerate eigenstate. To show this we will use the results of the paper 13 , where the following Hamiltonian transformation was developed to get a desirable set of states for the lowest eigenstates. Consider a Hamiltonian H 0 with eigenfunctions ψ 0 i and eigenvalues E 0 i H 0 ψ 0 i = E 0 i ψ 0 i . Consider also an arbitrary set of n orthonormal functions ψ k and an arbitrary set of real numbers E k (k = 1, · · · , n). In 13 the potential in the form of the separable potential V = 3n i,j=1 | f i V ij f j |,(38) was found so that the n lowest eigenfunctions and eigenvalues of the Hamiltonian H ′ = H 0 + V H ′ ψ ′ i = E ′ i ψ ′ i are exactly ψ i and E i ψ ′ i = ψ i , E ′ i = E i , i = 1, 2, · · · , n while the rest of the H 0 spectrum is not affected E ′ i = E 0 i , i = n + 1, · · · . The eigenfunctions ψ ′ i with i > n differ, of course, from ψ 0 i because former are orthogonal to ψ k , while later are orthogonal to ψ 0 k , k = 1, · · · , n. When only one eigenstate is to be transformed (n = 1) the functions f i and the matrix V ij , i, j = 1, 2, 3 can be chosen as follows f 1 = ψ 1 , f 2 = ψ 0 1 + ψ 1 , f 3 = H 0 f 2 , V 11 = E 1 − E 0 1 , V 22 = |t| 2 < f 2 |f 3 >, V 23 = V 32 = −t * , t = 1 1+ < f 1 |f 2 > All other matrix elements V ij are equal to zero, so the separable potential (38) has only four non-zero components. Using these results we can make the Neel state Ψ (1) N eel be the ground non-degenerate state. For this we take the ground state eigenfunction and eigenvalue of the model for ψ 0 1 and E 0 1 . When the model's ground state eigenvalue is degenerate we take the closest to Ψ V = |ψ (1) N eel > (E 1 − E 0 1 ) < ψ(1) N eel | In all other models the potential (38) has four nonzero components. The values of matrix elements V ij calculated for all considered above models, except (32) and (36) are given in the table I. Model Parameters Values CONCLUSIONS Considering the quasionedimensional spin chain the Neel state was analyzed and its time evolution in various models was examined. It was shown that in those models, where the Neel state is not the eigenstate of the Hamiltonian and, therefore, is not stationary, the time averaged magnetic properties of this state do not correspond to that commonly attributed to the Neel state. In those models, where the particular Neel state is an eigenstate of the Hamiltonian, and this eigenstate is degenerate, the Neel state is stationary, but another Neel state with opposite sublattices magnetization has the same energy. Therefore, small perturbations make the particular Neel state unstable. To be stable the Neel state must be the non-degenerate ground state of the system. The procedure is proposed to generate the Hamiltonian with the Neel state as its ground non-degenerate state. the orthonormality of wave functions of a complementary sublattice 3 − ℓ. M ℓ orthonormality the only non-zero term in the sum corresponds to M ℓ = S 0 = N/4. Hence, the only possible value of z is z = 0, and one has ψ(1) FIG. 1 : 1Time evolution of the Neel state in the system described by Heisenberg Hamiltonian b H = J1 P N−1 j=1 (b sj, b sj+1) + J2 P N−2 j=1 (b sj, b sj+2) with J1 = 1 and J2 = 0. FIG. 2 : 2Time evolution of the Neel state in the system described by Heisenberg Hamiltonian FIG. 3 : 3Time evolution of the Neel state in the XXZ-model. FIG. 4 : 4Time evolution of the Neel state with the Hamiltonian FIG. 5 : 5Time evolution of the Neel state with the Hamiltonian (37). Ising model with J > 0 and in the model with Hamiltonian (36) the Neel state is the ground state eigenfunction of the Hamiltonian, so ψ 1 11 = E 1 −E 0 1 , V 22 = E 0 1 /4, V 23 = V 32 = −1/4. Taking the sum of four components one can see that three components cancel each other and the potential (38) in this case becomes simply the energy level shift operator TABLE I : INon-zero matrix elements of the separable potential 38 for different models. AcknowledgmentsThe financial support from the RFBR (grant No 05-03-33243) is gratefully acknowledged. . S V Vonsovsky, M S Svirsky, Jetp, 57Vonsovsky S.V., Svirsky M.S., JETP, 57, 1(7) (1969) . J C Bonner, M E Fisher, Phys. Rev. 135640Bonner J.C., Fisher M.E., Phys. Rev., 135, A640 (1964) . R Orbach, Phys. Rev. 112309Orbach R., Phys. Rev., 112, 309 (1958) . P W Anderson, Phys. Rev. 86694Anderson P.W., Phys. Rev., 86, 694 (1952) . J M Ziman, Proc. Phys. Soc. 65548Ziman J.M., Proc. Phys. Soc., 65, 540, 548 (1952) . W Marshall, Proc. Roy. Soc. 48232Marshall W., Proc. Roy. Soc., 48, A232 (1955) . J Tessman, Phys. Rev. 881132Tessman J., Phys. Rev., 88, 1132 (1952) . C N Yang, C P Yang, Phys. Rev. 150327Yang C.N., Yang C.P., Phys. Rev., 150, 321, 327 (1966) . C N Yang, C P Yang, Phys. Rev. 15110Yang C.N., Yang C.P., Phys. Rev., 151, 258 (1966) 10 . R J Baxter, Ann, Physics. 70193Baxter R.J., Ann. Physics, 70, 193 (1972) . V A Fock, N Krylov, Journ. Phys. 112Fock V.A., Krylov N., Journ. Phys., 11, N2 (1947) Quantum Theory of Angular Momentum. D A Varshalovich, A N Moskalev, V K Khersonskii, World Scientific Pub Co IncVarshalovich D.A., Moskalev A.N., Khersonskii V.K., "Quantum Theory of Angular Momentum", World Scien- tific Pub Co Inc, 1988 . I V Abarenkov, I I Tupitsyn, J. Chem. Phys. 115Abarenkov I.V., Tupitsyn I.I., J. Chem. Phys, 115, N4, 1650-1660 (2001)
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[ "Euler-Lagrangian approach to stochastic Euler equations in Sobolev Spaces", "Euler-Lagrangian approach to stochastic Euler equations in Sobolev Spaces" ]
[ "Juan Londoño ", "Christian Olivera " ]
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The purpose of this paper is to establish the equivalence between Lagrangian and classical formulations for the stochastic incompressible Euler equations, the proof is based in Ito-Wentzell-Kunita formula and stochastic analysis techniques. Moreover, we prove a local existence result for the Lagrangian formulation in suitable Sobolev Spaces.
null
[ "https://export.arxiv.org/pdf/2303.07213v1.pdf" ]
257,496,288
2303.07213
274cfbf0c1d38c7f7e179ee870754a01c37dbf1e
Euler-Lagrangian approach to stochastic Euler equations in Sobolev Spaces 13 Mar 2023 Juan Londoño Christian Olivera Euler-Lagrangian approach to stochastic Euler equations in Sobolev Spaces 13 Mar 2023and phrases: Stochastic partial differential equationsEuler equationLagragian formulationIto-Wentzell-Kunita FormulaSobolev Spaces MSC2010 subject classification: 60H1535R6035F1060H30 The purpose of this paper is to establish the equivalence between Lagrangian and classical formulations for the stochastic incompressible Euler equations, the proof is based in Ito-Wentzell-Kunita formula and stochastic analysis techniques. Moreover, we prove a local existence result for the Lagrangian formulation in suitable Sobolev Spaces. Introduction We study a Lagragian formulation (following [7], [11] and [19] ) of the incompressible Euler equations on a domain T d . The Euler equations with transport noise model the flow of an incompressible inviscid fluid and are (classically) formulated in terms of a divergence-free vector field u (i.e. ∇ · u = 0) as follows: du t + (u t · ∇u t + ∇p t )dt + k L * σ k u t • dW k t = 0 (1.1) where p is a scalar potential representing internal pressure, L * σ j u := σ j · ∇u + (∇σ j ) * u (L is the Lie derivative ), W k t is a Wiener process and the integration is in the Stratonovich sense. The divergence-free condition reflects the incompressibility constraint. Equations related to fluid dynamics with multiplicative noise appeared in several other works, see for instance [1], [2] [5], [10], [11], [12], [13], [14] and many others. The main topic of this work, namely the Euler-Lagrangian formulation, called also Constantin-Iyer representation after [7], [8], among related works, see for instance [4], [9], [17], [18], [20]. First we show the Euler-Lagrangian formulation is equivalent to the stochastic Euler equations (1.1), see Proposition 2.1, the proof is based in Ito-Wentzell-Kunita formula and stochastic analysis techniques. We point that in [11] the authors show that the Lagrangian formulation verifies necessarily the equation (1.1), for d = 3, using the vorticity equation. We show that both formulations are equivalent for any dimension. Using this formulation we prove a local in time existence result for solutions in C 0 ([0, T ] ; (H s (T d ) d )) with s > d 2 + 1, new for equation (1.1). Equivalent formulations. Let L * σ j be the adjoint operator of L σ j with respect to the inner product in L 2 (T d , R d ): L * σ j v, w L 2 = − v, L σ j w L 2 . Since σ j is divergence free, we have L * σ j v = σ j · ∇v + ∇σ j * v. Definition 2.1. Given a (divergence free) velocity u, we define the material diferential D by Dγ := dγ + u · ∇γdt + σ i · ∇γ • dW i t Analogously to the deterministic case, see Proposition 2 in [19], by simple calculation we have the identities D (γβ) = (Dγ) β + γ (Dβ) , (2.2) D∇γ = ∇Dγ − (∇u) * ∇γdt − ∇σ i * ∇γ • dW i t . (2.3) We use the notation P for the Leray-Hodge projection onto the space of divergence-free functions. dX t = j σ j (X t ) • dW j t + u t (X t )dt (2.4) u t (x) = P [(∇A t ) * u 0 (A t )] (x),(2. 5) where * means the transposition of matrices and denote the back-to-labels map A by setting A (·, t) = X −1 (·, t). Proof. (⇒) We have u(t, x) = u 0 (x) − t 0 u∇uds − j t 0 L * σ j u • dW j s − t 0 ∇pds (2.6) and X t = x + t 0 u (s, X s ) ds + j t 0 σ j (X s ) • dW j s . (2.7) Then, from Itô-Wentzell-Kunita formula, see Theorem 8.3 in Chapter I of [15], the k-th component of the process u(t, X t ) is given by u k (t, X t ) =u 0 (x) − t 0 (u · ∇) u k (s, X s ) + (∇p) (X s ) ds − t 0 j ((σ j · ∇)u k (s, X s ) + ∂σ j ∂x k (X s ) u (s, X s )) • dB j s (2.8) + i t 0 ∂u k ∂x i (s, X s )u i (s, X s )ds + i,j t 0 ∂u k ∂x i (s, X s )σ j i (X s ) • dB j s . We observe that t 0 (u · ∇)u k (s, X s )ds = t 0 i u i ∂u k ∂x i Xs ds = i t 0 u i (s, X s ) ∂u k ∂x i (s, X s )ds = i t 0 ∂u k ∂x i (s, X s )u i (s, X s )ds and t 0 j (σ j · ∇)u k (s, X s ) • dW j s = j t 0 i σ j i ∂u k ∂x i Xs • dW j s = i,j t 0 ∂u k ∂x i (s, X s )σ j i (X s ) • dW j s . Making obvious cancellation we obtain u k (t, X t ) = u 0 (x) − t 0 (∇p) (X s )ds − j t 0 ∂σ j ∂x k (X s ) u k (X s ) • dW j s , i.e., u(t, X t ) = u 0 (x) − t 0 (∇p) (X s ) ds − j t 0 ∇σ j * u Xs • dW j s . Now, we observe that i ∂X i ∂x k u i is the k-th coordinate of (∇X) * u and ∂X i ∂x k = δ ik + t 0 ∇u i (s, X s ) ∂X s ∂x k ds + j t 0 ∇σ j i (X s ) ∂X s ∂x k • dW j s . Thus from Itô's formula for the product of two semimartingales we deduce ∂X i ∂x k u i (t, X t ) = δ ik u 0 (x) + t 0 u i (s, X s )∇u i (s, X s ) ∂X s ∂x k ds + t 0 u i (s, X s ) j ∇σ j i (X s ) ∂X s ∂x k • dW j s − t 0 ∂X i ∂x k (∇p) (X s )ds − t 0 ∂X i ∂x k j ∂σ j ∂x i (X s )u(s, X s ) • dW j s . Note that ∂X i ∂x k ∂σ j ∂x i u = ∂X i ∂x k ∂σ j 1 ∂x i , · · · , ∂σ d j ∂x i (u 1 , u 2 , · · · , u d ) * = ∂X i ∂x k α ∂σ j α ∂x i u α , hence i ∂X i ∂x k ∂σ j ∂x i u = i α ∂X i ∂x k ∂σ j α ∂x i u α . On the other hand, we have u i ∇σ j i ∂X s ∂x k = u i ∂σ j i ∂x 1 , ∂σ j i ∂x 2 , · · · , ∂σ j i ∂x d ∂X 1 ∂x k , · · · , ∂X d ∂x k * = u i β ∂σ j i ∂β ∂X β ∂x k , hence i u i ∇σ j i ∂Xs ∂x k = i β u i ∂σ j i ∂x β ∂X β ∂x k . Thus we obtain i t 0 u i (s, X s ) j ∇σ j i (X s ) ∂X s ∂x k • dW j s = i t 0 ∂X i ∂x k j ∂σ j ∂x i (X s )u(s, X s ) • dW j s Then the k−th term of (∇X t ) * u (t, X t ) is i ∂X i ∂x k u i (t, X t ) = i δ ik u 0 (x) − i t 0 ∂X i ∂x k (∇p) (X s )ds + i t 0 u i (s, X s ) ∇u i (s, X s ) ∂X s ∂x k ds, i.e., (∇X t ) * u (t, X t ) = u 0 (x) + t 0 (∇X s ) * (∇u(s, X s )) * u (s, X s ) ds − t 0 (∇X s ) * (∇p) (X s )ds = u 0 (x) + 1 2 t 0 ∇ |u s | 2 • X s ds − t 0 ∇ (p • X s ) ds = u 0 + ∇q, whereq := t 0 1 2 (|u s | 2 • X s ) − p • X ds. Then, if we denote M t := (∇X t ) * , follow that u t • X t = u (t, X t ) = M −1 t u 0 + M −1 t ∇q = (∇A t Xt ) * u 0 + (∇A t Xt ) * ∇q. Finally we conclude u t = (∇A t ) * u 0 • A t + (∇A t ) * (∇q)(A t ) = (∇A t ) * u 0 • A t + ∇q, where q :=q • A. Therefore, u t = P [(∇A t ) * u 0 • A t ]. (⇐) We follow directly from proposition 2.2 in [12]. We shall also show an alternative formal proof. We set v = u 0 • A, then by Theorem 2.3.2 of [6] we have DA = 0 and Dv = 0. Since u satisfies (2.5) there exists a function q such that u = (∇A) * v − ∇q. Then by (2.2) and (2.3) we have Du = D [(∇A) * v − ∇q] = D [(∇A) * v] − D∇q = [D (∇A) * ] v + (∇A) * Dv − D∇q = ∇DA − (∇u) * (∇A) dt − ∇σ i * ∇A • dW i t * v − ∇Dq + (∇u) * ∇qdt + ∇σ i * ∇q • dW i t . Hence, after a calculation and a rearrangement of the terms, we get Du = − (∇u) * (∇A) * vdt + (∇u) * ∇qdt − ∇Dq − ∇σ i * [(∇A) * v − ∇q] • dW i t = − (∇u) * udt − ∇Dq − ∇σ i * u • dW i t = −∇ |u| 2 2 dt − ∇Dq − ∇σ i * u • dW i t . Then, u(t, x) = u 0 (x) − t 0 u · ∇uds − t 0 σ i · ∇u • dW i s − ∇ t 0 |u| 2 2 ds − ∇ t 0 Dq s − t 0 ∇σ i * u • dW i s = u 0 (x) − t 0 u · ∇uds − t 0 L * σ i u • dW i s − t 0 ∇pdt, where formally we have p = |u| 2 2 + ∂ t q + u · ∇q + σ i · ∇q ∂ t W i t . Therefore we conclude that u is solution of the Euler equation (1.1). An Existence Theorem Decomposition of the Flow We use the idea of [11] to decompose the stochastic flow in the system (2.4)-(2.5). More precisely, we consider the stochastic equation without drift: dϕ t = k σ k (ϕ t ) • dW k t , ϕ 0 = I, where I is the identity diffeomorphism of T d . Under the assumption that k σ k 2 5,α ′ < ∞ for α ′ ∈ (0, 1), the above equation generates a stochastic flow {ϕ t } t≥0 of C 4,α −diffeomorphisms on T d , where α ∈ (0, α ′ ). We denote by ω a generic random element in a probability space Ω. For a given random vector field u : Ω × [0, T ] × T d → R d , we definẽ u (ω, x) = ϕ t (ω, ·) −1 * u t (ω, ·) (x) (3.9) which is the pull-back of the field u t (ω, ·) by the stochastic flow {ϕ t (ω, ·)} t≥0 . If we denote by K t (ω, x) = (∇ϕ t (ω, x)) −1 , i.e., the inverse of the Jacobi matrix, theñ u (ω, x) = K t (ω, x) u t (ω, ϕ t (ω, x)) . (3.10) From this expression we see that if u ∈ C 0 ([0, T ] , H s ) a.s., then one also a.s.ũ ∈ C 0 ([0, T ] , H s ). Moreover, if the process u is adapted, then so isũ. Now we consider the random ODĖ Y t =ũ t (Y t ) , Y 0 = I. (3.11) Applying the Itô-Wentzell-Kunita formula, we see that (cf. [11]) X t = ϕ t • Y t is the flow of C 3,α −diffeomorphisms associated to the SDE in (2.4)-(2.5). Sobolev Estimations. All notations and results in this subsection we follow from [19]. For r ≥ 0, we will use the notation H r variously for sacalar or vector valued functions in H r T d (componentwise), where this does not cause ambiguity. We will often consider functions in spaces of the norm C 0 ([0, T ] ; (H s (T d ) d )). To simplify notation we define Σ s (T ) (usually denoted Σ s ) for T ≥ 0 and s ≥ 0 by Σ s (T ) := C 0 ([0, T ] ; (H s (T d )) d ). We consider the natural norm on Σ s : u Σs = sup t∈[0,T ] u(t) s We begin by stating two inequalities concerning the advection term (u · ∇)v, using the notation B(u, v) := (u · ∇)v. The following two results are taken from Lemma 1 and Lemma 2 of [19]. Lemma 3.1. For s > d 2 there exists C 1 > 0 such that if u ∈ H s and v ∈ H s+1 then B (u, v) ∈ H s and B (u, v) s ≤ C 1 u s v s+1 . Lemma 3.2. If s > d 2 + 1 there exists C 2 > 0 such that for u ∈ H s , v ∈ H s+1 with divergence-free we have |(B (u, v) , v) s | ≤ C 2 u s v 2 s We use the following shorthand for closed balls in Σ s : B M = B · Σs (0, M), i.e., B M is the closed unit ball centred at the origin of radius M > 0 with respect to the norm · Σs . Where ambiguity could arise we write B M (T ) for the closed ball in Σ s (T ). We need the following key technical result, see Lemma 3 of [19]. Lemma 3.3. If s > d 2 + 1 and η, v ∈ Σ s (T ) then P [(∇η) * v] ∈ Σ s and there exists a constant C 3 > 0 (independent of η, v t and T ) such that for fixed t, P [(∇η) * v] r ≤ C 3 u s v r , (3.12) where r = s or r = s − 1. Furthermore, there exists C ′ 3 > 0 such that for any M > 0 and T > 0, the following bounds hold uniformly with respect to t ∈ [0, T ] for any η 1 , η 2 , v 1 , v 2 ∈ B M (T ): P [(∇η 1 ) * v 1 − (∇η 2 ) * v 2 ] X ≤ C ′ 3 ( η 1 − η 2 X + v 1 − v 2 X ) , (3.13) where X is L 2 T d or H s−1 . The next lemma gives uniform bounds on the H s norms of solutions to the transport equations (3.18) and (3.19). We will consider the following system:    ∂ t f + (u · ∇) f = g, f (0) = f 0 . (3.14) where f, g : [0, T ] × T d → T d and u is divergence-free. Lemma 3.4. Let s > d 2 + 1 and fix f 0 ∈ H s , g ∈ Σ s . If u ∈ Σ s is non-zero and divergence free then exists a unique solution f to (3.14). Furthermore, the solution f ∈ Σ s ∩ C 1 ([0, T ]; H s−1 ) ∩ C 1 ([0, T ] × T d ) and there exists C 4 > 0 (from Lemma 3.2) such that if r, t ∈ [0, T ] we have: f (t) s ≤ f (r) s + g Σs C 4 u Σs exp (C 4 |t − r| u Σs ) − g Σs C 4 u Σs . (3.15) Lemma 3.5. For s > d 2 + 1 fix u 1 , u 1 ∈ Σ s and f 0 ∈ H s . Let g 1 = g 2 = 0 or g i = −u i for i = 1, 2. If f 1 , f 2 are the solutions of (3.14) corresponding to u 1 , u 2 , g 1 , g 2 respectively, then in the case that g 1 = g 2 = 0, there exists C 5 > 0 depending only of s such that f 1 (t) − f 2 (t) L 2 ≤ C 5 f 1 + f 2 Σs u 1 − u 2 Σ 0 t (3.16) for all t ∈ [0, T ]. In the case that g i = −u i for i = 1, 2 we instead have f 1 (t) − f 2 (t) L 2 ≤ (C 5 f 1 + f 2 Σs + 1) u 1 − u 2 Σ 0 t (3.17) We consider the following transport equation: ∂ t B + (ũ · ∇) B = 0, (3.18) ∂ t v + (ũ · ∇) v = 0. (3.19) Given an initial divergence-free velocity u 0 for the classic equations, we choose initial conditions for the above system as follows: η (x, 0) = 0 (3.23) respectively. We do this because the identity map (hence B) does not have sufficient Sobolev regularity when considered as a function on the torus with values in R d (i.e. whitout accounting for the topology of the target torus). B (x, 0) = x, (3.20) u (x, 0) = v (x, 0) = u 0 (x) . Contraction The aim of the rest of this paper is to prove the following theorem. Theorem 3.1. If d ≥ 2, s > d 2 + 1 and u 0 ∈ H s is divergence free then there exists T (ω) > 0, such that the systems (3.18)-(2.5) has solution u ∈ Σ s (T ) Proof. For u ∈ Σ s , we consider the following system:             ũ t (x) = ϕ −1 t * u t (x) , Y t =ũ t (Y t ) , Y 0 = I, X t = ϕ t (Y t ) , Su t (x) = P ∇X −1 t * u 0 X −1 t (x) . (3.24) From Chapter 7 of [3] we have that exists positives constants C 0 := C 0 (t, ϕ), C 1 := C 1 (t, ϕ), C 2 := C 2 (t, ϕ) such that ϕ −1 s ≤ C 1 ; (3.25) ũ s = (∇ϕ −1 ) (ϕ)u(ϕ) s ≤ C 0 u s ; (3.26) η • ϕ −1 s ≤ C 2 η s and v • ϕ −1 s ≤ C 2 v s .C u 0 s exp (C 4 C 0 T M) C 2 2 C 1 C 4 (exp (C 4 C 0 T M) − 1) + 1 ≤ M, where C is a constant to be defined later. Let u ∈ B M (T ) be a divergence free function and let η = Y −1 t − x is unique solution of (3.22) with initial data η 0 = 0. Let v = u 0 (Y −1 t ) is the unique solution of (3.19) for initial data v 0 = u 0 . From (3.26) follow that ũ Σs = sup t∈[0,T ] ũ s ≤ C 0 M, Then by Lemma 3.4, v(t) s ≤ u 0 s exp (C 4 t ũ Σs ) ≤ u 0 s exp (C 0 C 4 T M) (3.28) and η(t) s ≤ 1 C 4 (exp (C 0 C 4 T M) − 1) . (3.29) Hence, with C = C 1 C 2 C 3 , (the constants C 1 , C 2 , C 3 are given by (3.25), (3.27) and Lemma 3.3 respectively) Su(t) s ≤ P (∇(η • ϕ −1 )) * v(ϕ −1 ) s + P (∇ϕ −1 ) * v(ϕ −1 ) s ≤ C 3 η • ϕ −1 s v(ϕ −1 ) s + C 3 ϕ −1 s v(ϕ −1 ) s ≤ C 3 C 2 2 η s v s + C 3 C 1 C 2 v s = C 3 C 2 v s (C 2 η s + C 1 ) , here the third inequality follows from (3.25) and (3.27). Then by (3.28) and (3.29) Su(t) s ≤ u 0 s exp (C 4 C 0 tM) C 3 C 2 2 C 4 (exp (C 4 C 0 tM) − 1) + C 3 C 1 C 2 = C u 0 s exp (C 4 C 0 tM) C 2 2 C 1 C 4 (exp (C 4 C 0 tM) − 1) + 1 ≤ M (3.30) for all t ∈ [0, T ]. Hence S : B M (T ) → B M (T ). Before proving that the map S is a contraction in a certain space, we need to prove a couple of inequalities. By Lemmas 3.3, 3.4 and 3.5 we have P ∇ η 1 • ϕ −1 * v 1 ϕ −1 − ∇ η 2 • ϕ −1 * v 2 ϕ −1 L 2 ≤ C 3 M η 1 • ϕ −1 − η 2 • ϕ −1 L 2 + v 1 • ϕ −1 − v 2 • ϕ −1 L 2 = C 3 M [ η 1 − η 2 L 2 + v 1 − v 2 L 2 ] ≤ C 3 M C 5 η 1 + η 2 Σs + 1 ũ 1 −ũ 2 Σ 0 t + C 5 v 1 − v 2 Σs ũ 1 −ũ 2 Σ 0 t ≤ C 3 M 2C 5 C 4 (exp (C 4 C 0 tM) − 1) + 1 t ũ 1 −ũ 2 Σ 0 + 2C 5 Mt u 0 s exp (C 4 C 0 tM) ũ 1 −ũ 2 Σ 0 ≤ C 3 M 2C 5 C 4 (exp (C 4 C 0 T M) − 1) + 1 t + 2C 5 t u 0 s exp (C 4 C 0 T M) ũ 1 −ũ 2 Σ 0 , and P ∇ϕ −1 * v 1 (ϕ −1 ) − v 2 (ϕ −1 ) L 2 ≤ C 3 M v 1 (ϕ −1 ) − v 2 (ϕ −1 ) L 2 = C 3 M v 1 − v 2 L 2 ≤ C 3 C 5 M v 1 − v 2 Σs ũ 1 −ũ 2 Σ 0 t ≤ 2C 3 C 5 Mt u 0 s exp (C 4 tC 0 M) ũ 1 −ũ 2 Σ 0 , where C 3 , C 4 , C 5 are the constants from Lemmas 3.3, 3.4 and 3.5 respectively. Now we will show that the map S is a contraction on B M (T ) in the L 2 norm if T is sufficiently small. For u 1 , u 2 ∈ B M (T ) we construct v i and η i from u i as above for i = 1, 2 with v 1 (·, 0) = v 2 (·, 0) = u 0 . Then by the inequalities above Su 1 − Su 2 L 2 ≤ P ∇ η 1 • ϕ −1 * v 1 ϕ −1 − ∇ η 2 • ϕ −1 * v 2 ϕ −1 L 2 + P ∇ϕ −1 * v 1 (ϕ −1 ) − v 2 (ϕ −1 ) L 2 ≤ C 3 M 2C 5 C 4 (exp (C 4 C 0 T M) − 1) + 1 t ũ 1 −ũ 2 Σ 0 (3.31) + 4C 3 C 5 Mt u 0 s exp (C 4 C 0 T M) ũ 1 −ũ 2 Σ 0 ≤ 2T MC 3 C 6 C 5 2 u 0 s + 1 C 4 exp (C 4 C 0 T M) + 1 2 − C 5 C 4 u 1 − u 2 Σ 0 = C(u 0 , ω, M, T ) u 1 − u 2 Σ 0 , where lasts inequality follows from a change of variables and Hölder's inequality ũ 1 −ũ 2 Σ 0 = sup t∈[0,T ] ũ 1 −ũ 2 L 2 = sup t∈[0,T ] (∇ϕ) −1 u 1 (ϕ) − (∇ϕ) −1 u 2 (ϕ) L 2 = sup t∈[0,T ] ∇ϕ −1 (ϕ) (u 1 − u 2 ) (ϕ) L 2 = sup t∈[0,T ] ∇ϕ −1 (u 1 − u 2 ) L 2 ≤ sup t∈[0,T ] ∇ϕ −1 L ∞ u 1 − u 2 L 2 ≤ C 6 u 1 − u 2 Σ 0 , where C 6 := sup t∈[0,T ] ∇ϕ −1 L ∞ . We observe that C(u 0 , ω, M, T ) is given by the formula C(u 0 , ω, M, T ) := 2T C 5 MC 3 C 6 2 u 0 s + 1 C 4 exp (C 4 C 0 T M) + C 3 C 6 M 1 2 − C 5 C 4 , . (3.32) We observe that ϕ ∈ C 0 ([0, T ], C 4,α (T d , T d )) then this implies that C 0 , C 1 , C 2 and C 6 are limited uniformly in time. Then taking the supreme of (3.31) with respect to t and choosing T > 0 small enough, we see that S is a contraction. We finished the proof as in [19]. We conclude that S has a unique accumulation point u, in the closure of B M with respect to · Σ 0 . Since B M (T ) is convex and closed in Σ s it is weakly closed, hence u ∈ B M is a fixed point of S. Proposition 2. 1 . 1Assume that u is C 3,α -continuos semimartingale. Then u is solution of the equation (1.1) if and only if verifies the Lagrangian formulation (x, t) := B (x, t) − x and replace (3.18) and (3.20) with the equations ∂ t η + (ũ · ∇) η +ũ = 0, (3.22) let C 3 , C 4 be the constants in (3.12), (3.15) (from Lemmas 3.3 and 3.4) respectively. Fix u 0 s < M and T > 0 so that AcknowledgementsThis study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001-on behalf of the first-named author. The second author is funded by the grant 2020/15691 − 2 and 2020/04426 − 6, São Paulo Research Foundation (FAPESP) On the well-posedness of stochastic Boussinesq equations with transport noise. D Alonso-Oran, Bethencourt-De-Leon, Journal of Nonlinear Science. 30D. Alonso-Oran, A Bethencourt-de-Leon, On the well-posedness of stochastic Boussinesq equa- tions with transport noise, Journal of Nonlinear Science 30, 2020. Modelling the climate and weather of a 2D Lagrangian-averaged Euler-Boussinesq equation with transport noise. D Alonso-Oran, A Bethencourt-De-Leon, D D Holm, S Takao, Journal of Statistical Physics. 179D. Alonso-Oran, A. Bethencourt-de-Leon, D.D.Holm, S. Takao.: Modelling the climate and weather of a 2D Lagrangian-averaged Euler-Boussinesq equation with transport noise, Journal of Statistical Physics 179, 5, 2020. . A Behzadan, M Holst, Sobolev-Slobodeckij Spaces on Compact Manifolds, Mathematics. 10A. Behzadan, and M. Holst, Sobolev-Slobodeckij Spaces on Compact Manifolds, Mathematics 10, 2022. Stochastic Lagrangian perturbation of Lie transport and applications to fluids, Nonlinear Analysis. N Besse, 2322023N. Besse, Stochastic Lagrangian perturbation of Lie transport and applications to fluids, Non- linear Analysis, 232, 2023. Stochastic navier-stokes equations with multiplicative noise. Z Brzeniak, M Capinski, F Flandoli, Stoch. Anal. Appl. 10Z. Brzeniak, M. Capinski, F. Flandoli, Stochastic navier-stokes equations with multiplicative noise, Stoch. Anal. Appl. 10, 1992. P L Chow, Stochastic partial differential equations. Boca Raton; Boca Raton, FLCRC Press2nd ed.P.L. Chow, Stochastic partial differential equations, 2nd ed., Advances in Applied Mathematics (Boca Raton), Boca Raton, FL: CRC Press, 2015. An Eulerian-Lagrangian approach for incompressible fluids: local theory. P Constantin, Journal of the American Mathematical Society. 14P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory, Journal of the American Mathematical Society, 14, 2001. A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations. P Constantin, G Iyer, 61P.Constantin, and G. Iyer, A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations, 61, 2008. Constantin and Iyer's representation formula for the NavierStokes equations on manifolds, Potential Anal. S Fang, D Luo, 48S. Fang, D. Luo, Constantin and Iyer's representation formula for the NavierStokes equations on manifolds, Potential Anal. 48, 2018. Particles and fields in fluid turbulence, Rev. Modern Phys. 73. G Falkovich, K Gawedzki, M Vergassola, G. Falkovich, K. Gawedzki, M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod- ern Phys. 73, 2002. Random Perturbation of PDEs and Fluid Dynamic Models. F Flandoli, Springer-VerlagEcole Saint Flour; BerlinF. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Ecole Saint Flour 2010, Springer-Verlag, Berlin, 2011. Euler-Lagrangian approach to 3D stochastic Euler equations. F Flandoli, D Luo, Journal of Geometric Mechanics. 18F. Flandoli, D. Luo, Euler-Lagrangian approach to 3D stochastic Euler equations, Journal of Geometric Mechanics, 18, 2019. Neklyudov Noise prevents infinite stretching of the passive field in a stochastic vector advection equation. F Flandoli, M Maurelli, M , J. Math. Fluid Mech. 16F. Flandoli, M. Maurelli, M. Neklyudov Noise prevents infinite stretching of the passive field in a stochastic vector advection equation, J. Math. Fluid Mech. 16, 2014. Well-posedness of the vector advection equations by stochastic perturbation. F Flandoli, C Olivera, J. Evol. Equ. 18F. Flandoli, C. Olivera, Well-posedness of the vector advection equations by stochastic pertur- bation J. Evol. Equ, 18, 2018. Stochastic differential equations and stochastic flows of diffeomorphisms. H Kunita, Lectures Notes in Mathematics. 1097Springer-VerlagH. Kunita. Stochastic differential equations and stochastic flows of diffeomorphisms, Lectures Notes in Mathematics, Springer-Verlag, Berlin, 1097, 143-303, 1982. H Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, Cambridge etc. Cambridge University Press24H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Ad- vanced Mathematics, Cambridge etc.: Cambridge University Press, 24, 1990. A local solution to the Navier-Stokes equations on manifolds via stochastic representation. D S Ledesma, D.S. Ledesma, A local solution to the Navier-Stokes equations on manifolds via stochastic representation, Nonlinear Anal. 198, 2020. Probabilistic representation for mild solution of the Navier-Stokes equations. C Olivera, Mathematical Research Letters. 282021C. Olivera, Probabilistic representation for mild solution of the Navier-Stokes equations, Math- ematical Research Letters, 28, 2021. An Eulerian-Lagrangian form for the Euler equations in Sobolev spaces. B Pooley, J Robinson, J. Math. Fluid Mech. 18B. Pooley, J. Robinson, An Eulerian-Lagrangian form for the Euler equations in Sobolev spaces, J. Math. Fluid Mech., 18, 2016. Stochastically symplectic maps and their applications to the Navier-Stokes equation, Annales de l'Institut Henri Poincare C, Analyse non lineaire. F Rezakhanlou, 33F. Rezakhanlou Stochastically symplectic maps and their applications to the Navier-Stokes equation, Annales de l'Institut Henri Poincare C, Analyse non lineaire, 33, 2016.
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[ "Nuclear Physics A Towards a hybrid strong/weak coupling approach to jet quenching", "Nuclear Physics A Towards a hybrid strong/weak coupling approach to jet quenching" ]
[ "Jorge Casalderrey-Solana \nDepartament d'Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos (ICCUB)\nUniversitat de Barcelona\nMartí i Franquès 108028BarcelonaSpain\n", "Doga Can Gulhan \nDepartment of Physics\nLaboratory for Nuclear Science\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n", "José Guilherme Milhano \nCENTRA\nInstituto Superior Técnico\nUniversidade de Lisboa\nAv. Rovisco PaisP-1049-001LisboaPortugal\n\nPhysics Department\nTheory Unit\nCERN\nCH-1211Genève 23Switzerland\n", "Daniel Pablos \nDepartament d'Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos (ICCUB)\nUniversitat de Barcelona\nMartí i Franquès 108028BarcelonaSpain\n", "Krishna Rajagopal \nDepartment of Physics\nLaboratory for Nuclear Science\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n" ]
[ "Departament d'Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos (ICCUB)\nUniversitat de Barcelona\nMartí i Franquès 108028BarcelonaSpain", "Department of Physics\nLaboratory for Nuclear Science\nMassachusetts Institute of Technology\n02139CambridgeMAUSA", "CENTRA\nInstituto Superior Técnico\nUniversidade de Lisboa\nAv. Rovisco PaisP-1049-001LisboaPortugal", "Physics Department\nTheory Unit\nCERN\nCH-1211Genève 23Switzerland", "Departament d'Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos (ICCUB)\nUniversitat de Barcelona\nMartí i Franquès 108028BarcelonaSpain", "Department of Physics\nLaboratory for Nuclear Science\nMassachusetts Institute of Technology\n02139CambridgeMAUSA" ]
[ "Nuclear Physics A" ]
We explore a novel hybrid model containing both strong and weak coupling physics for high energy jets traversing a deconfined medium. This model is based on supplementing a perturbative DGLAP shower with strongly coupled energy loss rate. We embed this system into a realistic hydrodynamic evolution of hot QCD plasma. We confront our results with LHC data, obtaining good agreement for jet R AA , dijet imbalance A J and fragmentation functions.
10.1007/jhep10(2014)019
[ "https://web.archive.org/web/20180722083812/http:/dspace.mit.edu/bitstream/handle/1721.1/108047/Towards%20a%20Hybrid%20Strong:Weak%20Coupling%20Approach%20to%20Jet%20Quenching.pdf;jsessionid=3299D32E62AC2C340143316A5B802328?sequence=1" ]
3,175,416
1405.3864
fbc7bc7b0cbe3ee39624bc60d44a6dfb4614ae69
Nuclear Physics A Towards a hybrid strong/weak coupling approach to jet quenching 2014 Jorge Casalderrey-Solana Departament d'Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos (ICCUB) Universitat de Barcelona Martí i Franquès 108028BarcelonaSpain Doga Can Gulhan Department of Physics Laboratory for Nuclear Science Massachusetts Institute of Technology 02139CambridgeMAUSA José Guilherme Milhano CENTRA Instituto Superior Técnico Universidade de Lisboa Av. Rovisco PaisP-1049-001LisboaPortugal Physics Department Theory Unit CERN CH-1211Genève 23Switzerland Daniel Pablos Departament d'Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos (ICCUB) Universitat de Barcelona Martí i Franquès 108028BarcelonaSpain Krishna Rajagopal Department of Physics Laboratory for Nuclear Science Massachusetts Institute of Technology 02139CambridgeMAUSA Nuclear Physics A Towards a hybrid strong/weak coupling approach to jet quenching Nuclear Physics A 002014jetsquenchingAdS/CFT We explore a novel hybrid model containing both strong and weak coupling physics for high energy jets traversing a deconfined medium. This model is based on supplementing a perturbative DGLAP shower with strongly coupled energy loss rate. We embed this system into a realistic hydrodynamic evolution of hot QCD plasma. We confront our results with LHC data, obtaining good agreement for jet R AA , dijet imbalance A J and fragmentation functions. Introduction We study the quenching of energetic jets traversing a strongly coupled plasma via a hybrid strong/weak coupling model based on treating physics processes occurring at different energy scales differently. Since very high energy jets are created at a scale Q Λ QCD their production spectrum is obtained perturbatively and is therefore under good theoretical control. Subsequent parton showering is described by DGLAP evolution which relaxes the initial virtuality down to the hadronization scale. In the presence of a medium, travelling partons also participate in soft momentum transfers with the plasma constituents of the order of the temperature T Λ QCD . Since this scale is not large, we assume non-perturbative effects to be important. The treatment of these is inspired by gauge/gravity duality insights. The latter technique determines the energy loss of probes travelling a strongly coupled medium of other non-Abelian theories which have holographical dual descriptions as gravitational theories. However, a description of the hard splittings is still lacking within this context, which means that a holistic explanation of the phenomena is at the moment not possible. For this reason we resort to a phenomenological model. We embed this hybrid model into a hydrodynamic description of expanding hot QCD matter and analyse several jet observables which we compare to LHC data. A Hybrid Model We generate hard events with PYTHIA [1], and assume that for these high virtuality processes modifications of nuclear parton distribution functions are small. To avoid unconstrained assumptions regarding hadronization and color reconnection we stay at partonic level, since fully reconstructed jets are expected to have little sensitivity to hadronization effects. We use a formation time argument (τ f = 2E/Q 2 ) to ascribe a space-time structure to the decay chain, for which the splitting probabilities remain as in vacuum. The system is embedded into a hydrodynamical model of expanding plasma with a realistic equation of state [2]. Since we are only interested in the strongly coupled phase, we stop subtracting energy from the jet when the local temperature of the plasma falls below T C . This temperature is not unambiguously defined, so we explore the range 180 < T C < 200 MeV to gauge our theoretical uncertainties. We neglect energy loss in the hadronic phase. Following [4] we explore an energy loss rate inspired by that of an energetic light quark traversing a slab of supersymmetric Yang-Mills plasma 1 E in dE dx = − 4x 2 πx 2 stop x 2 stop − x 2 (1) where E in is the initial energy of the excitation at its creation point and x stop the stopping distance. Keeping the energy and temperature dependence of the holographic calculation we parametrize this maximum distance by x stop = 1 2α S C E 1/3 in T 4/3 (2) where in the strong coupling calculation α S C = 1.05λ 1/6 , with λ the 't Hooft coupling. Since these calculations are not done in QCD itself we reinterpret α S C as a dimensionless free parameter that we fit to data. Note that the path dependence in Eq. (1) is not simply L 2 , but has a rather intricate dependence, with the presence of a Bragg-like peak. This energy loss rate is determined for quarks. For adjoint degrees of freedom, gluons, it is known that the parametric dependence is the same as for quarks, but no exact formula has been yet derived. Since the gluons are represented by two strings, we prescribe that the stopping distance and loss rate for a gluon are the same as those for a quark with half the energy of the gluon. We interpret this factor of 2 as the ratio of the Casimirs of the two color representations in the large N C limit. This reasoning leads to x G stop = x Q stop C F C A 1/3(3) To bracket the sensitivity of the various jet observables we analyse to the microscopic dynamics, we explore other energy loss mechanisms inspired by perturbative calculations which contain different path length dependences. In particular, we implement an energy loss rate given by dE dx = −α R C r C F T 3 x(4) which captures the path length dependence of radiative energy loss (to logarithmic accuracy), the dominant mechanism at weak coupling. The r in the Casimir stands for parton representation. We stress that this simplified implementation is not aimed at superseding other more sophisticated analyses of such processes. Finally, we also explore an energy loss rate with distinct path length dependence of the form dE dx = −α C C r C F T 2(5) which is inspired by collisional energy loss. For both benchmark models α C and α R are dimensionless fitting parameters. The complete description of the whole setup can be found in our main paper [6]. Comparison with Data Following the experimental procedure, we reconstruct jets using the FastJet [7] anti-k T algorithm with R = 0.3 within a rapidity range |y| < 2. We constrain our models by fitting the jet R AA for 100 < p T < 110 GeV in the 0-10% centrality class. In Figure 1 we show the jet R AA for the strongly coupled model compared to preliminary CMS data [3] for two different centrality classes. The shape is weakly dependent on p T . The model predicts the main features of the centrality dependence although the more peripheral bins deviate mildly from data (not shown); this may be due in part to the absence of any quenching at all in the hadron gas phase, whose relative importance becomes greater with diminishing centrality. In Figure 2 we show the model prediction for the dijet imbalance for the same centralities. Since current data [5,8] are not fully unfolded from resolution effects, a direct comparison with our model results is not possible. The CMS collaboration has shown that a simple smearing procedure [9] takes into account the main systematics of such effects. After applying this procedure, the strongly coupled calculation is in very good agreement with the data. This success is not exclusive of this model. A useful way of establishing the capability for the different models to simultaneously describe these two observables is scanning their parameter space as shown in the left panel in Figure 3. We plot the mean dijet imbalance as a function of jet suppression for central collisions for different choices of the fitting parameter, which describe a curve in parameter space. The theoretical uncertainties in T C transform these curves into bands. Current systematic uncertainties cannot discriminate between the different model implementations. However, the modest separation of the bands displayed in the figure provides a quantification of the level of precision in jet data needed to discriminate between these very different underlying dynamics. In the right panel in Figure 3 we have computed the ratio between in medium and vacuum partonic fragmentation functions. The points labelled as data correspond to three characteristic points of the measured ratio of hadronic fragmentation functions: the two crossing points at one and the minimum. The inspection of this plot strongly disfavours the collisional model. Admittedly, a direct comparison of our partonic calculations with data is less constraining for this observable. However, in general we expect that hadronization softens the fragmentation functions shifting all 3 curves to the right, reinforcing our previous conclusion. While the strongly coupled model provides the best fit to data in hard and medium regions, it fails to reproduce the softest part. Nevertheless it is precisely this sector that is more sensitive to hadronization effects and to the possible enhancement of soft particles by the lost energy. Conclusions Despite the simplifications of the approach explored we observe a satisfactory agreement between our hybrid strongly coupled model and jet data both at the qualitative and quantitative level. The extracted values of the fitting parameter α S C , 0.3 < α S C < 0.4 are of order one, as expected from holographic calculations. Even though this success is encouraging, we notice that some of the observables considered have limited sensitivity to the underlying dynamics at present. Their sensitivity will improve as the systematic uncertainties in the data are reduced. It is also important to investigate further observables, such as photon+jet observables and heavy quark suppression. Figure 1 . 1Jet R AA for two centralities. The only free parameter is fitted to the first bin of the left panel. Both the p T and centrality dependence are in agreement with the displayed data[3]. Figure 2 . 2Dijet imbalance for two centralities. The strongly coupled bands follow experimental points satisfactorily. Data taken from[5] (systematic error not shown) Figure 3 . 3Left:Parameter space scan < A J > vs R AA for central collisions. The chosen bin for R AA corresponds to the one used to fit the data Right: Ratio of partonic fragmentation functions for central collisions. The three data points shown correspond to the two crossings with unity and the location of the minimum of the hadronic fragmentation functions from CMS[10]. . T Sjostrand, S Mrenna, P Z Skands, arXiv:hep-ph/0603175JHEP. 060526T. Sjostrand, S. Mrenna, P. Z. Skands, JHEP 0605 (2006) 026. arXiv:hep-ph/0603175. . T Hirano, P Huovinen, Y Nara, arXiv:1012.3955Phys.Rev. 8411901T. Hirano, P. Huovinen, Y. Nara, Phys.Rev. C84 (2011) 011901. arXiv:1012.3955. Lai for the CMS collaboration. Y S , these proceedingsY. S. Lai for the CMS collaboration, these proceedings. P M Chesler, K , arXiv:1402.6756Jet quenching in strongly coupled plasma. P. M. Chesler, K. Rajagopal, Jet quenching in strongly coupled plasma arXiv:1402.6756. . S Chatrchyan, arXiv:1202.5022Phys.Lett. 712S. Chatrchyan, et al., Phys.Lett. B712 (2012) 176-197. arXiv:1202.5022. J Casalderrey-Solana, D C Gulhan, J G Milhano, D Pablos, K , arXiv:1405.3864A Hybrid Strong/Weak Coupling Approach to Jet Quenching. J. Casalderrey-Solana, D. C. Gulhan, J. G. Milhano, D. Pablos, K. Rajagopal, A Hybrid Strong/Weak Coupling Approach to Jet Quenching arXiv:1405.3864. . M Cacciari, G P Salam, G Soyez, arXiv:1111.6097Eur.Phys.J. C72. 1896M. Cacciari, G. P. Salam, G. Soyez, Eur.Phys.J. C72 (2012) 1896. arXiv:1111.6097. . G Aad, arXiv:1011.6182Phys.Rev.Lett. 105252303G. Aad, et al., Phys.Rev.Lett. 105 (2010) 252303. arXiv:1011.6182. Jet quenching in heavy-ion collisions at LHC with CMS detector. Y Yilmaz, PhD thesisY. Yilmaz, Jet quenching in heavy-ion collisions at LHC with CMS detector, PhD thesis. Yaxian for the CMS collaboration. M , these proceedingsM. Yaxian for the CMS collaboration, these proceedings.
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[ "LÉVY-TYPE OPERATORS WITH LOW SINGULARITY KERNELS: REGULARITY ESTIMATES AND MARTINGALE PROBLEM", "LÉVY-TYPE OPERATORS WITH LOW SINGULARITY KERNELS: REGULARITY ESTIMATES AND MARTINGALE PROBLEM" ]
[ "Eryan Hu ", "Guohuan Zhao " ]
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We consider the linear non-local operator L denoted byHere a(x, z) is bounded and J(z) is the jumping kernel of a Lévy process, which only has a low-order singularity near the origin and does not allow for standard scaling. The aim of this work is twofold. Firstly, we introduce generalized Orlicz-Besov spaces tailored to accommodate the analysis of elliptic equations associated with L, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with L. By utilizing analytic results, we prove the wellposedness of the martingale problem under mild conditions. Additionally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz-Besov spaces.
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[ "https://export.arxiv.org/pdf/2304.14056v1.pdf" ]
258,352,354
2304.14056
ba83b66276542e3f09ca4342dda3eed02fefbd20
LÉVY-TYPE OPERATORS WITH LOW SINGULARITY KERNELS: REGULARITY ESTIMATES AND MARTINGALE PROBLEM 27 Apr 2023 Eryan Hu Guohuan Zhao LÉVY-TYPE OPERATORS WITH LOW SINGULARITY KERNELS: REGULARITY ESTIMATES AND MARTINGALE PROBLEM 27 Apr 2023arXiv:2304.14056v1 [math.PR]Non-local operatorOrlicz spaceBesorv spacesMorrey inequalitySubordinate Brownian motionMartingale problemKrylov estimate AMS 2020 Mathematics Subject Classification: Primary 35B65; Secondary 60J76 35R09 We consider the linear non-local operator L denoted byHere a(x, z) is bounded and J(z) is the jumping kernel of a Lévy process, which only has a low-order singularity near the origin and does not allow for standard scaling. The aim of this work is twofold. Firstly, we introduce generalized Orlicz-Besov spaces tailored to accommodate the analysis of elliptic equations associated with L, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with L. By utilizing analytic results, we prove the wellposedness of the martingale problem under mild conditions. Additionally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz-Besov spaces. Introduction Consider an integrodifferential operator in R d of the form: Lu(x) =ˆR d (u(x + z) − u(x)) a(x, z) 1 B 1 (z) |z| d dz. (1.1) Here a : R d × R d → R is a bounded measurable function, and u ∈ C ∞ b . Under certain assumptions on a (such as smoothness), there exists a pure jump Markov process X generated by L, and a(x, z)1 B 1 (z)/|z| d is the intensity of jumps from x to x + z. An important question that arises is whether the solution u to the equation Lu = f exhibits additional smoothness under certain continuity conditions satisfied by a and f . This paper addresses this question, among other related topics. Additionally, we also aim to identify a strong Markov process X whose infinitesimal generator, when restricted to the class of smooth functions, coincides with L. This probability problem is intimately connected to the analytic one mentioned above, and we pursue it along the path of Stroock-Varadhan [SV07]. In this work, we present several analytic results for the equation Lu = f , including Schauder-type estimates. Additionally, to obtain Krylov-type estimates for the corresponding strong Markov jump process, we provide a priori estimates for solutions to the same equation in generalized Orlicz-Besov spaces. Recall that, in the case of an uniformly elliptic operator of second order Lu = i,j a ij ∂ ij u, the standard Schauder estimates and L p -estimates are as follows: Theorem 1.1. Let α ∈ (0, 1) and p ∈ (1, ∞). Assume a is uniformly elliptic. Then (a) there is a constant C such that for each u ∈ C 2,α , u C 2+α C ( Lu C α + u ∞ ) , provided that a ∈ C α ; (b) there is a constant C such that for each u ∈ W 2 p , u W 2 p C ( Lu p + u p ) , provided that a is uniformly continuous. This work establishes similar estimates for a broad class of integrodifferential operators that including L (see Theorem 1.2 and Lemma 4.4 below). Over the past few years, there has been a significant amount of research regarding the regularity of α stable-like operators (as described in (1.7)) with α ∈ (0, 2). Interested readers are referred to the brief discussion below. However, the present investigation focuses on the case where α = 0, which is particularly interesting because the standard scaling techniques used for stable-like operators are ineffective in this instance (see Appendix A for more details). Additionally, in [CW19], Chen and Weth demonstrated that d ds (−∆) s u(x) s=0 = log(−∆)u(x) = − c dˆR d (u(x + z) − u(x)) 1 B 1 (z) |z| d dz − c dˆ| z| 1 u(x + z) |z| d dz + ρ d u(x). This means the highest-order term of d ds (−∆) s s = 0 is −c d L defined in (1.1) with a = 1. So the study of L is crucial to the study of the asymptotic properties of the fractional Laplacian (−∆) s as s ∼ 0 + . From a probabilistic standpoint, the martingale problem for the α stable-like operator L defined in equation 1.7 has been extensively studied in some literature, but the endpoint case with α = 0 remains a challenging open problem. While the existence and uniqueness of martingale solutions associated with L appear as hypotheses in some works (such as [CZ14] and [KM17]), there is currently no paper that provides sufficient conditions for this assumption to hold. This work addresses this gap and presents a new Krylov-type estimate for the corresponding Markov process, under very mild conditions. These results are detailed in Theorem 1.4 and Corollary 1.5. 1.1. Assumptions, examples and main results. To state our main theorems, we must first introduce fundamental concepts from probability theory. Specifically, let S denote a subordinator lacking drift, which is a Lévy process featuring non-negative jumps. We assume that the Laplace exponent for S is given by φ(λ) = − log Ee −λS 1 . Let B be a d-dimensional Brownian motion independent of S. We define Z t = √ 2B St , which is a pure jump Lévy process known as the subordinate Brownian motion corresponding to S. Its jumping kernel J(z) is rotational symmetric, meaning that there is a function j : (0, ∞) → (0, ∞) such that J(z) = j(|z|). The Lévy exponent of Z is given by the formula ψ(ξ) := − log Ee iξ·Z 1 = φ(|ξ| 2 ). Given ψ, define ρ(r) = ρ ψ (r) := 1 ψ(r −1 ) . (1.2) In this work, we study the following spatial inhomogeneous non-local operator: Lu =ˆR d (u(x + z) − u(x)) a(x, z)J(z) dz. (1.3) Our assumption on L is Assumption 1. Let a : R d × R d → R be a measurable function. Let S be a pure jump subordinator with Laplace exponent φ, and let J(z) be the jumping kernel of the subordinate Brownian motion Z t = √ 2B St . (A 1 ) : φ is a slowly varying function at infinity (see Definition 2.7) and lim λ→∞ φ(λ) = ∞; (A 2 ) : For each λ > 0, the inverse Laplace transform of 1/(λ+φ) is a non-negative decreasing function; (A 3 ) : For any r ∈ (0, 1), there exists a constant C > 0 such that J(z) = j(|z|) C|z| −d−1 ψ ′ (|z| −1 ), (L j ) where ψ is the Lévy exponent of the subordinate Brownian motion corresponding to S; (A 4 ) : There are positive real numbers ρ 0 , c 0 ∈ (0, 1) such that a(x, z) c 0 , for every x ∈ R d , z ∈ B ρ 0 ; (U a ) (A 5 ) : There are real numbers α > 0, c 0 ∈ (0, 1) such that a(·, z) C α ρ c −1 0 , for every z ∈ R d , (H a ) where ρ is given by (1.2) and C α ρ is the generalized Hölder-type space defined in Section 3 (Definition 3.5). We now can formulate our first main result: Theorem 1.2 (Schauder-type estimate). Suppose that (A 1 )-(A 5 ) are satisfied, then there exists a constant λ 0 > 0 such that for each λ 2λ 0 and f ∈ C α ρ , the following Poisson equation λu − Lu = f,(PE) admits a unique solution in C 1+α ρ . Moreover, for any β ∈ [0, α], λ u C β ρ + u C 1+β ρ C f C β ρ , where C only depends on d, ψ, ρ 0 , c 0 , α and β. Corollary 1.3. Let c 0 ∈ (0, 1), α > 0 and ρ(r) = 1/ log(1 + r −1 ). Suppose that c 0 a c −1 0 and |a(x, z) − a(x ′ , z)| c −1 0 ρ α (|x − x ′ |), then for each β ∈ [0, α], it holds that u C 1+β ρ C Lu C β ρ + u ∞ . Here L is defined in (1.1), and C only depends on d, ρ 0 , c 0 , α and β. Example 1. Numerous significant examples satisfy the assumptions (A 1 )-(A 3 ). One notable example is the gamma subordinator, whose probability density function and Laplace exponent are given by P(S t ∈ du) = 1 Γ(t) u t−1 e −u du and φ(λ) = log(1 + λ), respectively. In this case the infinitesimal generator of Z = √ 2B St is Lu =ˆR d (u(x + z) − u(x)) dz |z| d ℓ(|z|) . where ℓ : (0, ∞) → (0, ∞) is a slowly varying function at 0 and ℓ(r) ∼ c d (r → 0 + ) (cf. [Mim14]). Another example is the infinitesimal generator of the solution to stochastic differential equation (SDE) driven by variance gamma process Z t = √ 2B St : dX t = σ(X t− ) dZ t , X 0 = x. (1.4) Here σ : R d → R d×d is invertible. In this case, the infinitesimal generator L σ of X is formulated by L σ u(x) =ˆR d (u(x + σ(x)z) − u(x)) dz |z| d ℓ(|z|) =ˆR d (u(x + z) − u(x)) |z| d ℓ(|z|) | det σ(x)| · |σ −1 (x)z| d ℓ(|σ −1 (x)z|) a(x,z) 1 |z| d ℓ(|z|) J(z) dz. Example 2. One of the main examples we have in mind is the operator L defined in (1.1). Rewrite L as Lu(x) =ˆR d (u(x + z) − u(x)) a(x, z)1 B 1 (z)ℓ(|z|) a ′ (x,z) 1 |z| d ℓ(|z|) J(z) dz, where J is the jump kernel of the variance gamma process given in Example 1. Obviously, a ′ satisfies (A 4 ) and (A 5 ) if a satisfies the same conditions. (a) for each x ∈ R d , the martingale problem (L, δ x ) has a unique solution P x , and that the family {X, P x , x ∈ R d } forms a Feller Markov process on R d ; (b) for any Young function A satisfying A(t) [ψ −1 (t 1+ε )] d (∀ε > 0), the following Krylov- type estimate is valid E xˆ∞ 0 e −λt f (X t ) dt C f A /λ. (1.5) Here f A is the Luxemburg norm of f (see Definition 2.3) with respect to A, and C does not depends on f . Corollary 1.5. Let c 0 ∈ (0, 1), α > 0. Let ρ(r) = 1/ log(1 + r −1 ). Suppose c 0 I d×d σ c −1 0 I d×d and |σ(x) − σ(x ′ )| c −1 0 ρ α (|x − x ′ |) . Then SDE (1.4) has a unique weak solution X. Moreover, for any λ > 0 and β > 1, X satisfies Eˆ∞ 0 e −λt f (X t ) dt C λ inf λ > 0 :ˆR d exp |f (x)/λ| β − 1 dx 1 . (1.6) Here the constant C does not depends on f . Remark 1.6. The inequalities (1.5) and (1.6) are commonly referred to as Krylov-type estimates. For Itô processes, the dominant term on the right-hand side of the inequality is C f L d /λ, which can be derived from the ABP maximal principle (see [Kry09]). On the other hand, for the Markov process associated with L, the "diffusion" rate of the process is exceedingly sluggish, causing the occupation time E 0´0 1 1 Br (X t ), dt to decay more slowly than r s , for any s > 0, as r → 0. Consequently, we require a stronger integrability condition for f in this case. 1.2. Some related works. The first class of non-local operators to be studied in depth is so called stable-like operators, which were introduced in [Bas88] are defined by L u(x) =ˆR d (u(x + z) − u(x) − ∇u(x) · z1 B 1 (z)) a(x, z) |z| d+α dz. (1.7) Regularity properties for linear and nonlinear equations corresponding to non-local operators have been extensively studied. Harnack inequalities for positive L -harmonic functions and Hölder continuity of L -harmonic functions were first established in [BL02] For the case where α = 0, Kassmann-Mimica [KM17] proved a growth lemma and a ρ-Hölder regularity estimate (where ρ is given by (1.2)) for L-harmonic functions under the assumption that the function a only satisfies some boundedness conditions (see also [Mim14] for the case a = 1). The Dirichlet problem for the logarithmic Laplacian was studied by Chen-Weth in [CW19]. Time-dependent equations with drifts were considered in [CZ14]. However, the following natural problem remains unresolved: whether u is ρ-continuous, if a satisfies some continuity conditions and u, Lu ∈ L ∞ . In the context of jump Markov processes, several studies have focused on the martingale problem for L , including [AK09, BT09, CZ16, MP14], among others. Related problems involving stochastic differential equations driven by cylindrical α-stable processes were also explored in [BC06] and [CZZ21]. However, to the best of our knowledge, the martingale problem for L has yet to be investigated by researchers. 1.3. Approaches and Difficulties. In Appendix A, we apply Littlewood-Paley theory and scaling techniques to reprove and extend the main result of [Bas09] for stable-like operators, as presented in [LZ22]. We emphasize that Schauder-type estimates in the Hölder-Zygmund space C s are more natural, where C s is defined by with ∆ j being the nonhomgeneous dyadic blocks (cf. [BCD11]). When s > 0 and s / ∈ N, C s coincides with the usual Hölder space. . When s > 0 and s / ∈ N, C s is the usual Hölder space. We extend this idea to the study of L, but immediately face difficulties. The dyadic decomposition is no longer applicable to the situation we are concerned with. For example, for the operator L = − log(1 − ∆), a naive substitute of C s is the function space X defined by X = u : sup j −1 (2 + j) s ∆ j f ∞ < ∞ . (1.8) However, as we demonstrate in Appendix A, X contains unbounded discontinuous functions even if s > 0. Thus, classical decomposition theory is no longer applicable to the problems we are concerned with. To overcome the aforementioned obstacle, we leverage the concept introduced in [KM17] and propose a novel decomposition for distributions, which we refer to as the "intrinsic dyadic decomposition" or " ψ-decomposition" (where ψ denotes any positive, increasing function, though we ultimately choose it as the Lévy exponent of Z for our purposes). This new decomposition replaces the role of ∆ j with the ψ-dyadic block Π ψ j , which is defined in (3.1). Additionally, the generalized Orlicz-Besov space B ψ,s A (as presented in Section 2.1) supplants the position of the Besov space B s p,∞ in classical theory. A key observation in proving Schauder-type estimates is Theorem 3.6, which serves as an analogy to Theorem A.2 and establishes that the generalized Hölder space C s ρ and B ψ,s ∞ are consistent. The second issue at hand pertains to the inadequacy of scaling methods in proving Schauder estimates for the very low-order operators (further details are provided in Appendix A). Consequently, we begin by focusing on a specific subset of non-local operators, namely the infinitesimal generators of subordinate Brownian motions with slowly varying symbols. Leveraging the favorable analytical properties of the subordinator S and the Gaussian kernel, we establish the desired a priori estimates in Orlicz-Besov spaces (as stated in Theorem 4.1). Subsequently, we gradually expand these estimates to encompass the general case using classical techniques developed for differential operators. Regarding the martingale problem associated with L, following [SV07], one can see that well-posedness is a consequence of the solvability of (PE) in generalized Hölder spaces. However, certain crucial properties of the Markov process corresponding to L cannot be inferred from estimates in Hölder-type spaces. For example, such estimates do not inform us whether the occupation time´T 0 1 B (X s )ds is nicely upper or lower bounded. To address this, we turn to L ∞ -estimates for (PE) under the assumption that f belongs to some Orlicz space. The L ∞ bounds for solutions to (PE) are derived via a Morrey-type inequality (3.23) and an a priori estimate for solutions in Orlicz-Besov spaces (as described in Lemma 4.4). We note that although L p -estimates may also be possible, they are inadequate for obtaining the probabilistic results we require since the following embedding result fails for any p < ∞: (1 + log(1 − ∆)) −1 L p ֒→ L ∞ . 1.4. Layout of this paper. The paper is organized into several sections. Section 2 provides a review of Orlicz spaces, slowly varying functions, and some important properties of subordinators with slowly varying Laplace exponents. In Section 3, we introduce the ψdecomposition, generalized Orlicz-Besov spaces, and present the proof of Theorem 3.6. This section also includes crucial Morrey-type inequalities. In Section 4, we present the proof of Theorem 1.2 and Corollary 1.3. Section 5 investigates the martingale problem associated with L. Finally, we include an appendix where we briefly discuss a theorem by Bass [Bas09]. We close this section by mentioning some conventions used throughout this paper: We use := as a way of definition. Reciprocal of f is denoted by 1 f and f −1 denote the inverse function of f . The letter c or C with or without subscripts stands for an unimportant constant, whose value may change in difference places. We use a ≍ b to denote that a and b are comparable up to a constant, and use a b (a b) to denote a Cb (a Cb) for some constant C. Preliminaries 2.1. Orlicz space. The notion of Orlicz space extends the usual notion of L p space. The function t p entering the definition of L p is replaced by a more general convex function A(t), which is called a Young function. A Young function A is said to satisfy the ∆ 2 -condition, or briefly A ∈ ∆ 2 if there exists a positive constant C such that A(2t) CA(t), for all t > 0. Typical examples of Young functions include A(t) = t p for 1 < p < ∞ and A(t) = e t β − 1 for β 1. The convex conjugate A * of a Young function A is defined by its Legendre transformation, namely A * (s) = sup t 0 [st − A(t)]. (2.1) By the definition, one can see that A * : [0, ∞) → [0, ∞) is also a Young function (cf. [HH19]). By A −1 we denote the inverse of A: A −1 (s) = inf {t : A(t) > s} , s 0. The following result can be founded in [HH19]. Proposition 2.2. Assume that A : [0, ∞) → [0, ∞) is a Young function, then s A −1 * (s) · A −1 (s) 2s. (2.2) The proof of the above conclusion in [HH19] is quite complicated. Here we give a simpler one. Proof. By (2.1) and the continuity of A * , we have s A * (A −1 * (s)) A −1 * (s)t − A(t), for each s, t 0. Letting t = A −1 (s), we obtain s A −1 * (s) · A −1 (s) − s, which implies the upper bound. It remains to prove the lower bound for s > 0. Given s > 0, set u = A −1 (s) and v = A −1 * (s). We need to prove uv A(u) = A * (v) = s. By D − f we denote the left derivative of a convex f . (i) If D − A(u) v, by the convexity of A, we have uv uD − A(u) A(u) − A(0) = s; (ii) Assume D − A(u) > v. We claim that u D − A * (D − A(u)). (2.3) If (2.3) holds, then by the monotonicity of D − A * (since A * is convex), we get u D − A * (D − A(u)) D − A * (v). Using the convexity of A * , we arrive uv vD − A * (v) A * (v) − A * (0) = s. Thus, it remains to prove (2.3). Let k = D − A(u). Since A * is the Legendre transform of A, one can see that A * (k) = ku − A(u). (2.4) Noting that the Legendre transform is an involution, i.e., A = (A * ) * , we have k ′ u − A * (k ′ ) A(u), ∀k ′ ∈ (0, ∞). (2.5) Combining (2.4) and (2.5), we get A * (k) − A * (k ′ ) (k − k ′ )u. (2.6) This yields D − A * (D − A(u)) = D − A * (k) = lim k ′ ↑k A * (k) − A * (k ′ ) k − k ′ u. Given a Young function A, for each measurable function on R d , we define I A (f ) :=ˆR d A(|f |(x)) dx. t s A(t) u A * (k) A * (k ′ ) (k − k ′ ) − (A * (k) − A * (k ′ ))f A := inf {a > 0 : I A (f /a) 1} ∈ [0, ∞]. The Orlicz spaces L A is the collection of all functions with finite Luxemburg norm. Orlicz spaces have good duality and interpolation properties, making them useful in many areas of mathematics, including probability theory, functional analysis, and PDEs. The following two propositions can also be found in [HH19]. Proposition 2.4. For all f ∈ L A and g ∈ L A * , it holds that |f g| 2 f A g A * . (2.7) Proposition 2.5. f A ≍ sup f g 1 :ˆR d A * (g(x)) dx 1 . The following Young's inequality is crucial. Proposition 2.6 ([O'N65, Theorem 2.5]). Assume that A, B and C are Young functions and that A −1 (t)B −1 (t) tC −1 (t), then f * g C 2 f A g B . (2.8) 2.2. Regular variation. Definition 2.7. A measurable function f : (0, ∞) → (0, ∞) is regularly varying at infinity with index α (denoted by f ∈ R α (∞)), if for all λ > 0, lim x→∞ f (λx) f (x) = λ α . If α = 0, then we say that f is slowly varying. Regular (slow) variation at zero can be defined similarly, by changing x → ∞ to x → 0 + . Let f be a slowly varying function, theń R 1 f (t)t −1 dt f (R) → ∞, as R → ∞ (2.9) (see [BGT87, Proposition 1.5.9a]). The following result (cf. [BGT87, Theorem 1.5.11]) will be used frequently. Lemma 2.8 (Karamata's theorem). Let f be a slowly varying function. Then (i) for any σ −1, lim λ→∞ λ σ+1 f (λ) λ M t σ f (t) dt = σ + 1; (ii) for any σ < −1, lim λ→∞ λ σ+1 f (λ) ∞ λ t σ f (t) dt = −(σ + 1). Subordinators. Recall that S is a subordinator without drift. φ, the Laplace exponent of S can be written in the form φ(λ) =ˆ∞ 0 (1 − e −λt ) Π(dt). (2.10) Here Π is a measure on (0, ∞), and´∞ 0 (1 ∧ t) Π(dt) < ∞. Since (λt)e −λt 1 − e −λt for each λ, t > 0, by (2.10), we get λφ ′ (λ) φ(λ). (2.11) Recall that B is a d-dimensional Brownian motion independent of S, and that Z t = √ 2B St , and that ψ(ξ) = − log Ee iξ·Z 1 = φ(|ξ| 2 ) is the Lévy exponent of the subordinate Brownian motion Z. J(z), the jumping kernel of Z, has the form J(z) = j(|z|), where j(r) =ˆ∞ 0 1 (4πt) d 2 e − r 2 4t Π(dt). (2.12) For any λ 0, let U λ (du) =ˆ∞ 0 e −λt P(S t ∈ du) dt be the λ-potential measure of S. Let T (f ) denote the Laplace transform of a function f : (0, ∞) → (0, ∞). Suppose that U λ , the λ-potential measure of S, has a density µ λ , then T (µ λ )(τ ) =ˆ∞ 0 e −τ u U λ (du) =ˆ∞ 0 e −τ uˆ∞ 0 e −λt P(S t ∈ du) dt =ˆ∞ 0 e −λt dtˆ∞ 0 e −τ u P(S t ∈ du) =ˆ∞ 0 e −(λ+φ(τ ))t dt = 1 λ + φ(τ ) . Here T (f ) is the Laplace transformation of f . Equivalently, µ λ (u) = T −1 1 λ + φ (u). (2.13) The following two propositions will be used in the proof for our main results. Proposition 2.9. Suppose that j satisfies (A 3 ), then for any r ∈ (0, 1), it holds that r 0 |z|J(z) dz rψ(r −1 ) (2.14) andˆ∞ r J(z) dz ψ(r −1 ). (2.15) Proof. For (2.14), using (L j ) and (2.11), one sees that r 0 |z|J(z) dz ≍ˆr 0 j(s)s d ds (L j ) ˆr 0 s −1 ψ ′ (s −1 ) ds (2.11) ˆr 0 ψ(s −1 ) ds rψ(r −1 ). For (2.15), using (L j ), we get ∞ r J(z) dz ≍ˆ∞ r j(s)s d−1 ds (L j ) ˆ∞ r s −2 ψ ′ (s −1 ) ds =ˆr −1 0 ψ ′ (u) du = ψ(r −1 ). The following result is proved in [Mim14, Theorem 4.1]. Proposition 2.10. Suppose the Lévy measure Π is infinite and has a decreasing density. Assume that φ ∈ R 0 (∞) and φ ′ ∈ R −1 (∞), then j(r) ∼ r −d−1 ψ ′ (r −1 ) (r → 0+). In particular, j satisfies (A 3 ). Intrinsic dyadic decomposition, function spaces and embedding theorems The contents of this section are purely analytical. Generalized Orlicz-Besov spaces, which is defined by a refined Littlewood-Palay decomposition called ψ-decomposition are introduced. Such function space is a natural extension of Besov space B s p,∞ (see [BCD11] for its definition). Crucial Morrey-type inequalities are also proved in this section. 3.1. Intrinsic dyadic decomposition. Let ψ : (0, ∞) → (0, ∞) be a strictly increasing function such that lim R→∞ ψ(R) = ∞. We point out that in this section, ψ need not be the Lévy exponent of a subordinate Brownian motion. Set ψ −1 (r) := inf {λ > 0 : ψ(λ) > r} (r > 0). Let χ be a rotational symmetric, nonnegative and smooth function with compact support such that χ(ξ) = χ(|ξ|) 1 when |ξ| 3/4, 0 when |ξ| 1. Let ̺ := F −1 (χ). We definē χ(ξ) = χ(2ξ) and χ(ξ) = χ(ξ/2);̺ = F −1 (χ) and ̺ = F −1 ( χ). Given j −1, set χ j (ξ) := χ ξ ψ −1 (2 j ) , ̺ j (x) := F −1 (χ j )(x) = ψ −1 (2 j ) d ̺ ψ −1 (2 j )x . Similarly, we definē χ j =χ ξ ψ −1 (2 j ) = χ 2ξ ψ −1 (2 j ) ,̺ j (x) = 1 2 ψ −1 (2 j ) d ̺ 1 2 ψ −1 (2 j )x and χ j = χ ξ ψ −1 (2 j ) = χ ξ 2ψ −1 (2 j ) , ̺ j (x) = 2ψ −1 (2 j ) d ̺ 2ψ −1 (2 j )x . The ψ-dyadic block Π ψ j is formulated by Π ψ j f := F −1 ((χ j+1 − χ j )F(f )), j 0; Π ψ −1 f := F −1 χ ξ ψ −1 (1) F(f )(ξ) . (3.1) Similarly, for each j 0, we define Π ψ j f := F −1 (( χ j+1 −χ j )F(f )). Obviously, for each j 0, it holds that Π ψ j Π ψ j = Π ψ j Π ψ j = Π ψ j . The following simple result will be used several times later. Lemma 3.1. Let A be a Young function or A = ∞. It holds that ∇Π ψ j f A Cψ −1 (2 j+1 ) Π ψ j f A , j −1. (3.2) Proof. Noting that Π ψ j f = ̺ j+1 * (Π ψ j f ), by Young's inequality, we get ∇Π ψ j f A ∇ ̺ j+1 * (Π ψ j f ) A ∇ ̺ j+1 1 Π ψ j f A Cψ −1 (2 j+1 ) Π ψ j f A . 3.2. Generalized Orlicz-Besov and Orlicz-Bessel potential spaces. Definition 3.2. Let s ∈ R and A be a Young function. The · ψ,s;A -norm of a distribution f is given by f ψ,s;A := sup j −1 2 js Π ψ j f A , and the collection of all distributions with finite · ψ,s;A -norm is denoted by B ψ,s A . If A = ∞, we define the space B ψ,s ∞ in the same way, and also denote it by C s ψ . For simplicity, we denote B ψ,1 A (C 1 ψ ) as B ψ A (C ψ ). By the definition of B ψ,s A , it is easy to get the following interpolation result: Lemma 3.3. Suppose s, s 0 , s 1 ∈ R, θ ∈ (0, 1) and s = θs 0 + (1 − θ)s 1 . Then f B ψ,s A f θ B ψ,s 0 A f 1−θ B ψ,s 1 A . We also introduce the definition of rlicz-Bessel potential spaces. Definition 3.4. Let A be a Young function. The space H ψ A := f ∈ S ′ (R d ) : F −1 [(1 + ψ(·))F (f )] ∈ L A with the norm f H ψ A := (1 + φ(−∆))f A is called generalized Orlicz-Bessel potential space. Definition 3.5. Let s > 0. Let ω : (0, ∞) → (0, ∞) be a strictly increasing function satisfying lim r↓0 ω(r) = 0. Assume ω s (t)/t → ∞ as t → 0. The · C s ω -norm of a measurable function f is given by f C s ω := f ∞ + sup x =y |f (x) − f (y)| ω s (|x − y|) . The collection of all function with finite · C s ω -norm is denoted by C s ω . Recall that ρ(r) = ρ ψ (r) := 1 ψ(r −1 ) . The following result gives a characterization of generalized Hölder spaces C s ρ in terms of ψ-decomposition. Theorem 3.6. Let s > 0. Let ρ be the function defined by (1.2). Assume r → r ψ s (r) is a increasing function, andˆR 1 dr ψ s (r) CR ψ s (R) for all R 1. (3.3) Then there is a constant c ∈ (0, 1) does not depend on f such that c f C s ψ f C s ρ c −1 f C s ψ . (3.4) Proof. We only prove the case that s = 1 (the proof for the general case is similar). Assume that f C ψ < ∞. For any |x − y| ≪ 1, |f (x) − f (y)| j −1 |Π ψ j f (x) − Π ψ j f (y)| j −1 Π ψ j f ∞ ∧ |x − y| ∇Π ψ j f ∞ . Thanks to (3.2), for each j −1, ∇Π ψ j f ∞ ψ −1 (2 j+1 ) Π ψ j f ∞ . So for each K ∈ N, it holds that |f (x) − f (y)| f C ψ j −1 2 −j ∧ |x − y|ψ −1 (2 j+1 )2 −j f C ψ ∞ K+1 2 −j + f C ψ |x − y| K j=−1 ψ −1 (2 j+1 )2 −j f C ψ 2 −K + |x − y|ˆK +1 0 ψ −1 (2 t )2 −t dt . (3.5) For R 1, by integration by parts and (3.3), we havê R 0 ψ −1 (2 t )2 −t dt ˆ2R 1 ψ −1 (r)r −2 dr = −ˆψ −1 (2 R ) ψ −1 (1) u d 1 ψ(u) u ψ(u) ψ −1 (1) ψ −1 (R) +ˆψ −1 (2 R ) ψ −1 (1) du ψ(u) ψ −1 (2 R )2 −R . Choosing R = K + 1 = [log 2 ψ(|x − y| −1 )], one sees that 2 −K + |x − y|ˆK +1 0 ψ −1 (2 t )2 −t dt 2 −K 1 ψ(|x − y| −1 ) . (3.6) Plugging (3.6) into (3.5), we see that for any |x − y| ≪ 1, |f (x) − f (y)| ρ(|x − y|) = |f (x) − f (y)|ψ(|x − y| −1 ) f C ψ . This together with the fact that f ∞ f C ψ imply f C ρ = f ∞ + sup x =y |f (x) − f (y)| ρ(|x − y|) f C ψ . On the other hand, suppose that M = sup x =y |f (x) − f (y)| ψ −1 (|x − y| −1 ) + f ∞ < ∞. By the definition of Π ψ j and the fact that´̺ j = χ j (0) = 1 (j 0) , we have |Π ψ j f (x)| = ˆR d (̺ j+1 − ̺ j )(z)f (x + z) dz = ˆR d (̺ j+1 − ̺ j )(z)(f (x + z) − f (x)) dz ˆR d |̺(z)| f x + z ψ −1 (2 j ) − f (x) + f x + z ψ −1 (2 j+1 ) − f (x) dz MˆR d |̺(z)|   1 ∧ 1 ψ ψ −1 (2 j ) |z|   dz + MˆR d |̺(z)|   1 ∧ 1 ψ ψ −1 (2 j+1 ) |z|   dz Mˆ∞ 0 ̺(r)r d−1 1 ∨ ψ ψ −1 (2 j ) r dr + Mˆ∞ 0 ̺(r)r d−1 1 ∨ ψ ψ −1 (2 j+1 ) r dr (j 0). (3.7) So we need to dominate the integration´∞ 0 ̺(r)r d−1 1∨ψ( λ r ) dr. Using the fact that |ρ(t)| 1 ∧ t −k (∀k > 0), one sees that for each λ > 0, ∞ 0 ̺(r)r d−1 1 ∨ ψ(λ/r) dr =λ dˆ∞ 0 ̺(λ/t) t d+1 [1 ∨ ψ(t)] dt λ d−kˆλ Choosing k = d + 2 and using (3.3), we get I 1 = λ −2ˆλ 0 t 1 ∨ ψ(t) dt λ −1ˆλ 0 dt 1 ∨ ψ(t) (3.3) 1 1 ∨ ψ(λ) . Moreover, I 2 = λ dˆ∞ λ 1 t d+1 [1 ∨ ψ(t)] dt 1 1 ∨ ψ(λ) λ dˆ∞ λ t −d−1 dt 1 1 ∨ ψ(λ) . Thus, for each λ > 0,ˆ∞ 0 ̺(r)r d−1 1 ∨ ψ(λ/r) dr 1 1 ∨ ψ(λ) . (3.8) Combining (3.7) and (3.8), we arrive Π ψ j f ∞ M 2 −j , for all j −1. So we complete our proof. 3.3. Morrey-type inequalities. In this section, we prove two Morrey-type inequalities, (3.12) and (3.23). The novelty of them is that both regularity and integrability of u are characterized by monotone functions ψ and A, respectively. Moreover, ψ can be a slowly varying function. From a perspective of theoretical completeness, exploring Sobolev-type inequalities for generalized Orlicz-Bessel potential spaces can provide a more comprehensive understanding of these function spaces. However, while this inequality is of interest, it is not directly utilized in the proof of our main results. Therefore, we have chosen to defer the investigation of this topic to a future paper. We need to make some necessary preparations. Let S be a subordinator with Laplace exponent φ. Recall Z is the subordinate Brownian motion corresponding to S. Here and below, the function ψ is given by the Lévy exponent of Z, i.e., ψ(ξ) = − log Ee iξ·Z 1 = φ(|ξ| 2 ). In this section, we assume S satisfies (A ′ 1 ) : φ ∈ R α 2 (∞) with α ∈ [0, 1); (A ′ 2 ) : The inverse Laplace transform of 1/(1 + φ) is a non-negative decreasing function. Lemma 3.7. (a) Assume that φ satisfies (A ′ 1 ), then R 1 dr φ(r 2 ) R φ(R 2 ) , for all R 1; (3.9) (b) Assume that S satisfies (A ′ 2 ), then µ 1 (u) = T −1 1 1 + φ (u) d du 1 1 + φ(u −1 ) , for all u > 0. (3.10) Proof. (a). If φ ∈ R α 2 (∞) with α ∈ [0, 1), then there exists a function ℓ ∈ R 0 (∞) such that φ(r) = r α 2 ℓ(r). For R 1, using Lemma 2.8, we havê R 1 dr φ(r 2 ) ˆR 1 dr r α ℓ(r 2 ) R 1−α ℓ(R 2 ) R φ(R 2 ) ; (b). To prove (3.10), taking λ, ε > 0, we have ε −1 1 1 + φ(λ) − 1 1 + φ(λ + ε) =ε −1 (T (µ 1 )(λ) − T (µ 1 )(λ + ε)) =ε −1ˆ∞ 0 (e −λu − e −(λ+ε)u )µ 1 (u) du λ −1ˆ1 0 1 − e −ετ /λ ε e −τ µ 1 (τ /λ) dτ. Noting that 1−e −ετ /λ ε τ /λ, using Fatou's lemma and letting ε → 0, we obtain λ −2ˆ1 0 τ e −τ µ 1 (τ /λ) dτ − d dλ 1 1 + φ(λ) . Since u → µ 1 (u) is decreasing, the above estimate implies (1 − 2/e)λ −2 µ 1 (1/λ) − d dλ 1 1 + φ(λ) . Letting λ −1 = u, we get µ 1 (u) −u −2 d dλ 1 1 + φ(λ) λ=u −1 = d du 1 1 + φ(u −1 ) . 3.3.1. Morrey-type inequality for Orlicz-Bessel potential spaces. The following Morrey-type inequality for Orlicz-Sobolev space W 1 A can be found in [AF03]. Theorem 3.8. Let A be a Young function. Assume that´∞ 1 A −1 (s d ) s 2 dt < ∞, then f Cρ f A + ∇f A , where W 1 A is the Orlicz-Sobolev space and ρ(r) = ˆ∞ r −1 A −1 (s d ) s 2 ds −1 . Remark 3.9. As far as the authors are aware, there is no result guaranteeing that the Riesz transform is bounded in the Orlicz space L A , so in general H 1 A does not coincide with W 1 A . Therefore, H 1 A ֒→ C ρ cannot be derived from theorem 3.8. The following result is about Morrey inequalities for generalized Orlicz-Bessel potential spaces. Theorem 3.10. Assume that S satisfies (A ′ 1 ) and (A ′ 2 ). Let A be a Young function satisfyinĝ ∞ 1 A −1 (t d )ψ ′ (t) ψ 2 (t) dt < ∞. Let Ψ(R) := ˆ∞ R A −1 (t d )ψ ′ (t) ψ 2 (t) dt −1 . (3.11) Then it holds that u C Ψ u H ψ A . (3.12) Our result implies the classical Morrey's inequality for Bessel potential space: Corollary 3.11. (a) Let α ∈ (0, 1). Assume ψ(R) = φ(R 2 ) = R α and A(t) = t p with p > d/α. Then u C α− d p u H ψ A = u H α p . (b) Let ψ(R) = log(1 + R 2 ) and A(t) = e t β − 1 with β > 1. It holds that u ∞ + sup |x−y|< 1 2 |u(x) − u(y)| · (− log |x − y|) 1− 1 β u H ψ A = u + log(1 − ∆)u A . Proof. (a) Let φ(r) = r α 2 . Since ρ Ψ (r) = 1 Ψ(r −1 ) =ˆ∞ r −1 A −1 (t d )ψ ′ (t) ψ 2 (t) dt ≍ˆ∞ r −1 t d p −α−1 ds ≍ r α− d p , the conclusion follows by Theorem 3.10 and Theorem 3.6. (b) By definition, A −1 (t) = log 1 β (1 + t). For any r ≪ 1, ρ Ψ (r) = 1 Ψ(r −1 ) =ˆ∞ r −1 A −1 (t d )ψ ′ (t) ψ 2 (t) dt =ˆ∞ r −1 2s log 1 β (1 + s d ) (1 + s 2 ) log 2 (1 + s 2 ) ds ≍ˆ∞ r −d t −1 log −2+ 1 β t dt ≍ (log r −1 ) 1 β −1 . Again by Theorem 3.10 and Theorem 3.6, we get the desired result. Now we give the proof for Theorem 3.10 below. Proof of Theorem 3.10. Recall that S is a subordinator with Laplace exponent φ, and that Z is the subordinate Brownian motion corresponding to S. Set P t f (x) = Ef (x + Z t ) and G λ f (x) =ˆ∞ 0 e −λt P t f (x) dt. By a standard approximation procedure, one can see that for each u ∈ H ψ A , it holds that G λ (λ + φ(−∆))u = u. Let f = u + φ(−∆)u. Then u = G 1 f =ˆ∞ 0 e −t P t f dt. We need to prove Π Ψ j u ∞ = Π Ψ jˆ∞ 0 e −t P t f dt ∞ C2 −j f A . (3.13) Denote h t and p t by the transition probability density functions of Brownian motion √ 2B t and Z t respectively. Let s t (du) be the distribution of S t . Let ζ j (x) = Ψ d (2 j )̺(Ψ(2 j )x), where ̺ is the smooth function in Section 2.1. Thanks to Hölder's inequality (2.7), we have Π Ψ j u ∞ Π Ψ jˆ∞ 0 e −t P t f dt ∞ ˆ∞ 0 e −t p t * (ζ j+1 − ζ j ) * f dt ∞ f Aˆ∞ 0 e −t p t * (ζ j+1 − ζ j ) A * dt. (3.14) Thus, to prove (3.13), we need to estimate p t * (ζ j+1 − ζ j ) A * . Let R ′ > R > 0 and ̺ R (x) = R d ̺(Rx). Using the fact that p t (x) =ˆ∞ 0 h u (x)s t (du), we get p t * (̺ R ′ − ̺ R )(x) =ˆR d p t (x + z) R ′d ̺(R ′ z) dz −ˆR d p t (x + z) R d ̺(Rz) dz =ˆR d p t x + z R ′ − p t x + z R ̺(z) dz =ˆ∞ 0ˆR d h u x + z R ′ − h u x + z R ̺(z) s t (du) dz =ˆ∞ 0ˆR d u − d 2 h 1 x √ u + z √ uR ′ − h 1 x √ u + z √ uR s t (du)̺(z) dz. (3.15) Here we use the fact that h u (x) = u − d 2 h 1 (x/ √ u). By mean value theorem, one sees that h 1 x √ u + z √ uR ′ − h 1 x √ u + z √ uR h 1 x √ u + z √ uR ′ + h 1 x √ u + z √ uR ∧ z √ uR ′ − z √ uR ˆ1 0 ∇h 1 x √ u + (1 − θ) z √ uR ′ + θ z √ uR dθ. This implies u − d 2 h 1 · √ u + z √ uR ′ − h 1 · √ u + z √ uR A * u − d 2 h 1 · √ u A * ∧ z √ uR ′ − z √ uR u − d 2 ∇h 1 · √ u A * 1 ∧ z √ uR h u A * + u − d 2 ∇h 1 · √ u A * . (3.16) Setting f A * (u) = h u A * + u − d 2 ∇h 1 · √ u A * ,(3.17) and combining (3.15) and (3.16), we get p t * (̺ R ′ − ̺ R ) A * ˆ∞ 0ˆR d 1 ∧ z √ uR ̺(z) f A * (u) dz s t (du) ˆ∞ 0 1 √ uRˆ√ uR 0 |̺(r)|r d dr +ˆ∞ √ uR |̺(r)|r d−1 dr f A * (u) s t (du). (3.18) Noting that |̺(r)| 1 ∧ r −k (∀k > 0), we have 1 NˆN 0 |̺(r)|r d dr 1 ∧ 1 N andˆ∞ N |̺(r)|r d−1 dr 1 ∧ 1 N , ∀N > 0. (3.19) Plugging estimate (3.19) into (3.18), we get P t (̺ R ′ − ̺ R ) A * ˆ∞ 0 1 ∧ 1 √ uR f A * (u) s t (du). (3.20) So the main problem comes down to the estimation of f A * (u). Let us estimate the first term on the right-hand side of (3.17). By the scaling property of h u , one sees that I A * (h u /λ) 1 ⇐⇒ˆR d A * h 1 (x/ √ u) ( √ u) d λ dx 1 ⇐⇒ˆR d A * h 1 (x) ( √ u) d λ dx u − d 2 . (3.21) Let K = (4πu) −d/2 . Basic calculation yieldŝ R d A * h 1 (x) ( √ u) d λ dx =ˆR d A *   e − |x| 2 4 ( √ 4πu) d λ   dx =c dˆ∞ 0 A * e −r 2 /4 ( √ 4πu) d λ r d−1 dr =c dˆK λ 0 A * (r) log K λ − log r d 2 −1 r −1 dr =c dˆK λ 0 log K λ − log r d 2 −1 r −1 drˆr 0 dA * (s) =c dˆK λ 0 dA * (s)ˆK λ s log K λ − log r d 2 −1 r −1 dr =c dˆK λ 0 log K λ − log s d 2 D − A * (s) ds. Noting that D − A * is increasing and D − A * (x)(y − x) A * (y) − A * (x), we get R d A * h 1 (x) ( √ u) d λ dx c dˆ∞ 0 u d 2 e −u du · D − A * K λ K λ c d A * 2K λ − A * K λ c d A * 2K λ . Using (3.21), we have I A * (h u /λ) 1 ⇐= A * 2K λ CK. This yields h u A * = inf {λ : I A * (h u /λ) 1} 2K A −1 * (CK) u −d/2 A −1 * (cu −d/2 ) . Similarly, we also have u − d 2 ∇h 1 · √ u A * u −d/2 A −1 * (cu −d/2 ) . Thus, there exists a constant c > 0 such that f A * (u) u −d/2 A −1 * (cu −d/2 ) . This together with (3.20) and (2.2) yield ∞ 0 e −t p t * (̺ R ′ − ̺ R ) A * dt ˆ∞ 0 e −tˆ∞ 0 1 ∧ 1 √ uR u d/2 A −1 * (cu −d/2 ) s t (du) dt ˆR−2 0 A −1 (cu −d/2 )µ 1 (u) du + 1 Rˆ∞ R −2 1 √ u A −1 (cu −d/2 )µ 1 (u) du = : I 1 + I 2 . where µ 1 (u) is the density of 1-potential measure U 1 of S t . Now using (3.10), we have I 1 ˆR−2 0 A −1 (cu −d/2 ) d 1 1 + φ(u −1 ) = − d 2ˆ∞ R d A −1 (ct) d 1 1 + φ(t 2/d ) =: I ′ 1 . For I 2 , again using (3.10), we have I 2 = − d 2RˆR d 0 t 1/d A −1 (ct) d 1 1 + φ(t 2/d ) = − d 2RˆR d 0 A −1 (ct) d t 1/d 1 + φ(t 2/d ) + 1 2RˆR d 0 A −1 (ct)t 1/d−1 1 + φ(t 2/d ) dt = : I ′ 2 + I ′′ 2 . Since A −1 is increasing, by definition, one can see |I ′ 2 | A −1 (cR d ) 1 + φ(R 2 ) I ′ 1 . For I ′′ 2 , using (3.9), we get I ′′ 2 A −1 (cR d ) RˆR 0 dr 1 + φ(r 2 ) A −1 (cR d ) 1 + φ(R 2 ) I ′ 1 . Therefore, for any R ≫ 1, ∞ 0 e −t P t (̺ R ′ − ̺ R ) A * dt I ′ 1 −ˆ∞ R d A −1 (ct) d 1 1 + φ(t 2/d ) = −ˆ∞ R A −1 (cs d ) d 1 1 + ψ(s) ˆ∞ R A −1 (s d )ψ ′ (s) ψ 2 (s) ds (3.11) = 1 Ψ(R) . (3.22) Letting R ′ = 2Ψ −1 (2 j+1 ) and R = 1 2 Ψ −1 (2 j ), and plugging (3.22) to (3.14), we obtain Π Ψ j u ∞ f A Ψ 1 2 Ψ −1 (2 j ) 2 −j f A . So we obtain (3.13) and complete our proof. 3.3.2. Morrey-type inequality for generalized Orlicz-Besov spaces. We also prove a Morrey inequality for generalized Orlicz-Besov spaces, which will be used in the proofs for our main results. Theorem 3.12. Assume S satisfies (A ′ 1 ) and (A ′ 2 ). If A is a Young function satisfies A(t) [ψ −1 (ct 1+ε )] d for some ε > 0, then u C ε 1+ε ψ u B ψ A . (3.23) Proof. Following the proof of (3.14), one can verify that Π ψ j u ∞ ˆ∞ 0 e −t P t Π ψ j (Π ψ j f ) dt ∞ ˆ∞ 0 e −t p t * ( ̺ j+1 −̺ j ) * (Π ψ j f ) dt ∞ Π ψ j f Aˆ∞ 0 e −t p t * ( ̺ j+1 −̺ j ) A * dt. By our assumption A(t) [ψ −1 (ct 1+ε )] d , we have A −1 (s) Cψ 1 1+ε (s 1 d ). Using this and following the proof for (3.22), we get ∞ 0 e −t p t * ( ̺ j+1 −̺ j ) A * dt ˆ∞ 1 2 ψ −1 (2 j ) A −1 (s d )ψ ′ (s) ψ 2 (s) ds ˆ∞ 1 2 ψ −1 (2 j ) ψ 1 1+ε (s)ψ ′ (s) ψ 2 (s) ds 1 ψ ε 1+ε ( 1 2 ψ −1 (2 j )) 2 − εj 1+ε . Thus, 2 εj 1+ε Π ψ j u ∞ Π ψ j f A u B ψ,1 A . So we complete the proof for (3.23). Studies of Poisson equations In this section, we study the Poisson equation (PE) in the generalized Orlicz-Besov space. Our analysis is divided into two parts: In the first part, we explore the case where the coefficient a is independent of x. Subsequently, we extended our findings to the more general case by using the method of frozen coefficients. 4.1. Spatially homogeneous case. Theorem 4.1. Let A be a Young function or A = ∞. Assume (A 1 )-(A 2 ) hold. Suppose that a : R d → [0, ∞) is measurable, R d (1 ∧ |z| 2 ) a(z)J(z) dz < ∞, (4.1) and there are positive numbers ρ 0 , c 0 ∈ (0, 1) such that a(z) c 0 , for all z ∈ B ρ 0 . (4.2) Then there exists a constant λ 0 > 0 such that for any λ λ 0 and any α ∈ R, λ u B ψ,α A + u B ψ,1+α A C λu − Lu B ψ,α A , (4.3) provided that u ∈ B ψ,1+α A . Here C is a constant only depends on d, ψ, ρ 0 , c 0 and α. Remark 4.2. We would like to emphasize that Theorem 4.1 does not require the assumption that α 0. Proof. Without loss of generality, we can assume that c 0 = 1 to simplify the analysis. By utilizing the condition in (4.1), we can identify a Lévy process Z whose infinitesimal generator corresponds to the operator L. Specifically, we define P t f (x) = Ef (x + Z t ) and G λ f (x) = ∞ 0 e −λt P t f (x) dt. To establish the validity of (4.3), it is enough to demonstrate its validity for u ∈ S (R d ). This is because the general case can be obtained using standard approximation techniques. For each u ∈ S (R d ), it holds that G λ (λ − L)u = u. Using this and letting f = λu − Lu, we obtain Π ψ j u A = Π ψ j G λ f A = Π ψ jˆ∞ 0 e −λt (P t f )dt A ˆ∞ 0 e −λt P t Π ψ j f A dt λ −1 Π ψ j f A . This yields λ u B ψ,α A f B ψ,α A = λu − Lu B ψ,α A . (4.4) Thus, it remains to show that the second term on the left-hand side of (4.3) can be dominated by C λu − Lu B ψ,α A . We divide the proof into several steps. Step 1. We first consider the case that a ≡ 1. In this case, L is the infinitesimal generator of a subordinate Brownian motion Z t = √ 2B St . It is enough to show Π ψ jˆ∞ 0 e −λt (P t f )dt A C2 −j Π ψ j f A , j 0. (4.5) Using the fact that Π ψ j Π ψ j = Π ψ j (j 0), we see that Π ψ j (P t f ) = P t (Π ψ j f ) = P t ( Π ψ j Π ψ j f ). Thus, Π ψ j (P t f ) A P t ( ̺ j+1 −̺ j ) 1 Π ψ j f A , (4.6) due to Young's inequality (2.8). So our problem boils down to estimating P t ( ̺ j+1 −̺ j ) 1 (j 0). For any R ′ > R ≫ 1, set ̺ R (x) = R d ̺(Rx), ̺ R ′ (x) = (R ′ ) d ̺(R ′ x) Employing the procedure deducing (3.18), for any t > 0, we have P t (̺ R ′ − ̺ R ) 1 ˆ∞ 0ˆR d 1 ∧ |z| √ uR |̺(z)|dz s t (du) ˆ∞ 0 1 √ uRˆ√ uR 0 |̺(r)|r d dr +ˆ∞ √ uR |̺(r)|r d−1 dr s t (du). (4.7) Noting that |̺(r)| 1 ∧ r −k (∀k > 0), one sees that 1 NˆN 0 |̺(r)|r d dr 1 ∧ 1 N andˆ∞ N |̺(r)|r d−1 dr 1 ∧ 1 N , ∀N > 0. (4.8) Combining (4.7) and (4.8), we obtain P t (̺ R ′ − ̺ R ) 1 ˆ∞ 0 1 ∧ 1 √ uR s t (du) P S t R −2 + R −1 E S − 1 2 t 1 {St R −2 } =:I 1 (t, R) + I 2 (t, R). (4.9) For I 1 , noting that e −R 2 St e −1 1 {St R −2 } , we have I 1 (t, R) = P S t R −2 E exp(1 − R 2 S t ) e −tψ(R) . Thus,ˆ∞ 0 e −λt I 1 (t, R) dt 1 λ + ψ(R) . (4.10) ∞ 0 e −λt I 2 (t, R) dt = R −1ˆ∞ 0 e −λt E S − 1 2 t 1 St R −2 dt =R −1ˆ∞ R −2 u − 1 2ˆ∞ 0 e −λt P(S t ∈ du) dt = R −1ˆ∞ R −2 u − 1 2 U λ (du). Using (A 2 ) and following the proof for (3.10), we havê ∞ 0 e −λt I 2 (t, R) dt R −1ˆ∞ R −2 u − 1 2 d 1 λ + φ(u −1 ) = −R −1ˆR 0 s d 1 λ + ψ(s) =R −1ˆR 0 sψ ′ (s) (λ + ψ(s)) 2 (2.11) R −1ˆR 0 1 λ + ψ(s) ds 1 λ + ψ(R) . (4.11) Here we use Karamata's theorem in the last inequality. By (4.9)-(4.11), we get ∞ 0 e −λt P t (̺ R ′ − ̺ R ) 1 dt 1 λ + ψ(R) . (4.12) Combining (4.6) and (4.12), and letting j 0, R ′ = 2ψ −1 (2 j+1 ) and R = 1 2 ψ −1 (2 j ) therein, we get Π ψ jˆ∞ 0 e −λt (P t f )dt A (4.6) Π ψ j f Aˆ∞ 0 e −λt P t ( ̺ j+1 −̺ j ) 1 dt (4.12) Π ψ j f A λ + ψ 1 2 ψ −1 (2 j ) Π ψ j f A λ + 2 j . In the last inequality above, we used the fact that for each a > 0, lim R→∞ ψ aψ −1 (R) R = lim R→∞ ψ aψ −1 (R) ψ ψ −1 (R) = 1. This completes the proof for (4.5) when a(x, z) = a(z) = 1. Step 2. Now assume that a satisfies (4.2) with c 0 = 1 and ρ 0 = ∞. Let ν(dz) = J(z)dz and ν(dz) = (a(z) − 1) 0 J(z)dz. Sinceν and ν are two Lévy measures, there are two pure jump Lévy processesZ and Z associated withν and ν, respectively. LetP t and P t be the semigroups corresponding toZ and Z, respectively. Noting that Π ψ j P t f = Π ψ jP t P t f , by Step 1, one sees that Π ψ jˆ∞ 0 e −λt (P t f )dt A (4.6) Π ψ j f Aˆ∞ 0 e −λt P t ( ̺ j+1 −̺ j ) 1 dt (4.12) Π ψ j f A λ + ψ 1 2 ψ −1 (2 j ) Π ψ j f A λ + 2 j . Thus, we get (4.5) in the case that a is bounded below by c 0 > 0. Step 3. Now suppose a only satisfies (4.2) with c 0 = 1. Set Lu :=ˆB ρ 0 (u(x + z) − u(x)) a(z)1 Bρ 0 (z) + 1 B c ρ 0 (z) J(z) dz. and Ru :=ˆR d (u(x + z) − u(x)) (a(z) − 1)1 B c ρ 0 J(z) dz. ThenL satisfies (4.2) with c 0 = 1 and ρ 0 = ∞. By Step 2 and noting that λu −Lu = λu − Lu + Ru, we get u B ψ,1+α A λu −Lu B ψ,α A λu − Lu B ψ,α A + Ru B ψ,α A . (4.13) Since Rf A f Aˆ| z| ρ 0 (1 + |a(z)|)J(z) dz f A , one sees that Ru B ψ,α A = sup j −1 2 jα Π ψ j Ru A = sup j −1 2 jα RΠ ψ j u A sup j −1 2 jα Π ψ j u A = u B ψ,α A . (4.14) Combining (4.4), (4.13) and (4.14), we obtain λ u B ψ,α A + u B ψ,1+α A C λu − Lu B ψ,α A + u B ψ,α A . Choosing λ 0 = 2C, we obtain (4.3) for all λ λ 0 . This completes our proof. 4.2. Spatially inhomogeneous case. Before proving our main results, we need to make some necessary preparations. Under assumptions (A 1 ), one can verify that there is an integer N 1 such that for all λ 1, ψ −1 (2 1−N λ) 3 8 ψ −1 (λ). (4.15) Define S ψ k f = l≺k Π ψ l f, T ψ f g = k S ψ k f · Π ψ k g, R ψ (f, g) = k∼l Π ψ k f · Π ψ l g. Here l ≺ k means l < k − N and k ∼ l means |k − l| N . Thus, f · g = k,l Π ψ k f · Π ψ l g = T ψ f g + T ψ g f + R ψ (f, g). By the definition of Π ψ j and estimate (4.15), we have (a) For any θ > 0, it holds that Π ψ j (S ψ k f · Π ψ k g) = 0, if |j − k| N ; Π ψ j k:k∼l Π ψ k f · Π ψ l g = 0, if j k + 2N.Lu B ψ,0 A C u B ψ,1 A · sup z∈R d a(·, z) C θ ψ , (4.17) where C only depends on d, c 0 , ψ, θ (b) For any α > 0 and θ ∈ (0, α), it holds that Lu B ψ,α A C u B ψ,1+α A · sup z∈R d a(·, z) C θ ψ + u B ψ,1+θ A · sup z∈R d a(·, z) C α ψ , (4.18) where the constant C only depends on d, c 0 , ψ, α, θ. (i) If α = 0. Combining inequalities (4.20), (2.14) and (2.15), we get R d I (1) j (z) A J(z) dz k≺l∼jˆR d δ z Π ψ k u · Π ψ l a z A J(z) dz (4.20) 2 −θj sup z a z C θ ψ u B ψ,1 A k≺l∼j 2 −k   ψ −1 (2 k+1 )ˆ| z| 1 ψ −1 (2 k+1 ) |z|J(z) dz +ˆ| z|> 1 ψ −1 (2 k+1 ) J(z) dz   (2.14),(2.15) u B ψ,1 A sup z a z C θ ψ 2 −θj k≺l∼j 2 −k 2 k ψ −1 (2 k+1 ) + 2 k u B ψ,1 A sup z a z C θ ψ j2 −θj . (4.21) Similarly, noting that θ > 0, we havê R d I (2) j (z) A J(z) dz l≺k∼jˆR d δ z Π ψ k u · Π ψ l a z A J(z) dz u B ψ,1 A sup z a z C θ ψ l≺k∼j 2 −θl 2 −k 2 k ψ −1 (2 k+1 ) + 2 k u B ψ,1 A sup z a z C θ ψ (4.22) and R d I (3) j (z) A J(z) dz u B ψ,1 A sup z a z C θ ψ l∼k;k>j−2N 2 −θl 2 −k 2 k ψ −1 (2 k+1 ) + 2 k u B ψ,1 A sup z a z C θ ψ 2 −θj . (4.23) Combining (4.19), (4.21), (4.22) and (4.23), we obtain (4.17). (ii) When α > 0, by (4.20) and similar discussions in (i), one can see that R d I (1) j (z) A J(z) dz 2 −αj sup z a z C α ψ u B ψ,1+θ A k≺l∼j 2 −(1+θ)k 2 k ψ −1 (2 k+1 ) + 2 k 2 −αj u B ψ,1+θ A sup z a z C α ψ , R d I (2) j (z) A J(z) dz 2 −αj u B ψ,1+α A sup z a z C θ ψ l≺k∼j 2 −θl 2 −k 2 k ψ −1 (2 k+1 ) + 2 k 2 −αj u B ψ,1+α A sup z a z C θ ψ and R d I (3) j (z) A J(z) dz u B ψ,1+α A l∼k;k>j−2N 2 −(1+α)k 2 k ψ −1 (2 k+1 ) + 2 k sup z Π ψ l a z ∞ 2 −αj u B ψ,1+α A sup x,z |a(x, z)|. Combining the above estimates, we get (4.18). Lemma 4.4. Let A be a Young function or A = ∞. Suppose that (A 1 )-(A 5 ) are satisfied, then there are universal constants ε 1 > 0 and λ 0 > 0 such that for each λ 2λ 0 and f ∈ B ψ,β A with β ∈ [0, α], the Poisson equation (PE) has a unique solution u ∈ B ψ,1+β A , provided that |a(x, z) − a(0, z)| ε 1 and λ λ 0 . Moreover, λ u B ψ,β A + u B ψ,1+β A C f B ψ,β A , for each β ∈ [0, α]. (4.24) In particular, it holds that λ u C α ρ + u C 1+α ρ C f C α ρ , (4.25) if |a(x, z) − a(0, z)| ε 1 and λ λ 0 , and f ∈ C α ρ . Remark 4.5. Although the above analytic result relies on a special assumption regarding the oscillation of a, it is sufficient for us to establish our Theorem 1.4. In the case where A(t) = ∞ and s > 0, the space B ψ,s A can be identified as the generalized Hölder space C s ρ , which exhibits the following localization property: f C s ρ ≍ ε sup y∈R d f η((· − y)) C s ρ , where η is some smooth cut-off function. The method of frozen coefficients, combined with the aforementioned result, can be employed to eliminate the requirement that the oscillation of a is small. We will present a detailed proof of this approach for Theorem 1.2. However, it should be noted that the localization property can not extend to B s p,∞ (see [Tri92]). As a result, it is unclear whether the assumption on the oscillation of a can be entirely eliminated. Proof of Lemma 4.4. When a ≡ 1, by the proof of Theorem 4.1, we see that if f ∈ B ψ,β A , then G λ f ∈ B ψ,1+β A is a solution to (PE) (∀λ > 0), and estimate (4.24) holds true. Through the continuity method, it is enough to prove (4.24), under the premise that u ∈ B ψ,1+β A is a solution to equation (PE). By interpolation theorem, we only need to show (4.24) for β = 0 and β = α. Assume u ∈ B ψ,1+α A . Set f = λu − Lu, a 0 (z) = a(0, z) and L 0 u(x) =ˆR d (u(x + z) − u(x))a 0 (z)J(z) dz. Thus, λu − L 0 u = (L − L 0 )u + f. Using Theorem 4.1, we have λ u B ψ,α A + u B ψ,1+α A C (L − L 0 )u B ψ,α A + f B ψ,α A . (4.26) Choosing θ ∈ (0, α), we have a(·, z) − a 0 (·, z) C θ ψ a(·, z) − a 0 (·, z) 1− θ α C 0 ψ a(·, z) − a 0 (·, z) θ α C α ψ c −1 0 ε 1− θ α 1 . By (4.18) and interpolation, we have (L − L 0 )u B ψ,α A Cε 1− θ α 1 u B ψ,1+α A + C u B ψ,1+θ A Cε 1− θ α 1 u B ψ,1+α A + C u B ψ,α A . (4.27) Combining (4.26) and (4.27), and choosing λ ≫ 1, we get λ u B ψ,α A + u B ψ,1+α A Cε 1− θ α 1 u B ψ,1+α A + C f B ψ,α A Choosing ε 1 sufficiently small, so that Cε 1− θ α 1 1/2, then we obtain (4.24) for β = α. The case β = 0 can be proved by using (4.17) and following the same procedure above. So we complete our proof. Now we are at the point of proving Theorem 1.2. Proof of Theorem 1.2. Let χ be the nonnegative smooth function with compact support defined in Section 2.1. Fixed x 0 ∈ R d , define χ x 0 ε (x) := χ x − x 0 ε and a x 0 ε (x, z) := [a(x, z) − a(x 0 , z)]χ ε (x). By definition, a x 0 ε satisfies that |a x 0 ε (x, z)| 1 (4.28) and for every z ∈ R d and |x − x ′ | < 1, |a x 0 ε (x, z) − a x 0 ε (x ′ , z)| C ε ρ α (|x − x ′ |). (4.29) For simplicity, we omit the superscript x 0 below. Let v = uχ ε , δ z f (x) = f (x + z) − f (x) and L 0 u(x) =ˆR d δ z f (x) a(x 0 , z)J(z) dz. We have λv − L 0 v =[f χ ε − uL 0 χ ε ] + (Lu − L 0 u)χ ε − [L 0 (uχ ε ) − (L 0 u)χ ε − u(L 0 χ ε )] = : 3 i=1 I (i) ε . (4.30) Obviously, I (1) ε C α ρ = f χ ε C α ρ + uL 0 χ ε C α ρ ε f C α ρ + u C α ρ . (4.31) Noting that I (2) ε (x) = (L 0 u(x) − Lu(x))χ ε (x) = −ˆR d δ z u(x)a ε (x, z) J(z) dz, by (4.28), (4.29) and (4.18), for any θ ∈ (0, α) I (2) ε C α ρ c 1 ρ α (ε) u C 1+α ρ + C u C 1+θ ρ , (4.32) and the constant c 1 is independent with ε. For I ε , by definition I (3) ε (x) =ˆR d δ z χ ε (x) δ z u(x) a(x 0 , z)J(z) dz (4.33) and I (3) ε (x) − I (3) ε (y) =ˆR d δ z χ ε (x) δ z u(x) − δ z u(y) a(x 0 , z)J(z) dz +ˆR d δ z χ ε (x) − δ z χ ε (y) δ z u(y) a(x 0 , z)J(z) dz. (4.34) By (4.33), we have I (3) ε (x)| u ∞ ˆ| z| 1 ∇χ ε ∞ |z|a(x 0 , z)J(z) dz + 2ˆ| z|>1 χ ε ∞ a(x 0 , z)J(z) dz ε u ∞ . By (4.34), one can see that I (3) ε (x) − I (3) ε (y) ε ρ α (|x − y|) u C α ρ ∇χ ε ∞ˆ| z| 1 |z|J(z) dz + χ ε ∞ˆ| z|>1 J(z) dz + |x − y| ∇ 2 χ ε ∞ u ∞ˆ| z| 1 |z|J(z) dz + ∇χ ε ∞ u ∞ˆ| z|>1 J(z) dz ε ρ α (|x − y|) u C α ρ . Therefore, I (3) ε C α ρ ε u C α ρ . (4.35) Using Theorem 4.1, and combining (4.31), we obtain that (4.32) and (4.35), λ u C α ρ (B ε/2 (x 0 )) + u C 1+α ρ (B ε/2 (x 0 )) C v C 1+α ρ C v C α ρ + C L 0 v C α ρ c 2 ρ α (ε) u C 1+α ρ + C u C 1+θ ρ + C u C α ρ + C f C α ρ , for all λ λ ′ 0 (d, α, ψ, c 0 , ρ 0 ) > 0. Using the fact that u C 1+α ρ sup x 0 ∈R d u C 1+α ρ (B ε/2 (x 0 )) + C ε u ∞ , we obtain u C 1+α ρ (B ε/2 (x 0 )) c 2 ρ α (ε) sup x 0 ∈R d u C 1+α ρ (B ε/2 (x 0 )) + C u C 1+θ ρ + u C α ρ + f C α ρ . We then fixed ε 0 sufficiently small, such that c 2 ρ α (ε) 1/2, so arrive sup x 0 ∈R d λ u C α ρ (B ε 0 /2 (x 0 )) + u C 1+α ρ (B ε 0 /2 (x 0 )) C u C 1+θ ρ + u C α ρ + f C α ρ . This yields λ u C α ρ + u C 1+α ρ C ε 0 sup x 0 ∈R d λ u C α ρ (B ε 0 /2 (x 0 )) + u C 1+α ρ (B ε 0 /2 (x 0 )) u C 1+θ ρ + u C α ρ + f C α ρ . By Interpolation, one sees that λ u C α ρ + u C 1+α ρ f C α ρ , for all λ λ 0 (d, α, ψ, c 0 , ρ 0 ) > 0. Martingale problem In this section, we study the martingale problem associated with L. We need to introduce some necessary terminology. Let D = D([0, ∞), R d be the Skorokhod space of càdlàg R d -valued trajectories and let X t = X t (ω) = ω t (ω ∈ D), be the canonical process on it. Set D t = ε>0 σ(X s , 0 s t + ε), D = σ(D t , t 0). Definition 5.1 (Martingale problem). Let µ be a probability measure on R d . A probability measure P on (D, D) is said to be a solution to the martingale problem for (L, µ) if P•X −1 0 = µ and for each f ∈ C ∞ b , M f t := f (X t ) − f (x) −ˆt 0 Lf (X s ) ds is a D t -martingale under P. Below we briefly introduce the SDE corresponding to the martingale problem (L, µ). Let (Ω, P, F) be a probability space and N (dr, dz, ds) be a Poisson random measure on R + × R d × R + with intensity measure dr J(z)dz ds. Consider the following SDE driven by Poisson random measure N : X t = x +ˆt 0ˆR dˆ∞ 0 z1 [0,a(X s− ,z)) (r)N (dr, dz, ds). (5.1) If |a(x, z) − a(x, z)| C|x − y| for any x, y ∈ R d , then R dˆ∞ 0 |z| 2 1 [0,a(x,z)) (r) − 1 [0,a(y,z)) (r) 2 dr J(z)dz C|x − y| 2 . This implies (5.1) admits a unique strong solution (see [App09]). By Itô's formula, we see that f (X t ) − f (x) =ˆt 0ˆR dˆ∞ 0 f X s− + z1 [0,a(X s− ,z)) (r) − f (X s− ) N (dr, dz, ds) =ˆt 0ˆR d (f (X s− + z) − f (X s− )) a(X s− , z) J(z)dz ds +ˆt 0ˆR dˆ∞ 0 (f (X s− + z) − f (X s− )) 1 [0,a(X s− ,z)] (r) N (dr, dz, ds), =ˆt 0 Lf (X s− )ds + M f t , where N(dr, dz, dt) = N (dr, dz, dt) − drJ(z)dzdt and M f t is a martingale. Therefore, when x → a(x, z) is Lipchitz continuous (uniformly in z), P • X −1 is a martingale solution to (L, δ x ). Proof of Theorem 1.4. Existence: The proof for the existence is standard, for the convenience of the reader, we give the details here. Let η ∈ C ∞ c (B 1 ) satisfying´η = 1. Set η n (x) = n d η(nx) and a n (x, z) =´a(x − y, z)η n (y) dy. Let X n t be the unique strong solution to (5.1) (a is replaced by a n ), and P n x := P•(X n ) −1 . By the discussion above, P n x is a martingale solution associated with (L n , δ x ), and L n f (x) =ˆR d (f (x + z) − f (x)) a n (x, z)J(z) dz. We claim that {P n x } is tight in P(D). Then upon taking a subsequence, still denote by n, we can assume that P n x ⇒ P x . For any 0 s 1 · · · s k s t, f ∈ C ∞ c (R d ) and h i ∈ C ∞ c (R d ), i ∈ {1, 2 · · · , k}, we have E Px f (X t ) − f (X s ) −ˆt s Lf (X u ) du k i=1 h i (X t i ) = lim n→∞ E f (X n t ) − f (X n s ) −ˆt s Lf (X n u ) du k i=1 h i (X n t i ) = lim n→∞ E f (X n t ) − f (X n s ) −ˆt s L n f (X n u ) du k i=1 h i (X n t i ) + lim n→∞ E ˆt s (L n − L)f (X n u ) du k i=1 h i (X n t i ) = 0. Here we use the fact that P • (X n ) −1 is a solution to (L n , δ x ), and (L n − L)f → 0 uniformly (since a n → a uniformly). Thus, P x is a martingale solution to (L, δ x ). It remains to show {P n x } is tight. Given a bounded stopping time τ , we define N τ (dz) =ˆτ +1 τˆc −1 0 0 N (dr, dz, ds). N τ (dz) is a Poisson random measure on R d with intensity measure c −1 0 J(z)dz. P(|X n τ +δ − X n τ | > ε) P(|X n τ +δ − X n τ | > ε; N τ (B c M ) = 0) + P(N τ (B c M ) 1) ε −1 E ˆτ +δ τˆB Mˆa (X s− ,z) 0 zN (dr, dz, ds) + P(N τ (B c M ) 1) δ c 0 εˆ| z| M |z|J(z)dz + c −1 0ˆ| z|>M J(z)dz. Letting δ → 0 and then M → 0, we get sup n,τ P(|X n τ +δ − X n τ | > ε) → 0 (δ → 0). Similarly, one can also see that sup n P(|X n t | > M ) → 0 (M → ∞). By Aldous tightness criterion, we get the tightness of {P n x }. Uniqueness: For any f ∈ C ∞ b , consider equation (PE) in C 1+α ρ . By Theorem 1.2, equation (PE) admits a unique solution u ∈ C 1+α ρ . Suppose that (P i x , X t ) (i = 1, 2) are two solutions to martingale problem (L, δ x ), one can easily verify that M u t = u(X t ) − u(x) −ˆt 0 Lu(X t ) dt is a martingale under P i x . This implies d(e −λt u(X t )) = − e −λt (λ − L)u(X t ) dt + e −λt dM u t = − e −λt f (X t ) dt + e −λt dM u t . Taking expectation, one sees u(x) = E i xˆ∞ 0 e −λt f (X t ) dt, i = 1, 2. Thanks to [EK09,Theorem 4.2], we obtain P 1 x = P 2 x . Krylov-type estimate: Let δ ∈ (0, 1 ∧ ρ 0 /10) such that ρ α (2δ) < ε 1 , where ε 1 is the same constant in Lemma 4.4. For each y ∈ R d , let a y (x, z) =      a(x, z) if x ∈ B δ (y) a y + δ 2 (x − y) |x − y| 2 , z if x ∈ B c δ (y). Then a y satisfies (U a ), (H a ) and |a y (x, z) − a y (y, z)| < ε 1 . Let L y u(x) =ˆR d (u(x + z) − u(x)) a y (x, z)J(z) dz. Since a y also satisfies conditions (A 4 ) and (A 5 ), by Theorem 1.4 (a), for each y ∈ R d , the martingale problem (L y , δ y ) admits a unique solution (Q y , X t ). Moreover, for each y ∈ R d and f ∈ C ∞ c , by the proof for Theorem 1.4 (a), we have E Qyˆ∞ 0 e −λt f (X t ) dt = u a y (y), (5.2) where u a y is the solution to (PE) with L replaced by L y . Noting that a y meets all the assumptions in Lemma 4.4, we have u a y B ψ,1 A C f B ψ,0 A C f A . (5.3) Combining and (5.2) and (5.3) and using Theorem 3.12, we get E Qyˆ∞ 0 e −λt f (X t ) dt (5.2) C u a y C ε 1+ε ψ (3.23) C u a y B ψ,1 A (5.3) C f A . (5.4) Now let τ 1 = inf{t > 0 : |X t − X 0 | > δ} and τ k+1 = τ k + τ 1 • θ τ k . It holds that Q y | Fτ 1 = P y | Fτ 1 , due to the fact that a y (·, z)| B δ (y) = a(·, z)| B δ (y) and the uniqueness of martingale problem (L, δ y ). Suppose that (P µ , X t ) is the unique martingale solution to (L, µ), then for any non-negative function f ∈ C ∞ c , E µˆτ 1 0 e −λt f (X t ) dt =ˆR d µ(dy) E yˆτ 1 0 e −λt f (X t ) dt =ˆR d µ(dy) E Qyˆτ 1 0 e −λt f (X t ) dt sup y E Qyˆ∞ 0 e −λt f (X t ) dt f A . By strong Markov property, E µˆτ k+1 τ k e −λt f (X t ) dt =E µˆτ k+1 τ k e −λt f (X t ) dt = E µˆτ 1 •θτ k 0 e −λ(τ k +t) f (X t • θ τ k ) dt =E µ e −λτ k ˆτ 1 0 e −λt f (X t ) dt • θ τ k = E µ e −λτ k E Xτ kˆτ 1 0 e −λt f (X t ) dt f A E µ e −λτ k f A sup y E y e −λτ k . To estimate sup y E y e −λτ k , using strong Markov property again, we have E x e −λτ k+1 =E x e −λ(τ k +τ 1 •θτ k ) = E x e −λτ k E x e −λτ 1 • θ τ k |F τ k =E x e −λτ k E Xτ k e −λτ 1 sup y E y e −λτ 1 E x e −λτ k · · · sup y E y e −λτ 1 k+1 . Thus, E µˆ∞ 0 e −λt f (X t ) dt ∞ k=0 E µˆτ k+1 τ k e −λt f (X t ) dt f A ∞ k=1 sup y E y e −λτ k f A ∞ k=1 sup y E y e −λτ 1 k . So the desired conclusion follows if E y e −λτ 1 1/2 for all y. To achieve this, we choose g ∈ C ∞ b , which satisfies g(x) = 0 if x ∈ B δ/2 , g(x) = 1 if x ∈ B δ and ∇g ∞ Cδ −1 . Let g y (x) = g(x − y). It is easy to verify that there exists a constant K δ < ∞ such that Lg y (x) − λg y (x) Lg y ∞ K δ . E y e −λτ 1 g y (X τ 1 ) − g y (y) = E yˆ∞ 0 e −λs (Lg y − λg y )(X s ) ds. Since g y (y) = g(0) = 0 and g y (X τ 1 ) = 1, P y -a.s., we have E y e −λτ 1 K δ /λ. Choosing λ 2K ∨ λ 0 , we have sup y E y e −λτ 1 1/2. Therefore, E µˆ∞ 0 e −λt f (X t ) dt f A ∞ k=1 1 2 k f A , for all λ 2K ∨ λ 0 . For any λ > 0 and µ ∈ P(R d ), E µˆ∞ 0 e −λt f (X t ) dt sup y E yˆ1 0 f (X t ) dt · ∞ k=0 e −λk λ −1 sup y E yˆ∞ 0 e −(2K∨λ 0 )t f (X t ) dt f A /λ. So we complete our proof. Proof of Corollary 1.5. Let S be the gamma subordinator, whose Laplace exponent is given by φ(λ) = log(1 + λ). By Proposition 2.10, one sees j(r) ≍ r −d , r → 0. Let a ′ (x, z) = |z| −d j −1 (|z|)1 B 1 (z) a(x, z). By our assumptions on a, it holds that a ′ (·, z) C α ρ a(·, z) C α ρ C, where ρ(r) = 1 ψ(r −1 ) = 1 log(1+r −2 ) . Noting that Lu(x) =ˆB 1 (u(x + z) − u(x)) a(x, z) |z| d dz =ˆR d (u(x + z) − u(x)) a ′ (x, z)J(z) dz, by Theorem 1.4, the martingale problem (L, δ x ) admits a unique solution (P x , X t ). Letting A(t) = e t β − 1 with β > 1. Then [ψ −1 (t 1+ε )] d = (e t 1+ε − 1) d/2 A(t) , for all t ≫ 1 and some ε ∈ (0, β − 1). Thanks to Theorem 1.4 (inequality (1.5)), we obtain E xˆ∞ 0 e −λt f (X t ) dt C λ inf λ > 0 :ˆR d A (f (x)/λ) dx 1 = C λ inf λ > 0 :ˆR d exp (f (x)/λ) β − 1 dx 1 . Appendix A. A Remark on a theorem by Bass This section extends the main results of [Bas09] on stable-like operators using Littlewood-Paley theory and scaling techniques, and emphasizes that it is more natural to study a priori estimates in the C s space. We also observe that the function space X mentioned in Section 1.3 (aimed to replace C s ) contains unbounded and discontinuous functions, so it is not suitable for the problems we care about. We will also show that techniques that have proven successful for stabe-like operators can not yield satisfactory results for our main problem of concern. To begin, we review some fundamental concepts from classical Littlewood-Paley theory. Let χ be the same smooth function defined in Section 3. (A.1) Operators ∆ j is defined by ∆ j f := F −1 (χ(2·)Ff ), j = −1, F −1 (ϕ(2 −j ·)Ff ), j 0. Definition A.1 (Hölder-Zygmund space). Assume that s ∈ R, let C s denote the collection of all distribution f satisfying f C s := sup j −1 2 js ∆ j f ∞ < ∞. The following result is well-known Theorem A.2 ([BCD11, Theorem 2.36]). Assume s > 0 and s / ∈ N, then C s is the usual Hölder space and f C s ≍ f ∞ + sup x =y |∇ [s] f (x) − ∇ [s] f (y)| |x − y| s−[s] . Let ν be a non-degenerate α-stable measure, i.e., for each ξ ∈ R d \{0}, S d−1 |σ · ξ| 2 Σ(dσ) > 0. Here Σ is the spectral measure of ν, i.e. ν(dz) = dr r 1+α Σ(dσ), z = rσ. Let a is a bounded positive function on R d × R d . The operator L a is defined by L a u(x) =ˆR d u(x + z) − u(x) − ∇u(x) · z1 α∈(1,2) a(x, z)ν(dz) when α ∈ (0, 2) and α = 1. If α = 1, we always assume´S d−1 σ a(x, rσ)Σ(dσ) = 0 and define L a u(x) =ˆR d (u(x + z) − u(x) − ∇u(x) · z1 B 1 (z)) a(x, z)ν(dz). The following result is a generalization of Proposition 4.2 and Proposition 4.3 in [Bas09]. Note that here we do not need to assume ν is absolutely continuous with respect to Lebesgue measure. Lemma A.3. Let c 0 ∈ (0, 1). Suppose ν is a non-degenerate α-stable measure. Assume that a only depends on z and c 0 a c −1 0 , then for each λ 1, β ∈ R, there is a constant C = C(d, ν, α, β, c 0 ) such that λ u C β + u C α+β C λu − L a u C β . (A.2) Proof. We only prove the case α = 1 here. Step 1. Let A = a ∈ L ∞ (R d ) : c 0 a(z) c −1 0 , z ∈ R d . We first prove that for any v ∈ S ′ (R d ) satisfying suppv ⊆ C, it holds that v ∞ C inf a∈A L a v ∞ . (A.3) Here C is a constant does not depends on v. Assume (A.3) does not hold. Then, there is a sequence v n such that v n is supported on C, and a sequence a n ∈ A such that 1 = v n ∞ n L an v n ∞ . (A.4) Let h = F −1 (χ(·/2) − χ(4·)), where χ is the same function in Section 3. Noting that (χ(·/2) − χ(4·)) v n = v n , we have v n (x) =ˆR d h(x − y)v n (y)dy. So for any k ∈ N, ∇ k v n ∞ = ∇ k h * v n ∞ ∇ k h 1 v n ∞ C k . (A.5) By Ascoli-Azela's lemma and diagonal argument, there is a subsequence of {v n } (still denoted by v n for simple) and v ∈ C ∞ b such that ∇ k v n converges to ∇ k v uniformly on any compact set. Let χ R (·) = χ(·/R). For any φ ∈ S (R d ), ˆφ (v n − v) ˆ| φχ R · (v n − v)| +ˆ|φ(1 − χ R )(v n − v)| φ L 1 v n − v L ∞ (B 3R/2 ) + 2 sup |x|>R |φ(x)|. Letting n → ∞ and then R → ∞, we get φ, v n → φ, u , ∀φ ∈ S (R d ). Here , denotes the dual pair of S (R d ) and S ′ (R d ). That is to say v n → v in S ′ (R d ) and consequently, v n →v in S ′ (R d ). For any φ ∈ S (R d ) supported on R d \C, we have φ,v = lim n→∞ φ, v n = 0, which meansv is also supported on C. On the other hand, note that a n ν is a sequence of Radon measure on R d \{0}, and sup n´K a n (z)ν(dz) c −1 0 ν(K) < ∞ for each compact subset K in R d \{0}. Thus, there is a subsequence of a n (still denoted by a n for simple) such that a n ν This implies 0 = L a 0 v = ψ a 0v . Since for ξ = 0, Re(ψ a 0 (ξ)) =ˆR d (1 − cos(z · ξ)) a 0 (z)ν(dz) |σ · ξ| 2 Σ(dσ) > 0, we get that the support ofv is the origin. This contradicts the previous conclusion that suppv ⊆ C. So we complete the proof for (A.3). Step 2. Suppose u ∈ S ′ (R d ) and suppû ⊆ λC. Let v(x) := u(λ −1 x) and a λ (z) = a(λ −1 z). By the scaling property of ν, we get L a λ v(x) =ˆR d u(λ −1 x + λ −1 z) − u(λ −1 x) − ∇u(λ −1 (x)) · (λ −1 z)1 α∈(1,2) a λ (z)ν(dz) =ˆR d u(λ −1 x + z ′ ) − u(λ −1 x) − ∇u(λ −1 (x)) · z ′ 1 α∈(1,2) a(z ′ )λ −α ν(dz ′ ) =λ −α (L a u)(λ −1 x). Thus, by (A.3), we get u ∞ = v ∞ C L a λ v ∞ = Cλ −α L a u ∞ , (A.7) provided that suppû ∈ λC. Step 3. Like proof for Theorem 4.1, it is easy to see that for each j −1, λ ∆ j u ∞ λ∆ j u − L a ∆ j u ∞ . By (A.7), for any j 0, 2 jα ∆ j u ∞ C L a ∆ j u ∞ Cλ ∆ j u ∞ + C λ∆ j u − L a ∆ j u ∞ λ∆ j u − L a ∆ j u ∞ . Thus, for all λ 1 and β ∈ R, we get (A.2). Theorem A.4. Assume that ν is a non-degenerate α-stable measure, and that c 0 a c −1 0 , then for each λ > 0, β ∈ [0, α], there is a constant C = C(d, ν, α, β, c 0 ) such that λ u C β + u C α+β C λu − L a u C β . With Lemma A.3 at hand, utilizing the method of frozen coefficients as before, we can prove the case that a is a positive function on R d × R d . Since the proof for this result is just a repetition of the proof of Theorem 1.2, we omit it here. Below we provide an example to demonstrate that the function space X defined in equation (1.8) encompasses unbounded, discontinuous functions. φ(2 j ·) = φ ∞ < ∞. On the other hand, since φ ∈ S and φ(0) =ˆR d F(φ)(ξ)dξ > 0, there are constants ε 0 , δ 0 ∈ (0, 2 −10 ) such that φ(x) ε 0 if |x| < δ 0 . Let δ := δ 10 0 and j x := [− log 2 |x|]. Fix x ∈ B δ \{0}. It holds that C log j x j jx+log 2 δ 0 j −1 φ(2 j x) ε 0 (log j x − 1). Noting that |φ(2 j x)| C(2 j |x|) −N if j j x , we have j>jx |j −1 φ(2 j x)| Cj −1 x . Moreover, jx+log 2 δ 0 <j jx |j −1 φ(2 j x)| −C log 2 δ 0 . Thus, for any x ∈ B δ \{0}, f (x) ε 0 log j x − C ε 0 2 log log 2 |x| −1 − C and |f (x)| C log log 2 |x| −1 . This implies f / ∈ L ∞ and f ∈ L 1 (B δ ). To sum up, f ∈ (X ∩ L 1 loc )\L ∞ . To conclude, we emphasize that the approach used in the proof of Lemma A.3 is not straightforward to extend to the case of ν(dz) = 1 B 1 (z) |z| d dz, and it may not be possible. One significant hurdle arises from the need to obtain the desired regularity estimate, which requires the use of the ψ-decomposition with ψ(R) ≍ log R (R → ∞). When attempting to apply the same procedure as in the previous case, it becomes apparent that every convergent subsequence of ν n converges to the zero measure. The research of E. Hu is supported by the National Natural Science Foundation of China (No. 12171354); The research of Guohuan is supported by the National Natural Science Foundation of China (No. 12288201). Our second main result concerns the weak well-posedness of equation (1.4) and relies on the martingale problem associated with non-local operators of the form (1.3), which includes L σ . Theorem 1.4. Suppose (A 1 )-(A 5 ) are satisfied. Assume A : [0, ∞) → [0, ∞) is a Young function (see Definition 2.1). Then Definition 2.1. A function A : [0, ∞) → [0, ∞) is called a Young function if A is a increasing convex function that satisfies A(0) = 0 and lim t→∞ A(t) = ∞. Figure 1 . 1A Geometric Proof of (2.6) Definition 2.3 (Luxemburg norm). Let A be a Young function. For any measurable function f , . 3 . 3Let A be a Young function or A = ∞. Let a : R d × R d → R be a bounded measurable function (need not be positive). Assume the linear operator L is given by (1.3), and (A 1 )-(A 3 ) are satisfied. ϕ(ξ) := χ(ξ) − χ(2ξ).It is easy to see that ϕ 0, supp ϕ ⊂ C := B 1 0 and a 0 := dν 0 /dν ∈ [c 0 , c −1 0 ]. (A.6) Using (A.4)-(A.6) and the fact that ∇ k v n → ∇ k v uniformly on each compact subset, we haveL a 0 v(x) =ˆR d v(x + z) − v(x) − ∇v(x) · z1 α∈(1,2) a(z)ν(dz) = lim n→∞ ˆR d v n (x + z) − v n (x) − ∇v n (x) · z1 α∈(1,2) a n (z)ν(dz) Example 3 . 3Let φ ∈ S (R d ) such that F(φ) ∈ C ∞ c Due to (A.1), F(φ)ϕ = F(φ). Thus, F(f )(ξ)ϕ(2 −j ξ) =j −1 2 −j (F(φ)ϕ)(2 −j ξ) =j −1 2 −j F(φ)(2 −j ξ), which yields sup j −1 max{1, j} ∆ j f ∞ = sup j 1 t k t d+1 [1 ∨ ψ(t)] dt + λ dˆ∞ λ 1 t d+1 [1 ∨ ψ(t)] dt =I 1 + I 2 . AcknowledgementsThe authors would like to thank Huyuan Chen and Jaehoon Kang for many useful conversations.Proof. Set δ z u(x) = u(x + z) − u(x), a z (x) = a(x, z).By definition,(4.19)We drop the index x below for simplicity. By (4.16),By mean value theorem, we getThus,This and (3.2) imply that(4.20) Sobolev spaces. A Robert, John Jf Adams, Fournier, ElsevierRobert A Adams and John JF Fournier. Sobolev spaces. Elsevier, 2003. 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L p -maximal regularity of nonlocal parabolic equations and applications. Xicheng Zhang, Ann. Inst. H. Poincaré C Anal. Non Linéaire. 304Xicheng Zhang. L p -maximal regularity of nonlocal parabolic equations and applications. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 30(4):573-614, 2013. China Email address: eryan. [email protected] Institute of Applied Mathematics, Academy of Mathematics and Systems Science. 300072Center for Applied Mathematics, Tianjin UniversityCAS, Beijing. China Email address: [email protected] for Applied Mathematics, Tianjin University, Tianjin, 300072, China Email address: [email protected] Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Bei- jing, 100190, China Email address: [email protected]
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[ "Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy", "Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy" ]
[ "Siavash Golkar \nCenter for Computational Neuroscience\nFlatiron Institute\n\n", "Tiberiu Tesileanu \nCenter for Computational Neuroscience\nFlatiron Institute\n\n", "Yanis Bahroun \nCenter for Computational Neuroscience\nFlatiron Institute\n\n", "Anirvan M Sengupta \nCenter for Computational Mathematics\nFlatiron Institute\n\n\nCenter for Computational Quantum Physics\nFlatiron Institute\n\n\nDepartment of Physics and Astronomy\nRutgers University\n\n", "Dmitri Chklovskii [email protected]@gmail.com \nCenter for Computational Neuroscience\nFlatiron Institute\n\n\nNeuroscience Institute\nNYU Medical Center\n\n" ]
[ "Center for Computational Neuroscience\nFlatiron Institute\n", "Center for Computational Neuroscience\nFlatiron Institute\n", "Center for Computational Neuroscience\nFlatiron Institute\n", "Center for Computational Mathematics\nFlatiron Institute\n", "Center for Computational Quantum Physics\nFlatiron Institute\n", "Department of Physics and Astronomy\nRutgers University\n", "Center for Computational Neuroscience\nFlatiron Institute\n", "Neuroscience Institute\nNYU Medical Center\n" ]
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Predictive coding (PC) has emerged as an influential normative model of neural computation, with numerous extensions and applications. As such, much effort has been put into mapping PC faithfully onto the cortex, but there are issues that remain unresolved or controversial. In particular, current implementations often involve separate value and error neurons and require symmetric forward and backward weights across different brain regions. These features have not been experimentally confirmed. In this work, we show that the PC framework in the linear regime can be modified to map faithfully onto the cortical hierarchy in a manner compatible with empirical observations. By employing a disentangling-inspired constraint on hidden-layer neural activities, we derive an upper bound for the PC objective. Optimization of this upper bound leads to an algorithm that shows the same performance as the original objective and maps onto a biologically plausible network. The units of this network can be interpreted as multi-compartmental neurons with non-Hebbian learning rules, with a remarkable resemblance to recent experimental findings. There exist prior models which also capture these features, but they are phenomenological, while our work is a normative derivation. Notably, the network we derive does not involve one-to-one connectivity or signal multiplexing, which the phenomenological models required, indicating that these features are not necessary for learning in the cortex. The normative nature of our algorithm in the simplified linear case allows us to prove interesting properties of the framework and analytically understand the computational role of our network's components. The parameters of our network have natural interpretations as physiological quantities in a multi-compartmental model of pyramidal neurons, providing a concrete link between PC and experimental measurements carried out in the cortex.
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[ "https://export.arxiv.org/pdf/2210.15752v2.pdf" ]
253,224,349
2210.15752
3d3546bc4fbe116fc147e91ae09d48aa646f43b6
Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy Siavash Golkar Center for Computational Neuroscience Flatiron Institute Tiberiu Tesileanu Center for Computational Neuroscience Flatiron Institute Yanis Bahroun Center for Computational Neuroscience Flatiron Institute Anirvan M Sengupta Center for Computational Mathematics Flatiron Institute Center for Computational Quantum Physics Flatiron Institute Department of Physics and Astronomy Rutgers University Dmitri Chklovskii [email protected]@gmail.com Center for Computational Neuroscience Flatiron Institute Neuroscience Institute NYU Medical Center Constrained Predictive Coding as a Biologically Plausible Model of the Cortical Hierarchy Predictive coding (PC) has emerged as an influential normative model of neural computation, with numerous extensions and applications. As such, much effort has been put into mapping PC faithfully onto the cortex, but there are issues that remain unresolved or controversial. In particular, current implementations often involve separate value and error neurons and require symmetric forward and backward weights across different brain regions. These features have not been experimentally confirmed. In this work, we show that the PC framework in the linear regime can be modified to map faithfully onto the cortical hierarchy in a manner compatible with empirical observations. By employing a disentangling-inspired constraint on hidden-layer neural activities, we derive an upper bound for the PC objective. Optimization of this upper bound leads to an algorithm that shows the same performance as the original objective and maps onto a biologically plausible network. The units of this network can be interpreted as multi-compartmental neurons with non-Hebbian learning rules, with a remarkable resemblance to recent experimental findings. There exist prior models which also capture these features, but they are phenomenological, while our work is a normative derivation. Notably, the network we derive does not involve one-to-one connectivity or signal multiplexing, which the phenomenological models required, indicating that these features are not necessary for learning in the cortex. The normative nature of our algorithm in the simplified linear case allows us to prove interesting properties of the framework and analytically understand the computational role of our network's components. The parameters of our network have natural interpretations as physiological quantities in a multi-compartmental model of pyramidal neurons, providing a concrete link between PC and experimental measurements carried out in the cortex. Initially proposed as an unsupervised learning paradigm in the retina [3], it has since been expanded to the supervised regime [4] with arbitrary graph topologies [5,6]. The PC framework has been analyzed in many contexts [7,8] and has found many applications, from clinical neuroscience [9,10] to memory storage and retrieval [11]. We refer the reader to [12,13] for recent reviews. [4], figure adapted from [13]. The intra-layer connectivity is to be one-toone, while the inter-layer connectivity is symmetric. (b) Our BioCCPC network. There is no requirement for symmetric weights across layers, and the connectivity within layers can be diffuse. Predictive coding is viewed as a possible theory of cortical computation, and many parallels have been drawn with the known neurophysiology of cortex [12]. While the initial works proposed a biologically plausible network [1,2], the connection with cortex was more closely examined in [14], where the PC module was mapped onto a cortical-column microcircuit. However, there are aspects of this mapping that have proved controversial. Among these are the requirement of multiple redundant cortical operations, the symmetric connectivity pattern, the one-to-one connectivity of value and error neurons, and also the requirement that feedback connections be inhibitory [12,15,16], as sketched in Fig. 1a . The presence of separate error and value neurons has itself been called into question [15]. The PC-based neural circuits also do not account for more recent experimental findings highlighting the details of computation in the cortex [17][18][19][20][21][22][23][24][25][26]. For example, the learning dynamics of multicompartmental pyramidal neurons has been closely investigated. In particular, it was observed that the plasticity of the synapses of the basal compartment is driven by the activity in the apical tuft of the neuron by so-called calcium plateau potentials [19,20,24], leading to non-Hebbian learning rules [25]. These experiments have motivated the development of several models of microcircuits with multi-compartmental neurons [27][28][29][30][31][32][33]. In a number of cases, it has been shown that these models can replicate learning similar to the backpropagation algorithm under specific assumptions [30,34,35]. However, because of their rather phenomenological nature, detailed analysis of these models is in many cases challenging and one must resort to numerical simulations, rendering the task of understanding the role of various neurophysiological quantities difficult. To this date, a normative framework (PC or otherwise) that can explain these experimental findings is still lacking. In this work, we show how the PC framework can be made compatible with the aforementioned experimental observations. Inspired by prior work which explored the effects of finding decorrelated representations [36][37][38][39][40][41], we add a decorrelating inequality constraint to the covariance of the latent representations of PC. Using this constraint, we derive an upper bound for the PC objective. By working in the linear regime, we can prove interesting properties of our algorithm. We show that the learning algorithm derived from this upper bound does not suffer from the issues of prior implementations and naturally maps onto the hierarchical structure of the cortical pyramidal neurons. Contributions • By imposing a decorrelation-inspired inequality constraint on the latent space, we find an analytic upper bound to the PC objective. • We introduce BioCCPC, a novel biologically plausible algorithm derived from this upper bound and show that it closely matches the known physiology of the cortical hierarchy. • We interpret the different parameters of the algorithm in terms of the conductances and leaks of the separate compartments of the pyramidal neuron. We find that the neural compartmental conductances encode the varainces of the PC framework, and the somatic leak maps onto a thresholding mechanism of the associated eigenvalue problem. Related work and review of predictive coding The backpropagation algorithm [34,42] is the predominant tool in the training of deep neural networks but is generally not considered biologically plausible [43]. Over the years, many authors have explored biologically plausible approximations or alternatives to backpropagation [30,31,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58] (for a more complete review see [59]). These approaches generally fall into two categories. First are normative approaches, such as Predictive Coding [1][2][3], Target Propagation and variations [44,45,47], Equilibrium Propagation [49,53] and others [51,[54][55][56], where one starts from a mathematically motivated optimization objective. These methods, by virtue of their normative derivation, have a firm grounding in theory; however, they do not fully conform to the experimental observations in the brain (see below and Sec. 4). The second approach is driven by biology, with network structures and learning rules inspired by experimental findings [28,30,31,48,60]. While these works mostly conform to the experimentally observed findings, they are more challenging to analyze because of their conjectured phenomenological nature. The goal of the present paper is not to propose yet another biologically plausible alternative to backpropagation. It is rather to demonstrate that the normative framework of predictive coding, when combined with a constraint, can indeed closely match experimental observations. For this reason, in this work, we focus on comparing our method with previous implementations of predictive coding and do not concern ourselves with other biologically plausible alternatives to backpropagation. The relationship between the PC framework and backpropagation was explored in [4,[61][62][63][64]. The advantages of PC over backpropagation were highlighted in [13]. Notation. Bold upper case M and lower case variables v denote matrices and vectors, respectively. By upper case letter X, Y, Z, we denote data matrices in R d×T , where d and T are the dimensions of the relevant variable and the number of samples. Lower case x, y, z, denote the relevant quantities of a single sample, and x t , y t , z t , denote the t th sample. M 2 F denotes the Frobenius norm of M. Review of predictive coding Probabilistic model. In this section, we review the supervised predictive coding algorithm [4]. The derivation starts from a probabilistic model for supervised learning, which parallels the architecture of an artificial neural network (ANN) with n + 1 layers. In this model, the neurons of each layer are random variables z (l) (denoting the vector of activations in the l th layer) with layers 0 and n, respectively, denoting the input and output layers of the network. We assume that the joint probability of the latent variables factorizes in a Markovian structure p(z (0) , z (1) , · · · , z (n) ) = p(z (n) |z (n−1) ) × p(z (n−1) |z (n−2) ) × · · · × p(z (0) ) with the relationship between the random variables of adjacent layers given by: p z (l) |z (l−1) = N z (l) ; µ (l) , Σ (l) , with µ (l) = W (l−1) f (z (l−1) ), Σ (l) = σ (l)2 I . (1) The mean of the probability density on layer l mirrors the activity of the analogous ANN given by µ (l) = W (l−1) f (z (l−1) ), where W (l−1) are the weights connecting layers l − 1 and l. The objective function is then given by the negative log-likelihood of the joint distribution function: L = − t log p(z (0) t , . . . , z (n) t ) = 1 2 l Z (l) − W (l−1) f (Z (l−1) ) 2 F σ (l)2 + const ,(2) where we have switched to the data matrix notation for brevity and assumed the variances σ (l)2 are fixed hyperparameters. In the following, we refer to L, Eq. (2), without the constant term. Learning. Learning takes place in two steps. First, the values of the random variables are determined by finding the most probable configuration of the joint distribution function when both the input and output layers are conditioned on the given input and output (z (0) = x and z (n) = y): z * (1) , . . . , z * (n−1) = arg min z (1) ,...,z (n−1) L(z (0) = x, z (n) = y).(3) The solution to this minimization problem can be found via gradient descent, which we evaluate component-wise for clarity as: z (l) j = −η∂ zj L = η(−ε (l) j + i ε (l+1) i W (l) ij f (z (l) j )), with ε (l) i = z (l) i − µ (l) i σ (l)2 ,(4) where η is the gradient descent step size. The second step is then to minimize the objective with respect to the weights while keeping the previously obtained neuron values fixed. This corresponds to optimizing the value of the loss at the MAP estimate and can also be carried out by gradient descent. This algorithm can be implemented by a biologically plausible network as described in [4]; see Figure 1a. However, as discussed in Section 1, its mapping onto the cortex has proved controversial. For further details regarding these steps see [4]. A constrained predictive coding framework In this section, we introduce and discuss our novel, covariance-constrained predictive coding (CCPC) model within the supervised PC paradigm of [4]. Our model also straightforwardly extends to the unsupervised learning paradigm discussed in [1,2]. For simplicity we work in the linear regime (f (x) = x) which allows us to prove different properties of our framework. Derivation of upper bound objective Reduction to a sum of objectives. We start by reducing the optimization problem, Eq. (2), into a set of overlapping sub-problems, which will allow us to break the symmetry between feedforward and feedback weights. To do so, we first introduce a copy of the terms containing the weights W (1) to W (n−2) , denoted by W Z (l) − W (l−1) Z (l−1) 2 F σ (l)2 = min Z,Wa,W bL L = min Z,Wa,W b 1 2 Z (1) − W (0) b Z (0) 2 F σ (1)2 + 1 2 Z (n) − W (n−1) a Z (n−1) 2 F σ (n)2 + 1 4 n−1 l=2 Z (l) − W (l−1) b Z (l−1) ) 2 F σ (l)2 + Z (l) − W (l−1) a Z (l−1) ) 2 F σ (l)2 .(5) For consistency, we rename W (0) , W (n−1) to W (0) b , W (n−1) a , respectively. Introducing these copies does not change the optimization 2 but will help us avoid weight sharing in the steps below. We now pair the terms two by two as min Z,Wa,W bL = 1 2 n−1 l=1 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + g (l) a Z (l+1) − W (l) a Z (l) 2 F ,(6) with g (n−1) a = 1/σ (n)2 , g (1) b = 1/σ (1)2 , and g (l−1) a = g (l) b = 1/(2σ (l)2 ) for l = 2, . . . , n − 1. Weight sharing occurs here from terms like z (l+1) Wz (l) , obtained from expanding the squared norms in Eq. (6). Indeed, the gradient descent dynamics with respect to z (l+1) (resp. z (l) ) leads to terms of the form Wz (l) inż (l+1) (resp. W z (l+1) inż (l) ), which use the same weights W. Thanks to the doubling of the weights, we can avoid this problem by optimizing each term in the sum in Eq. (6) separately. In other words, min Z,Wa,W bL ≤ n−1 l=1 min Z (l) ,W (l) a ,W (l−1) bL (l) ,(7)whereL (l) ≡ 1 2 n l=0 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + g (l) a Z (l+1) − W (l) a Z (l) 2 F . 2 This can be directly verified by finding the optima for W's before and after the change. We have W (l) = W (l) a = W (l) b = Z (l+1) Z (l) (Z (l+1) Z (l) ) −1 . Plugging these back into Eq. (5) we see that the equality holds. However, in the next step, since we treat Wa's and W b 's differently, Wa = WB will no longer hold. This equality holds simply because we are no longer finding the minimum of the full objectiveL. We are instead finding the minimum of each component separately, and then evaluating lL (l) =L. This splits the (n + 1)-layer optimization problem into a set of 3-layer optimizations, in each of which only the middle layer is being optimized. Note, however, that these are overlapping, so the different optimization problems need to be solved self-consistently. We make this precise in the supplementary materials section and show that it provides an upper bound for our objective L (SM Sec. A). Separating the objective function in this manner eliminates the weight sharing problem for W b , but the problem remains for W a . We address this problem in the following. Whitening constraint. The idea of decorrelating internal representation has been widely used for unsupervised tasks, often motivated by neuroscience [36][37][38]. In the case of deep learning, the main motivations were improved convergence speed and generalization [39][40][41]. Decorrelation has also been used to circumvent the weight transport problem [55]. Inspired by these observations, we introduce the constraint 1 T Z (l) Z (l) I, imposing an upper bound on the eigenvalues of the covariance matrix. The inequality can be implemented by using a positive-definite Lagrange multiplier Q (l) Q (l) (for details see SM Sec. E): minL (l) ≤ min Z (l) ,W (l) a ,W (l−1) b max Q (l) 1 2 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + g (l) a T W (l) a W (l) a + g (l) a Tr −2Z (l+1) W (l) a Z (l) + Tr Q (l) Q (l) Z (l) Z (l) − T I + c (l) Z (l) 2 F ,(8) where we have added an additional quadratic term in Z as a regularizer. Neural dynamics and learning rules. Similar to the PCN of [4], the dynamics of our network during learning proceeds in two steps. First the neural dynamics is derived by taking gradient steps ofL (l) from Eq. (8) with respect to z (l) : z (l) = g (l) b W (l−1) b z (l−1) + g (l) a W (l) a z (l+1) − g (l) b + c (l) z (l) − g (l) a Q (l) n (l) ,(9) where we have defined the variables n (l) = (1/g (l) a )Q (l) z (l) for each layer of the network. The weight updates are derived via stochastic gradient descent of the loss given in Eq. (8) after the neural dynamics have reached equilibrium. These are given by δW (l) b ∝ g (l+1) a W (l+1) a z (l+2) − Q (l+1) n (l+1) − c (l+1) z (l+1) z (l) , (10a) δW (l) a ∝ z (l+1) z (l) − W (l) a ,(10b)δQ (l) ∝ n (l) z (l) − Q (l) .(10c) We used the neural dynamics equilibrium equation for z (l) to simplify the weight update for W (l) b . This yields our online algorithm (Alg. 1), with the architecture shown in Fig. 1b. The algorithm can be implemented in a biologically plausible neural network as in Fig. 2; see Sec. 4. Algorithm 1: Covariance Constrained predictive coding algorithm (BioCCPC) input: x t , y t new sample and previous weight matrices W (l) a , W (l−1) b , Q (l) ∀l ∈ {1, . . . , n − 1} z (0) t ← x t , z (l) t ← W (l−1) b z (l−1) t initialize latents via forward pass run until convergence for all l ∈ {1, n − 1} neural dynamics n (l) ← (1/g (l) a )Q (l) z (l) t v (l) a ← W (l) a z (l+1) t − Q (l) n (l) , v (l) b ← W (l−1) b z (l−1) t z (l) t ← z (l) t + τ −1 −g (l) lk z (l) t + g (l) a (v (l) a − z (l) t ) + g (l) b (v (l) b − z (l) t ) update for all l ∈ {1, n − 1} synaptic weight updates W (l) a ← W (l) a + η z (l+1) t z (l) t − W (l) a W (l−1) b ← W (l−1) b + η g (l) a W (l) a z (l+1) t − Q (l) n (l) − c (l) z (l) t z (l−1) t Q (l) ← Q (l) + η n (l) z (l) t − Q (l) output: z (l) t ∀l ∈ {0, . . . , n} internal representations and updated weights W (l) a , W (l−1) b , Q (l) ∀l ∈ {1, . . . , n − 1} Biological implementation and comparison with experimental observations In this section, we introduce a biologically plausible neural circuit that implements the CCPC algorithm, denoted by BioCCPC. We also demonstrate that the details of this circuit resemble the neurophysiological properties of pyramidal cells in the neocortex and the hippocampus. Neural architecture The algorithm for BioCCPC (Alg. 1) summarized by the neural dynamics from Eq. (9) and weight update rules from Eqs. (10) can be implemented by a neural circuit with schematic shown in Fig. 2. The activity of the n − 1 hidden layers, {z (l) } n−1 l=1 , is encoded as the outputs of n − 1 sets of neurons, representing pyramidal neurons of different cortical regions. The matrices W ) is the efficacy of the synapse connecting z (l+1) (resp. z (l−1) ) to the pyramidal neurons z (l) . Because of the disjoint nature of these two inputs, we model these as synapsing respectively onto the distal (apical tuft) and proximal (mostly basal) dendrites of the pyramidal neurons, respectively; see Fig. 2. This is reminiscent of cortical pyramidal neurons, which also have two integration sites, the proximal compartment comprised of the basal and proximal apical dendrites providing inputs to the soma, and the distal compartment comprised of the apical dendritic tuft [22,65]. These two compartments receive excitatory inputs from two separate sources [18,66]. Similarly, the auxiliary variables n (l) are represented by the activity of interneurons in each cortical region. 3 The Q (l) synaptic weights are encoded in the weights of synapses connecting n (l) to z (l) , while Q (l) models the weights of synapses from z (l) to n (l) . In a biological setting, the implied equality of weights of synapses from z (l) to n (l) and the transpose of those from n (l) to z (l) can be guaranteed approximately by application of the same Hebbian learning rule [55]. However, note that our learning rules, unlike previous work, do not require interneurons of one layer to be connected to the pyramidal neurons of another layer. Neural dynamics and canonical components The neural dynamics of the pyramidal neurons of our circuit given by Eq. (9) can be recast as the dynamics of a three compartment neuron: τż (l) = −g (l) lk z (l) + g (l) a (v (l) a − z (l) ) + g (l) b (v (l) b − z (l) ),(11) where we have defined the apical and basal compartmental membrane potentials v (l) a = W (l) a z (l+1) − Q (l) n (l) , v (l) b = W (l−1) b z (l−1) .(12) Here we defined the leak conductance g (l) lk = c (l) − g (l) a . Interestingly, this is the same neural dynamics of a three-compartment neuron posited previously in [28,30], which we derived in a normative way. Here, g a (resp. g b ) are conductances between the apical (resp. basal) compartments and the soma, and g lk is the somatic leak. Our derivation clarifies the relationship between the physiological quantities and the objective function. Indeed, the basal and apical conductances play the role of the inverse variances in the predictive coding objective. These inverse variances, generally referred to as 'precisions' in the PC literature, have previously been argued to be encoded in lateral inhibitions [2] or top-down attention [69,70]. In our model, they are simply encoded in the compartmental conductances. In Sec. 5 we show that the leak term also has a functional role and determines a threshold for dynamically learning more diverse latent variables [71,72]. Synaptic weight updates Matching observations in the cortex, in our circuit [18,22,65,73,74], the basal and apical weights (W b and W a ) of our algorithm are updated differently. The basal synaptic weights of the pyramidal neurons, given by the elements of W δW (l−1) b ∝ g (l) a v (l) a − c (l) z (l) z (l−1) .(13) The first term in parentheses, g (l) a v (l) a , is proportional to the total apical current, given by the difference between the excitatory synaptic current in the apical tuft, W (l) a z (l+1) , and the inhibitory current induced by interneurons synapsing onto the distal compartment, Q (l) n (l) . Biologically, this factor can be approximated by the calcium plateau potential traveling down the apical shaft. The calcium plateau potential has been experimentally seen to drive plasticity of the basal synapses matching the derived update rule of our circuit [18,20,[23][24][25]. The second factor is simply the neural output. Because the total update is not purely dependent on the action potentials of the pre-and postsynaptic neurons, such plasticity is called non-Hebbian [25]. The synaptic learning rule for the apical weights W (l) a (Eq. 10b) and the synapses connecting the interneurons and the pyramidal neurons Q (l) (Eq. 10c) are simply Hebbian, also matching experimental observations in the cortex [19,75]. Features and limitations Here we summarize the main differences between the biological implementations of CCPC and PC. In PC, the forward and backward weights (in our notation W b and W a ) are symmetric, whereas in CCPC, they are not constrained to be symmetric and are generically not so (see Sec. 6). Another difference is that in PC, the connection between value neurons and error neurons is one-to-one and fixed; this constraint does not exist between the pyramidal and interneurons of our model, which are no longer one-to-one, and have plastic connectivity. For the relationship between our interneurons and the PC error neurons see Sec. 5. While we have shown the improvement in biological realism of our model over the traditional PC network, this was done in the linear case. However, recent work on deep linear networks [76], has provided many insights into the learning dynamics of deep networks. Some of the properties discovered in the deep linear network, like "balancedness" of weights [77,78], generalize for certain nonlinear networks [78]. We believe several of our observations will generalize in a similar fashion. There are other aspects that can also be improved. For example, the connectivity between pyramidal neurons and interneurons of CCPC is required to be symmetric. This symmetry can be achieved via Hebbian learning rules [55]; however, it would be interesting to explore whether such symmetry is indeed required. Note that in CCPC, symmetric connectivity is only required between neurons of the same cortical region, which is a much less stringent biological requirement than the that of symmetric weights between different cortical regions, as in PC. Another shortcoming of our model is that while the teaching signal for the basal synapses in CCPC is signed and graded, in the cortex these signals are generally believed to be stereotypical [18]. Graded calcium-mediated signals were recently observed [32]. Theoretical arguments We summarize some of the theoretical features of our framework. For proofs see SM Sec. B.) The errors are implicitly computed in v (l) a . In backpropagation as well as in PC, the learning algorithm computes a loss (or a local error) which is then used to compute updates to the weights. In CCPC, no error or loss is explicitly computed. So how does CCPC learn? In the following proposition, we show that in the g a → 0 limit, the algorithm implicitly computes an error in its weight updates. This quasi feed-forward or weak nudging limit, has been used to explore the learning dynamics of biologically plausible networks [30] and is related to the 'fixed prediction' assumption [4,61,62]. Proposition 1. Assume that the learning rules (Eqs. 10) are at equilibrium and we receive a new datapoint given by x T +1 , y T +1 . For c (l) = 0 and in the limit of ≡ g a /g b → 0, the leading term in for the forward weight updates δW (l−1) b for this new sample is given by δW (l−1) b ∝ v (l) a,T +1 z (l−1) T +1 = n−l W (l) a · · · W (n−1) a (y T +1 −ỹ T +1 ) z (l−1) T +1 + O( n−l+1 ), whereỹ T +1 = YZ (l−1) Z (l−1) Z (l−1) −1 z (l−1) T +1 is the optimal linear inferred value for y T +1 . This implies that v (l) a , which we argued can be implemented as the calcium plateau potential, implicitly encodes an error signal. This is in spirit similar to difference target propagation [47], where the difference between the forward pass and the target is explicitly computed and backpropagated. Adaptive latent dimension discovery for a single module. For optimization within a single hidden layer, we show that the system performs adaptive latent dimension discovery by thresholding eigenvalues of a Gramian including contributions from both adjacent layers. If we take the loss function in (8), setting g a = 1, g b = , and optimizing over W a , W b , we get the objective Tr[(c + )Z Z − Z Z X (XX ) −1 X + Y Y for Z. The next theorem indicates the solution: Theorem 1. Let the concatenated matrix Ξ = [ 1/2 C −1/2 x X, Y] ∈ R (d X +d Y )×T have SVD Ξ = (d X +d Y ) α=1 u α λ α v α with {u α ∈ R (d X +d Y ) } and {v α ∈ R T } both being sets of orthonormal vectors, with the convention that (λ α ) are sorted in decreasing order. We consider the minimization min Z∈R d×T , ZZ I d Tr (c + )Z Z − Z Z X (XX ) −1 X + Y Y .(14) Then one of the optimal solutions is given byẐ = D α=1 w α 1(λ α > c + )v α where {w α ∈ R d } is an arbitrary set of orthonormal vectors, with D = min(d, d X + d Y ). Thus, the latent modes are found by thresholding the eignevalues [72] of the sum of Gramians [71] in Eq. (14). This implies that the quantity c (l) that we showed is encoded in the somatic compartment leak conductance, biologically sets an adaptive threshold for the dimensionality of the latent representation. We tested our BioCCPC algorithm on the MNIST dataset [79], with one-hot labels. We centered the input images and normalized by the standard deviation. We compared our results with a standard predictive coding network [4] with linear activation (PC) and with a version that includes a covariance constraint (PC+cons; Sec. C). We used unit variances, σ (l)2 = 1, in the predictive-coding networks, and matching conductances in our algorithm, g (1) b = g (n) a = 1, g (l) a = g (l) = 1/2 for other l. Our method finds solutions that are almost optimal in terms of predictive-coding loss (see Fig. 3a) and are generally better than the solutions obtained from simply adding the covariance constraint to a predictive-coding network. Learning is slower in our network than in unconstrained PCN, as it takes time for the covariance constraint to be enforced. It is possible that different initialization schemes could improve the learning speed, but we leave this for future work. The covariance constraint is saturated in the narrowest layer-in our example, the second hidden layer, of size 5-see Fig. 3c. The activation in the other hidden layer is rank-restricted by the narrowest layer, as described above, and so it is only whitened within a 5D subspace, Fig. 3b. In the supplementary materials, we provide further numerical experiments, where we look at the effects of the thresholding, the size of the network, and performance on other datasets. We also show that the forward and backward weights, W b and W a , are not symmetric, in accordance with biological observations, and we explore their relation. See SM Sec. C for these and other simulation details. Conclusion We have derived a biologically plausible algorithm for covariance-constrained predictive coding and have shown that it avoids the criticisms of PC and has many features in common with recent experimental observations. Our algorithm does not have symmetric forward and backward connectivity matrices, and does not need one-to-one connectivity between any neurons. Furthermore, we showed that the learning rules of our algorithm closely match experimental observations, and we connected our parameters to neurophysiological quantities. Using the simplicity afforded to us because of the linearity, we showed a number of interesting properties of our algorithm, including the effect of the somatic leak term, which acts as an adaptive thresholding mechanism. We also showed how our algorithm implicitly computes a difference term that it uses for learning. We argued that this term could be encoded in the calcium plateau potential, which is known to affect the learning of basal synapses. We hope that this concrete connection between our normative model and observation can further our understanding of cortical computation by guiding future experiments. Broader impact We do not foresee any potential societal harm caused by our work. We hope our contribution moves forward the discussion on the relationship between multi-layer networks and cortical structure. Supplementary Materials A Derivation of upper bound In this section we give a more detailed derivation of our framework, using a slightly modified order of steps compared to the main text. This should help clarify some aspects of the final algorithm. Similar to the main text, we start with the predictive coding objective and introduce the weight doubles. We arrive at min Z,W L = min Z,Wa,W b 1 2 n−1 l=1 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + g (l+1) a Z (l+1) − W (l) a Z (l) 2 F + c (l) Z (l) Z (l) 2 F ,(15) where we have also introduced a quadratic cost (a prior on Z) with coefficient c (l) . We now impose the constraint on the empirical covariance, 1 T Z (l) Z (l) ρ (l) I, of the internal variables Z (l)4 : min Z,W L ≤ min Z,Wa,W b 1 T Z (l) Z (l) ρ (l) I 1 2 n−1 l=1 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + g (l+1) a Z (l+1) − W (l) a Z (l) 2 F + c (l) Z (l) Z (l) 2 F . (16) Expanding this and using the constraint we again upper bound the objective: min Z,W L ≤ min Z,Wa,W b 1 T Z (l) Z (l) ρ (l) I 1 2 n−1 l=1 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + g (l+1) a Z (l+1) Z (l+1) 2 F + g (l+1) a Tr −2Z (l+1) W (l) a Z (l) + T ρ (l) W (l) a W (l) a + c (l) Z (l) Z (l) 2 F ,(17) We next implement the inequality constraint using positive-definite Lagrange multipliers: min Z,W L ≤ min Z,Wa,W b max Q n−1 l=1L (l) ,(18)whereL (l) = 1 2 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + c (l) q Tr Q (l) Q (l) Z (l) Z (l) − T ρ (l) I + g (l+1) a Tr −2Z (l+1) W (l) a Z (l) + T ρ (l) W (l) a W (l) a + c (l) Z (l) Z (l) 2 F + g (l+1) a Z (l+1) Z (l+1) 2 F ,(19) where we have added a positive constant c A final upper bound to (18) can be found if eachL (l) is optimized individually. Explicitly, we perform gradient descent of each variable, Z (l) , W (l) a , W (l) b , only with respect toL (l) and not with respect to the entire objective L: δZ (l) = ∇ Z (l) L → ∇ Z (l)L (l) , δW (l) a = ∇ W (l) a L → ∇ W (l) aL (l) , δW (l) b = ∇ W (l) b L → ∇ W (l) bL (l) .(20) This affords us an upper bound because we are no longer finding the actual minimum of L. Symbolically this can be written as in (8): min Z,W L ≤ n−1 l=1 min Z,Wa,W b max QL (l) ,(21) where the precise definition of this equation is given by the update equations (20). Note that these updates are no longer curl-free and hence cannot be found as the gradient of a single objective function. B Theoretical section proofs Note that the definition of in Proposition 1 is different from the definition in other sections. B.1 Proof of Proposition 1 To prove Proposition 1, we first show that the latent variables z (l) have a specific form. As in the main text, we define (l) = g (l) a /g (l) b and, for simplicity, we assume (l) = is the same for all layers. For simplicity also, we assume c (l) (equivalently d (l) in previous section) is equal to zero, and use c (l) q = g (l) a as in the previous section. This choice makes the notation simpler without changing our results. Equation (9) becomeṡ z (l) = g (l) b W (l−1) b z (l−1) + g (l) a W (l) a z (l+1) − g (l) b z (l) − g (l) a Q (l) Q (l) z (l) .(22) Proposition 2. Assume the neural dynamics given in Eq. (22) are at equilibrium. Then the latent variables take the form z (n−k) = B k W (n−k−1) b z (n−k−1) + k A k W (n−1) a y ,(23) where B k and A k are independent of y and defined recursively via A k+1 = W (n−k−1) a B k A k , B k+1 = 1 − F (n−k−1) W (n−k−1) a B k W (n−k−1) b −1 F (n−k−1) , with F (l) = I + Q (l) Q (l) −1 and B 1 = F (n−1) , A 1 = 1. Proof. We prove this via induction. Starting from the neural dynamics Eq. (22), at equilibrium (ż (l) = 0) we have: z (l) = F (l) W (l−1) b z (l−1) + W (l) a z (l+1) . Setting l = n − 1 we get: z (n−1) = F (n−1) W (n−2) b z (n−2) + W (n−1) a y , which satisfies the proposed form in Eq. (23) for k = 1. Now, we assume that the form in Eq. (23) holds for z (n−k) , and show that it also holds for z (n−k−1) . We have z (n−k−1) =F (n−k−1) W (n−k−2) b z (n−k−2) + W (n−k−1) a z (n−k) =F (n−k−1) W (n−k−2) b z (n−k−2) + W (n−k−1) a B k W (n−k−1) b z (n−k−1) + k A k W (n−1) a y . Collecting z (n−k−1) and multiplying by the inverse on the left we arrive at: z (n−k−1) = 1 − F (n−k−1) W (n−k−1) a B k W (n−k−1) b −1 F (n−k−1) × × W (n−k−2) b z (n−k−2) + k+1 W (n−k−1) a B k A k W (n−1) a y , as proposed. Note that, whereas Proposition 1 holds only in the → 0 limit, Proposition 2 holds for general . Using this result we show that the relationship between the activity, feedback and feedforward inputs of the l th module has a simple form given in the following corollary: Corollary 1. The leading order in of the dependence of z (l) and z (l+1) on z (l−1) and y, which has non-zero dependence on y, is given by Proof. The result follows directly from applying Eq. (23) twice and then dropping all but the leading dependence on in the coefficient of the y term. Now, we restate and prove Proposition 1, which can be viewed as a generalization of the results in [55]. Note that in this proposition, the exact form of C (l) 1 and C (l) 2 is not important. All that matters is that both z (l) and z (l+1) have a linear dependence on z (l−1) . Since the dependence of z (l) on y is subleading compared to the dependence of z (l+1) on y, this can be dropped in the limit of → 0. Proposition 1. Assume that the learning rules (Eqs. 10) are at equilibrium and we receive a new datapoint given by x T +1 , y T +1 . For c (l) = 0 and in the limit of ≡ g a /g b → 0, the leading term in for the forward weight updates δW z (l) = C (l) 1 z (l−1) + O( n−l ), z (l+1) = C (l) 2 z (l−1) + n−l−1 W (l+1) a · · · W (n−1) a y + O( n−l ),(l−1) b for this new sample is given by δW (l−1) b ∝ v (l) a,T +1 z (l−1) T +1 = n−l W (l) a · · · W (n−1) a (y T +1 −ỹ T +1 ) z (l−1) T +1 + O( n−l+1 ), whereỹ T +1 = YZ (l−1) Z (l−1) Z (l−1) −1 z (l−1) T +1 is the optimal linear inferred value for y T . Proof. The statement follows from two applications of the update rule for W b given in Eq. (10a) for W (l−1) b . Putting c (l) = 0 we have: δW (l−1) b ∝ W (l) a z (l+1) − Q (l) Q (l) z (l) z (l−1) .(24) The equilibrium condition given by δW b = 0 is given by averaging over the entire dataset: Q (l) Q (l) Z (l) Z (l−1) = W (l) a Z (l+1) Z (l−1) .(25) Now we plug the derived values in Corollary 1. After right-multiplying by (Z (l−1) Z (l−1) ) −1 we get: Q (l) Q (l) C 1 = W (l) a C 2 + n−l−1 W (l) a · · · W (n−1) a YZ (l−1) Z (l−1) Z (l−1) −1 .(26) Now, we look at the update for a new sample T + 1. Plugging the above as well as the equations from Corollary 1 into the update rule we see that the C (l) 2 dependent term cancels and we are only left with the term depending on y: δW (l−1) b ∝ n−l W (l) a z (l+1) T +1 − Q (l) Q (l) z (l) T +1 z (l−1) T +1 = n−l W (l) a · · · W (n−1) a (y T +1 −ỹ T +1 ) z (l−1) , withỹ T +1 as defined in the proposition. B.2 Proof of Theorem 1 and consequences In this section we look at some properties of the within-layer computations of CCPC. We will focus on the computations within the l th layer and for clarity of notation, for this subsection only, we rename Z (l) → Z, Z (l+1) → Y, and Z (l−1) → X. We will assume ρ (l) = 1 for brevity. The results have straight-forward generalizations for the case where ρ (l) = 1. Theorem 1. Let the concatenated matrix Ξ = UΛV = [ 1/2 C −1/2 x X, Y] ∈ R (d X +d Y )×T have singular value decomposition Ξ = (d X +d Y ) α=1 u α λ α v α , with {u α ∈ R (d X +d Y ) } and {v α ∈ R T } both being sets of orthonormal vectors, and singular values (λ α ) sorted in decreasing order. We consider the minimization min Z∈R d×N ZZ I d Tr (c + )Z Z − Z Z X (XX ) −1 X + Y Y .(27) Then one of the optimal solutions is given byẐ = D α=1 w α 1(λ α > c + )v α where {w α ∈ R d } is an arbitrary set of orthonormal vectors, with D = min(d, d X + d Y ). Proof. Let the SVD of Z be WΛ Z V Z . Minimizing over Z amounts to finding the matrices W, Λ Z , and V Z that give the smallest value of the objective. Because of the constraint ZZ I d , Λ Z satisfies 0 ≤ λ Zα ≤ 1. Plugging this form into the optimization objective we have Tr (c + )Z Z − Z Z X (XX ) −1 X + Y Y = Tr (c + )Λ Z Λ Z − Λ Z Λ ZṼ Λ ΛṼ whereṼ = V V Z . Note that W does not appear in this expression and therefore remains undetermined in this optimization objective. V only appears in the second term and hence can be found by optimizing the second term alone. This term is of the form − α,β λ 2 Zα |Ṽ αβ | 2 λ 2 β . TheṼ that minimizes the term is the one that pairs λ β and λ Zα , in the sorted order. Assuming (λ Zα ) are also sorted in the decreasing order, this is achieved by setting V Z = V. Plugging this in, the entire expression simplifies to − α (λ α − c − )λ Zα . Since 0 ≤ λ Zα ≤ 1, it is clear that, for a minimum, if (λ α − c − ) < 0, λ Zα = 0, while (λ α − c − ) > 0 implies λ Zα = 1. In case of equality, the choice of λ Zα does not matter. Hence From this theorem, we derive two corollaries, describing the different limits of → 0 and → ∞. The proofs of these corollaries follow directly from Theorem 1 which maps the within-layer computation of CCPC onto a PCA problem, and the known properties of PCA. Corollary 2. If the input sample covariance matrix, C x , is full rank, the internal representation Z in the limit → 0 is driven by the principal components (scores) of Y, i.e., Z * = W a Y + W b X,(28) with W a the top principal directions of Y and W b the maximally correlated directions between X and the columns of W a . The BioCCPC network uses an inequality constraint on the covariance of the hidden variables, 1 T Z (l) Z (l) ρ (l) I, and this constraint is saturated at convergence. 6 The scale, ρ (l) , is therefore important: a scale that is too small or too big forces the hidden-layer activities into a range that can hinder learning, as seen in Fig. 4a. Interestingly, the BioCCPC algorithm with a well-chosen constraint scale converges faster than a plain PC network, that is, it can reach a lower PC loss given the same number of training samples (Fig. 4a). The predictive-coding loss, Eq. (5), penalizes prediction errors in all layers. If we think about the task as supervised learning, however, it might make more sense to focus only on the accuracy of the prediction in the last layer. The best output prediction error may require a different constraint scale compared to the lowest PC loss, cf. Fig. 4b and Fig. 4d. C.2 Effect of dataset choice While we derived our algorithm by starting from the supervised PC framework of [4], the same derivation would hold for the unsupervised or self-supervised PC networks. In these settings, PC can be thought of as a framework which combines information from the inputs at the first and last layer, rather than a method for predicting an output from an input. In the linear case, this has much in common with biologically plausible circuits that perform canonical correlation analysis (CCA) [55,71]. We therefore tested the algorithm on Mediamill [81], a dataset consisting of two views of the same scene, one given by the video and the other by text annotation. The results (Figs. 4b-d and Fig. 5) show similar trends as those obtained on MNIST. In particular, an appropriate choice of constraint scale leads to results that are competitive with less biologically plausible implementations of PC. In Fig. 6 we show the evolution of the validation-set loss for four other datasets: FashionMNIST, CIFAR10, CIFAR100, and the LFW (Labeled Faces in the Wild). The first three are supervisedlearning scenarios and we use a network similar to the one we employed for MNIST. The LFW dataset contains pairs of views of the same subject, and so we employ a symmetric autoencoder-like architecture for this self-supervised task. As before, we see that the BioCCPC algorithm converges to a solution that has predictive-coding loss comparable to the minimum obtained from traditional PC implementations (see Fig. 6). C.3 Network architecture We tested BioCCPC across different architectures, such as a shallow network with a small hidden-layer dimension (Fig. 4) and a deeper network with two hidden layers, one of which is large-dimensional ( Fig. 4a,b,d). In both cases BioCCPC compares well to PC. C.4 Lack of symmetry in feedforward and feedback weights One of the difficulties with a biological implementation of predictive coding is that it requires symmetric weights across layers. This is not so much an issue of plausibility-indeed, symmetric weights can be obtained simply by using the same Hebbian rules to train both feedforward and feedback connections-but rather a problem with realism: weights in many areas of the brain are not in fact seen to be symmetric. MNIST hidden sizes [10,5] MNIST hidden sizes [25,5] Our algorithm explicitly breaks the symmetry between the weights, leading to solutions that generically alleviate this difficulty. Indeed, in some cases there is very little correlation between feedforward and feedback weights; see Fig. 7. However, in some cases we see that the forward and backward weights converge to similar values even though the learning rules are not symmetric. We suspect that this can occur in cases where there is a large imbalance between the width of adjacent layers, the larger the imbalance (i.e. the lower rank the wider layer is) the more correlation we see in our experiments; see Fig. 8. In biologically relevant neural networks, we do not generally see this large imbalance in the width of adjacent layers and therefore we do not expect to see high correlation between the forward and backward weights. MediaMill C.5 Predictive coding with constraint In the main text we considered a predictive-coding network that imposes a covariance constrained but we did not make explicit what this entails. Explicitly, we augmented the Whittington-Bogacz objective, Eq. and optimized using stochastic gradient descent. This lets us distinguish the effect of the constraint from the effect of the approximations we used in the derivation. C.6 Details of simulation runs We initialized our the weights in our networks using uniform random numbers in the range [−a, a], where a = 1/ √ number of columns. This is similar to Xavier initialization [82] except the scale of initial weights depends only on the number of columns instead of both rows and columns. We have found this initialization to provide slightly better results. We optimized hyperparameters using Optuna [83] with the tree-structured Parzen Estimator (TPE) algorithm [84]. We ran the optimization for 100 trials and repeated it 10 times with different random seeds for every combination of dataset, network architecture, and constraint scale ρ (with the exception of the plain PC algorithm which does not have a constraint). In each case we optimized the learning rate η used in the weight updates and the learning rate τ −1 of the fast dynamics (see Algorithm 1). For networks that have a constraint, we allowed a different learning rate f η for the Q dynamics, and optimized for the factor f . Finally, for BioCCPC network, we allowed for different learning rates for the forward and backward weights, W b and W a , and optimized for the ratio between these learning rates. Hyperparameter optimization runs were trained for 500 batches with batch size 100 and were evaluated on a held-out validation set comprising 500 samples. Each run was repeated 4 times with different random initialization to account for stochasticity in the learning dynamics. The simulation runs that were used to make the figures in the paper were based on the hyperparameter optimization results that yielded the lowest predictive-coding loss. Since hyperparameter optimization tends to push the learning rate up to the brink of instability, the learning rate used in the simulations was lowered by multiplying by a certain "safety" factor α. We generally used α = 0.8. We ran our simulations on the CPU. Each run took from 1 to 4 minutes when run single-threaded on a 24-core Intel Xeon CPU E5-2643 at 3.40GHz running Linux. We also used a cluster to run the hyperparameter optimization for different architectures in parallel. D Generalization error CIFAR10 CIFAR100 FashionMNIST LFW MediaMill MNIST (a) (b) (c) (f ) (e) (d) Figure 9: There is little overfitting in the linear setting that we are investigating. The plots compare the training-and validation-set predictive-coding losses for both the less biologically plausible predictive-coding implementation from [4] (PC) and for our algorithm (BioCCPC). The linear constraint that we impose on our network is a strong regularizer. Figure 9 compares the training and validation losses during training on six different datasets (as described above). We see that there is very little overfitting in all cases. ≤ min Z (l) ,W (l) a ,W (l) b Z (l) Z (l) T ×I 1 2 g (l) b Z (l) − W (l−1) b Z (l−1) 2 F + c (l) Z (l) 2 F + g (l) a Tr −2Z (l+1) W (l) a Z (l) + g (l) a T W (l) a W (l) a The first inequality results from the fact that the parameters of the new optimization problem are a subset of the parameters of the original optimization problem. In the second inequality we have expanded the second term ofL (l) and used the constraint 1 T Z (l) Z (l) I to replace Z (l) Z (l) with T , making the term larger, resulting again in an upper bound. Figure 1 : 1Schematic architecture of the predictive coding network (PCN) and our covariance-constrained network (BioCCPC). (a) PCN from encoded as the weights of synapses between the pyramidal neurons of adjacent layers. Explicitly, the matrix W Figure 2 : 2Schematic of the biological implementation of the BioCCPC algorithm. Left: overall connectivity pattern, showing all-to-all connections between pyramidal neurons in consecutive layers, and between pyramidal and interneurons in the same layer. Right: zoom in on one neuron and its immediate neighbors, showing the membrane potentials and synaptic weights that are relevant for our algorithm. are updated by the product of two factors represented in the corresponding post-and pre-synaptic neurons (Eq. 10a): From this theorem, we derive the computational limits of → ∞ and → 0 in the supplementary materials. We also show that the neurons of the narrowest layer get whitened and the other layers become low-rank. For further details of these statements and all proofs see SM Sec. B. Figure 3 : 3Our algorithm (BioCCPC) compares well to less biologically plausible predictive-coding (PC) implementations, while enforcing an inequality constraint on the covariance of hidden-layer activations. Simulations run on the MNIST dataset, using a network with two hidden layers, sizes 50 and 5. Shaded area represents the 95% range out of 40 runs. (a) Evolution of the predictive-coding loss, Eq. (5), on a validation set during learning. (b), (c) Evolution of the eigenvalues of the covariance matrix of hidden-layer activations in the two hidden layers. The dashed lines show the scale of the constraint; see SM Sec. A. Inset: covariance matrix at end of training. Note how activity is whitened in the second hidden layer (c), and low-rank in the first hidden layer (b). the coefficient of the inequality constraint. This constant does not affect the solution of the optimization problem but does affect the details of the neural dynamics. Motivated by biological arguments 5 , we set this constant equal to g (l)a . In doing so, the interneuron variables no longer need a (1/g (l) a ) in their definition, as in the main text, and are instead given by n (l) = Q (l) z (l) . λ Zα = 1(λ α > c + ) is a valid minimum. The vectors {w α } are the columns of W, while {v α } are the columns of V. Hence, the result. Figure 4 : 4Dependence of BioCCPC performance on constraint scale. Unless otherwise noted, this is the predictive-coding loss (Eq. 5) after 500,000 iterations. (a) A large network with two hidden layers of sizes 50 and 5 trained on MNIST. (b) The same large network as in (a) run on the Mediamill dataset. See text. (c) A small net (single hidden layer of size 5) on Mediamill. (d) Mean-squared prediction error for the large net on Mediamill, showing that the best prediction performance may occur at a different choice of ρ compared to the lowest predictive-coding loss. Figure 5 : 5Comparison of our algorithm with PC on the Mediamill dataset. Simulations run using a network with two hidden layers, sizes 50 and 5. Shaded area represents the variation over 10 runs. (a) Evolution of the predictive-coding loss, Eq. (5), on a validation set during learning. (b) and (c) Evolution of the eigenvalues of the covariance matrix of hidden-layer activations in the two hidden layers. The dashed lines show the scale of the constraint; see SM Sec. A. Inset: covariance matrix at end of training. Note how activity is whitened in the second hidden layer (c), and low-rank in the first hidden layer (b). Note also that the network performance is good even though the constraint is not yet fully converged. Figure 6 : 6Comparison of our algorithm with PC on a variety of datasets. The name of the dataset and the architecture we employed are shown below each figure. Shaded area represents the variation over 10 runs. Figure 7 : 7Comparison of feedforward and feedback weights after learning on a network with hidden layers of sizes 50 and 5 trained on the Mediamill dataset. Figure 8 : 8Comparison of feedforward and feedback weights after learning for different network sizes. This is trained on MNIST. (a) There is less symmetry in a smaller network with hidden layers of sizes 10 and 5. (b) There is more symmetry in a larger network, with hidden layers 25 and 5. ( 2 )σ 2, with a Lagrange multiplier enforcing an inequality constraint on the covariance of hidden variables, similar to what we do in BioCCPC: (l)2 + Tr Q (l) Q (l) (Z (l) Z (l) − T ρ (l) I), There are multiple types of interneurons targeting pyramidal cells[67,68]. The interneurons of BioCCPC most closely resemble the somatostatin-expressing interneurons, which preferentially inhibit the apical dendrites. The factors ρ (l) which set the scale of the constraint were set to 1 in the derivation in the main text for brevity. However, it is sensible for this scale factor to vary from layer to layer and possibly depend on the layer width. In Sec. C, we explore the effects of this parameter on the performance of the network.5 Since the feedback of the interneurons is received and integrated in the apical compartment, we expect the apical-to-soma conductance to also be present here. More precisely, the constraint is saturated only within the subspace that Z (l) occupies. In general, this is only the case in the layer with the smallest dimension, as the activity in the other layers is restricted to a subspace of dimension given by the smallest layer; see Corollary 5. Corollary 3. If the input sample covariance matrix, C x , is full rank, the internal representation Z in the limit → ∞ is driven by the maximally correlated directions between C −1/2 x X and Y, i.e.,with W a , W b projecting onto a common lower-dimensional subspace so that the projections are maximally correlated.B.3 Rank structure of CCPCIn this section again we set c (l) = 0 and ρ (l) = 1, and assume g a = 1, g b = is the same for all layers. The generalization to other cases is straightforward. We demonstrate a series of rank constraints on the intermediate layers of CCPC. The first of these is very general and requires no assumptions. The following statements become gradually more tight but also require more assumptions.In this section, we define r x and r y to be the ranks of the the covariances of X and Y, and r (l) to be the rank of the covariance of Z (l) at the equilibrium of the CCPC update equations. We take r (0) = r x and r (n) = r y .Corollary 4. The rank of the l th layer is bounded by r (l+1) + r (l−1) :Proof. This follows directly from Theorem 1 which describes Z (l) as the whitened PCA of the concatenation of Z (l−1) and Z (l+1) .When the adjacent layers are whitened, i.e., the non-zero eigenvalues of Z (l+1) Z (l+1) and Z (l−1) Z (l−1) are all one, we can prove a tighter form of the rank constraint. This is relevant for the intermediate layers of CCPC other than the final hidden layer. This proposition is relevant for the final hidden layer only if the supervisory input Y is white.c be the rank of the covariance matrix Z (l−1) Z (l+1) . The rank of intermediate layers l ≤ n − 2 is given by:Proof. This is a slight generalization of[71]. It follows from the eigenvalue structure of the concatenated data matrix given in Theorem 1, where only a subset of the eigenvalues exceed the threshold given by . The number of eigenvalues that exceed this threshold is given by Eq.(31).Applying this proposition to all the layers of a CCPC network, we arrive at the following corollary:Assume that Y is white and ≥ 1. In a CCPC network with c (l) ≥ 0, ρ (l) = 1, and layer widths w (l) , the rank of the latent variables z (l) is less than or equal to min{r x , r y , w(1), · · · , , w (n−1) }.In other words, the rank of the latent variables is bounded by the minimum of the input/output ranks and the bottleneck of the network.B.4 Alignment and relation to CCAIn the limit where the internal loss terms in Eq. (5) are zero, we effectively have a deep linear network[76]. A lot is known about deep linear networks, trained by gradient descent of weights. One of the remarkable properties of such networks is that the weights remain balanced, namely W (l) W (l) −W (l−1) W (l−1) remains a constant, under gradient descent. 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[ "five rules for friendly rivalry in direct reciprocity", "five rules for friendly rivalry in direct reciprocity" ]
[ "Yohsuke Murase \nRIKEN Center for Computational Science\n650-0047KobeHyogoJapan\n", "&amp; Seung ", "Ki Baek \nDepartment of Physics\nPukyong National University\n48513BusanKorea\n" ]
[ "RIKEN Center for Computational Science\n650-0047KobeHyogoJapan", "Department of Physics\nPukyong National University\n48513BusanKorea" ]
[]
Direct reciprocity is one of the key mechanisms accounting for cooperation in our social life. According to recent understanding, most of classical strategies for direct reciprocity fall into one of two classes, 'partners' or 'rivals'. A 'partner' is a generous strategy achieving mutual cooperation, and a 'rival' never lets the co-player become better off. They have different working conditions: For example, partners show good performance in a large population, whereas rivals do in head-to-head matches. By means of exhaustive enumeration, we demonstrate the existence of strategies that act as both partners and rivals. Among them, we focus on a human-interpretable strategy, named 'cApRi' after its five characteristic ingredients, i.e., cooperate, accept, punish, recover, and defect otherwise. Our evolutionary simulation shows excellent performance of CAPRI in a broad range of environmental conditions.
10.1038/s41598-020-73855-x
null
214,743,569
2004.00261
67834f86ed852fbb1111d3e9672b4a0ecdea9385
five rules for friendly rivalry in direct reciprocity 0123456789 Yohsuke Murase RIKEN Center for Computational Science 650-0047KobeHyogoJapan &amp; Seung Ki Baek Department of Physics Pukyong National University 48513BusanKorea five rules for friendly rivalry in direct reciprocity 012345678910.1038/s41598-020-73855-x Direct reciprocity is one of the key mechanisms accounting for cooperation in our social life. According to recent understanding, most of classical strategies for direct reciprocity fall into one of two classes, 'partners' or 'rivals'. A 'partner' is a generous strategy achieving mutual cooperation, and a 'rival' never lets the co-player become better off. They have different working conditions: For example, partners show good performance in a large population, whereas rivals do in head-to-head matches. By means of exhaustive enumeration, we demonstrate the existence of strategies that act as both partners and rivals. Among them, we focus on a human-interpretable strategy, named 'cApRi' after its five characteristic ingredients, i.e., cooperate, accept, punish, recover, and defect otherwise. Our evolutionary simulation shows excellent performance of CAPRI in a broad range of environmental conditions. Theory of repeated games is one of the most fundamental mathematical frameworks that has long been studied for understanding how and why cooperation emerges in human and biological communities. Even when cooperation cannot be a solution of a one-shot game, repetition can enforce cooperation between the players by taking into account the possibility of future encounters. A spectacular example is the prisoner's dilemma (PD) game: It describes a social dilemma between two players, say, Alice and Bob, in which each player has two options 'cooperation' (c) and 'defection' (d). The payoff matrix for the PD game is defined as follows: where each entry shows (Alice's payoff, Bob's payoff) with T > R > P > S and 2R > T + S . If the game is played once, mutual defection is the only equilibrium because Alice maximizes her payoff by defecting no matter what Bob does. However, if the game is repeated with sufficiently high probability, cooperation becomes a feasible solution because the players have a strategic option that they can reward cooperators by cooperating and/or they can punish defectors by defecting in subsequent rounds (see, e.g., Table 1). This is known as direct reciprocity, one of the most well-known mechanisms for the evolution of cooperation 1 . Through a series of studies, recent understanding of direct reciprocity proposes that most of well-known strategies act either as partners or as rivals 2,3 . Partner strategies are also called 'good strategies' 4,5 , and rival strategies have been described as 'unbeatable' 6 , 'competitive' 2 , or 'defensible' 7,8 . Derived from our everyday language, the 'partner' and 'rival' are defined as follows. As a partner, Alice aims at sharing the mutual cooperation payoff R with her co-player Bob. However, when Bob defects from cooperation, Alice will punish Bob so that his payoff becomes less than R. In other words, for Alice's strategy to be a partner, we need the following two conditions: First, π A = π B = R when Bob applies the same strategy as Alice's, where π A and π B represent the long-term average payoffs of Alice and Bob, respectively. Second, when π A is less than R because of Bob's defection from mutual cooperation, π B must also be smaller than R, whatever Bob takes as his strategy. It means that one of the best responses against a partner strategy is choosing the same partner strategy so that they form a Nash equilibrium. If a player uses a rival strategy, on the other hand, the player aims at a payoff higher than or equal to the co-player's regardless of the co-player's strategy. Thus, as long as Alice is a rival, it is guaranteed that π A ≥ π B . Note that these two definitions impose no restriction on Bob's strategy, which means that the inequalities are unaffected even if Bob remembers arbitrarily many previous rounds. ( www.nature.com/scientificreports/ Which of these two traits is favoured by selection depends on environmental conditions, such as the population size N and the elementary payoffs R, T, S, and P. For instance, a large population tends to adopt partner strategies when R is high enough. A natural question would be on the possibility that a single strategy is both a partner and a rival simultaneously: The point is not to gain an extortionate payoff from the co-player in the sense of the zero-determinant (ZD) strategies 9 but to provide an incentive to form mutual cooperation. Let us call such a strategy a 'friendly rival' hereafter. Tit-for-tat (TFT) or Trigger strategies can be friendly rivals in an ideal condition that the players are free from implementation error due to "trembling hands". However, this is not the case in a more realistic situation in which actions can be misimplemented with probability e > 0 . Here, the apparent contradiction between the notions of a partner and a rival is seen as the most acute form. That is, Alice must forgive Bob's erroneous defection to be a partner and punish his malicious defection to be a rival, without knowing Bob's intention. This is the crux of the matter in relationships. In this work, by means of massive supercomputing, we show that a tiny fraction of friendly rival strategies exist among deterministic memory-three strategies for the iterated PD game without future discounting. Differently from earlier studies 9-17 , our strategies are deterministic ones, which makes each of them easy to implement as a behavioural guideline as well as a public policy without any randomization device 18 . In particular, we focus on one of the friendly rivals, named CAPRI, because it can be described in plain language, which implies great potential importance in understanding and guiding human behaviour. We also argue that our friendly rivals exhibit evolutionary robustness 13 for any population size and for any benefit-to-cost ratio. This property is demonstrated by evolutionary simulation in which CAPRI overwhelms other strategies under a variety of environmental parameters. Methods Despite the fundamental importance of memory in direct reciprocity, combinatorial explosion has been a major obstacle in understanding the memory effects on cooperation: Let us consider deterministic strategies with memory length m, which means that each of them chooses an action between c and d as a function of the m previous rounds. The number of such memory-m strategies expands as N = 2 2 2m , which means N m=1 = 16 , N m=2 = 65536 , and N m=3 ≈ 1.84 × 10 19 . The number of combinations of these strategies grows even more drastically, which renders typical evolutionary simulation incapable of exploring the full strategy space. Here, we take an axiomatic approach 7,8,20 to find friendly rivals. That is, we search for strategies that satisfy certain predetermined criteria, and the computation time for checking those criteria scales as O(N) instead of O(N 2 ) or greater. More specifically, we begin with the following two criteria 7,8 : 1. Efficiency: Mutual cooperation is achieved with probability one as error probability e → 0 + , if both Alice and Bob use this strategy. 2. Defensibility: If Alice uses this strategy, she will never be outperformed by Bob when e = 0 regardless of initial actions. This is a sufficient condition for being a rival, i.e., lim e→0 + (π A − π B ) ≥ 0. The efficiency criterion requires a strategy to establish cooperation in the presence of small e when both the players adopt this strategy. This criterion is satisfied by many generous strategies such as unconditional cooperation (AllC), generous TFT (GTFT), Win-Stay-Lose-Shift (WSLS) and Tit-for-two-tats (TF2T). Partner strategies constitute a sub-class of efficient ones by limiting the co-player's payoff to be less than or equal to R regardless of the co-player's payoff 2,3,5 . On the other hand, a defensible strategy must ensure that the player's long-time average payoff will be no less than that of the co-player who may use any possible strategy, and this idea is equivalent to the notion of a 'rival strategy' 2,3 . Defensible strategies include unconditional defection (AllD), Trigger, TFT, and extortionate ZD strategies. Figure 1a schematically shows how these two criteria narrow down the list of strategies to consider. The overlap of efficient and defensible strategies means a set of friendly rivals because it is a subset of partner strategies. It assigns the most strict limitation on the co-player's payoff among the partner Table 1. Description of well-known strategies in the iterated PD game. Whenever possible, each strategy is represented as a tuple of five probabilities, i.e., (p 0 , p R , p S , p T , p P ) , where p 0 means the probability to cooperate in the first round, and p β means the probability to cooperate after obtaining payoff β in the previous round (see Eq. 1). Here, a zero-determinant (ZD) strategy has a positive parameter φ , and its other parameter η lies in the unit interval 9,13,19 . Strategy Description AllC (1, 1, 1, 1, 1) AllD (0, 0, 0, 0, 0) Tit-for-tat (TFT) (1, 1, 0, 1, 0) Generous TFT (1, 1, q, 1, q) with 0 < q < 1 Tit-for-two-tats (TF2T) Defect if the co-player defected in the previous two rounds. Win-Stay-Lose-Shift (WSLS) (1, 1, 0, 0, 1) generous ZD (1, 1, 1 − φ[(1 − η)(S − R) + T − S], φ[(1 − η)(R − T) + T − S], φ(1 − η)(R − P)) extortionate ZD (0, 1 − φ(1 − η)(R − P), 1 − φ[(1 − η)(S − P) + T − S], φ[(1 − η)(P − T) + T − S], 0) Trigger Defect if defection has ever been observed. Scientific RepoRtS | (2020) 10:16904 | https://doi.org/10.1038/s41598-020-73855-x www.nature.com/scientificreports/ strategies as shown in Fig. 1b. Indeed, the overlap region between these two criteria is extremely tiny: It is pure impossibility for m = 1 , and we find only 8 strategies out of N = 65536 for m = 2. To further narrow down the list of strategies, we impose the third criterion 7,8 : 3. Distinguishability: The strategy has a strictly higher payoff than the co-player's when its strategy is AllC in the small-error limit, i.e., lim e→0 + (π A − π AllC ) > 0. This requirement originates from evolutionary game theory: If this criterion is violated, the number of AllC players may increase due to neutral drift, which eventually makes the population vulnerable to invasion of defectors such as AllD. We check these criteria for each strategy by representing it as a graph and analysing its topological properties (see Supplementary Methods S1 at the end of this manuscript). If a strategy satisfies all those three criteria, it will be called 'successful' . Among deterministic memory-two strategies, it is known that only four strategies satisfy these three criteria 7 . They have minor differences from each other, and one of them is called TFT-ATFT, which is a combination of TFT and anti-tit-for-tat (ATFT). It usually behaves as TFT, but it takes the opposite moves after mistakenly defecting from mutual cooperation. Similar analysis has been conducted for the three-person public-goods (PG) game: At least 256 successful strategies exist when m = 3 , whereas no such solution exists when m < 3 8 . It has also been shown that a friendly rival strategy must have m ≥ n for the general n-person PG game, although such a strategy for n > 3 is yet to be found. These results suggest that a novel class of strategies may appear as the memory length exceeds a certain threshold. For memory-three strategies, we have obtained an exhaustive list of successful strategies by massive supercomputing (see Supplementary Methods S1 at the end of this manuscript). The efficiency and defensibility criteria find 7, 018, 265, 885, 034 friendly rivals out of N m=3 = 2 64 ≈ 1.84 × 10 19 strategies. If the distinguishability criterion is additionally imposed, 4, 261, 844, 305, 281 strategies are found. There are four actions commonly prescribed by all these successful strategies: Let A t and B t denote Alice's and Bob's actions at time t, respectively. When their memory states are (A t−3 A t−2 A t−1 , B t−3 B t−2 B t−1 ) = (ccc, ccc) , (ccc, ddd), (cdd, ddd), and (ddd, ddd), all the successful strategies tell Alice to choose c, d, d, and d, respectively. The first one is absolutely required to maintain mutual cooperation. The latter three are needed to satisfy the defensibility criterion: If c was prescribed at any of these states, Alice would be exploited by Bob's continual defection. Except for these four prescriptions, we see a wide variety of patterns. For example, let us assume that both Alice and Bob adopt one of these strategies. When Bob defects by error, they must follow a recovery path from state (ccc, ccd) to (ccc, ccc). We find 839 different patterns from our successful strategies ( Table 2). The most common one is also the shortest, in which only two time steps are needed to recover mutual cooperation. It cannot be shorter because Alice must defect at least once to assure defensibility. It is even shorter than that of TFT-ATFT, which is identical to the third entry of Table 2. This finding disproves a speculation that friendly rivals are limited to variants of TFT even if m > 2 7 . This shortest recovery path is possible only when m ≥ 3 , indicating a pivotal role of memory length in direct reciprocity. Results cApRi strategy. The shortest recovery path in Table 2 shows that Bob can recover his own mistake simply by accepting Alice's punishment provided that he has m = 3 . Among the strategies using this recovery pattern, we have discovered a strategy which is easy to interpret, named 'CAPRI' , after the first letters of its five constitutive rules listed below: 1. Cooperate at mutual cooperation. This rule prescribes c at (ccc, ccc). The first rule is clearly needed for efficiency. In addition, mutual cooperation must be robust against one-bit error, i.e., occurring with probability of O(e), when both Alice and Bob use this strategy. This property is provided by the second and the third rules. In addition, for this strategy to be efficient, the players must be able to escape from mutual defection through one-bit error so that the stationary probability distribution does not accumulate at mutual defection, which is handled by the fourth rule. Note that these four rules for efficiency do not necessarily violate defensibility when m > 2 , as we have already seen in Table 2. Actually, due to the fifth rule, both efficiency and defensibility are satisfied by CAPRI. The action table and its minimized automaton representation 21 are given in Table 3 and Fig. 2a, respectively. The self-loop via dc at state '2' in Fig. 2a proves that this strategy also satisfies distinguishability. CAPRI requires m = 3 because otherwise it violates defensibility: If CAPRI were a memory-two strategy, (cd, dc) → c and (dc, cd) → c must be prescribed to recover from error. However, with these prescriptions, Bob can repeatedly exploit Alice by using the following sequence: TFT-ATFT and its variants are the only friendly rivals when m < 3 . Compared with TFT-ATFT, CAPRI is closer to Grim trigger (GT) rather than to TFT. Alice keeps cooperating as long as Bob cooperates, but she switches to defection, as prescribed by the fifth rule, when Bob does not conform to her expectation. Due to the similarity of CAPRI to GT, it also outperforms a wider spectrum of strategies than TFT-ATFT. Figure 2b shows the distribution of payoffs of the two players when Alice's strategy is CAPRI, and Bob's strategy is sampled from the 64-dimensional unit hypercube of memory-three probabilistic strategies. Alice's payoff is strictly higher than (2) . . . c c d c c . . . . . . c d c d c . . . . Table 2. Recovery paths to mutual cooperation for the memory-three successful strategies. Only the most common five patterns are shown in this table. The upper and lower rows represent the sequences of actions taken by Alice and Bob, respectively, when Bob defected from mutual cooperation by error. The right column shows the number of strategies having each pattern, as well as its fraction with respect to the total number of successful strategies. A t−3 A t−2 A t−1 B t−3 B t−2 B t−1 ccc ccd cdc cdd dcc dcd ddc ddd ccc 1 c 3 d d d 2,4 c d d d ccd 2 c d 3 c d d d d d cdc d 2 c d d 3 c d d d cdd d d d d d d d d dcc 3,4 c d 2 c d 4 c d 4 c d dcd d d d d d d d d ddc d d d d 4 c d 4 c 4 c ddd d d d d d d 4 c d Scientific RepoRtS | (2020) 10:16904 | https://doi.org/10.1038/s41598-020-73855-x www.nature.com/scientificreports/ Bob's in most of the samples. On the other hand, when Alice uses TFT-ATFT, payoffs are mostly sampled on the diagonal because it is based on TFT, which equalizes the players' payoffs. However, we also note that CAPRI is significantly different from GT in two ways. First, CAPRI is error-tolerant: Even when Bob makes a mistake, Alice is ready to recover cooperation after Bob accepts punishment, as described in the second and the third rules. Second, whereas GT is characterized by its irreversibility, CAPRI lets the players escape from mutual defection according to the fourth rule. evolutionary simulation. Although defensibility assures that the player is never outperformed by the coplayer, it does not necessarily guarantee success in evolutionary games, where everyone is pitted against every other in the population. For example, extortionate ZD strategies perform poorly in an evolutionary game 12,13,22 . In this section, we will check the performance of CAPRI in the evolutionary context. When we consider performance of a strategy in an evolving population, the most famous measure of assessment is evolutionary stability (ES) 23 . Although conceptually useful, ES is too strong a condition, requiring that when a sufficient majority of population members apply the strategy, every other strategy is at a selective disadvantage. Evolutionary robustness has thus been introduced as a more practical notion of stability 13 : A strategy is called evolutionary robust if no other strategy has fixation probability greater than 1/N, which is the fixation probability of a neutral mutant. In other words, an evolutionary robust strategy cannot be selectively replaced by any mutant strategy. Evolutionary robustness of a strategy depends on the population size: Partner strategies have this property when N is large enough, whereas for rival strategies, it is when N is small 13 . Friendly rivals have the virtue of both: They keep evolutionary robustness regardless of N, as will be shown below. As in the standard stochastic model 24 , let us consider a well-mixed population of size N in which selection follows an imitation process. At each discrete time step, a pair of players are chosen at random, and we will call their strategies X and Y, respectively. The probability for one of the players to replace her strategy X with Y is given as follows: where s x and s y denote the average payoffs of X and Y against the entire population, respectively, and σ is a parameter which denotes the strength of selection. In population dynamics, we assume that the mutation rate µ is low enough: That is, when a mutant strategy X appears in a resident population of Y, no other mutant will be introduced until X reaches fixation or goes extinct. The dynamics is formulated as a Moran process, under which the fixation probability of X is given in a closed form 13 : where s xy denotes the long-term payoff of player X against player Y. Using Jensen's inequality, we see that www.nature.com/scientificreports/ When Y is a partner strategy, it satisfies s yy ≥ s xy and s yy ≥ s xx . When Y is also a rival strategy, it has another inequality, s yx ≥ s xy . Therefore, the fixation probability of an arbitrary mutant ρ ≤ 1/N regardless of N and σ. (3) f x→y = 1 1 + exp σ s x − s y ,(4)ρ = 1 N−1 i=0 i j=1 e σ [(N−j−1)syy+jsyx−(N−j)sxy−(j−1)sxx] ,(5) We have conducted evolutionary simulation to assess the performance of friendly rivals. First, we run simulation without CAPRI and TFT-ATFT. This simulation adopts the setting of a recent study 3 and serves as a baseline of performance. A mutant strategy is restricted to reactive memory-one strategies, according to which the player's action depends only on the co-player's last action. The reactive strategies are characterized by a pair of probabilities ( p c ,p d ), where p α denotes the probability to cooperate when the co-player's last move was α . www.nature.com/scientificreports/ parameters 3,13,14 . Rival strategies have higher abundance when the benefit-to-cost ratio is low, population size N is small, and error rate e is high. Otherwise, partner strategies are favoured. Let us now assume that a mutant can also take TFT-ATFT in addition to the reactive memory-one strategies. Figure 3b shows that TFT-ATFT occupies significant fractions across a broad range of parameters. The situation changes even more remarkably when CAPRI is introduced instead of TFT-ATFT. As seen in Fig. 3c, CAPRI overwhelms the other strategies for almost the entire parameter ranges. The low abundance at N = 2 or e = 10 −5 does not contradict with the evolutionary robustness of CAPRI because it is still higher than the abundance of a neutral mutant. Although the abundance of partners is higher than CAPRI when e = 10 −5 , the reason is that it is an aggregate over many partner strategies. If we compare each single strategy, CAPRI is still the most abundant one for the entire range of e. The qualitative picture remains the same even if we choose a different value of σ , and CAPRI tends to be more favoured as σ increases. Furthermore, by comparing Fig. 3b,c, we see that CAPRI shows better performance than TFT-ATFT. The evolutionary advantage of CAPRI over TFT-ATFT is directly observed in Fig. 3d), where both CAPRI and TFT-ATFT are introduced into the population. As we have seen in Fig. 2b, it tends to earn strictly higher payoffs against various types of co-players, whereas TFT-ATFT, based on TFT, aims to equalize the payoffs except when it encounters naive cooperators. This observation shows a considerable amount of diversity even among evolutionary robust strategies 25 . Summary To summarize, we have investigated the possibility to act as both a partner and a rival in the repeated PD game without future discounting. By thoroughly exploring a huge number of strategies with m = 3 , we have found that it is indeed possible in various ways. The resulting friendly rivalry directly implies evolutionary robustness for any population size, benefit-to-cost ratio, and selection strength. We observe its success even when e is of a considerable size (Fig. 3). It is also worth noting that a friendly rival can publicly announce its strategy because it is guaranteed not to be outperformed regardless of the co-player's prior knowledge. Rather, it is desirable that the strategy should be made public because the co-player can be advised to adopt the same strategy by knowing it from the beginning to maximize its payoff. The resulting mutual cooperation is a Nash equilibrium. The deterministic nature offers additional advantages because the player can implement the strategy without any randomization device. Moreover, even if uncertainty exists in the cost and benefit of cooperation, a friendly rival retains its power because it is independent of (R, T, S, P). This is a distinct feature compared to the ZD strategies, whose cooperation probabilities have to be calculated from the elementary payoffs. Furthermore, the results are independent of the specific payoff ordering T > R > P > S of the PD. These are valid as long as mutual cooperation is socially optimal ( R > P and 2R > T + S ) and exploiting the other's cooperation pays better than being exploited ( T > S ). This condition includes other well-known social dilemma, such as the snowdrift game (with T > R > S > P ) and the stag-hunt game (with R > T > P > S). This work has focused on one of friendly rivals, named CAPRI. We speculate that it is close to the optimal one in several respects: First, it recovers mutual cooperation from erroneous defection in the shortest time. Second, it outperforms a wide range of strategies. Furthermore, its simplicity is almost unparalleled among friendly rivals discovered in this study. CAPRI is explained by a handful of intuitively plausible rules, and such simplicity greatly enhances its practical applicability because the required cognitive load will be low when we humans play the strategy 11,26,27 . It is an interesting research question whether this statement can be verified experimentally. In particular, we would like to stress the importance of memory length in theory and experiment, considering that much research attention has been paid to the study of memory-one strategies 2,5,13,14,[28][29][30] . Besides the combinatorial explosion of strategic possibilities, one can argue that a memory-one strategy, if properly designed, can unilaterally control the co-player's payoff even when the co-player has longer memory 9 . It has also been shown that m = 1 is enough for evolutionary robustness against mutants with longer memory 13 . However, the payoff that a strategy receives against itself may depend on its own memory capacity 13,25 , and this is the reason that a friendly rival is feasible when m > 1 . We can gain some important strategic insight only by moving beyond m = 1 . Related to the above point, one of important open problems is how to design a friendly-rival strategy for multi-player games. Little is known of the relationship between a solution of an n-person and that of an (n − 1)-person game of the same kind. For example, it is known that TFT-ATFT for the PD game 7 is not directly applicable to the three-person PG game 8 . We nevertheless hope that the five rules of CAPRI may be more easily generalized to the n-person game, considering that its working mechanism seems more comprehensible than that of TFT-ATFT to the human mind. In a broader context, although 'friendly rivalry' sounds self-contradictory, the term captures a crucial aspect of social interaction when it goes in a productive way: Rivalry is certainly ubiquitous between artists, sports teams, firms, research groups, or neighbouring countries [31][32][33][34] . At the same time, they are subject to repeated interaction, whereby they eventually become friends, colleagues, or business partners to each other. Our finding suggests that such a seemingly unstable relationship can readily be sustained just by following a few simple rules: Cooperate if everyone does, accept punishment for your mistake, punish defection, recover cooperation if you find a chance, but in all the other cases, just take care of yourself. These seem to be the constituent elements for such a sophisticated compound of rivalry and partnership. Data availability The source code for this study is available at https ://githu b.com/yohm/sim_exhau stive _m3_PDgam e. Received: 8 May 2020; Accepted: 26 August 2020 Figure 1 . 1(Left) A schematic diagram of the strategy space. Strategies that tend to cooperate (defect) are shown on the left (right). The blue ellipse represents a set of efficient strategies, which are cooperative to sustain mutual cooperation, and its subset of partner strategies is denoted by the dashed blue curve. On the other hand, the red ellipse represents a set of defensible strategies, which often defect to defend themselves from malicious co-players. In general, their intersection is small. When m = 2 , for instance, the sizes of efficient and defensible sets are 7639 and 2144, respectively, whereas the intersection contains only eight strategies. (Right) The diamond depicts the region of possible average payoffs for Alice and Bob. The blue triangle shows the feasibility region when Alice uses a defensible strategy. If Alice and Bob both use the same strategy satisfying efficiency, they will reach (R, R) (the blue dot). Accept punishment when you mistakenly defected from mutual cooperation. This rule prescribes c at (ccd, ccc), (cdc, ccd), (dcc, cdc), and (ccc, dcc). 3. Punish your co-player by defecting once when he defected from mutual cooperation. This rule prescribes d at (ccc, ccd), and then c at (ccd, cdc), (cdc, dcc), and (dcc, ccc). 4. Recover cooperation when you or your co-player cooperated at mutual defection. This rule prescribes c at (ddd, ddc), (ddc, dcc), (dcc, ccc), (ddc, ddd), (dcc, ddc), (ccc, dcc), (ddc, ddc), and (dcc, dcc). 5. In all the other cases, defect. i[(2N−i−3)s yy +(i+1)s yx −(2N−i−1)s xy −(i−1)s xx ]/2 (6) ≥Ne σ (N−1)[(N−2)(s yy −s xx )+(N+1)(s yx −s xy )+(N−2)(s yy −s xy )]/6 . Figure 2 . 2(a) Automaton representation of CAPRI. Its prescribed actions are denoted by the node colours (blue for c and red for d). The labels on the edges indicate the players' actions. The transition caused by erroneous defection at mutual cooperation ('0') is depicted by an orange dashed arrow. (b) Distribution of payoffs when Alice's strategy is CAPRI (left) or TFT-ATFT (right), whereas Bob adopts one of probabilistic memory-three strategies uniformly at random. The elementary payoffs are (R, T, S, P) = (3, Figure 3 . 3Rival strategies are represented by p d = 0 , and partners are by p c = 1 and p d < p * d , where p * d ≡ min{1 − (T − R)/(R − S), (R − P)/(T − P)} . Mutant strategies may be randomly drawn from [0, 1] × [0, 1], but we have discretized the unit square in a way that each p α takes a value from {0, 1/10, 2/10, . . . , 9/10, 1} . We have run the simulation until mutants are introduced 10 7 times, and measured how frequently partner or rival strategies are observed. As shown inFig. 3a, evolutionary performance of strategies depends (a) Abundance of memory-one partners, rivals, and the other strategies. We consider a simplified version of the PD game, parametrized by b, the benefit to the co-player when a player cooperates with a unit cost. In terms of the elementary payoffs, this corresponds to R = b − 1 , T = b , S = −1 , and P = 0 . The Moran process is simulated with selection strength σ in a population of size N, where the product Nσ is fixed as 10. Three parameters (benefit-cost-ratio b, population size N, and error rate e) are varied one by one3 . Their default values are b = 3, N = 50, and e = 10 −3 unless otherwise stated. We also show the simulation results with (b) TFT-ATFT, (c) CAPRI, and (d) both TFT-ATFT and CAPRI, introduced with probability µ = 0.01 . These are average results over 10 Monte-Carlo runs. Scientific RepoRtS | (2020) 10:16904 | https://doi.org/10.1038/s41598-020-73855-x 1 ) open 1 1openRIKEN Center for Computational Science, Kobe, Hyogo 650-0047, Japan. 2 Department of Physics, Pukyong National University, Busan 48513, Korea. * email: [email protected] RepoRtS | (2020) 10:16904 | https://doi.org/10.1038/s41598-020-73855-x . . . c c d d c . . . . . . c d c d c . . . . . . c c d d c . . . . . . c d d c c . . . . . . c c d c d c . . . . . .c d c c d c . . . . . . c c d c d d c . . . . . . c d c c d d c . . . . . . c c d d d c . . . . . . c d c c d c .. .Action sequence # of strategies . . . c c d c . . . . . . c d c c . . . 905, 772, 524, 235 (21.3%) 522, 061, 013, 252 (12.2%) 437, 671, 509, 356 (10.3%) 409, 458, 612, 318 (9.6%) 227, 113, 898, 468 (5.3%) 184, 052, 002, 852 (4.3%) Table 3 . 3Action table of CAPRI. The superscript on the upper left corner of each element indicates which rule is involved. © The Author(s) 2020 AcknowledgementsThe authors would like to thank C. Hilbe for his careful reading and insightful comments on the manuscript. OACIS was used for the simulations in this study35. We acknowledge the hospitality at APCTP where part of this work was done.Author contributionsY.M. designed the research, carried out the computation, and analysed the results. S.K.B. verified the method. Y.M. and S.K.B. wrote and reviewed the manuscript.competing interestsThe authors declare no competing interests.Additional informationSupplementary information is available for this paper at https ://doi.org/10.1038/s4159 8-020-73855 -x.Correspondence and requests for materials should be addressed to S.K.B.Reprints and permissions information is available at www.nature.com/reprints.Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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[ "Simultaneous sorting many quDits using different input ports", "Simultaneous sorting many quDits using different input ports" ]
[ "Iulia Ghiu " ]
[]
[]
Quantum sorter has gained a lot of attention during the last years due to its wide application in quantum information processing and quantum technologies. A challenging task is the construction of a quantum sorter, which collect many high-dimensional quantum systems, which are simultaneously incident on different input ports of the device. In this paper we give the definition of the general quantum sorter of multi-level quantum systems. We prove the impossibility of the construction of the perfect quantum sorter, which works for many particles incident on any input port, while keeping their states unmodified. Further we propose an approximate multi-particle multi-input-port quantum sorter, which performs the selection of the particles in a certain output port according to the properties of the initial states, but changing the final states. This method is useful for those situations which require high speed of quantum state sorting. Thus, the information contained in the initial states of the particles is revealed by the click statistics of the detectors situated in each output port.Keywords Quantum sorter · High-dimensional quantum systems · Quantum gatesIntroductionMulti-input multi-output quantum devices have attracted an increasing interest for many applications in quantum technologies during the last years. They have opened or developed branches of research in the fields of quantum information theory and quantum optics: Boson Sampling [1], [2], [3], Bell multi-port beam splitter [4], quantum Fourier transform [5], generalized interference [6],
10.1007/s11128-019-2395-2
[ "https://arxiv.org/pdf/1909.02465v1.pdf" ]
201,261,300
1909.02465
287b76ca59dc8054371f07f5fc9ec63cd580cd23
Simultaneous sorting many quDits using different input ports 5 Sep 2019 Iulia Ghiu Simultaneous sorting many quDits using different input ports 5 Sep 2019Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Quantum sorter · High-dimensional quantum systems · Quantum gates Quantum sorter has gained a lot of attention during the last years due to its wide application in quantum information processing and quantum technologies. A challenging task is the construction of a quantum sorter, which collect many high-dimensional quantum systems, which are simultaneously incident on different input ports of the device. In this paper we give the definition of the general quantum sorter of multi-level quantum systems. We prove the impossibility of the construction of the perfect quantum sorter, which works for many particles incident on any input port, while keeping their states unmodified. Further we propose an approximate multi-particle multi-input-port quantum sorter, which performs the selection of the particles in a certain output port according to the properties of the initial states, but changing the final states. This method is useful for those situations which require high speed of quantum state sorting. Thus, the information contained in the initial states of the particles is revealed by the click statistics of the detectors situated in each output port.Keywords Quantum sorter · High-dimensional quantum systems · Quantum gatesIntroductionMulti-input multi-output quantum devices have attracted an increasing interest for many applications in quantum technologies during the last years. They have opened or developed branches of research in the fields of quantum information theory and quantum optics: Boson Sampling [1], [2], [3], Bell multi-port beam splitter [4], quantum Fourier transform [5], generalized interference [6], [7] , [8], [9], [10], [11], quantum metrology [12], [13], [14], [15], entanglement between spatially separate multi-mode quantum systems [16], [17], [18], [19], [20]. In quantum information protocols, a key step for performing a task is the measurement of the states of the quantum system. Measuring the states is a challenging problem since one has to determine the possible orthogonal states of the quantum system. A possible way of measuring the state can be performed with the help of the so-called quantum sorter, which selects the states in different output ports. The quantum sorter became of paramount importance for protocols in quantum information processing, as many implementations require the extraction and manipulation of the information stored by quantum systems [21], [22], [23], [24], [25]. Most of the proposals regarding the sorting of quantum systems are related to the properties of the photons: polarization, orbital angular momentum (OAM) [26], [27], [28], [29], [30], [31], the total angular momentum [32], or radial modes [33], [34]. A more general quantum sorter was proposed in Ref. [35], which is universal, i.e. it enables the selection of the states for an arbitrary degree of freedom of a single quDit (D-level quantum system). This device is characterized by D-input and D-output ports. We show in the present work that the universal quantum sorter of Ref. [35] fails in sorting many quantum systems, which are incident on different input ports. In other words, this universal quantum sorter works properly only in the case when all the particles to be sorted are injected through the same (unique) input port. Simultaneous sorting of many high-dimensional quantum systems, which are incident on different input ports of the device is still an open and challenging problem. In this work, we introduce the definition of a general multiinput-port quantum sorter of quDits. The main contribution is written as a theorem, which provides the expression of the unitary operator of the approximate multi-input-port quantum sorter in terms of the well-known quantum gates of quDits. This sorter can be used to obtain the information of the initial states by measuring the number of particles in each output port. For example, the photocounting measurements were recently employed for the estimation of the expectation values of different observables [36]. Therefore, the measurements of the number of particles play an essential role for extracting different kinds of information or properties of the initial states. This article is organized as follows. Section 2 introduces the concept of the general quantum sorter of quDits. The motivation of our work is presented in Sect. 3, where we show that the quantum sorter given in Ref. [35] cannot perform a simultaneously selection of many particles injected through any input port. This sorter is restricted to select the states only in the case when all the particles are incident on a unique input port. This fact leads us to the motivation to construct a multi-input-port quantum sorter. The two-inputport sorter of the polarization of photons is presented in details in Section 4. In Section 5 we present the classification of the quantum sorters and prove that the perfect multi-input-port quantum sorter of quDits, which keeps the state of the particles unmodified, does not exist. Further, we focus on the notion of the multi-input-port quantum sorter, that enables the selection of the particles according to their initial properties. But this device performs an approximate sorting since the final states of the particles are changed. Section 6 is devoted to a theorem, which presents a formula of the unitary operator of the multi-input-port quantum sorter expressed in terms of the quantum gates of quDits. By using this theorem, we construct the quantum circuit of this quantum sorter. In addition, we describe the physical implementation of the four-input-port quantum sorter for photons carrying four-dimensional OAM. Section 7 summarizes our main results of this paper. In Appendix A we present some results regarding the quantum gates for quDits. Definition of the general quantum sorter of quDits In this section, we introduce the definition of a general quantum sorter. Suppose that we have a D-level quantum system, whose states are described by the computational basis {| 0 , | 1 ,..., | D−1 } in the Hilbert space H system . In addition, one has to consider the states associated with D ports, also denoted by | j , with j= 0, 1, ..., D − 1 acting in the Hilbert space H port . Sorting the quantum states of the system consists of a device that collects a given state of the system through a unique output port. This fact requires that the quantum sorter must have D output ports, one associated with each possible state of the quantum system. In the following we will use the notation | system-state | port-state for the total state in the Hilbert space H system ⊗ H port , i.e. for the state of the quantum system and for the spatial quantum modes, through which the particles travel. The quantum sorter is a device that selects the particles characterized by the property | s of the quantum system by collecting them to the same output port-state, which is denoted by | s . Definition 1. Consider two D-dimensional Hilbert spaces H system and H port , the first one associated with a quDit system, while the second one to the states of the D ports. A device acting in the Hilbert space H system ⊗ H port described by the unitary operator U U | s | * = | # | s , with s = 0, 1, ..., D − 1 is called quantum sorter. The symbols * and # are allowed to take the values 0, 1, 2, ..., D − 1. The purpose of the quantum sorter is to send the particle, which initially was in the state | s , to the output port-state | s : U | in-system-state | in-port-state = | out-system-state | out-port-state=in-system-state .(2) Remark: the symbol * in Eq. (1) may be unique for all s = 0, 1, ..., D − 1 or not, i.e. the particles could be incident on a unique input port or not. The only requirement imposed by the definition is that the operator U to be unitary. Motivation A particular quantum sorter of a D-level quantum system was introduced in Ref. [35] as a device with D input ports and D output ports, whose purpose is to sort the D possible states of the quDit. The states of the quantum system exist in the Hilbert space H system , while the states of the ports belong to the Hilbert space H port . The quantum sorter of Ref. [35] is defined by the unitary transformation acting on H system ⊗ H port as follows | s | k −→ | s | k ⊕ s , s, k = 1, 2, . . . D,(3) By ⊕ we denote the addition modulo D. The D output ports are used in order to collect the D states of the quDit. It may be worth noting that the quantum sorter given by Eq. (3) depends on the input port, i.e., there is a correspondence between the input and the output ports, the emerging one being influenced by the input. For example, let us consider that we inject two qutrits (three-level systems), which are in the same initial state | 1 , on the quantum sorter of Eq. (3). If one qutrit is incident on the mode | 0 and the other one on the mode | 1 , then, according to Eq. (3), one obtains | 1 | 0 −→ | 1 | 1 , | 1 | 1 −→ | 1 | 2 .(4) The two initial qutrits found in the same state | 1 entering different input ports will exit through different output port-states (denoted by | 1 and | 2 ). But the purpose of the quantum sorter is to collect particles with the same property denoted by s through the same output port, which should be unique for the particles found initial in the state | s . This means that the quantum sorter of Eq. (3) fails in sorting many quantum systems, which are incident on different input ports. The impossibility of a proper simultaneous selection of the quantum states of many particles by using the sorter (3) was our motivation for constructing the general quantum sorter which works for many particles incident on different input ports of a such device, as we have defined in Sect. 2. The two-input-port quantum polarization sorter of photons In this section we investigate the possibility of sorting two photons, which are incident on a polarizing beam splitter (PBS), according to their polarizations. The PBS is characterized by two-input modes and by two-output modes. If one sends two H-polarized photons through the two different input ports of a PBS, then they will exit through different output ports. The PBS cannot be used as a quantum sorter when the two photons are injected through different input ports, because a proper quantum sorter would collect photons with identical polarization through the same output port. This means that the PBS works as a quantum polarization sorter of two photons only in the case when the two photons are incident on the same input port. In a setup of a quantum circuit, one expects that many particles are injected on a multi-port device. This leads us to the motivation to find a scheme, which allows the simultaneous selection of many photons according to their initial polarizations, regardless of the input port. We impose the following requirement: the output mode | 0 of this two-input two-output device will collect all the photons, which were initially H-polarized, while the output mode | 1 collects the initially V -polarized photons, regardless of the final state of the photons. In this way, by photocounting measurements, we obtain the information stored in the initial state of the photons. Suppose that we insert a half-wave plate (HWP) in the arm of the input mode | 1 of the PBS, as it is shown in Fig. 1. Again we sent two H-polarized photons on different input ports of this device. The action of the HWP is given by the X gate, i.e. it flips the polarization state | H ←→ | V of the photons, which are incident on the input | 1 . Then, the two photons exit through the same output-state, which is | 0 . The disadvantage of this device is that it flips the polarization of one of the photons, such that the emerging photons will be one H-polarized and the other one V -polarized. 00 00 00 11 11 11 00 00 00 00 11 11 11 11 PBS HWP |H>|0> |H>|1> |0> Fig. 1 The scheme of the two-input-port quantum polarization sorter of photons. The labels are as follows: PBS represents the polarizing beam splitter and HWP is the half-wave plate. The two H-polarized photons, which are incident on the input ports | 0 and | 1 of the quantum sorter, will exit through the output port | 0 . The device shown in Fig. 1 is called the two-input-port quantum polarization sorter of photons and consists of a PBS and a HWP placed in one arm of the PBS. The action of this quantum sorter of photons is described by the relations: (i) for the incident H-polarized photons | H | 0 −→ | H | 0 (5) | H | 1 −→ | V | 0 ;(6) (ii) for the incident V -polarized photons | V | 0 −→ | V | 1 (7) | V | 1 −→ | H | 1 .(8) We can obtain the information stored in the initial states of the photons by measuring the photocounting distribution, i.e. from the click statistics of the detectors. In the example shown in Fig. 1, we can infer that two H-polarized photons were injected on the quantum sorter, since we obtain both photons measured by the detector in the output port-state | 0 . If we denote | H = | 0 and | V = | 1 , we can rewrite the above four expressions (5), (6), (7) and (8) in the compact form: | s | k −→ | s ⊖ k | s , s, k = 0, 1,(9) where ⊖ denotes the subtraction modulo 2. Classification of the quantum sorters of quDits The purpose of this section is to introduce and to make a classification of different kinds of quantum sorters, starting from the general definition (1). A quantum sorter would be perfect if it sends the particle, which is initially in the state | s , through the output port-state | s , regardless of the input port-state, while keeping the state of the particle unchanged. Definition 2. Consider two D-dimensional Hilbert spaces H system and H port , the first one associated to a quDit system, while the second one to the states of the D ports. A device acting in the Hilbert space H system ⊗ H port , that performs the unitary transformation | s | k −→ | s | s , with s, k = 0, 1, ..., D − 1 (10) is called the perfect multi-input-port quantum sorter. An equivalent way of writing Eq. (10) is as follows: U | in-system-state | in-port-state = | out-system-state=in-system-state | out-port-state=in-system-state . In the following, we prove that the perfect multi-input-port quantum sorter is forbidden. Suppose that we sent two particles found in the same state | s on this device, one through the input port-state | m , while the second one through | n : | s | m −→ | s | s ; | s | n −→ | s | s , with m = n.(11) The proof is straightforward, since there is no unitary operator satisfying both Eqs. (11). Definition 3. Consider two D-dimensional Hilbert spaces H system and H port , the first one associated with a quDit system, while the second one to the states of the D ports. A device that performs the following unitary transformation acting in the Hilbert space H system ⊗ H port U SQS | s | k = | s | s ⊕ k = | s | s , if k = 0 | s | j , with j = s ⊕ k = s, if k = 0,(12) with k = 0 and s = 0, 1,..., D − 1, is called the single-input-port quantum sorter (SQS). The incident particles must be injected only on the input port-state | 0 . The action of the quantum sorter of Eq. (12) can be rewritten as: U SQS | in-system-state | in-port-state=0 = | out-system-state=in-system-state | out-port-state=in-system-state . The unitary operator U SQS satisfies the condition of the general quantum sorter given by Eq. (1) only for the input port s = 0; therefore it is called single-input-port quantum sorter. The device of Eq. (12) was introduced in Ref. [35] and it was analyzed in Sect. 3, where we have emphasized that this quantum sorter fails in selecting many particles, which are incident on different input ports. In Ref. [35], it was proved that: U SQS = C(X D ) = (I ⊗ F † ) C(Z D ) (I ⊗ F ),(13) where F is the quantum Fourier transform (QFT) defined by Eq. (23), while C(X D ) and C(Z D ) are the controlled-X D gate and controlled-Z D gate, respectively, given by Eqs. (24) and (25). Since the perfect multi-input-port quantum sorter of many quDits is forbidden, we try to construct a quantum sorter, which may change the state of the particles, while the particles exit through the proper output port dependent on the initial state. In the following, we give the definition of a such device. Definition 4. Consider two D-dimensional Hilbert spaces H system and H port , the first one associated with a quDit system, while the second one to the states of the D ports. A device acting in the Hilbert space H system ⊗H port , that performs the unitary transformation U MQS | s | k = | s ⊖ k | s , with s, k = 0, 1, ..., D − 1(14) is called the multi-input-port quantum sorter (MQS). This device sends the particle, initially found in the state | s , through the output state | s , regardless of the input port state, while the state of the particle is changed to | s ⊖ k , with ⊖ the subtraction modulo D: U MQS | in-system-state | in-port-state = | out-system-state=in-system-state − in-port-state ⊗| out-port-state=in-system-state . Or in other words, it performs an approximate sorter action of many incident particles, where the task is only to count the number of the particles which were initial in the state | s , with no interest in the state of the output particles. Therefore, we can extract the information stored in the initial states from the click statistics of the detectors. The MQS is useful in the case when high speed of quantum state sorting is required, since the sorting efficiency is increased by a factor of D. In Table 1, we present the classification of the quantum sorters, by emphasizing on which input port-states they work, as well as the equation which defines each sorter. Table 1 The classification of quantum sorters. The incident particle is found in the state | s . The word "YES" means that the condition of the general quantum sorter (1) is satisfied, i.e. the particle exits through the output port-state | s . The word "NO" has the meaning that the particle emerges through the output port-state | j , with j = s. We have proved that the perfect quantum sorter cannot be constructed. Quantum Input port-states Equation which Sorter | 0 | 1 , | 2 ,..., | D − 1 defines the sorter Perfect YES YES Eq. (10) Single-input-port YES NO Eq. (12) Multi-input-port YES YES Eq. (14) Let us return now to the discussion regarding the PBS by using the new definitions introduced above. Since the PBS works as a quantum sorter only if all the photons are injected on the same input mode, it represents the singleinput-port polarization sorter of the photons according to Definition 3. The device presented in Fig. 1 in Sect. 4 constructed with the help of a PBS and a half-wave plate is an example of two-input-port quantum sorter of qubits due to Eq. (9) and Definition 4. Let us discuss again the example of Eq. (4) (presented in Sec. 3), by considering now the scenario when the two qutrits are incident on the multi-inputport quantum sorter given by Eq. (14): | 1 | 0 −→ | 1 | 1 , | 1 | 1 −→ | 0 | 1 .(15) Both particles initially found in the state | 1 incident on different modes will exit through the output port | 1 , as one requires for a quantum sorter. 6 The quantum circuit of the multi-input-port quantum sorter In the following, we find an equivalent expression of the unitary operator U MQS in terms of the well-known quantum gates of quDits, as well as the quantum circuit associated to the multi-input-port quantum sorter. Theorem. The unitary operator that satisfies condition (14) of the multiinput-port quantum sorter is given by U MQS = U SQS (X † D ) k ⊗ I ,(16) where k is the input port of the incident particle, while the expression of U SQS is given by Eq. (13). Proof. By using the action of the operator X † D according to Eq. (22) of Appendix A, one obtains (X † D ) k ⊗ I | s | k = | s ⊖ k | k .(17) Further, by applying the single-input-port quantum sorter (12), we have U SQS | s ⊖ k | k = | s ⊖ k | s ,(18) i.e. the desired outcome. It may be worth noting that in the particular case D = 2, one has X † = X and this property was used in Sect. 4, when we have discussed the two-inputport quantum polarization sorter for photons (see also Fig. 1). The quantum circuit of the multi-input-port quantum sorter is depicted in Fig. 2. Firstly, the state of a particle injected on the input-state | k is modified by the (X † D ) k gate. Secondly, according to the Theorem described by Eq. (16), one uses the scheme of the SQS. In Ref. [35], it was shown that the SQS consists of a QFT, which has the role of coupling all the modes with the same probability 1/D, followed by a path-dependent Z k D gate and further an inverse of the QFT. The realization of experimental high-dimensional gates for any degrees of freedom of the quantum systems is a challenging task and still an open problem. Schemes for the implementation of the Z D gate defined by Eq. (21) for photons carrying OAM were proposed in Ref. [37] by using a Dove prism, as well as for the Z k D gate were found by employing also the Dove prisms rotated by a specific angle [35]. The generalized Pauli gate X D of Eq. (20) seems to be more difficult to be generated in the lab. Recently it was experimentally demonstrated the construction of the X D gate and all of its integer powers on single photons with OAM for D = 4 in Ref. [38]. The physical implementation of the four-input-port quantum sorter for photons carrying four-dimensional OAM can be realized as it is shown in Fig. 3. On the input mode | 1 , one applies the gate X † 4 , whose scheme is shown in Fig. 1(c) of Ref. [38], consisting of a Mach-Zehnder interferometer, two The quantum circuit of the multi-input-port quantum sorter of quDits. The input and the output port-states are indexed by | 0 , | 1 ,..., | D − 1 . A particle found in the quantum state | s , which is injected to the input port-state | k , with k arbitrary, will exit through the output port-state | s , while its state is modified to | s ⊖ k , according to Eq. (14) and the Theorem described by Eq. (16). This device sorts simultaneously many particles found in the states | s 1 , | s 2 ,... regardless of the input port-state. parity splitters, and a spiral phase plate placed in the output mode. By using the fact that (X † 4 ) 2 = X 2 4 , we place in the input mode | 2 the gate X 2 4 , whose representation is given in Fig. 1(b) of Ref. [38], i.e. a Mach-Zehnder interferometer, two parity splitters, and a spiral phase plate placed in the even arm of the interferometer. On the last input mode, it is sufficient to use the gate X 4 due to the fact (X † 4 ) 3 = X 4 -the diagram of this gate is drawn in Fig. 1(a) of Ref. [38]: again a Mach-Zehnder interferometer, two parity splitters, and a spiral phase plate situated in the input mode. Further, a quantum Fourier gate is applied, followed by the action of a Dove prism placed in the modes k = 1, 2, 3 rotated with the angles α 1 , α 2 , α 3 , respectively: α k = k π D .(19) Finally, the inverse of the QFT is used on all the modes. Conclusions In conclusion, in this paper we have introduced the definition of the general quantum sorter of quDits given by Eq. (1). With the help of this definition, we have made a classification of the quantum sorters and have shown the impossibility of construction of the perfect multi-input-port quantum sorter, Fig. 3 The four-input-port quantum sorter of four-dimensional OAM of photons. The input and the output ports are indexed by | 0 , | 1 , | 2 , | 3 . The schematic diagrams of the gates X † 4 , X 2 4 , and X 4 are given in Fig. 1 of Ref. [38], being constructed with the help of a Mach-Zehnder interferometer, two parity splitters, and a spiral phase plate. QFT is the quantum Fourier transform. Further, in the modes | 1 , | 2 , | 3 is placed a Dove prism rotated by the angle α k of Eq. (19). This device sorts simultaneously many photons carrying OAM for D = 4, which are injected on any input port. + X 4 X 4 X 4 |0> |1> |2> |3> 2 α α α 1 2 3 QFT + QFT which simultaneously selects many quDits, while keeping their states unmodified. This fact led us to the proposal of the multi-input-port quantum sorter that performs an approximate selection. Importantly, in contrast to the SQS, which works properly only in the case when all the particles are incident on the input port-state | 0 , our MQS has the ability to simultaneously collect many particles, which initially were in the same state, to the same output port-state. Therefore, the speed of quantum state sorting is increased by a factor of D. The price to be paid is the modification of the state of the output particles. Our scheme of the MQS may be suited to further applications to quantum information processes, because it performs the sorting of many multi-level particles by measuring the number of the particles in each output port. This means that from the click statistics of the detectors one can reconstruct the information stored in the initial states of the particles. If {| 0 , | 1 ,..., | D − 1 } is the basis of the D-dimensional Hilbert space of the quDit, then the generalized Pauli operators are defined as [39], [40] X D = D−1 j=0 | j ⊕ 1 j |; (20) Z D = D−1 j=0 ω j | j j |,(21) where addition is taken modulo D and ω = exp(2πi/D) is the Dth root of unity. The particular case D = 2 corresponds to the regular Pauli operators. After applying the inverse of X D k times, one obtains X † D k | s = | s ⊖ k .(22) The quantum Fourier transform (QFT) has the expression [41]: F | k = 1 √ D D−1 j=0 ω k j | j .(23) The controlled-U gate acting on two quDits is defined as [35]: C(U ) | s | k = | s U s | k , which in the particular cases of X D and Z D gates reads: C(X D )| s | k = | s | k ⊕ s (24) C(Z D )| s | k = ω s k | s | k .(25) Fig. 2 The quantum circuit of the multi-input-port quantum sorter of quDits. The input and the output port-states are indexed by | 0 , | 1 ,..., | D − 1 . A particle found in the quantum state | s , which is injected to the input port-state | k , with k arbitrary, will exit through the output port-state | s , while its state is modified to | s ⊖ k , according to Eq. (14) and the Theorem described by Eq. (16). This device sorts simultaneously many particles found in the states | s 1 , | s 2 ,... regardless of the input port-state. Acknowledgements I wish to thank one of the two anonymous referees for his/her valuable suggestions, which led to an improved version of this paper. This work was supported by the funding agency CNCS-UEFISCDI of the Romanian Ministry of Research and Innovation through Grant PN-III-P4-ID-PCE-2016-0794 within PNCDI III.A Quantum gates acting on quDitsIn this appendix, we provide a brief review of the definitions of the most important quantum gates acting on one and two quDits, such as the generalized Pauli operators, the quantum Fourier transform, and the controlled-U gate. The computational complexity of linear optics. 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[ "CP-violating Higgs Di-tau Decays: Baryogenesis and Higgs Factories", "CP-violating Higgs Di-tau Decays: Baryogenesis and Higgs Factories" ]
[ "Shao-Feng Ge \nTsung-Dao Lee Institute & School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nShanghai Key Laboratory for Particle Physics and Cosmology\nShanghai Jiao Tong University\n200240ShanghaiChina\n", "Gang Li \nAmherst Center for Fundamental Interactions\nDepartment of Physics\nUniversity of Massachusetts\n01003AmherstMAUSA\n", "† ", "Pedro Pasquini ‡[email protected]§[email protected] \nTsung-Dao Lee Institute & School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina\n", "Michael J Ramsey-Musolf \nTsung-Dao Lee Institute & School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nShanghai Key Laboratory for Particle Physics and Cosmology\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nAmherst Center for Fundamental Interactions\nDepartment of Physics\nUniversity of Massachusetts\n01003AmherstMAUSA\n\nKellogg Radiation Laboratory\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n" ]
[ "Tsung-Dao Lee Institute & School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina", "Shanghai Key Laboratory for Particle Physics and Cosmology\nShanghai Jiao Tong University\n200240ShanghaiChina", "Amherst Center for Fundamental Interactions\nDepartment of Physics\nUniversity of Massachusetts\n01003AmherstMAUSA", "Tsung-Dao Lee Institute & School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina", "Tsung-Dao Lee Institute & School of Physics and Astronomy\nShanghai Jiao Tong University\n200240ShanghaiChina", "Shanghai Key Laboratory for Particle Physics and Cosmology\nShanghai Jiao Tong University\n200240ShanghaiChina", "Amherst Center for Fundamental Interactions\nDepartment of Physics\nUniversity of Massachusetts\n01003AmherstMAUSA", "Kellogg Radiation Laboratory\nCalifornia Institute of Technology\n91125PasadenaCAUSA" ]
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We demonstrate how probes of CP-violating observables in Higgs di-tau decays at prospective future lepton colliders could provide a test of weak scale baryogenesis with significant discovery potential. Measurements at the Circular Electron Positron Collider, for example, could exclude a CP phase larger than 2.9 • (5.6 • ) at 68% (95%) C.L. assuming the Standard Model value for magnitude of the tau lepton Yukawa coupling. Conversely, this sensitivity would allow for a 5 σ discovery for 82% of the CP phase range [0, 2π). The reaches of the Future Circular Collider -ee and International Linear Collider are comparable. As a consequence, future lepton colliders could establish the presence of CP violation required by lepton flavored electroweak baryogenesis with at least 3 σ sensitivity. Our results illustrate that Higgs factories are not just precision machines but can also make O(1) measurement of the new physics beyond the Standard Model. *
10.1103/physrevd.103.095027
null
229,677,893
2012.13922
8d17189c94afc02152a863a55677eea98b3ca8a7
CP-violating Higgs Di-tau Decays: Baryogenesis and Higgs Factories December 29, 2020 27 Dec 2020 Shao-Feng Ge Tsung-Dao Lee Institute & School of Physics and Astronomy Shanghai Jiao Tong University 200240ShanghaiChina Shanghai Key Laboratory for Particle Physics and Cosmology Shanghai Jiao Tong University 200240ShanghaiChina Gang Li Amherst Center for Fundamental Interactions Department of Physics University of Massachusetts 01003AmherstMAUSA † Pedro Pasquini ‡[email protected]§[email protected] Tsung-Dao Lee Institute & School of Physics and Astronomy Shanghai Jiao Tong University 200240ShanghaiChina Michael J Ramsey-Musolf Tsung-Dao Lee Institute & School of Physics and Astronomy Shanghai Jiao Tong University 200240ShanghaiChina Shanghai Key Laboratory for Particle Physics and Cosmology Shanghai Jiao Tong University 200240ShanghaiChina Amherst Center for Fundamental Interactions Department of Physics University of Massachusetts 01003AmherstMAUSA Kellogg Radiation Laboratory California Institute of Technology 91125PasadenaCAUSA CP-violating Higgs Di-tau Decays: Baryogenesis and Higgs Factories December 29, 2020 27 Dec 2020ACFI-T20-18 1 We demonstrate how probes of CP-violating observables in Higgs di-tau decays at prospective future lepton colliders could provide a test of weak scale baryogenesis with significant discovery potential. Measurements at the Circular Electron Positron Collider, for example, could exclude a CP phase larger than 2.9 • (5.6 • ) at 68% (95%) C.L. assuming the Standard Model value for magnitude of the tau lepton Yukawa coupling. Conversely, this sensitivity would allow for a 5 σ discovery for 82% of the CP phase range [0, 2π). The reaches of the Future Circular Collider -ee and International Linear Collider are comparable. As a consequence, future lepton colliders could establish the presence of CP violation required by lepton flavored electroweak baryogenesis with at least 3 σ sensitivity. Our results illustrate that Higgs factories are not just precision machines but can also make O(1) measurement of the new physics beyond the Standard Model. * Introduction The discovery of the Higgs boson at the Large Hadron Collider (LHC) [1,2] and subsequent measurements of its properties strongly favor the mechanism of electroweak symmetry-breaking (EWSB) given by the Standard Model (SM) of particle physics. In particular, the SM predicts each Higgs boson-fermion Yukawa coupling to be purely real, with magnitude proportional to the fermon mass. However, LHC measurements have confirmed this prediction up to only O(10)% precision [3][4][5][6] , leaving considerable room for physics beyond the Standard Model (BSM) in Higgs boson-fermion interactions. Precision Higgs boson studies aim to explore these BSM possibilities. Proposed future lepton colliders, including the Circular Electron-Positron Collider (CEPC) [7], Future Circular Collider (FCC)-ee [8], and International Linear Collider (ILC) [9], are designed for this purpose. The motivations for BSM Higgs interactions are well known, including solutions to the hierarchy problem, generation of neutrino mass, dark matter, and the cosmic baryon asymmetry (BAU). In what follows, we focus on the possibility that the observation of CP-violating effects in Higgs-tau lepton interactions at a future lepton collider could provide new insight into the BAU problem. As pointed out by Sakharov [10], a dynamical generation of the BAU requires three ingredients in the particle physics of the early universe: : (1) non-conserving baryon number (2) out-of-equilibrium dynamics (assuming CPT conservation) and (3) C and CP violation. While the SM contains the first ingredient in the form of electroweak sphalerons, it fails to provide the needed out-of-equilibrium conditions and requisite CP-violation. The presence of BSM physics in the dynamics of EWSB could remedy this situation -the electroweak baryogenesis (EWBG) scenario (for a recent review, see, Ref. [11]). While flavor-diagonal CP-violating (CPV) interactions relevant to EWBG are strongly constrained by limits on permanent electric dipole moments (EDMs) of the electron, neutron, and neutral atoms [12,13] , the landscape for flavor non-diagonal CPV is less restricted. Here, we show that searches for CPV effects in the Higgs di-tau decays at future lepton colliders could provide an interesting probe of "flavored EWBG" [14][15][16][17][18] in the lepton sector. In addition to being theoretically well-motivated in its own right, EWBG has the additional attraction of experimental testability. Modifications of the SM scalar sector necessary for the EWBG out-of-equilibrium conditions provide a rich array of signatures accessible at the LHC and prospective future colliders [19]. The signatures of CP violation could appear in either low-or high-energy experiments. Our present focus is on the possible modification of the the τ Yukawa coupling by a nonzero CP phase ∆ as defined in Eq. (2.1) below and the resulting impact on Higgs decay into a pair of τ leptons. In this context, it has been known for some time that the ∆ phase can be measured at colliders [20][21][22][23][24][25][26][27][28][29][30][31][32]. An O(1) modulation in the relevant differential distribution allows very accurate determination of ∆. The measurements at LHC can achieve a precision around 10 • [33][34][35][36][37][38] at 95% confidence level (C.L.) by using the full data with 3 ab −1 of integrated luminosity. Future lepton colliders [39] can further improve the measurements with higher integrated luminosity, optimized energy for Higgs production, and cleaner environment. Indeed, it was shown that the projected sensitivity for the ILC can reach 4.3 • [40] and 2.9 • for the CEPC [41] at 1 σ level. To our knowledge, a detailed study connecting this sensitivity to the BAU has yet to appear in the literature. In this work we thus study the capability of the CEPC, FCC-ee and ILC in probing ∆ and the resulting prospects of testing lepton flavored EWBG scenario. The projected sensitivities at future lepton colliders are much better than the current LHC results [42]. In Sec. 2, we first make detailed comparison of several observables (the neutrino azimuthal angle difference δφ ν , the polarimeter δφ r , the acoplanarity φ * , and the Θ variable) to show that the polarimeter [32,43] is not just the optimal choice for probing ∆ but can also apply universally to both the τ → πν and τ → ρν decay channels. Then we use a simplified smearing scheme to simulate the detector responses and use χ 2 minimization to find the physical solution of neutrino/tau momentum in Sec. 3. Based on these, we find that the future lepton colliders can make 5 σ discovery of a nonzero CP phase for 82% of the allowed range. With a combination of these channels, the 1 σ sensitivities can reach 2.9 • , 3.2 • , and 3.8 • at the CEPC, FCC-ee, and ILC, respectively. Our result is better than the previous study for the ILC [40] and the same as Ref. [41] for the CEPC. Notice that although the leptonic decay mode of τ is also considered in addition to the two meson decay modes with more usable events, a matrix element based observable is adopted [41] instead of the polarimeter δφ r as we do here, leading to accidentally the same result as ours. Finally, in Sec. 4, we follow Ref. [14] in analyzing the implications for lepton flavored EWBG scenario with 3 σ sensitivity for the presence of CP violation at the CEPC, FCC-ee and ILC. We summarize our findings in Sec. 5. CP Phase and Azimuthal Angle Distributions The SM predicts the Higgs couplings with other SM particles to be proportional to their masses, including the τ lepton, −y τ / √ 2τ L τ R h + h.c. = −m τ /vτ τ h , where h denotes the SM-like Higgs boson, v = 246 GeV is the vacuum expectation value (VEV) and m τ is the tau lepton mass. Although y τ is in general complex, its CP phase can be rotated away without leaving any physical consequences. However, this is not always true when going beyond the SM. Any deviation from the SM prediction, for either the τ Yukawa coupling magnitude or the CP phase, indicates new physics. We first study the CP phase measurement by explicitly comparing various definitions of differential distributions in this section and then the detector response behaviors in Sec. 3. The influence of the Yukawa coupling magnitude deviation from the SM prediction will be discussed in later parts of this paper. The τ Yukawa coupling can be generally parametrized as L hτ τ = −κ τ m τ vτ (cos ∆ + iγ 5 sin ∆)τ h (2.1) with κ τ being real and positive by definition, and ∆ ∈ [0, 2π) in general. We will consider ∆ ∈ [0, π] since the CP measurement is insensitive to the multiplication of κ τ by −1 [33]. The SM prediction for τ Yukawa coupling can be recovered with κ τ → 1 and ∆ → 0. A non-zero value of the CP phase ∆ indicates CP violation in the τ Yukawa coupling and can be connected to baryogenesis in the early Universe [14]. Because of the P-and T-violating nature of the second term in Eq. (2.1), the spin correlation among the two τ leptons from a Higgs decay is an especially interesting probe for constraining its value [23]. In practice, one cannot measure the τ lepton directly but must rely on its decay products. The two most promising channels are the τ decay into π ± (τ ± → π ± ν τ ± ) and into ρ ± (τ ± → ρ ± ν τ ± → π ± π 0 ν τ ± ) with ν τ − (ν τ + ) being the neutrino (anti-neutrino) from the decay of τ − (τ + ). These two channels contribute 10.82% and 25.49% of a single τ decay branching fractions [44], respectively. Observables For each τ decay, one decay plane can be formed by its decay products. The generic azimuthal angle φ difference between the two decay planes is then a good observable for probing ∆, which can be expressed as 1 Γ dΓ dδφ = 1 2π [1 + A cos(2∆ − δφ)] ,(2.2) where the coefficient A depends on the choice of observable. Note that only for ∆ differing from integer multiples of π/2, this distribution will contain a term odd in δφ. There exist a variety of observables that afford access to ∆, which appears as the azimuthal angle difference in two decay planes. Here we review several possibilities and discuss the rationale for our choice of one of them. Neutrino azimuthal angle difference. For both τ leptons decaying into a single charged pion, τ ± → π ± ν τ ± , the differential distribution of the neutrino momentum azimuthal angle difference [43] is 1 Γ dΓ(h → π + π − ν τ ν τ ) dδφ ν = 1 2π 1 − π 2 16 cos(2∆ − δφ ν ) ,(2.3) where δφ ν ≡ φ ν − φ ν and φ ν (φ ν ) are defined in the τ − (τ + ) rest frame. On the other hand, if both τ 's decay to rho mesons, τ ± → ρ ± (→ π ± + π 0 )ν τ ± , the differential distribution of the neutrino azimuthal angle difference δφ ν becomes, 1 Γ dΓ(h → ρ + ρ − ν τ ν τ ) dδφ ν = 1 2π   1 − π 2 16 m 2 τ − 2m 2 ρ m 2 τ + 2m 2 ρ 2 cos(2∆ − δφ ν )   (2.4) with a non-negligible suppression factor, (m 2 τ − 2m 2 ρ ) 2 /(m 2 τ + 2m 2 ρ ) 2 ∼ 0.2. As shown in Fig. 1, this significantly reduces the sensitivity to the CP phase ∆. The neutrino azimuthal angle difference δφ ν is a good observable for the τ → πν τ decay, but not for the τ → ρν τ channel. Polarimeter. Since the azimuthal angle difference is not necessarily the optimized choice and multiple definitions of azimuthal angle have been invented. In Refs. [32,40,43], the azimuthal angle difference between the polarimeter vectors r ± was studied. For the τ decays, the polarimeter vectors are defined as τ ± → π ± ν τ ± : r ± ≡ −p ν τ ± , (2.5a) τ ± → ρ ± (→ π ± π 0 )ν τ ± : r ± ≡ − 1 N ±  p ν τ ± + 2m τ m 2 ρ − 4m 2 π E π ± − E π 0 ± E π ± + E π 0 ± p π ± − p π 0 ±   , (2.5b) where r ± is calculated in the corresponding τ ± rest frame, (E π 0 ± , p π 0 ± ) is the π 0 momentum in the τ ± decay, and N ± is a normalization factor to ensure |r ± | = 1. Then the differential distribution in Eq. (2.2) becomes 1 Γ dΓ dδφ r = 1 2π 1 − π 2 16 cos(2∆ − δφ r ) ,(2.6) for both decay channels including the mixed mode, h → ρ ± π ∓ ν τ ν τ . From the neutrino azimuthal angle difference δφ ν in Eq. (2.4) to the one of the polarimeter in Eq. (2.6), the amplitude gets amplified by a factor of 5 which is a significant improvement. The azimuthal angle difference δφ r ≡ φ r + − φ r − is defined with respect to the z direction, z ≡p τ − . In the Higgs rest frame, tan δφ r =p τ − · (r + × r − ) r − · r + − (r + ·p τ − )(r − ·p τ − ) . (2.7) For the τ ± → π ± ν τ ± decay channel, the polarimeter is along the neutrino momentum direction, namely, r ± = −p ν τ ± as shown in Eq. (2.5a). When di-tau decay into pions it is the azimuthal angle difference φ ν . In contrast, the polarimeter for the τ ± → ρ ± ν τ ± decay channel does not coincide with any momentum of the final-state particles. For illustration, the distribution of Fig. 1. δφ r for h → τ + (→ ρ +ν τ )τ − (→ ρ − ν τ ) is shown in The 4-vector r ± = (0, r ± ) serves as the effective spin of the corresponding τ ± leptons. This becomes evident in the total matrix element of the Higgs decay chain, |M total | 2 ∝ Tr / p τ − + m τ 1 + γ 5 / r − O / p τ + − m τ 1 − γ 5 / r + O , (2.8) with O ≡ cos ∆ + iγ 5 sin ∆, O ≡ γ 0 O † γ 0 and p τ ± ≡ (E τ ± , p τ ± ) being the momentum of τ ± . If the Higgs boson decays to polarized τ leptons, it should be the τ spin vector s ± = (|p τ ± |/m τ , E τ ± /m τpτ ± ) that appears in place of the polarimeter r ± . But since the Higgs decay chain also contains contribution from the τ decays, s ± is replaced by r ± to take the extra effects into consideration. Acoplanarity. This observable was introduced in Ref. [22,23] for the τ → ρν τ decay mode. In the rest frame of the ρ + ρ − system, the ρ momenta are back to back. The decay products of ρ ± form two decay planes and the angle difference between them is defined as acoplanarity φ * , tan φ * ≡p ρ − · [(p π + × p π 0 + ) × (p π − × p π 0 − )] (p π + × p π 0 + ) · (p π − × p π 0 − ) . (2.9) −180 • −135 • −90 • −45 • 0 • 45 • 90 • 135 • 180 • δφ ν , φ * , Θ∆ = 0 • h → τ + (→ ρ + ν τ )τ − (→ ρ − ν τ ) δφ ν δφ r φ * Θ Fig. 1: The differential distributions of h → τ + (→ ρ +ν τ )τ − (→ ρ − ν τ ) for neutrino momentum δφ ν (red) , polarimeter δφ r (black), acoplanarity φ * (blue), and the Θ variable (green) at the truth level for ∆ = 0 • . This interesting variable requires only the knowledge of the directly observable momenta of π ± and π 0 . However, the oscillation amplitude of the distribution is suppressed by around 30% in comparison with the polarimeter as shown in Fig. 1. The Θ Variable. For τ → ρν τ decay, a fourth observable similar to the usual acoplanarity angle, can be defined as, tan Θ ≡p τ + · (E + × E − ) E − · E + − (E + ·p τ + )(E − ·p τ + ) , (2.10) with E ± taking analogy to the electromagnetic fields. In the τ ± rest frames, the E ± vector can be expressed as [33], E ± ≡ m 2 ρ − 4m 2 π 2m τ m 2 τ − m 2 ρ m 2 τ + m 2 ρp ν τ ± + 2m τ m 2 ρ − 4m 2 π (E π ± − E π 0 ) (E π ± + E π 0 ) p π ± − p π 0 ± . (2.11) Note that Eq. (2.10) is slightly more general than the one presented in Ref. [33], where they take the approximation (E ± ·p τ ± ) ≈ 0. It is very interesting to see that Eq. (2.10) has very similar form as Eq. (2.7) with the only difference of a proportional factor (m 2 ρ − 4m 2 π )/2m τ . Due to these similarities, the Θ variable has roughly the same sensitivity as polarimeter, see The comparison in Fig. 1 shows that the polarimeter δφ r and the Θ variable are the optimal ones. However, the Θ variable needs both momenta of π ± and π 0 , limiting its scope to only the τ → ρν τ decay mode. In contrast, the polarimeter method applies for both channels by matching r ± with different combination of final-state particle momenta as shown in Eq. (2.5). So we adopt the polarimeter scheme in the following part of this paper. Measurements at Future Lepton Colliders Future lepton colliders [45] are designed to produce millions of Higgs events. The three prominent candidate colliders are the CEPC [7], FCC-ee [8] and ILC [46]. The CEPC experiment [7] is expected to have around 1.1 × 10 6 Higgs events. This comes from an integrated luminosity of 5.6 ab −1 with two interaction points (IP) and 7 years of running at √ s = 240 GeV. The FCC-ee has a higher luminosity and 4 interaction points, but runs in the Higgs factory mode for only 3 years resulting in a 5 ab −1 of integrated luminosity or equivalently 1.0 × 10 6 Higgs events [8]. The ILC, on the other hand, has a significantly lower integrated luminosity at 2 ab −1 , but is able to produce polarized electrons/positrons which increases the cross section significantly, effectively raising its number of Higgs production to 0.64 × 10 6 [9]. The configuration of these three experiments and the expected numbers of Higgs events at the benchmark luminosities have been summarized in Table 1 In this section, we study the detector responses, including the smearing effects, selection cuts, and momentum reconstruction ambiguities. With around 650 ∼ 1100 events, the uncertainty at the level of 14% ∼ 18% is much smaller than the expected 60% modulation in the CP measurement. This allows a 5 σ discovery potential for approximately 80% of the allowed range in [0, 2π) of the CP phase ∆ and a determination of ∆ with the accuracy of 2.9 • ∼ 3.8 • . Simulation and Detector Responses At lepton colliders, the Higgs boson is mainly produced in the so-called Higgsstrahlung process, e + e − → Zh, with an associated Z boson. This channel allows a model-independent measurement of the Higgs properties thanks to the recoil mass reconstruction method [47]. The Higgs event is first selected by reconstructing the Z boson without assuming any Higgs coupling with the SM particles. The Higgs boson momentum can be either derived from the Z boson momentum using energy-momentum conservation or reconstructed from the Higgs decay products. Since there are always two neutrinos in the final state of h → τ τ events, the Z boson momentum is needed to reconstruct the Higgs momentum as the initial condition of the Higgs decay kinematics to fully recover the two neutrino momenta. We use MadGraph [48] and TauDecay [49] packages to simulate the spin correlation in the Higgs decay chains. For a realistic simulation, both detector response and statistical fluctuations have to be taken into consideration. In order to perform fast detector simulation we construct a simplified smearing algorithm which is validated by comparing with Delphes [50] output. Using the recoil mass method, the smearing should in principle be applied to the Z momentum. Nevertheless, since the Higgs and Z bosons are back to back in the center of mass frame, we can directly smear the Higgs momentum. Defining the z-axis along the Higgs momentum, only its P z component is affected by the Z boson decay modes while the other two, P x and P y , have independent smearing behaviors. To select the Higgsstrahlung events, those with the reconstructed Z invariant mass outside the range 80 GeV < p 2 Z < 100 GeV are discarded. The momentum uncertainties of Higgs smearing have been summarized in the left part of Table 2. The pion momentum smearing is performed by randomly sampling the azimuthal angle φ and the pseudo-rapidity η according to Gaussian distribution [50,53]. In addition, the transverse momentum |p T | is sampled with a Log-normal like distribution from Ref. [50], Higgs Smearing Observables Uncertainty P x,y 1.82 GeV P z (Z → jj) 2.3 GeV P z (Z → ll) 0.57 GeV Pion Smearing Observables Uncertainty φ 0.0002|η|+ 0.000022 η 0.000016|η| + 0.00000022 |p T | 0.036|p T ||p rec T | = exp   log |p T | − 2 1 + σ 2 |p T | 2   ,(3.1) with being a random number following a Gaussian distribution centered in 0 with error 1 and N a normalization factor. For τ decay into ρ ± , the reconstructed ρ invariant mass is required to be within the range of 0.3 GeV < p 2 ρ < 1.2 GeV. The uncertainties of (φ, η, |p T |) for pions are summarized in the right part of Table 2. Although our simplified smearing algorithm is admittedly less sophisticated, our results are broadly compatible with those commonly adopted in the literature. A complete analysis in momentum reconstruction and detailed cuts was performed in Refs. [41,50]. For validation, we compare our smearing algorithm to the Delphes simulation with the configurations cards delphes card CircularEE.tcl [51] for the CEPC/FCC-ee [54] and delphes card ILD.tcl [52] for the ILC. Fig. 2 shows the smeared distributions of the pion kinematic variables simulated with Delphes (black) vs our simplified smearing (blue). We can see that the results of these two simulations agree with each other quite well. In this work, we take the simplified smearing algorithm for a fast simulation. To obtain the total number of expected events, one needs to consider several branching ratios. First, the Z boson can only be reconstructed if it decays into either leptons or jets with 80% of branching ratio in total [44]. Also, since the decay branching ratio of Higgs decaying into two τ leptons is 6.64% [44], only around 5.3% of the actual Higgs events associated with Z production are available for the CP measurement. Further suppression comes from the branching fraction of the decay of τ into π or ρ. And we arrive at 7704 events at the CEPC, 7003 events at the FCC-ee, and 4482 events at the ILC. Taking into account the identification of τ jets and tagging of the Higgs boson and other selection cuts [40], we obtain an overall efficiency, = 0.145, 0.144, 0.142 for (π, π), (π, ρ) and (ρ, ρ) decay modes, respectively. Table 3: Left: Branching fractions associated with the entire reaction. The values were obtained from [44]. Right: Expected event numbers at the CEPC [7] with the integrated luminosity L = 5.6 ab −1 , FCC-ee [8] with L = 5 ab −1 and ILC [46] with L = 2 ab −1 . The expected numbers of events before and after selection cuts are shown in the columns "before" and "after", respectively, with the overall cut efficiencies taken from Ref. [40]. The expected event numbers before and after applying the selection efficiencies are shown in Table 3 for comparison. In total, roughly 1105, 1004, and 643 events of the h → τ + τ − , τ ± → π ± /ρ ± ν τ ± decay chains can be reconstructed at the CEPC, FCC-ee, and ILC, respectively. Ambiguities in Momentum Reconstruction Experimentally, in order to reconstruct the τ momentum, it is unavoidable to first obtain the neutrino momentum which is not directly detectable. With two neutrinos in the final state, we need to constrain two 4-vector momenta. Since the Higgs momentum can be fully reconstructed from the Z boson counterpart, only one neutrino momentum is independent due to energy-momentum conservation. The 4 degrees of freedom can be constrained by the on-shell conditions of the two neutrinos and the two τ leptons. Unfortunately, the solutions have a two-fold ambiguity. Since on-shell conditions are in quadratic forms, one sign can not be uniquely fixed. For completeness, we summarize the solution here in terms of the τ − momentum defined in the Higgs rest frame, p τ − = p 2 h − 4m 2 τ [sin θ τ (cos φ τn1 + sin φ τn2 ) ± cos θ τn3 ] . (3.2) The unit base vectorsn i are constructed in terms of the primary decay mesons, X ± ≡ π ± , ρ ± , n 1 =p X + ,n 2 =p X − − (p X + ·p X − )p X + 1 − (p X + ·p X − ) 2 ,n 3 =p X + ×p X − 1 − (p X + ·p X − ) 2 . (3. 3) The first base vectorn 1 is along the momentum of π + or ρ + while the third onen 3 is perpendicular to the momentum of both primary mesons. Finally,n 2 is simply the one perpendicular to bothn 1 andn 3 . The polar angles of the τ momentum can be reconstructed as, sin θ τ cos φ τ = m 2 τ + m 2 X − m h E X + |p X + | m 2 h − 4m 2 τ , (3.4a) sin θ τ sin φ τ = m h E X − − m 2 τ − m 2 X − |p X − ||s X − X + | m 2 h − 4m 2 τ + m h E X + − m 2 τ − m 2 X + |p X + ||s X − X + | m 2 h − 4m 2 τ c X − X + , (3.4b) where (s X − X + , c X − X + ) ≡ (sin θ X − X + , cos θ X − X + ) and θ X − X + is the angle between the momentum of X + and X − . However, in Eq. (3.2) the ± sign in front ofn 3 reflects the fact that both solutions obey all the constraints from energy-momentum conservation and the correct solution cannot be unambiguously obtained. This sign ambiguity can significantly decrease the CP sensitivity, especially for the neutrino azimuthal angle distribution. Using momentum conservation, the result in Eq. (2.7) for tan δφ ν can be written in the same form by substituting p ν by p X ± , hence, tan δφ ν ∝p τ − · (p X + × p X − ) = ± cos θ τ / 1 − (p X + ·p X − ) 2 . In other words, δφ ν can have both positive and negative solutions with the same magnitude. This would not be a big problem for the symmetric distribution of δφ ν around its origin, such as those curves in Fig. 1 with ∆ = 0 • . But it causes significant issues for other ∆ values and effectively flattens the curve for ∆ = ±45 • . This ambiguity can be solved by measuring other decay information. An especially useful quantity is the impact parameter [55,56], the minimum distance of charged meson trajectory to the τ leptons production point. The impact parameter measurement essentially removes the two-fold ambiguity for the τ Yukawa CP measurement at future lepton colliders [24,53]. A more recent study with spatial resolution of 5 µm can be found in Refs. [40,41]. Another ambiguity comes from the detector resolutions. The τ momentum is reconstructed from the smeared Higgs and meson momentum. This reconstruction is realized with energy and momentum conservation, assuming narrow width approximation for the τ momentum, p 2 τ ± = m 2 τ . Both smearing and finite width could lead to nonphysical solutions in Eq. (3.4), for example sin θ τ sin φ τ > 1. For those events, we follow a similar procedure introduced in Ref. [41]. We try to find the solution for the (anti-)neutrino momenta optimally consistent with all the information we have on each event (including four-momentum conservation) by minimizing the function χ 2 rec = 3 i=0 (p rec τ + ) i + (p rec τ − ) i − (p rec h ) i σ h 2 + (p rec τ + ) 2 − m 2 τ σ τ 2 + (p rec τ − ) 2 − m 2 τ σ τ 2 ,(3.5) where i = 0, · · · , 3 runs over the 4-momentum components of each particle momentum. We adopt the uncertainties as σ h = 4.0 GeV and σ τ = 0.2 GeV [41]. The χ 2 rec function is minimized over the 6 kinematic parameters of the unmeasured neutrino momentum: the pseudo-rapidity, azimuthal angle, and absolute value of the momentum for both neutrino/anti-neutrino. Then the τ momentum is then obtained with energy momentum conservation, p rec τ ± = p ν ± + p rec X ± . The best fit at the minimum of χ 2 rec approximates the physical solution. We keep the event if the minimum solution is consistent with the mass cuts. Otherwise, the event is discarded. Events/18 Events/18 The final result of the differential distribution for the h → τ + (→ ρ + ν τ )τ − (→ ρ − ν τ ) process is plotted in Fig. 3. The left panel shows the differential distributions for ∆ = 0 • (red), ∆ = 45 • (blue), and ∆ = 90 • (green), respectively. Being divided into 20 bins [33,40], there are 25 ∼ 35 events in each bin on average. The corresponding statistical uncertainty at the level of 17% ∼ 20% is much smaller than the oscillation amplitude, π 2 /16 ≈ 62%. The event rate at the CEPC are large enough to constrain the modulation pattern as elaborated in Sec. 3.3. The right panel shows the spectrum at the three future candidate lepton colliders, CEPC (red), FCC-ee (blue), and ILC (green), respectively, for comparison. While CEPC and FCC-ee have comparable spectrum, ILC has much lower event rate and hence larger fluctuations. • CEPC h → τ + (→ ρ + ν τ )τ − (→ ρ − ν τ ) ∆ =0 • ∆ =45 • ∆ =90 • −180 • −135 • −90 • −45 • 0 • 45 • 90 • 135 • 180 •• h → τ + (→ ρ + ν τ )τ − (→ ρ − ν τ ) ∆ = 0 • CEPC FCC-ee ILC It is interesting to see that for ∆ = 90 • , the differential distribution of δφ r has only cos δφ r but no sin δφ r in Eq. (2.6). In other words, the observable that we measure has only CP conserving contribution that does not change under CP transformation. However, the distributions in the left panel of Fig. 3 show that the difference between ∆ = 0 • and ∆ = 90 • is maximal. This is because cos 2∆ = ±1 take the two extreme values with opposite signs. Discovery Potential and Sensitivity of the CP Phase To evaluate the CP measurement sensitivities, we adopt a χ 2 function defined according to the Poisson distribution, χ 2 ≡ i 2(N test i − N true i ) + 2N true i log(N true i /N test i ),(3.6) where i = 1, · · · , 20 runs over all the 20 bins of the δφ r differential distribution. Since we are studying the projected sensitivity at future lepton colliders, there is no real data available yet. Instead, we simulate the measurement with some assumed true values of the CP phase ∆ to 5 σ τ ± → ρ ± τ ± → ρ ± and π/ρ All LHC Current 95% C. L. The CP phase discovery potential at the CEPC for κ τ = 1. The green shaded regions represent the results from various decay modes: only (ρ, ρ) (light green), (ρ, ρ) + (π, ρ) (green) and the fully combined one (ρ, ρ)+(π, ρ)+(π, π) (dark green), with the boundaries describing the χ 2 CPV values according to Eq. (3.7) given ∆ true . The black dashed lines with χ 2 CPV = 1.96, 5 mark the sensitivities at 95% C.L. and 5σ, respectively. Right: Sensitivity of all the channels at the CEPC (green), FCC-ee (blue), and ILC (red). In both panels, the region outside of the gray bands are excluded at 95% C.L. by the current CP measurement at the LHC [42]. The discovery ability of a nonzero CP phase can be parametrized as the smaller one of the two χ 2 values between the given ∆ true and the CP conserving cases ∆ test = 0 • or ∆ test = 180 • , 0 • 45 • 90 • 135 • 180 •χ 2 CPV (∆ true ) ≡ min[χ 2 (∆ true , ∆ test = 0 • ), χ 2 (∆ true , ∆ test = 180 • )]. (3.7) Fig. 4 shows the χ 2 CPV distribution as a function of ∆ assuming κ τ = 1. The sensitivities for the di-τ decay into (ρ, ρ), (ρ, ρ) + (π, ρ), and the full combination (ρ, ρ) + (π, ρ) + (π, π) are depicted in light green, green, and dark green regions, respectively. For different decay channels, the differential distributions have the same amplitude π 2 /16 as indicated in Eq. (2.6). So the main difference in the sensitivities is due to the event rates: the branching ratio of the τ → πν τ is only 10.8%, in comparison with the 25% for the τ → ρν τ channel. As indicated by the black dashed lines, 95% of the values of ∆ can be tested above 95% C.L. and 82% of the parameter space can be tested at even more than 5σ. The sensitivity peaks at ∆ = ±90 • where Eq. (2.6) takes the most different value from that of ∆ = 0 • or 180 • with more than 10 σ significance. In the right panel, we also show the comparison of the sensitivities at the CEPC (green), FCC-ee (blue), and ILC (red). As expected, the CEPC has the highest sensitivity due to the higher number of events. For completeness, Table 4 summarizes the expected precision of the ∆ measurement at future lepton colliders at 68% C.L. and 95% C.L. for m = 1 parameter (∆), or m = 2 parameters (∆ and κ τ ). Notice that our estimation at 1 σ level is slightly better than the 4.4 • with 1 ab −1 in only the τ → ρν τ decay channel [33] or 4.3 • with 2 ab −1 in both τ decay channels [40] at the ILC. For the CEPC, our result is the same as the 2.9 • in Ref. [41]. Notice that in addition to the 68% C.L. for m = 1 95% C.L. for m = 1 95% C.L. for m = 2 CEPC 2. two mesonic decay channels, the leptonic decay channel τ → νν is also considered in Ref. [41] with the matrix element based observable that is different from our polarimeter δφ r . We can clearly see from Table 4 that the future lepton colliders can differentiate the CPV scenario from the CP-conserving one very well. Prospects of Constraining New Physics As the aforementioned analysis shows, there remains significant potential for discovering CP violation in the h → τ + τ − decay at prospective future lepton colliders. We now draw the connection with the lepton flavored EWBG scenario, following the treatment given in Ref. [14] for concrete illustration (see Refs. [15][16][17][18]).This discussion exemplifies future lepton colliders are not only precision machines but can also make an O(1) measurement of BSM physics effects. Two Higgs Doublet Model The set up in Ref. [14] relies on the type III Two Higgs Doublet Model (THDM) [57,58], wherein the two scalar doublet fields before EWSB are denoted as Φ 1,2 . Both neutral scalars inside Φ 1,2 acquire nonzero VEVs, v 1 and v 2 , respectively, with v ≡ v 2 1 + v 2 2 = 246 GeV. The neutral components can mix with each other to form three neutral massive scalar fields after one neutral Goldstone boson is eaten by the Z boson. We assume a CP-invariant scalar potential, namely, only the real parts of the two neutral scalars can mix with each other but not with the imaginary parts, H ≡ c α Re[Φ 0 1 ] + s α Re[Φ 0 2 ], h ≡ −s α Re[Φ 0 1 ] + c α Re[Φ 0 2 ], A ≡ −s β Im[φ 0 1 ] + c β Im[φ 0 2 ], (4.1) where s α ≡ sin α, c α ≡ cos α, tan β ≡ v 2 /v 1 , and Re and Im denote the real and imaginary parts, respectively. Note that α is the mixing angle from the neutral scalar mass matrix diagonalization. The neutral particle masses are ordered as m H , m A > m h ≈ 125 GeV, so that h is the SM-like Higgs boson. In the Type-III THDM, the Yukawa interaction for each doublet field has the same structure as the SM Yukawa interaction; L Y = −LY 1 R Φ 1 − LY 2 R Φ 2 + h.c. (4.2) In this way, both Higgs doublets can contribute its neutral components to couple with the τ lepton [29], − m τ v τ L τ R s β−α + N τ τ m τ c β−α h + c β−α − N τ τ m τ s β−α H + iAN τ τ , (4.3) where N τ τ is a complex parameter related to the matrix elements of Y 1,2 . Following the parametrization of Eq. (2.1), the τ Yukawa coupling becomes κ τ (cos ∆ + i sin ∆) = s β−α + N τ τ m τ c β−α . (4.4) Notice that CP violation arises due to the imaginary part of N τ τ . Moreover, for the particular texture, Y j,22 = Y j,23 = 0, Y 1,33 = Y 2,33 = Y 33 and Y 1,32 = r 32 Y 2,32 , one can write the imaginary part of the Jarskog invariant J A of the Yukawa interaction in Eq. (4.2) as, Im[J A ] = −Im[r 32 ]|Y 2,32 | 2 = 2m 2 τ v 2 c β−α κ τ sin ∆ . (4.5) It is the imaginary part of the Jarlskog invariant that controls the size of the BAU in early universe through lepton flavored baryogenesis [14]. Rewriting Eq. (4.5) gives sin ∆ = v 2 c β−α 2m 2 τ κ τ Im[J A ]. (4.6) Thus, one may connect the τ Yukawa CP phase ∆, which can be measured at future lepton colliders, with CPV source for baryogenesis during the era of EWSB in the early universe. Sensitivity to the Baryogenesis Scenario To make this connection concrete, we plot in Fig. 5 the 95% C.L. constraints on κ τ cos ∆ and κ τ sin ∆ from present and future collider probes and from lepton flavored EWBG. For generality, we also set κ test τ to be free to obtain a full picture on a two-dimensional plot. The CP sensitivity is then depicted as the contours around the true value ∆ true = 0 • and κ true τ = 1 in the left pannel, with the green, blue (dashed), and red (dot dashed) contours indicating the 95% C.L. sensitivities. The green dotted lines from the origin κ τ = 0 are added to show that the contour size corresponds to roughly 7 • at 95% C.L. Consistent with the previous observation, the CEPC and FCC-ee have comparable precision while that of the ILC is slightly weaker due to different luminosities. For all three cases, the pink region allowing for successful explanation of BAU is outside the 95% C.L. contour. In other words, the lepton flavored BAU mechanism as given in Ref. [14] could be excluded at better than than 95% C.L. . For comparison, we also show the projected τ Yukawa CP measurement at the High Luminosity (HL-)LHC [38] with the integrated luminosity of 3 ab −1 as the black contour, which will be further elaborated below. It is clear that even with the HL-LHC, the THDM BAU mechanism can only be tested with barely 95% C.L. The CP measurements at future lepton colliders can significantly improve the situation. We also include the constraints from the measurement of the h → τ τ decay signal strength µ τ τ , which is proportional to κ 2 τ . The current data at the LHC indicate µ τ τ = 1.09 +0.35 −0.30 at ATLAS [5] and µ τ τ = 0.85 +0.12 −0.11 at CMS [6], which are depicted as the gray region. In other words, the current measurement at LHC is still quite crude with at least 10% uncertainty. At the HL-LHC, the 1 σ uncertainty of µ τ τ can be further improved to 5% [59,60], which is further combined with the CP measurement [38] that is shown as the black contour. The future lepton colliders can significantly improve the sensitivities to 0.8% at the CEPC [61], 0.9% at the FCCee [8], and 1.09% at the ILC [61], which are shown as the rings in the left panel of Fig. 5. Note that these rings with inclusive τ decays are much narrower than the width of the contours or equivalently the marginalized sensitivity on κ τ after integrating out the CP phase ∆ from the original two-dimensional distributions. The discrepancy comes from the fact that the τ → πν τ and τ → ρν τ channels contribute only a very small fraction (∼ 13%) of the inclusive decay events. The strength measurement can provide very important complementary info and reduce the parameter space to be explored. Instead of assuming the SM values κ true τ = 1 and ∆ true = 0 • , it is interesting to ask the whether the lepton flavored EWBG scenario can explain the BAU and at the same time produce a signal that is distinguishable from the SM. To address this question, we show in the right panel of Fig. 5 the similar contours around ∆ true = ±13.3 • and κ true τ = 1 that is at the boundary of the BAU region. Under this assumption, the CEPC and FCC could establish the presence of CPV in the τ Yukawa interaction with 3σ significance, while for the ILC the significance The green region represents the space parameter where the sensitivity is below 95% C.L. The pink region represents the parameter space that can explain the BAU in the lepton flavored EWBG scenario [14]. would be somewhat weaker. It is also interesting to investigate the behavior of the CP violation sensitivity when one varies the assumed true values of κ τ . This can be observed from Fig. 6 where we show the sensitivity as a function of the CP phase ∆ and the coupling strength κ τ . The dashed gray lines give several typical sensitivities χ 2 CPV = 5, 10, 15, 20. Note that the dashed gray lines expand with larger τ Yukawa coupling due to event number enhancement. This is especially significant for small κ τ while for large values of κ τ the CP sensitivity does not change substantially. The BAUcompatible region has a lower limit at κ τ ≈ 0.25 due to the lower limit on κ τ sin ∆ according to Fig. 5 and most of the BAU-compatible region falls inside the χ 2 CPV = 5 curve, corresponding to 5 σ discovery. Conclusions Explaining the origin of the baryon asymmetry of the Universe is a key open problem at the interface of particle and nuclear physics with cosmology. An essential ingredient in the explanation is the presence of BSM CP violation. In the electroweak baryogenesis scenario, the relevant CPV interactions would have generated the BAU during the era of EWSB. The corresponding mass scale makes these interactions in principle experimentally accessible. While null results for permanent EDM searches place strong constraints on new flavor diagonal, elec-troweak scale CPV interactions, flavor changing CPV effects are significantly less restricted. Lepton flavored EWBG draws on this possibility, with interesting implications for CPV in the tau-lepton Yukawa sector. In this work, we have shown how measurements of CPV observable in Higgs di-tau decays at prospective future lepton colliders could test this possibility, with significant discovery potential if it is realized in nature. After making a detailed comparison of the four differential distributions of the neutrino azimuth angle δφ ν , polarimeter δφ r , acoplanarity φ * , and the Θ variable for the first time as well as various detector responses, we explore the prospects of CP measurement in the τ Yukawa coupling at future lepton colliders. With (5.6, 5, 2) ab −1 luminosity, the 1 σ uncertainty can reach 2.9 • , 3.2 • , 3.8 • at the CEPC, FCC-ee, and ILC, respectively. This allows the possibility of distinguishing the attainable EWBG from the CP conserving case with 3 σ sensitivity. The future lepton colliders are not just precision machines for detailing our understanding of the Higgs boson, but can also make O(1) measurement of the possible new physics beyond the SM. Fig. 1 . 1 Fig. 2 : 2The pion smearing effects simulated by Delphes (black) and our simplified algorithm (blue). Notice that in the panel (b) the numbers in the horizontal axis are multiplied by a factor ×10 5 for better visualization. Fig. 3 : 3Smeared differential distributions of δφ r in the decay mode h → τ + (→ ρ + ν τ )τ − (→ ρ − ν τ ). Left:The distribution for CP phases ∆ = 0 • (red), ∆ = 45 • (blue), ∆ = 90 • (green) at the CEPC; Right: A comparison of the distributions at the CEPC (red), FCC-ee (blue), and ILC (green) for ∆ = 0 • . In both panels, the error bars indicate the statistical uncertainties. Fig. 4 : 4Left: a set of pseudo-data N true i and then fit these pseudo-data with some test values N test i . The event numbers N true i and N test i are functions of the true value ∆ true and ∆ test , respectively. Fig. 5 : 5Left: The 95% C.L. constraints on the τ Yukawa coupling at the CEPC (green), FCC-ee (blue), and ILC (red) assuming the true value ∆ true = 0 • and κ true τ = 1. The gray band gives 95% C.L. constraints from the current LHC signal strength measurements[5,6] while the black contour denotes the expected 95% C.L. constraint from the combined measurements of µ τ τ[59, 60] and ∆[38] at the HL-LHC. Right: The 3σ contours for each collider assuming central values ∆ true = ±13.3 • and κ true τ = 1 corresponding to the minimum |κ τ sin ∆| compatible with the BAU. Fig. 6 : 6The CP discovery capability of the CEPC as a function of the ∆ and κ τ true values. The black dashed lines represents several typical values of the significance, χ 2 CPV = 5, 10, 15, 20. for comparison.Integrated luminosity √ s Number of Higgs bosons CEPC [7] 5.6 ab −1 240 GeV 1.1 × 10 6 FCC-ee [8] 5 ab −1 240 GeV 1.0 × 10 6 ILC [9] 2 ab −1 250 GeV 0.64 × 10 6 Table 1 : 1Configurations (integrated luminosity, energy √ s, and Higgs production rate) at the future lepton colliders CEPC, FCC-ee, and ILC. Table 2 : 2Left: Uncertainties of the Higgs boson[41] and Right: pion momentum smearing parameters to be consistent with the Delphes configurations delphes card CircularEE.tcl[51] for the CEPC/FCC-ee and delphes card ILD.tcl[52] for the ILC. Table 4 : 4The CP phase precision at the CEPC, FCC-ee, and ILC for m parameter(s) . AcknowledgementsSFG is sponsored by the Double First Class start-up fund (WF220442604) provided by Tsung-Dao Lee Institute, Shanghai Jiao Tong University and the Shanghai Pujiang Program (20PJ1407800). SFG is also grateful to Kai Ma for sharing his PhD thesis with derivations on the differential distribution of the τ → πν τ decay as well as Manqi Ruan, Xin Chen, and Dan Yu for useful discussions. GL would like to thank Shou-hua Zhu for helpful discussions. MJRM was supported in part under National Natural Science Foundation of China grant number 19Z103010239. GL and MJRM were supported in part under U.S. Department of Energy contract number DE-SC0011095. 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[ "https://github.com/delphes/delphes/blob/master/cards/delphes_" ]
[ "Spectroscopic observation of a transition region network jet", "Spectroscopic observation of a transition region network jet" ]
[ "J Gorman \nMax Planck Institute for Solar System Research\nJustus-von-Liebig-Weg 337077GöttingenGermany\n", "L P Chitta \nMax Planck Institute for Solar System Research\nJustus-von-Liebig-Weg 337077GöttingenGermany\n", "H Peter \nMax Planck Institute for Solar System Research\nJustus-von-Liebig-Weg 337077GöttingenGermany\n" ]
[ "Max Planck Institute for Solar System Research\nJustus-von-Liebig-Weg 337077GöttingenGermany", "Max Planck Institute for Solar System Research\nJustus-von-Liebig-Weg 337077GöttingenGermany", "Max Planck Institute for Solar System Research\nJustus-von-Liebig-Weg 337077GöttingenGermany" ]
[]
Aims. Ubiquitous transition region (TR) network jets are considered to be substantial sources of mass and energy to the corona and solar wind. We conduct a case study of a network jet to better understand the nature of mass flows along its length and the energetics involved in its launch. Methods. We present an observation of a jet with the Interface Region Imaging Spectrograph (IRIS), while also using data from the Solar Dynamics Observatory (SDO) to provide further context. The jet was located within a coronal hole close to the disk center. Results. We find that a blueshifted secondary component of TR emission is associated with the jet and is persistent along its spire. This component exhibits upward speeds of approximately 20-70 km s −1 and shows enhanced line broadening. However, plasma associated with the jet in the upper chromosphere shows downflows of 5-10 km s −1 . Finally, the jet emanates from a seemingly unipolar magnetic footpoint.Conclusions. While a definitive magnetic driver is not discernible for this event, we infer that the energy driving the network jet is deposited at the top of the chromosphere, indicating that TR network jets are driven from the mid-atmospheric layers of the Sun. The energy flux associated with the line broadening indicates that the jet could be powered all the way into the solar wind.
10.1051/0004-6361/202142995
[ "https://arxiv.org/pdf/2202.11375v1.pdf" ]
247,058,320
2202.11375
d4f29bdb086c7557c74e1ae5af7e8bf312b25172
Spectroscopic observation of a transition region network jet February 24, 2022 J Gorman Max Planck Institute for Solar System Research Justus-von-Liebig-Weg 337077GöttingenGermany L P Chitta Max Planck Institute for Solar System Research Justus-von-Liebig-Weg 337077GöttingenGermany H Peter Max Planck Institute for Solar System Research Justus-von-Liebig-Weg 337077GöttingenGermany Spectroscopic observation of a transition region network jet February 24, 2022Received DD Month YYYY / Accepted DD Month YYYYAstronomy & Astrophysics manuscript no. mainSun: atmosphere -Sun: chromosphere -Sun: transition region -Line: profiles Aims. Ubiquitous transition region (TR) network jets are considered to be substantial sources of mass and energy to the corona and solar wind. We conduct a case study of a network jet to better understand the nature of mass flows along its length and the energetics involved in its launch. Methods. We present an observation of a jet with the Interface Region Imaging Spectrograph (IRIS), while also using data from the Solar Dynamics Observatory (SDO) to provide further context. The jet was located within a coronal hole close to the disk center. Results. We find that a blueshifted secondary component of TR emission is associated with the jet and is persistent along its spire. This component exhibits upward speeds of approximately 20-70 km s −1 and shows enhanced line broadening. However, plasma associated with the jet in the upper chromosphere shows downflows of 5-10 km s −1 . Finally, the jet emanates from a seemingly unipolar magnetic footpoint.Conclusions. While a definitive magnetic driver is not discernible for this event, we infer that the energy driving the network jet is deposited at the top of the chromosphere, indicating that TR network jets are driven from the mid-atmospheric layers of the Sun. The energy flux associated with the line broadening indicates that the jet could be powered all the way into the solar wind. Introduction The study of the transfer of energy from the cool, lower solar atmosphere through the transition region (TR) in order to sustain the hot corona remains an active field of research within solar physics. In particular, localized energetic events such as spicules and jets are believed to be important contributors of mass and energy to both coronal heating and the solar wind. Jets are seen at a broad range of temperatures and spatial scales, from hotter (≈ 1 MK) coronal jets with broader spires (≈ 10 Mm) (reviewed in e.g., Raouafi et al. 2016) to much cooler (10 4 K) and narrower (down to a few hundred kilometers) chromospheric anemone jets (Nishizuka et al. 2011). Recent observational works have been able to explore the ubiquity of jets in the TR using the Interface Region Imaging Spectrograph (IRIS;De Pontieu et al. 2014b). Transition region network jets are seen as collimated and propagating intensity fronts. These jets are ubiquitous and are generally rooted to the boundaries of network magnetic patches in the photosphere. They were first identified by Tian et al. (2014) with near-limb observations using the 1330 Å channel of IRIS' slit-jaw imager (SJI). These jet events were found to have characteristic velocities of 80-250 km s −1 , widths up to 300 km, and lifetimes of 20-80 s, reaching temperatures up to 10 5 K. A similar study done by Narang et al. (2016) compared network jets in the quiet Sun (QS) to their counterparts in coronal holes (CHs). The authors found that jets in CHs were longer and faster than QS jets. These differences were due to the effect of (interchange) reconnection with open field lines in CHs compared to that with the smaller, closed loops found in the QS. Various observational and simulation studies have been designed to better understand the primary driving mechanisms behind small-scale jets, often pointing to the involvement of magnetic reconnection at different heights above the solar surface and/or upward moving shocks. Panesar et al. (2016Panesar et al. ( , 2018 and McGlasson et al. (2019) have recently reported on the magnetic driver of on-disk QS coronal jets. They determined that flux cancelation at the neutral line of mini-filaments led to the eruption of these mini-filaments. This, in turn, resulted in the ejection of the jets as seen in TR plasma. A recent observational study conducted by Qi et al. (2019) examined the link between network jets and coronal plumes, concluding that jets are more energetic in regions with visible plumes. This led them to surmise that the stronger magnetic convergence witnessed in the regions with plumes creates an environment with faster shocks or more smallscale reconnection that would drive the more dynamic jets. From a simulation standpoint, Yang et al. (2018) show that flux emergence can trigger both warm (≈ 10 5 K) network jets and cool (≈ 10 4 K) spicules through reconnection and pressure gradients, respectively. While the presence of TR jets has been well-established, a discussion regarding the veracity of their high-speed mass flows remains open. Previous studies (such as e.g., Tian et al. 2014;Narang et al. 2016;Chen et al. 2019) relied on narrow-band imagery and sit-and-stare spectroscopic scans, which made it difficult to do more than track intensity movements along the jets or transverse jet motions. There is a debate about whether or not the assumed shooting upward movement of plasma is mostly just propagating thermal fronts, wave motions, or some weighted combination of the aforementioned (Chintzoglou et al. 2018;De Pontieu et al. 2017). Specifically, Rouppe van der Voort et al. (2015) point to the need for further study of the individual line profiles to better understand noted blueward asymmetries in jet spectra. In this context, capturing an ongoing network jet event while it is happening, along with its spectroscopic properties from its photospheric footpoints into the TR, would shed light on the nature of driving mechanisms and potential mass motions. However, such a clear spectroscopic observation of a jet event is rarely captured. We have investigated a network jet in both narrow-band imagery and IRIS spectroscopic rasters, tracing it from its signatures in the TR down through the chromosphere to its magnetic footpoint in the photosphere. Due to the advantageous timing and spatial coverage of the raster slit, we were able to follow the spectral evolution of the jet along its (projected) length. Here, we report on the results of this analysis and provide insights into the nature of the flow of mass and energy within small-scale jets. Observations and data analysis IRIS observed an equatorial CH at disk-center bounded by two active regions (ARs) on November 23, 2016, from 07:12:43-07:46:33 UT. This observation, which is centered at [x, y] = [110 , −134 ], is comprised of a very-large dense raster scan with accompanying contextual slit-jaw imagery. The raster scanned the CH in 64 steps with 30 s exposure times (the step cadence is about 32 s, and the step size is 0.35 in the scan direction; the observation is spatially binned such that the image scale along the slit is 0.33 pixel −1 ). The raster covered a field of view (FOV) of 22 × 175 . The entire far-UV (FUV) and near-UV (NUV) detector windows were read out. All four of the slit-jaw channels (i.e., 1330, 1400, 2796, and 2832) were operating. For the purposes of this study, only the 1400 and 2796 SJI channels are discussed. These channels recorded 16 images each, at a cadence of 127 s. The total SJI FOV was 167 × 175 . For the SJI, similar to the raster, the data are binned in the y-direction. We also binned the SJI data in the x-direction during processing, resulting in a final pixel-scale of [x, y] = [0.33 , 0.33 ]. Within each IRIS SJI channel, we normalized the images by exposure time and aligned them to one another. We complemented IRIS data with extreme ultraviolet (EUV) observations from the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012). In particular, we used AIA 193 Å EUV filter images for context. These data have a cadence of 12 s and an image scale of 0.6 pixel −1 . They were normalized by exposure time and co-registered to an AIA reference map taken at 07:27:54 UT on the day of the observation. To study the magnetic field distribution underlying IRIS' FOV, we used line of sight (LOS) magnetic field maps from the Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012) on board SDO. These HMI data have a cadence of 45 s, an image scale of about 0.5 pixel −1 , and were co-registered to the reference AIA image. Using a cross-correlation technique, we aligned the SDO and IRIS data to enable easier feature location and comparison. The final product shown here is an aligned set of AIA, HMI, and IRIS SJI maps that share the same pixel size [0.33 , 0.33 ] and orientation, with solar north pointed up. In addition to the co-aligned imagery, the original, highresolution spectra acquired by the IRIS raster scan were analyzed. To determine TR plasma properties, we fit a single-Gaussian (SG) model to the TR Si iv 1394 Å line. This was done using the procedure iris_auto_fit.pro available in the So-larSoft 1 (SSW) library. In regions of the jet where strong deviations from a SG profile were detected, double-Gaussian (DG) fits were also performed using the SSW procedure dgf_1lp.pro, which was modified slightly to suit our needs. Results Overview imagery reveals the solar environment in which the jet originates. Figure 1 shows the full Sun as imaged by the 193 Å channel of AIA at the time of the jet observation. A crescentshaped CH is flanked on opposite sides by two bright ARs. The jet is located at the border between the CH and the lower AR. This general environment of the jet is more readily seen in Fig. 2, which is zoomed into the slit-jaw FOV. The jet emanates from a magnetic network region, as outlined by bright patches in IRIS SJI1400 intensity (panel b) and HMI LOS magnetic field concentrations (panel c). Jet in the transition region The jet's immediate surroundings across different layers of the solar atmosphere, as shown in Fig. 3, provide insight into the dynamics at play for this event. The clearest signal from the jet is seen in the SG-fitted Dopplergram as a strip of upflowing material (blueshifted region in panel b). Based on these fits, we find that the observed jet has a line of sight velocity reaching −30 km s −1 . This strip appears in the intensity imagery taken by the SJI1400 channel (panel a), and it resembles the jets seen by Tian et al. (2014). However, in our observation, the jet is only found in a single SJI frame. This is due to the SJI's low cadence of 2 min. The jet is also not easily detectable in any of the other SJI channels or in AIA. Below the jet, there is an area of increased activity in the chromospheric SJI2796 channel (panel c), and this region overlies clumps of positive polarity network magnetic field concentration seen in the HMI magnetogram (panel d). Since the jet is only seen in one image frame, the jet's lifetime and apparent velocity within the plane-of-sky cannot be well constrained using slit-jaw images. With a cadence of 127 s, the maximum lifetime is estimated as the time between three frames, or 254 s. However, the minimum duration of observed upward-flowing material can also be constrained as the time it takes the raster to sweep across the jet. There are 7 slit positions with blueshifts recorded along the jet, with a cadence of 32 s between each position. The minimum duration of upflows is then 224 s. Together with the above estimates, this implies that the jet lifetime should be somewhere between 224 s and 254 s, that is on the order of about 4 min. Parameters derived from the SG fits of the Si iv 1394 Å line (formation temperature log T [K] = 4.8) outline the shape and extent of the jet within the transition region. Maps of the peak intensity, Doppler velocity, and full-width at half maximum (FWHM) derived from the SG fits are included in Fig. 4. They show that the jet consists of a narrow spire of increased intensity with broadened line widths. In comparison, directly beneath this spire are downflows (redshifted pixels) whose SG-fits have a lower peak intensity and FWHM. The Dopplergram shows a gradual increase in blueshift magnitude along the jet's spire from roughly 5-10 km s −1 at its base up to 30 km s −1 near the top. While the parameter maps provide an overview of the event, the individual spectra are better at showing the flow structure within the jet. Spectra from three pixels representative of the upper, middle, and bottom portions of the jet are shown in panels (d)-(i) of Fig. 4, with either SG or DG fits applied. The SG fits plotted in the middle row seem to visually represent the observed intensities fairly well. However, most profiles along the jet show a two-component flow behavior that is best described by a DG fit (bottom row). This is supported by the reduced-chi-square metric, which is improved in virtually all cases where the DG fit is applied, with an average reduction by a factor of four. Further examples of DG-like intensity profiles are shown in the Appendix (Fig. A.1). The spectral profiles along the length of the blueshifted segment of the jet (as seen in the SG Dopplergram) exhibit a secondary blueshifted component. Such profiles with a secondary blueshifted component are observed even at those locations that are redshifted when fit with a SG. At this base, the secondary component has a blueshift of −13 km s −1 , which increases to almost −80 km s −1 farther up along the jet spire. The average secondary component velocity is −51 km s −1 (see Figs. 4 and A.1). When comparing blueshifts, Doppler velocities of the secondary component of the DG fit are always larger than those derived from the SG fits. Panels (e) and (h) of Fig. 4 show the SG and DG curves for the same pixel in the middle of the jet. The DG fit secondary component has a velocity equal to −71 km s −1 , yet the SG fit velocity only has a line shift of −10 km s −1 . Comparable discrepancies can be seen all along the jet's spire. In addition to velocity, the secondary component has excess line widths (ξ2; non-thermal velocity) that exceed typical non-thermal widths of 20 km s −1 in the TR at ≈ 0.1 MK . We find that most of the secondary component non-thermal widths exceed this typical value, even by up to a factor of two or more. On average, the non-thermal width of the primary component is less than that for the secondary. Jet as seen in the chromosphere The jet's chromospheric activity is investigated using the Mg ii 2796 Å line, which is formed in the upper chromosphere within a temperature range of roughly log T [K] = 3.7-4.2. The normalized Mg ii k intensity profiles from along the jet and at its base, as plotted in panels (d)-(g) in Fig. 5, show that, while the Mg ii intensities along the upper spire of the jet are comparable to the QS profile, they become enhanced at its base (i.e., south of the region exhibiting blueshifts in Si iv). These spectra located in the bright patch of activity beneath the jet can get to be around four times as intense as the QS (panel g). The intensity maps obtained around the k2r and k3 features of the Mg ii line (panels (a) and (b) in Fig. 5) represent the intensities in the middle to top of the chromosphere at the base of transition region. There are still traces of the jet spire in these chromospheric layers. However, signatures of the spire are absent in the deeper layers of the chromosphere sampled with the Mg ii k1r feature (see panel c). Rather, only the bright patches (associated with the magnetic network) at the jet's footpoint are visible. It is possible to make inferences about the speed within the chromosphere by comparing the relative shifts and intensity variations along the Mg ii k line. Using the analysis procedure described in Pereira et al. (2013), the pixels near the jet's footpoint show redshifts of the k3 depression that are consistent with downflows at the top of the chromosphere of about 5-10 km s −1 . Intensity asymmetries between the k2r and k2v peaks can also be exploited to reveal the presence of up-or downflows. When the k2v peak is enhanced compared to k2r, as is seen for this observation at the base of the jet, this signifies that plasma above the k2 formation height is moving downwards (see Leenaarts et al. 2013;Pereira et al. 2013). The utility of chromospheric diagnostics using shifts of the k3 depression and comparison of the k2v to k2r peak intensities has also been analyzed and confirmed by Bose et al. (2019Bose et al. ( , 2021. Thus, both of these diagnostics consistently and coherently reveal that there are downflows present in the upper chromosphere below or at the base of the jet. Magnetic structure at base The features detected in the LOS component of the magnetic field can be linked to the overlying jet structure. Comparing panels (c) and (d) from Fig. 3 shows that the increased chromospheric activity (i.e., radiation) beneath the jet is spatially aligned with strong concentrations of magnetic field. The raster maps from Fig. 5 agree with this spatial overlapping of bright Mg ii intensity and magnetic field patches. The time sequence of this region reveals that the resolved clumps of positive polarity magnetic field rearrange themselves, separating and coalescing, in the period of time surrounding the jet (see online movie associated with Fig. 3d). A more quantitative view of the magnetic evolution is provided by the time evolution of the magnetic flux. Figure 6 illustrates how the magnetic flux, of both positive and negative polarity patches, at the footpoint of the jet changes with time. Here, the footpoint is defined by the white box in panel (d) of Fig. 3. To avoid noise, only pixels with magnetic flux density greater than 10 G were integrated to derive magnetic flux (this was done separately for positive and negative polarity regions). Although, selecting different cut-off strengths of 5 G, 10 G, 15 G, or 20 G showed no significant difference in the time series of magnetic flux, underlining that our results are not affected by noise. Magnetic flux contained by negative polarity patches in this region is two orders of magnitude weaker than that of the positive polarity patches. In addition, variations in the magnetic flux of negative polarity magnetic concentrations are generally within the computed root-mean-square-error (RMSE) for the whole time series. For the magnetic flux of positive polarity fields, however, a few things are noted. First, a general, linearly increasing trend is detected over the observing period of three hours. The fit to this trend was used to calculate the RMSE. Around the time of the jet, the positive flux shows strong fluctuations above the level of the RMSE. In the ten minutes preceding the jet, magnetic flux rises sharply and then decreases significantly after the jet is seen in SJI1400. There are a few other instances of such strong changes in positive flux outside of the IRIS observing time, though none are of the same net magnitude seen during the jet's lifetime. Discussion Most TR jet studies have been conducted using slit-jaw imagery or sit-and-stare rasters. As a result of this, any actual mass motions along a jet's trajectory are not easily identified. Propagating intensity fronts in SJIs could correspond to moving heat fronts or wave motions. Stationary-slit rasters, when at the limb, would only likely pick up transverse motions. When near disk-center, such rasters typically only cross small sections of a jet, thereby making it difficult to determine if upward flows are persistent along the rest of the structure. With this observation, however, we have the good luck that the raster scan was both co-temporal and co-spatial with the jet motions along its length, allowing for verification of the presence of mass flows and deeper inspection of spectral profiles. Jet properties Various jet properties, including the propagation speed and lifetime, can be estimated but not fully constrained, owing to various limitations of the observation. Due to the unknown inclination of the jet, exact speeds along its trajectory cannot be precisely calculated. Using the plane-of-sky propagation of the jet based only on the spatial extent of the blueshifted pixels as seen in Figs. 3b and 4b (≈ 20 pixels amounting to a total length of 5 Mm) and the time it takes the raster to scan across them (7 scans at a cadence of 32 s amounting to 224 s total), it can be estimated as a lower bound that the jet propagates laterally at 22 km s −1 . If we further assume a simplistic triangular geometry, where the plane-of-sky speed calculated above is used for the horizontal speed and the average LOS velocity of the jet material represents the vertical velocity (51 km s −1 , see Sect. 3.1), then we can calculate a propagation speed of 55 km s −1 . This speed is 1.5-4.5 times lower than that determined for typical CH jets by Tian et al. (2014) and Narang et al. (2016), but that leads us to presume the jet is traveling with speeds greater than the observational cadences are able to resolve. The projected length of the blueshifted material does agree with previous estimates Narang et al. 2016), rounding out to be between 4.5 and 5.2 Mm. As we stated previously, with only one slit-jaw image showing the jet, it is impossible to provide accurate estimates of the jet's lifetime. An upward estimate of 4 min 14 s, or the time between three slit-jaw images, grossly outlasts the average established jet lifetime of 20-80 s, as given by Tian et al. (2014). This number can be further winnowed, albeit slightly, by considering how long the raster takes to scan across the jet. In that case, a lifetime of 3.75 min is calculated (see Sect 3.1). Multiple flow components Plasma outflows (blueshifted region along the spire) confirm the presence of real mass motions. The physical extent of the jet actually extends beyond what is inferred solely from SG blueshifts of Si iv (see e.g., Fig. 4). The jet is traced down to its region of origin by analyzing the individual spectra, whose intensities reveal asymmetric profiles. These asymmetries are well-described by double-Gaussian fits to the observed Si iv line profiles. The primary component of the DG is representative of the typical TR redshifted material emitting beneath (i.e., below) the source region of the secondary component, which corresponds to the blueshifted jet material. By evaluating the secondary component, the behavior of the jet is elucidated. Previous work has investigated the common appearance of asymmetrical emission lines formed in the corona and TR (see e.g., Peter 2001;Hara et al. 2008;Tian et al. 2011;Yardley et al. 2021). As is the conclusion of this work, such line asymmetries are mostly attributed to the overlapping of background emission (primary component) and high-speed flows (secondary component). Tian et al. (2011) discovered a link between the secondary emission component and upward propagating disturbances seen in AIA. Ubiquitous, propagating torsional motions associated with dynamic features within the TR and chromosphere were first observed at small scales with IRIS and have been reported by De Pontieu et al. (2014a). Kayshap et al. (2018) attribute a change in Doppler shift of the secondary component from red to blue across network jets to twisting motions. For the event we describe here, however, the DG profiles are not interpreted as signatures of large-scale twist. One reason for this is that the secondary component remains blueshifted compared to the primary component, regardless of which side of the feature we regard. While there may be unresolved plasma motions that can be Each map shares the same axis-scaling, spatial extent, and pixel markers as that in Fig. 4. Mg ii k spectra from these pixels are shown in the bottom four rows, panels (d)-(g). The black curve in each panel shows the same normalized, average intensity profile for Mg ii for this observation, calculated using a large portion of quiet Sun. The k3, k2r, and k1r wavelength points were selected based on the shape of this profile and are marked by the vertical solid, dotted, and dashed lines, respectively. The k2v peak of this curve is used as the normalization factor for both the averaged profile and the individual pixel profiles (red curves). See Sect. 3.2. interpreted as small-scale torsional motions within the jet as evidenced by the non-thermal widths, a general, coherent overturning rotation of the structure is not supported by the observational data. The blueward flow component associated with the jet material is fast enough that it might contribute to the solar wind if the magnetic fields that confine the jet open into the heliosphere. This secondary component exhibits vertical speeds up to 50-75 km s −1 (see Fig. 4). Comparing this to the sound speed in the source region of Si iv that forms in ionization equilibrium at around 80 000 K (≈ 43 km s −1 ), these outward flows are supersonic. Such high-speed flows accelerate further as they ascend into progressively less dense plasma, and, if connected to an open field line within the CH, could escape into the solar wind. However, in order to be comparable with the solar wind speeds of a few hundred km s −1 recorded at 1 AU, these TR velocities would need to accelerate by at least a factor of six or more (and then the plasma has to be lifted out of the gravity field of the Sun). Hence, further acceleration would be required to eject this plasma as solar wind. An extra source of energy to propel the jet into the solar wind could be derived from its non-thermal velocities. The jet has nonthermal widths of up to 45 km s −1 along its spire (see Fig. 4). If these unresolved motions are assumed to be wavelike in nature, caused by an Alfvén wave, an energy flux can be calculated and compared to what is needed in terms of extra energy. The Alfvén wave energy flux, f , can be estimated from: f = ρ ξ 2 V A ,(1) where ρ is the mass density, ξ is the non-thermal velocity, and V A is the Alfvén speed. We use the average calculated nonthermal velocity of 34 km s −1 and the propagation speed estimated in Sect. 4.1, as well as a mass density of 1.8×10 −14 g cm −3 , which is calculated assuming a typical TR density of 10 10 cm −3 (see Supplementary Materials in Tian et al. 2014). This results in an energy flux of 1.2 kW m −2 , which is an order of magnitude greater than the 100-200 W m −2 needed on average to drive the solar wind (Schwenn 1990). Certainly, this is a significant amount of energy that could travel upward and boost the jet material at higher altitudes, thereby powering the solar wind. However, it could also be the case that the energy is not carried up high enough and is dissipated before it reaches a regime favorable for acceleration into the wind. Magnetic driver We can now look towards determining the jet's driver. We see signatures of increased activity in the Mg ii line extending below the jet's line of expression in Si iv. This intense patch is the presumed region of origin for the jet as it is located above strong concentrations of vertical magnetic field. Unfortunately, the behavior of the magnetic field at the jet's footpoint on both long and short timescales does not point unambiguously to a specific driving mechanism. Overall, no obvious signs of systematic flux cancelation are detected, but the positive flux linearly increases by around 9% across a period of three hours, which is potentially a sign of flux emergence (see Sect. 3.3 and Fig. 6). Interesting fluctuations, which are significant above the threshold set by the RMSE-level, are seen over the timescale during which the jet evolves. The sharp increase in positive magnetic flux immediately preceding the jet followed by a stark decrease during the jet's activity might be a signature of local reconnection (see the evolution in Fig. 6a from 07:20-07:25 UT indicated by the vertical lines). Similar magnetic field fluctuations before and after the event, while not quite as intense, could also be potential signatures of persistent jet activity in this region. However, a lack of IRIS coverage at those times makes it impossible to say whether or not this is true. Though large-scale flux cancelation is not obvious from the HMI data due to a lack of noticeable available negative field concentrations, the movie of the magnetic field at the base of the jet shows merging and separation of positive magnetic clumps (see online movie associated with Fig. 3d). This could further support the theory of small-scale reconnection between the strong, resolved positive magnetic flux and nearby, unresolved areas of negative flux. The plausibility of such a scenario is demonstrated by Chitta et al. (2017a) who show that, using the balloon-borne mission SUNRISE (Solanki et al. 2010(Solanki et al. , 2017, small jets at the base of coronal loops are driven by flux cancelation and reconnection between large-scale, dominant polarity patches and small-scale elements of the opposite polarity. When comparing the same observational area using HMI, however, the non-dominant polarity is not clearly seen. Similarly, Chitta et al. (2019) observe transient magnetic flux emergence and cancelation events using the Swedish 1-m Solar Telescope (SST; Scharmer et al. 2003), which has five times the resolution as HMI. These transient magnetic events are seen at granular spatial scales and are found to appear in photospheric plage regions with a predominantly unipolar vertical magnetic field. The authors compare these observations with 3D radiation magnetohydrodynamic (rMHD) simulations, verifying the presence of these magnetic events wherein emerging, small-scale and nondominant polarity magnetic elements cancel with the surrounding dominant unipolar magnetic field. They find that these events can appear on even smaller spatial and temporal scales than that resolved by observations. Chen et al. (2019) discuss the relationship between explosive events (EEs) and network jets. Two relevant categories of EEs that the authors define are for line profiles displaying either two peaks of comparable magnitude or enhancement at the blue wing, and they label these as categories (ii) and (iii), respectively. They find that EEs of the latter group are mainly located on network jets, away from the jet footpoints, while those of the former are either seen at the footpoints of jets or on transient compact brightenings not associated with network jets. For the jet evaluated in our study, the spectral profiles along the jet's spire show strong blue wing enhancements similar to those of category (iii), while closer to the base, some profiles exhibit category (ii)-type double peaks of commensurate intensity (see e.g., Appendix Fig. A.1, panel k). We find here that not only are the locations of the different profile types consistent with the previous work, but they also further the proposal of reconnection stemming from the shifting magnetic elements identified at the footpoint. Location of energy release In our observation, we find that the chromospheric plasma mainly exhibits downflows at the jet's base, whereas plasma sampling the TR at 0.1 MK is propelled upwards and thus blueshifted (see Sects. 3.1 and 3.2). This suggests that the actual location of energy release is located between the up-and downflows; in response to the energy deposition, the gas is propelled away from the release site and is channeled by the magnetic field in roughly up-and downward directions. This is essentially the same setup as in an EE (Dere et al. 1989), where spectroscopic observations showed a bi-directional outflow from a reconnection site (Innes et al. 1997). In those EEs, the up-and downflows were seen in the same spectral line, meaning they happened at the same temperature. This might be due to the fact that the reconnection process is happening in a low-beta regime (well above the chromosphere) where the plasma is locally heated by plasmoid-mediated reconnection . Observations suggest that in (some) EUV bursts the X-type neutral point where the reconnection occurs is located within the chromosphere some 500 km above the photosphere, closer to where plasma beta is near unity (Chitta et al. 2017b). Still, if the energy deposition is strong enough, those EUV bursts could show up-and downflows in a TR line like Si iv (Peter et al. 2014), even if the reconnection is located almost as deep in the atmosphere as it is for Ellerman bombs (Georgoulis et al. 2002). Our scenario that these network jets are driven from the top of the chromosphere is similar to earlier concepts, albeit in a different context. In previous works, a dichotomy of red-and blueshifts is seen in the QS TR (red) and corona (blue) in terms of temperature -below about 0.5 MK the plasma is redshifted, while above 0.5 MK it is blueshifted (Peter 1999;Peter & Judge 1999). This can be understood in terms of the 3D MHD model of Hansteen et al. (2010). In their work, the authors showed that the highest energy deposition per particle was found in the middle TR through episodic heating events. In response, the plasma expands and produces the redshifts of the TR and blueshifts of the coronal lines. Applying this model to our jet scenario, where energy is deposited at the top of the chromosphere, localized plasma heating creates a plug of increased pressure. The gas expands along field lines on both sides, which leads to downflows into the upper chromosphere seen in Mg ii and upflows towards the corona seen in Si iv. While the ultimate driver might be found in the photospheric stretching of the magnetic field (e.g., Priest et al. 2002), the actual energy release that drives the jet we observe might be released at the top of the chromosphere. Conclusion We have observed a network jet with mass motions as inferred from secondary blueshifted components (20-70 km s −1 ) in the spectral profiles. By comparing the magnetic field data with spectral information from the Mg ii and Si iv lines, the jet has been followed from its footpoint location in the photosphere up through the chromosphere and into the TR. We find that the energy that drives the network jet is released at the top of the chromosphere such that it drives parts of the plasma downwards into the chromosphere and other parts upwards into the upper atmosphere. The line broadening suggests that there should be sufficient surplus energy available to drive the jet plasma to join the solar wind originating from the coronal hole wherein this jet is embedded. This work is serendipitous in that the raster slit happened to be scanning across roughly the length of a jet just as it was accelerating. However, the findings are limited to a single jet that is only seen in one SJI frame. Thus, future work should attempt to obtain similar observations that round out these shortcomings. Such similar observations would be dense raster scans located within or at the boundary of CHs at disk center with accompanying slit-jaw imagery. Preferably, the slit-jaw imagery would have a higher cadence than the observation presented herein (127 s) in order to capture any jet events in more than one frame; however, the cadence should not be too high so as to degrade the S/N beyond the point of utility. In addition to having multiple, similarly well-aligned raster observations of small-scale jets, higher resolution magnetic field data would help to better constrain possible driving mechanisms. Also, higher cadence SJIs would be useful for directly comparing the jet motions in the image plane with its motions along the LOS. Fig. 2 . 2Co-aligned maps of observation for AIA193, IRIS slit-jaw imager (SJI) 1400 channel, and HMI line of sight (LOS) magnetogram. Each image is selected at the nearest observation time to when the jet is seen in IRIS SJI1400 at 07:21 UT. The FOV for the IRIS raster scan is outlined in yellow and the red box marks the region of interest for the network jet, which is also the FOV for panel (b) inFig. 3. See Sect. 3. Fig. 3 . 3Close-up view of jet. The region of interest for the network jet is shown from four different instruments: (a) IRIS SJI1400, (b) IRIS raster Dopplergram for Si iv 1394 Å, (c) IRIS SJI2796, (d) HMI LOS magnetogram. For (b), the FOV is the same as that demarcated in red inFig. 2. However, for panels (a), (c), and (d), the FOV has been extended 3.3 (10 pixels) to the east (which is outside of the raster scan's spatial coverage) in order to fully encompass the observed activity at the jet's magnetic footpoint. For each panel, the jet is outlined in black. (a), (c), and (d) are co-aligned images taken at 07:21 UT and the location of the raster slit at that time is overplotted with white whiskers. The time evolution for panel (d) is available as an online movie. Panel (b) is calculated from single-Gaussian fits applied to the original IRIS raster data and has a smaller FOV due to the limited spatial extent of the scan. See Sect. 3 for discussion. Fig. 4 . 4Si iv 1394 Å Gaussian parameters and selected spectra from jet. Panels (a)-(c) show the single-Gaussian (SG) parameter maps for peak intensity[DN], Doppler velocity [km s −1 ], and full-width at half maximum (FWHM) [km s −1 ] for Si iv, respectively. The colored diamonds mark the three individual pixels whose spectral profiles are shown in the bottom two rows, while the gold square marks, for reference, a fourth pixel whose spectrum is shown inFig. 5. Panels (d)-(f) display the individual pixels' spectra in gray [DN] with the SG fitted curve overlaid in black. The Doppler velocity (Vel) and non-thermal width (ξ) are listed in the upper left. Panels (g)-(i) show the same three spectra, except a double-Gaussian (DG) fitted curve is overlaid in black and its two components are drawn in red (primary component) and blue (secondary component). The primary and secondary Doppler velocities (Vel1, Vel2) and non-thermal widths (ξ1, ξ2) are listed in the upper left. For all panels (d)-(i), a vertical black line in the center marks the location of zero-velocity, for reference. See Sects. 3.1 and 4.2. Fig. 5 . 5Chromospheric activity at base of jet. Maps of the intensity [DN] at three wavelength points along the Mg ii line are shown in the top row. The wavelength locations are: (a) k3: 2796.37 Å, (b) k2r: 2796.52 Å, and (c) k1r: 2796.78 Å. Fig. 6 . 6Magnetic flux vs. time at jet's footpoint. Magnetic flux of positive (panel a) and negative (panel b) polarity magnetic concentrations within the footpoint region of the jet (white box in Fig. 3d) for pixels stronger than 10 G are shown. The dotted vertical lines mark the start and stop times for the entire observation period of the IRIS slit-jaws: 07:13:15-07:44:58 UT. The solid vertical lines mark the period when the raster is scanning across the jet: 07:21:43-07:25:46 UT. The SJI1400 frame wherein the jet is seen occurs at 07:21:43 UT. In panel (a), the black line shows the linear trend. The gray shading covers the root-mean-square error (RMSE) from this fit. Panel (b) shows the root-mean-square (RMS) and the RMSE for the calculated values from the RMS by the black horizontal line and the gray shading, respectively. See Sects. 3.3 and 4.3. Article number, page 1 of 9 arXiv:2202.11375v1 [astro-ph.SR] 23 Feb 2022 A&A proofs: manuscript no. mainFig. 1. Context image. AIA 193 Å full-disk overview of the Sun on November 23, 2016, at 07:12:41 UT. The fields of view (FOV) for the IRIS raster and slit-jaw imagery are outlined in yellow and blue, respectively. Solar north is pointed up. See Sect. 3. https://www.lmsal.com/solarsoft Article number, page 9 of 9 AcknowledgmentsWe thank the anonymous referee for providing helpful comments and suggestions. This work was supported by the International Max-Planck Research School (IMPRS) for Solar System Science at the University of Göttingen. IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre. SDO/AIA and HMI imagery and data are courtesy of NASA/SDO and the AIA, EVE, and HMI science teams.Appendix A: Further spectra examples from jetHere, we include more examples of double-Gaussian fitted spectra from along the jet. These are discussed in Sect. 4.2.Fig. 4, except for the colored diamond markers, which mark nine different pixels whose spectral profiles are shown in the bottom three rows. The style and layout of the spectra in panels (d)-(l) also match those for panels (g)-(i) inFig. 4. . S Bose, V M J Henriques, J Joshi, &amp; Rouppe Van Der, L Voort, A&A. 6315Bose, S., Henriques, V. M. J., Joshi, J., & Rouppe van der Voort, L. 2019, A&A, 631, L5 . 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[]
[ "Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n log log n)", "Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n log log n)" ]
[ "Michael Elkin ", "Idan Shabat " ]
[]
[]
Given an n-vertex undirected graph G = (V, E, w), and a parameter k ≥ 1, a pathreporting distance oracle (or PRDO) is a data structure of size S(n, k), that given a query (u, v) ∈ V 2 , returns an f (k)-approximate shortest u − v path P in G within time q(k) + O(|P |). Here S(n, k), f (k) and q(k) are arbitrary (hopefully slowly-growing) functions. A distance oracle that only returns an approximate estimated(u, v) of the distance d G (u, v) between the queried vertices is called a non-path-reporting distance oracle.A landmark PRDO due to Thorup and Zwick [56] has S(n, k) = O(k · n 1+ 1 k ), f (k) = 2k − 1 and q(k) = O(k). Wulff-Nilsen [59] devised an improved query algorithm for this oracle with q(k) = O(log k). The size of this oracle is Ω(n log n) for all k. Elkin and Pettie. These oracles (of[30,46]) can be much sparser than O(n log n) (the oracle of[46]can have linear size), but their stretch is polynomially larger than the optimal bound of 2k − 1. On the other hand, a long line of non-path-reporting distance oracles culminated in a celebrated result by Chechik [14], in which S(n, k) = O(n 1+ 1 k ), f (k) = 2k − 1 and q(k) = O(1). In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. In particular, we devise a PRDO with size S(n, k) = O k·log log n log n · n 1+ 1 k , stretch f (k) = O(k) and query time q(k) = O log k·log log n log n . As h = k·log log n log n = O(log k), its size is always at most O(log k · n 1+ 1 k ), and its query time is O(log log k). Moreover, for k = O log n log log n , we have h = O(1), i.e., S(n, k) = O(n 1+ 1 k ), f (k) = O(k), and q(k) = O(1). For k = Θ(log n), our oracle has size O(n log log n), stretch O(log n) and query time O(log (3) n). We can also have linear size O(n), stretch O(log n·log log n) and query time O(log (3) n).These trade-offs exhibit polynomial improvement in stretch over the PRDOs of[30,46]. For k = Ω log n log log n , our trade-offs also strictly improve the long-standing bounds of[56,59]. Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any ϵ > 0, any k = 1, 2, ..., and any graph G = (V, E, w) and a collection P of p vertex pairs, there exists a (1 + ϵ)-approximate preserver for G, P with O(γ(ϵ, k) · p + n log k + n 1+ 1 k ) edges, where γ(ϵ, k) = log k ϵ O(log k). These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter [41].
10.48550/arxiv.2304.04445
[ "https://export.arxiv.org/pdf/2304.04445v3.pdf" ]
258,048,770
2304.04445
392d6fa827cf885bb2634fb192be3e106b0d0d8b
Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n log log n) 25 May 2023 Michael Elkin Idan Shabat Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n log log n) 25 May 2023* Ben-Gurion University of the Negev, Beer-Sheva, Israel. [email protected] † Ben-Gurion University of the Negev, Beer-Sheva, Israel. [email protected] Given an n-vertex undirected graph G = (V, E, w), and a parameter k ≥ 1, a pathreporting distance oracle (or PRDO) is a data structure of size S(n, k), that given a query (u, v) ∈ V 2 , returns an f (k)-approximate shortest u − v path P in G within time q(k) + O(|P |). Here S(n, k), f (k) and q(k) are arbitrary (hopefully slowly-growing) functions. A distance oracle that only returns an approximate estimated(u, v) of the distance d G (u, v) between the queried vertices is called a non-path-reporting distance oracle.A landmark PRDO due to Thorup and Zwick [56] has S(n, k) = O(k · n 1+ 1 k ), f (k) = 2k − 1 and q(k) = O(k). Wulff-Nilsen [59] devised an improved query algorithm for this oracle with q(k) = O(log k). The size of this oracle is Ω(n log n) for all k. Elkin and Pettie. These oracles (of[30,46]) can be much sparser than O(n log n) (the oracle of[46]can have linear size), but their stretch is polynomially larger than the optimal bound of 2k − 1. On the other hand, a long line of non-path-reporting distance oracles culminated in a celebrated result by Chechik [14], in which S(n, k) = O(n 1+ 1 k ), f (k) = 2k − 1 and q(k) = O(1). In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. In particular, we devise a PRDO with size S(n, k) = O k·log log n log n · n 1+ 1 k , stretch f (k) = O(k) and query time q(k) = O log k·log log n log n . As h = k·log log n log n = O(log k), its size is always at most O(log k · n 1+ 1 k ), and its query time is O(log log k). Moreover, for k = O log n log log n , we have h = O(1), i.e., S(n, k) = O(n 1+ 1 k ), f (k) = O(k), and q(k) = O(1). For k = Θ(log n), our oracle has size O(n log log n), stretch O(log n) and query time O(log (3) n). We can also have linear size O(n), stretch O(log n·log log n) and query time O(log (3) n).These trade-offs exhibit polynomial improvement in stretch over the PRDOs of[30,46]. For k = Ω log n log log n , our trade-offs also strictly improve the long-standing bounds of[56,59]. Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any ϵ > 0, any k = 1, 2, ..., and any graph G = (V, E, w) and a collection P of p vertex pairs, there exists a (1 + ϵ)-approximate preserver for G, P with O(γ(ϵ, k) · p + n log k + n 1+ 1 k ) edges, where γ(ϵ, k) = log k ϵ O(log k). These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter [41]. Contents Computing shortest paths and distances, exact and approximate ones, is a fundamental and extremely well-studied algorithmic problem [53,4,3,60,16,21,22,61,17,62,63,8,51,25]. A central topic in the literature on computing shortest paths and distances is the study of distance oracles [56,55,59,48,13,28,14,30,52]. Given an undirected weighted graph G = (V, E) with weights w(e) ≥ 0 on the edges, and a pair of vertices (u, v) ∈ V 2 , let d G (u, v) denote the distance between them in G, i.e., the weight w(P u,v ) of the shortest u − v path P u,v . When the graph G is clear from the context, we write d(u, v) = d G (u, v). A path-reporting distance oracle (shortly, PRDO) is a data structure that given a query vertex pair (u, v) ∈ V 2 returns an approximately shortest u − v path P ′ u,v and its lengthd(u, v) = w(P ′ u,v ). The latter value is also called the distance estimate. An oracle that returns only distance estimates (as opposed to paths and estimates) is called a non-path-reporting distance oracle. Three most important parameters of distance oracles are their size, stretch and query time 1 . The size of a distance oracle is the number of computer words required for storing the data structure. The stretch of the distance oracle is the minimum value α such that for any query (u, v) ∈ V 2 to the oracle, it is guaranteed that the path P ′ u,v returned by it satisfies d(u, v) ≤ w(P ′ u,v ) ≤ α · d(u, v). The time required for the PRDO to handle a query (u, v) can typically be expressed as O(q +|P ′ u,v |). The (worst-case) overhead q is referred to as the query time of the PRDO. First (implicit) constructions of distance oracles were given in [5,15]. The authors devised hierarchies of neighborhood covers, which can be viewed as PRDOs with stretch (4 + ϵ)k (for any parameter k = 1, 2, ...), size O ϵ (k · n 1+ 1 k · log Λ) (where Λ = maxu,v d G (u,v) min u̸ =v d G (u,v) is the aspect ratio of the input graph), and query time O(log log Λ). 2 Another variant of Cohen's construction [15] provides a PRDO with stretch (2 + ϵ)k. However, the query time of this distance oracle isÕ ϵ (k · n 1 k · log Λ). The aforementioned distance oracles (of [5,15]) are path-reporting. Matousek [43] came up with an ℓ ∞ -embedding, that can be viewed as a non-path-reporting distance oracle with stretch 2k − 1, sizeÕ(k · n 1+ 1 k ) and query timeÕ(k · n 1 k ). A landmark path-reporting distance oracle was devised by Thorup and Zwick [56]. It provides stretch 2k − 1, size O(k · n 1+ 1 k ), and query time O(k). Wulff-Nilsen [59] devised an improved query algorithm for this PRDO with query time O(log k). Assuming Erdős girth conjecture [31] holds, it is easy to see that any distance oracle with stretch 2k − 1 requires Ω(n 1+ 1 k ) bits (see [56]). Moreover, Chechik [14] showed that for path-reporting distance oracles, a stronger lower bound of Ω(n 1+ 1 k ) words applies. Mendel and Naor [44] devised a non-path-reporting distance oracle with stretch 128k, size O(n 1+ 1 k ) and query time O(1). Naor and Tao [45] improved the stretch of this oracle to 33k, and argued that the approach of [44,45] can hardly lead to a stretch smaller than 16k. A path-reporting variant of Mendel-Naor's oracle was devised by Abraham et al. [1]. Their PRDO has size O(kn 1+ 1 k ), stretch O(k · log log n), and query time O (1). This PRDO has, however, size Ω(n log n). Wulff-Nilsen [59] came up with a non-path-reporting distance oracle with stretch (2 + ϵ)k (for any parameters ϵ > 0 and k = 1, 2, ...), size O(k · n 1+ 1 k ), and query time O ϵ (1). This result was improved by Chechik [13,14], who devised a non-path-reporting distance oracle with stretch 2k − 1, size O(n 1+ 1 k ) and query time O(1). To summarize, for non-path-reporting distance oracles, Chechik's construction [14] provides near-tight bounds. For path-reporting distance oracles, the construction of [56,59] is tight up to the factor k in the size, and the factor log k in the query time. In particular, the PRDO of [56,59] always has size Ω(n log n), regardless of the choice of the parameter k. The first PRDO with size o(n log n) was devised by Elkin et al. [28]. For a parameter t ≥ 1, the oracle of [28] provides stretch O(t · n 1 t ), size O(n · t · log n Λ) and query time O(log(t · log n Λ)). Note, however, that the stretch of this oracle is prohibitively large. Elkin and Pettie [30] devised two constructions of PRDOs with size o(n log n). Their first construction provides, for parameters k = 1, 2, ... and ϵ > 0, a PRDO with stretch O 1 ϵ c · k , size O(n 1+ 1 k ) and query time O(n ϵ ). Here c = log 4/3 7 is a universal constant. Their second construction provides stretch O(log c n), size O(n log log n) and query time O(log log n). Based on ideas from [26], one can improve the constant c in these results to log 4/3 5. Recently, Neiman and Shabat [46] further improved this constant to c ′ = log 4/3 4. Specifically, their PRDO provides, for a parameter k = 1, 2, ..., stretch O(k c ′ ), size O(n 1+ 1 k ) and query time O(log k). Moreover, for unweighted graphs, the stretch of their PRDO is O(k 2 ). Our Results Note that the state-of-the-art PRDOs that can have size o(n log n) [30,46] suffer from either prohibitively high query time (the query time of the first PRDO by [30] is polynomial in n), or have stretch k c ′ (for a constant c ′ = log 4/3 4 > 4.81, or c ′ = 2 in the case of unweighted graphs). This is in a stark contrast to the state-of-the-art non-path-reporting distance oracle of Chechik [14], that provides stretch 2k − 1, size O(n 1+ 1 k ) and query time O(1). In this paper we devise PRDOs that come much closer to the tight bounds for non-path-reporting distance oracles than it was previously known. Specifically, one of our PRDOs, given parameters k = 1, 2, ... and an arbitrarily small constant ϵ > 0, provides stretch (4 + ϵ)k, size O n 1+ 1 k · k · log log n · log (3) n log n , and query time 3 O(log k). For k ≤ log n log log n·log (3) n , the size bound is O(n 1+ 1 k ). Note that the stretch here is linear in k, and in fact, it exceeds the desired bounds of 2k − 1 by a factor of just 2 + ϵ. At the same time, the size of this PRDO in its sparsest regime is O(n log log n · log (3) n), i.e., far below n log n. Recall that the only previously-existing PRDOs with comparable stretch-size trade-offs [56,59] have size Ω(n log n). Also, our query time (O(log k)) is the same as that of [59,30,46]. We can also have a slightly larger stretch O(k) (i.e., still linear in k), size O n 1+ 1 k · k · log log n log n , and query time O log k·log log n log n = O(log log k). The query time of this oracle is exponentially smaller than that of the state-of-the-art PRDOs [56,59,30,46]. Moreover, for k = O log n log log n , the query time is constant. At the same time, its stretch is polynomially better than in the oracles of [30,46]. Its size is slightly worse that that of [46]. However, we can trade size for stretch and get stretch O k · k·log log n log n = O(k log k), size O(n 1+ 1 k ) and query time O(log log k), consequently outperforming the oracle of [46] in all parameters. In particular, for k = log n, we get a PRDO with stretch O(log n · log log n), size O(n) and query time O(log (3) n). It is instructive to compare our PRDO to that of Thorup and Zwick [56,59]. For stretch α = t · log n log log n , where 1 ≤ t ≤ log log n, the PRDO of [56,59] has size O nt · log 1+ 2 t n log log n and query time O(log log n), while our PRDO (with stretch C · k, size O k·log log n log n · n 1+ 1 k and query time O log k·log log n log n , for a universal constant C) has size O nt · log C t n . For t greater than a sufficiently large constant, the size of our PRDO is therefore strictly smaller than that of [56,59] (for the same stretch), and at the same time our query time O log k·log log n log n = O(log (3) n) is at least exponentially smaller then the query time of [56,59] (which is O(log log n)). In addition to the two new PRDOs that were described above (the one with stretch (4 + ϵ)k and query time O(log k), and the one with stretch C · k, for a much larger constant C, and query time O log k·log log n log n ), we also present a variety of additional PRDOs that trade between these two extremes. Specifically, we can have stretch C ′ · k, for several different constant values C ′ , 4 + ϵ < C ′ < C, and query time o(log k) (though larger than O log k·log log n log n ). See Section 6.2 and Table 5 for full details. Tables 1 and 2 provide a concise summary of main previous and new results on distance oracles. 3 Note that the total query time is q(k) + O(|P ′ |), where P ′ is the returned path. Thus decreasing the query time overhead at the expense of increased stretch may result in increasing the total query time. From this viewpoint our two strongest bounds are stretch (4 + ϵ)k, size O k·log log n·log (3) n log n · n 1+ 1 k , and stretch (12 + ϵ)k, size O k·log log n log n · n 1+ 1 k (for the latter, see Table 5). In both of these bounds, the query time is O(log k), like in [56,59]. Nevertheless, we believe that it is of interest to study how small can be the query time overhead, even at the expense of increased stretch, for two reasons. First, it can serve as a toolkit for future constructions with yet smaller (hopefully, optimal) stretch and query time. Second, there might be queries u, v for which the number of edges in the returned path P ′ u,v = P ′ is smaller than the query time overhead. [44,45] Table 1: A summary of results on non-path-reporting distance oracles. We mostly omit Onotations. (2 + ϵ)k k · n 1+ 1 k O ϵ (1) [59] 2k − 1 k · n 1+ 1 k 1 [13] 2k − 1 n 1+ 1 k 1 [14] Ultra-Sparse PRDOs Neiman and Shabat [46] also devised an ultra-sparse (or, an ultra-compact) PRDO: for a parameter t ≥ 1, their PRDO has size n + O n t , stretch O(t · log c ′ n), and query time O(t · log c ′ n) (recall that c ′ = log 4/3 4). Our PRDO can also be made ultra-sparse. Specifically, for a parameter t ≥ 1, the size of our PRDO is n + O n t (the same as in [46]), the stretch is O(t · log n · log log n), and the query time is O(t · log log n). Related Work The study of very sparse distance oracles is a common thread not only in the context of general graphs (as was discussed above), but also in the context of planar graphs. In particular, improving upon oracles of Thorup [58] and Klein [40], Kawarabayashi et al. [37,38] came up with (1 + ϵ)approximate distance oracles with size O ϵ (n log log n). Very sparse spanners and emulators were extensively studied in [49,50,26,24]. Ultra-sparse sparsifiers, or shortly, ultra-sparsifiers, are also very well-studied [18,39,42,54]. Distance Preservers All our PRDOs heavily exploit a novel construction of distance preservers that we devise in this paper. Pairwise distance preservers were introduced by Coppersmith and Elkin in [19]. Given an n-vertex graph and a collection P ⊆ V 2 of vertex pairs, a sub-graph G ′ = (V, H), H ⊆ E, is called a pairwise preserver with respect to G, P if for every (u, v) ∈ P, it holds that d G ′ (u, v) = d G (u, v) .(1) We often relax the requirement in Equation (1) to only hold approximately. Namely, we say that the sub-graph G ′ is an approximate distance preserver with stretch α, or shortly, an α-preserver, if for every (u, v) ∈ P, it holds that d G ′ (u, v) ≤ α · d G (u, v) . It was shown in [19] that for every n-vertex undirected weighted graph and a set P of p vertex pairs, there exists a (1-)preserver with O(n + √ n · p) edges. They also showed an upper bound of O( √ p · n), that applies even for weighted directed graphs. Stretch Size Query Time Paper (4 + ϵ)k k · n 1+ 1 k · log 1+ϵ Λ log log Λ [5,15] (2 + ϵ)k k · n 1+ 1 k · log 1+ϵ Λ k · n 1 k log Λ [15] (2) 2k − 1 k · n 1+ 1 k k [56] 2k − 1 k · n 1+ 1 k log k [56, 59] k · log log n k · n 1+ 1 k 1 [1] t · n 1 t t · n · log n Λ log(t · log n Λ) [28] k log 4/3 7 log k · n 1+ 1 k log k [30] (1) k · ( 1 ϵ ) log 4/3 7 n 1+ 1 k n ϵ [30] (2) k log 4/3 5 log k · n 1+ 1 k log k [30, 27] k log 4/3 4 n 1+ 1 k log k [46] k 2 n 1+ 1 k log k [46], unweighted (4 + ϵ)k n 1+ 1 k · k·log log n·log (3) n log n log k This paper, ϵ = Ω(1) O(k) n 1+ 1 k · k·log log n log n log log k This paper k log k n 1+ 1 k log log k This paper log n · log log n n log (3) n This paper, k = log n , which is always O(log log k), but for k = O log n log log n it is O(1). Improved exact preservers (i.e., 1-preservers) for unwieghted undirected graphs were devised by Bodwin and Vassilevska-Williams [12]. Specifically, the upper bound in [12] is O((n·p) 2/3 +n·p 1/3 ). Lower bounds for exact preservers were proven in [19,12,10]. Preservers for unweighted undirected graphs, that allow a small additive error were studied in [49,47,20,36]. In particular, Kavitha [36] showed that for any unweighted undirected graph and a set of p pairs, there exists a preserver withÕ(n · p 2/7 ) (respectively, O(n · p 1/4 )) edges and with purely 4 additive stretch of 4 (respectively, 6). Note, however, that the sizes of all these preservers are super-linear in n when p is large. One of the preservers of [19] has linear size for p = O( √ n), and this was exactly the preserver that [30] utilized for their PRDO. To obtain much better PRDOs, one needs preservers with linear size, or at least near-linear size, for much larger values of p. For unweighted graphs, a very strong upper bound on (1 + ϵ)-preservers follows directly from constructions of near-additive spanners. Given a graph G = (V, E), an (α, β)-spanner is a sub- graph G ′ = (V, E ′ ), E ′ ⊆ E, such that for every pair of vertices u, v ∈ V , d G ′ (u, v) ≤ α · d G (u, v) + β . When α = 1 + ϵ, for some small positive parameter ϵ > 0, an (α, β)-spanner is called a near-additive spanner. Constructions of near-additive spanners [29,22,57,49,26] give rise to constructions of (1 + ϵ)preservers (see, e.g., [2]). Specifically, one builds a (1 + ϵ, β H )-spanner H [29,49,25], and adds it to the (initially empty) preserver. Then for every pair (u, v) ∈ P with d G (u, v) ≤ β H ϵ , one inserts a shortest u − v path P u,v into the preserver. The preserver employs O(|H| + β H ϵ · p) edges. For the stretch bound, observe that for any (u, v ) ∈ P with d G (u, v) > β H ϵ , the spanner H provides it with distance d H (u, v) ≤ (1 + ϵ)d G (u, v) + β H < (1 + 2ϵ)d G (u, v) . Plugging here the original construction of (1 + ϵ, γ 2 )-spanners with O(γ 2 · n 1+ 1 k ) edges, where γ 2 = γ 2 (k) = O log k ϵ log k , from [29], one obtains a (1 + ϵ)-preserver with O(γ 2 · n 1+ 1 k + γ 2 ϵ · p) edges. Plugging instead the construction from [49,26] of (1 + ϵ, γ 4/3 )-spanners with O(n 1+ 1 k + n · log k) edges, where γ 4/3 = γ 4/3 (k) = O log k ϵ log 4/3 k , one obtains a (1 + ϵ)-preserver with O(n 1+ 1 k + n · log k + γ 4/3 ϵ · p) edges. For many years it was open if these results can be extended to weighted graphs. The first major progress towards resolving this question was recently achieved by Kogan and Parter [41]. Specifically, using hierarchies of hopsets they constructed (1 + ϵ)-preservers for undirected weighted n-vertex graphs with O(n log n) 1+ 1 k · log n + p · log n·log k ϵ log k edges, for any parameters k = 1, 2, ... and ϵ > 0. Note, however, that this preserver always has size Ω(n · (log n) log log n ), and this makes it unsuitable for using it for PRDOs of size o(n log n) (which is the focus of our paper). In the current paper we answer the aforementioned question in the affirmative, and devise constructions of (1 + ϵ)-preservers for weighted graphs, that are on par with the bounds of [29,49,26,2] for unweighted ones. Specifically, we devise two constructions of (1 + ϵ)-preservers for weighted graphs. The first one has size O(n 1+ 1 k + n · log k + γ 4/3 · p), and the second one has size O(n 1+ 1 k + n · log k +γ 2 · p), wherẽ γ 2 = log k + log log 1 ϵ ϵ log k+O(log (3) k)+log log 1 ϵ . When ϵ ≥ 2 −k c ′ , for a specific constant c ′ > 0, the second construction is better than the first one. Note that for p ≥ n 1+ 1 k , for some constant k, our preserver uses O ϵ,k (p) edges (i.e., only a constant factor more than a trivial lower bound), while the overhead in the construction of [41] depends polylogarithmically on n, with a degree that grows with k (specifically, it is O log n ϵ log k ). Also, for p ≤ O(log log n) , our construction provides a preserver of near-linear size (specifically, O(n log log n)), where the size of the preserver of [41] is always Ω(n · (log n) log log n ). This property is particularly useful for constructing our PRDOs. We also present a construction of (3+ϵ)-preservers (for weighted graphs) with O(n 1+ 1 k +n·log k+ k O(log 1 ϵ ) · p) edges. By substituting here k = log n, we obtain a (3 + ϵ)-preserver, for an arbitrarily small constant ϵ > 0, with size O(n log log n + p · log O(1) n). In addition, all our constructions of approximate distance preservers directly give rise to pairwise PRDOs with the same stretch and size. A pairwise PRDO is a scheme that given a graph G and a set P of p vertex pairs, produces a data structure, which given a query (u, v) ∈ P returns an approximate shortest u−v path. The size and the stretch of pairwise PRDOs are defined in the same way as for ordinary distance oracles. Pairwise PRDOs with stretch 1 were studied in [11], where they were called Shortest Path Oracles (however, in [11] they were studied in the context of directed graphs). Neiman and Shabat [46] have recently showed that the (1 + ϵ)-preservers of [41] with O(n log n) 1+ 1 k · log n + p · log n·log k ϵ log k edges can be converted into pairwise PRDOs with the same stretch and size. We improve upon this by presenting (1 + ϵ)-stretch pairwise PRDOs of size O(n 1+ 1 k + n · log k + γ 4/3 · p) and of size O(n 1+ 1 k + n · log k +γ 2 · p), and (3 + ϵ)-stretch pairwise PRDOs of size O(n 1+ 1 k + n · log k + k O(log 1 ϵ ) · p). Interestingly, to the best of our knowledge, the unweighted (1 + ϵ)-preservers of [29,49,25,2] do not give rise to partial PRDOs with similar properties. See Table 3 for a concise summary of existing and new constructions of (1 + ϵ)-preservers and pairwise PRDOs. Weighted/ Stretch Size Paper Partial Unweighted PRDO? Weighted 1 n + √ n · p [19] [30] Weighted 1 n · √ p [19] NO Unweighted 1 (n · p) 2/3 + n · p 1/3 [12] NO Unweighted 1 + ϵ γ 2 · n 1+ 1 k + γ 2 ϵ · p [29, 2] NO Unweighted 1 + ϵ n 1+ 1 k + n · log k + γ 4/3 ϵ · p [49, 25, 2] NO Weighted 1 + ϵ (n log n) 1+ 1 k · log n + p · log n·log k ϵ log k [41] [46] Weighted 1 + ϵ n 1+ 1 k + n · log k + γ 4/3 · p This paper YES Weighted 1 + ϵ n 1+ 1 k + n · log k +γ 2 · p This paper YES Weighted 3 + ϵ n 1+ 1 k + n · log k + k O(log 1 ϵ ) · p This paper YES Table 3: A summary of most relevant existing and new results about pairwise preservers and PRDOs. The first column indicates if a result applies to unweighted or weighted graphs. The Size omits O-notations (except for an exponent in the last row). The last column indicates whether there is a known PRDO-analogue of this particular preserver. In case there is, a reference is provided. The PRDO-analogue of the exact preserver by [19], devised in [30], provides somewhat weaker bounds on the size than those of the original preserver. See [30] for details. A Technical Overview Preservers and Hopsets with Small Support Size The construction of (1 + ϵ)-preservers in [41] is obtained via an ingenious black-box reduction from hierarchies of hopsets. Our constructions of significantly sparser preservers is obtained via a direct hopset-based approach, which we next outline. Given a graph G = (V, E), and a pair of positive parameters α, β, a set H ⊆ V 2 is called an (α, β)-hopset of G if for every pair of vertices u, v ∈ V , we have d (β) G∪H (u, v) ≤ α · d G (u, v) . Here G∪H denotes the weighted graph that is obtained by adding the edges H to G, while assigning every edge (x, y) ∈ H the weight d G (x, y). Also, d (β) G∪H (u, v) stands for β-bounded u − v distance in G ∪ H, i.e., the weight of the shortest u − v path in G ∪ H with at most β edges. Hopsets were introduced in a seminal work by Cohen [16]. Elkin and Neiman [25] showed that for any ϵ > 0 and k = 1, 2, ..., and any n-vertex undirected weighted graph G = (V, E), there exists a (1 + ϵ, γ 2 )-hopset, where γ 2 (k) = O log k ϵ log k , withÕ(n 1+ 1 k ) edges. Elkin and Neiman [27] and Huang and Pettie [34] showed that the Thorup-Zwick emulators [57] give rise to yet sparser (1 + ϵ, γ 2 )-hopsets. Specifically, these hopsets have size O(n 1+ 1 k ). In this paper we identify an additional (to the stretch α, the hopbound β, and the size |H|) parameter of hopsets, that turns to be crucially important in the context of PRDOs. Given a hopset H for a graph G = (V, E), we say that a subset E H of G is a supporting edge-set of the hopset H, if for every edge (u, v) ∈ H, the subset E H contains a shortest 5 u − v path in G. The minimum size |E H | of a supporting edge-set for the hopset H is called the support size of the hopset. We demonstrate that for any ϵ > 0, k = 1, 2, ..., there exists a γ 4/3 = γ 4/3 (ϵ, k) (defined above) such that any n-vertex graph admits a (1 + ϵ, γ 4/3 )-hopset with size O(n 1+ 1 k ) and support size O(n 1+ 1 k + n · log k). Moreover, these hopsets are universal, i.e., the same construction provides, in fact, a (1 + ϵ, γ 4/3 (ϵ, k))-hopsets for all ϵ > 0 simultaneously (this is a property of Thorup-Zwick emulators [57], and of hopsets obtained via their construction of emulators [27,34]). Furthermore, the same hopset serves also as a (3 + ϵ, k O(log 1 ϵ ) )-hopset (with the same size and support size, and also for all ϵ > 0 simultaneously). Next we explain how such hopsets directly give rise to very sparse approximate distance preservers. Below we will also sketch how hopsets with small support size are constructed. Given a hopset H as above, and a set P ⊆ V 2 of vertex pairs, we construct the preserver E P in two steps. First, the supporting edge-set E H of the hopset H, with size O(n 1+ 1 k + n · log k), is added to the (initially empty) set E P . Second, for each pair (u, v) ∈ P, consider the u − v path P u,v in G ∪ H, with weight d (β) G∪H (u, v) ≤ (1 + ϵ)d G (u, v) , and at most β = γ 4/3 edges. For every original edge e = (x, y) ∈ P u,v (i.e., e ∈ E, as opposed to edges of the hopset H), we add e to E P . This completes the construction. The size bound |E P | ≤ |E H | + γ 4/3 · p = O(n 1+ 1 k + n · log k + γ 4/3 · p) is immediate. For the stretch bound, consider a vertex pair (u, v) ∈ P. For every edge e ∈ P u,v ∩ E, the edge belongs to E P . For every hopset edge e ′ = (x ′ , y ′ ) ∈ P u,v ∩ H, the shortest x ′ − y ′ path in G belongs to the supporting edge-set E H of the hopset, and thus to E P as well. Hence d E P (x ′ , y ′ ) = d G (x ′ , y ′ ). Therefore, d E P (u, v) ≤w(P u,v ), wherew is the weight function in G ∪ H. It follows that d E P (u, v) ≤w(P u,v ) = d (β) G∪H (u, v) ≤ (1 + ϵ)d G (u, v) . To convert this preserver into a pairwise PRDO, we store a β = γ 4/3 -length path P u,v in G∪H for every pair (u, v) ∈ P, with weight d G∪H (u, v) ≤ (1 + ϵ)d G (u, v) . In addition, we build a dedicated 5 In some occasions, an approximately shortest path may suffice. pairwise PRDO D H , that given a hopset-edge (x, y) ∈ H, produces a (possibly approximate) shortest path P x,y in G. The oracle D H implicitly stores the union of these paths P x,y . Its size is at most the support size of the hopset, i.e., it is small. We note that the naive construction of (unweighted) (1 + ϵ)-preservers, which is based on near-additive spanners [29,49,26,2], does not give rise to a pairwise PRDO. The problem is that the paths in the underlying spanner may be arbitrarily long. Next, we shortly outline the construction of hopsets with small supporting size. Since they are closely related to the construction of Thorup-Zwick universal emulators [57] and universal hopsets [27,34], we start by overviewing the latter constructions. An (α, β)-emulator for a graph G = (V, E) is a weighted graph G ′ = (V, E ′ , w ′ ), where E ′ is not necessarily a subset of E (that is, G ′ may consist of entirely different edge-set than that of G), such that for every two vertices u, v ∈ V , d G (u, v) ≤ d G ′ (u, v) ≤ α · d G (u, v) + β . Consider a hierarchy of subsets V = A 0 ⊇ A 1 ⊇ A 2 ⊇ · · · ⊇ A l−1 ⊇ A l = ∅, where l is some integer parameter. For every index i ∈ [0, l − 1] and every vertex v ∈ A i , we define B i (v) -the i-th bunch of v -to be the set of A i -vertices which are closer to v than the closest vertex u ∈ A i+1 to v. The latter vertex u is called the (i + 1)-th pivot of v, and is denoted by p i+1 (v). The emulator H is now defined by adding an edge from v to all of its pivots and bunch members (for every i ∈ [0, l −1], and this is done for any v ∈ V ). Thorup and Zwick [57] showed that H is (in any unweighted graph) a (1 + ϵ, γ 2 = γ 2 (ϵ, k))emulator, for all ϵ > 0 simultaneously, when choosing l ≈ log 2 k. Elkin and Neiman [27] and Huang and Pettie [34] showed that the very same construction provides (1 + ϵ, γ 2 )-hopsets in any weighted graph (again, for all ϵ > 0 simultaneously). In addition, [57,27,34] showed that if each subset A i+1 is sampled independently at random from A i with an appropriate sampling probability, then the size of H is O(n 1+ 1 k ). Pettie [49] made a crucial observation that one can replace the bunches in this construction by "one-third-bunches", and obtain sparse near-additive spanners (as opposed to emulators). For a parameter 0 < ρ ≤ 1, a ρ-bunch B ρ i (v) of a vertex v ∈ A i (for some i ∈ [0, l − 1]) contains all vertices u ∈ A i that satisfy d G (v, u) < ρ · d G (v, p i+1 (v)) . Note that under this definition, a bunch is a 1-bunch. One-third-bunch (respectively, half-bunch) is ρ-bunch with ρ = 1 3 (respectively, ρ = 1 2 ). On one hand, the introduction of factor 1 3 makes the additive term β somewhat worse. On the other hand, one can now show that the set of all v − u shortest paths, for v ∈ V , i ∈ [0, l − 1], u ∈ B 1/3 i (v) is sparse. As a result, one obtains a spanner rather than an emulator. To establish this, Pettie [49] analysed the number of branching events 6 induced by these paths. This approach is based on the analysis of exact preservers from [19]. The same approach (of using one-third-bunches as opposed to bunches) was then used in the construction of PRDOs by [30]. Intuitively, using one-third-bunches rather than bunches is necessary for the path-reporting property of these oracles. This comes, however, at the price of increasing the stretch (recall that the stretch of the PRDO of [30] is O(log log 4/3 7 n), instead of the desired bound of O(log n)). Elkin and Neiman [27] argued that half-bunches can be used instead of one-third-bunches in the constructions of [49,30]. This led to improved constructions of near-additive spanners and PRDOs (in the latter context, the stretch becomes O(log log 4/3 5 n)). In this paper we argue that for any weighted graph, the set H 1/2 (achieved using the same construction as H, but with half-bunches instead of bunches) is a (1 + ϵ, γ 4/3 )-hopset with size O(n 1+ 1 k ) and support size O(n 1+ 1 k + n log k). The analysis of the support size is based on the analysis of near-additive spanners (for unweighted graphs) of [49,26]. The proof that H 1/2 is a sparse hopset can be viewed as a generalization of related arguments from [27,34]. There it was argued that H = H 1 is a sparse hopset, while here we show that this is the case for H ρ , for any constant 7 0 < ρ < 1. Naturally, its properties deteriorate as ρ decreases, but for any constant ρ > 0, the set H ρ is quite a good hopset, while for ρ ≤ 1 2 , we show that its support size is small. PRDOs Our constructions of PRDOs consist of three steps. In the first step, we construct h = O(log k) layers of the Thorup-Zwick PRDO [56], while using the query algorithm of [59]. The result is an oracle with stretch 2h − 1 = O(log k), size 8 O(h · n 1+ 1 k ) = O(log k · n 1+ 1 k ) and query time O(log h) = O(log log k). However, this oracle is a partial PRDO, rather than an actual PRDO. It provides approximate shortest paths only for queries of a certain specific form (see below). For the rest of the queries (u, v) ∈ V 2 , it instead provides two paths P u,u ′ , P v,v ′ , from u to some vertex u ′ and from v to some vertex v ′ , such that u ′ , v ′ are members in a relatively small set S of size O(n 1− h k ). The set S is actually exactly the set A h , from the hierarchy of sets that defines Thorup-Zwick PRDO (the vertices u ′ , v ′ are the corresponding pivots of u, v, respectively). The two paths P u,u ′ , P v,v ′ have the property that w(P u,u ′ ), w(P v,v ′ ) ≤ h · d G (u, v) . Intuitively, given a query (u, v), the partial PRDO provides a u − v path P u,v (rather than paths P u,u ′ , P v,v ′ ) if and only if the Thorup-Zwick oracle does so while employing only the first h levels of the oracle. We note that using the first h levels of the TZ oracle indeed guarantees almost all the aforementioned properties. The only problem is that the query time of such partial oracle is O(h), rather than O(log h). To ensure that this oracle has query time O(log h), we generalize the argument of Wulff-Nilsen [59] from full oracles to partial ones. In the second step, to find a low-stretch path between u ′ and v ′ , we construct an interactive emulator for the set S. An interactive emulator is an oracle that similarly to PRDOs, provides approximate shortest paths between two queried vertices. However, as opposed to a PRDO, the provided path does not use actual edges in the original graph, but rather employs virtual edges that belong to some sparse low-stretch emulator of the graph. Our construction of interactive emulator is based on the distance oracle of Mendel and Naor [44]. Recall that the Mendel-Naor oracle is not path-reporting. Built for N -vertex graph and a 7 In fact, we only argue this for ρ ≥ 1 2 , but the same argument extends to any constant ρ > 0. 8 The actual choice of the parameter h is more delicate than h = O(log k), and as a result, our oracle ultimate size is typically much smaller than O(log k · n 1+ 1 k ). However, for the sake of clarity, in this sketch we chose to set parameters in this slightly sub-optimal way. parameter k 1 , this oracle provides stretch O(k 1 ), has size O(N 1+ 1 k 1 ), and has query time O(1). We utilize the following useful property of the Mendel-Naor oracle: there is an O(k 1 )-emulator H ′ of the original graph, of size O(N 1+ 1 k 1 ), such that upon a query (u ′ , v ′ ), the oracle can return not only a distance estimated(u ′ , v ′ ), but also a u ′ − v ′ path -in H ′ -of lengthd(u ′ , v ′ ). We explicate this useful property in this paper. In fact, any other construction of interactive emulator could be plugged instead of the Mendel-Naor oracle in our construction. Indeed, one of the versions of our PRDO uses the Thorup-Zwick PRDO [56] for this step as well, instead of the Mendel-Naor oracle. This version has the advantage of achieving a better stretch, at the expense of a larger query time. Yet another version of our PRDO employs an emulator based upon a hierarchy of neighborhood covers due to Awerbuch and Peleg [5] and to Cohen [15]. As a result, this PRDO has intermediate stretch and query time: its stretch is larger than that of the PRDO that employs interactive emulators based on the construction of Thorup and Zwick [56] (henceforth, TZ-PRDO), but smaller than that of the PRDO that employs Mendel-Naor's interactive emulator (henceforth MN-PRDO). On the other hand, its query time is smaller than that of the TZ-PRDO, but larger than that of the MN-PRDO. Our sparse interactive emulator H ′ is built for the complete graph K = (S, S 2 ), where the weight of an edge (x, y) in K is defined as d G (x, y). Note that since we build this interactive emulator on K rather than on G, it is now sparser -it has size O(|S| After the first two steps, we constructed a path between the original queried vertices u, v, that passes through the vertices u ′ , v ′ ∈ S. Note, however, that this path is not a valid path in the original graph, as its edges between u ′ and v ′ are virtual edges in the emulator H ′ (of size O(|S| 1+ 1 k 1 )). To solve this problem, in the third step, we employ our new sparse pairwise PRDO on the graph G and the set of pairs P = H ′ ⊆ V 2 . Notice that since |S| = O(n 1− h k ), we can choose a slightly smaller k 1 for the Mendel-Naor oracle, such that |H ′ | = O(|S| 1+ 1 k 1 ) is still relatively small. We choose k 1 slightly larger than k h , so that |H ′ | = O((n 1− h k ) 1+ 1 k 1 ) = O(n 1+ 1 k 1 − h k ) will be smaller than n 1+ 1 k γ 4/3 . Then, we use our novel (1 + ϵ)-pairwise PRDO (which is also a (1 + ϵ)preserver) that was described above, that has size O(γ 4/3 ·|P|+n log k +n 1+ 1 k ) and query time O(1), on P = H ′ . The size of this pairwise PRDO is therefore O(n log k + n 1+ 1 k ). Recall that the size of the partial TZ oracle is O(log k · n 1+ 1 k ). Thus the total size of our construction is O(log k · n 1+ 1 k ). The stretch of this new PRDO is the product between the stretches of the partial Thorup-Zwick oracle, the interactive emulator, and the pairwise preserver, which is O(h · k 1 ) = O(k). The query time also consists of the query times of these three oracles, which is O(log h) = O(log log k). In case we use the Thorup-Zwick PRDO instead of Mendel-Naor oracle, we get a small constant coefficient on the stretch, while the query time increases to O(log k) (which is the running time of the query algorithm by [59], used in the Thorup-Zwick PRDO). One final ingredient of our PRDO is the technique that enables one to trade overheads in the size by overheads in the stretch. Specifically, rather than obtaining stretch O(k) and size O(log k·n 1+ 1 k ), we can have stretch O(k log k) and size O(n 1+ 1 k ) (the query time stays basically the same). To accomplish this, we build upon a recent result of Bezdrighin et al. [9] about stretch-friendly partitions. Given an n-vertex weighted graph G = (V, E), a stretch-friendly partition C of G is a partition of V into clusters, where each cluster C ∈ C is equipped with a rooted spanning tree T C for C. Moreover, these clusters and their spanning trees must satisfy the following properties: 1. For every edge e = (u, v) ∈ E with both endpoints belonging to the same cluster C ∈ C, and every edge e ′ in the unique u − v path in T C , w(e ′ ) ≤ w(e). 2. For every edge e = (u, v) ∈ E such that u ∈ C and v / ∈ C for some cluster C ∈ C, and every edge e ′ in the unique path from u to the root of T C , w(e ′ ) ≤ w(e). In [9], the authors prove that for every n-vertex graph, and every positive parameter t, there is a stretch-friendly partition with at most n t clusters, where the spanning tree of each cluster is of depth O(t). We use stretch-friendly partitions with parameter t = log k, and then apply our PRDO on the resulting cluster-graph. Naturally, the stretch grows by a factor of t, but the number of vertices in the cluster-graph is at most n t . Thus, the size of this PRDO becomes O(log k · ( n log k ) 1+ 1 k ) = O(n 1+ 1 k ). There are some details in adapting stretch-friendly partitions to the setting of PRDOs that we suppress here. See Section 6.2.1 for the full details. We note that originally, stretch-friendly partitions were devised in [9] in the context of ultra-sparse spanners. A similar approach in the context of PRDOs for unweighted graphs was recently applied by [46]. For weighted graphs, the construction of [46] is based on a clustering with weaker properties than that of stretch-friendly partitions. Once the clustering is constructed, [46] invoke a black-box PRDO (of Thorup and Zwick [56,59]) on the graph induced by this clustering. As a result, the stretch in [46] is at least the product of the maximum radius of the clustering by the stretch of Thorup-Zwick's oracle. The product of these quantities is at least Ω(log 2 n). Indeed, in its sparsest regime, the oracle of [56] has size Ω(n log n) and stretch Θ(log n). To have overall size of O(n), one needs to invoke the oracle on a cluster-graph with N = O n log n nodes, implying that the maximum radius of the clustering is Ω(log n). As a result, the approach of [46] is doomed to produce PRDOs with stretch Ω(log 2 n). Neiman and Shabat [46] show how this composition of clustering with a black-box PRDO can produce ultra-sparse PRDOs. To produce our ultra-sparse PRDOs we follow their approach. However, we use a stretch-friendly partition instead of their clustering, and we use our novel PRDO of size O(n log log n) instead of the black-box PRDO of [56,59]. As a result, we obtain ultra-sparse PRDOs with stretchÕ(log n) (as opposed to O(log c ′ n) in [46]), and query time O(log log n) (as opposed to at least O(log n) in [46]). New Combinatorial and Algorithmic Notions In this paper we introduce and initiate the study of a number of new combinatorial and algorithmic notions. We have already mentioned them in previous sections, but we believe that they are important enough for highlighting them again here. The first among these notions is the support size of hopsets (see Section 2.1 for definition). We present a number of constructions of sparse hopsets with small hopbound and small support size. We have also showed that they directly lead to significantly improved preservers. We believe that these new hopsets could also be useful in many other applications of hopsets, in which the ultimate objective is to compute short paths, rather than just distance estimates. Another class of new notions that we introduce are interactive structures. These include inter-active distance preservers, interactive spanners, and interactive emulators 9 . Intuitively, interactive distance preserver is a pairwise PRDO with the additional property that all of its output paths belong to a sparse approximate distance preserver. We show that our new sparse preservers give rise directly to interactive distance preservers with asymptotically the same size. Interactive spanner is a stronger notion than that of PRDO, because it adds an additional requirement that the union of all output paths needs to form a sparse spanner. Nevertheless, all known PRDOs for general graphs (including those that we develop in this paper) are, in fact, interactive spanners with asymptotically the same size. Finally, interactive emulator is a stronger notion than that of non-path-reporting distance oracle. It generalizes the distance oracle of Mendel-Naor [44]. Note that even though interactive emulators are not PRDOs by themselves, we show that in conjunction with interactive distance preservers, they are extremely helpful in building PRDOs. Organization After some preliminaries in Section 3, in Section 4 we devise our two new constructions of distance preservers. One of them is fully described in Section 4, while for the other we provide a short sketch (full details of it appear in Appendix C). Section 5 is dedicated to the construction of our interactive emulator that is based on the Mendel-Naor distance oracle [44]. Finally, in Section 6 we present our new constructions of PRDOs. In Appendix A we show that H 1/2 is a (1 + ϵ, log k ϵ O(log k) )-hopset, and also a (3 + ϵ, k O(log 1 ϵ ) )hopset. Appendix B is devoted to a few technical lemmas. In Appendix D we argue that the first h levels (for a parameter h ≤ k) of the TZ oracle can be used as a partial PRDO (in the sense described in Section 2.2), and extend Wulff-Nilsen's query algorithm [59] to apply to this PRDO (with query time O(log h)). Appendix E is devoted to trading stretch for size in a general way, via stretch-friendly partitions (based on [9]). In Appendix F we devise one more construction of an interactive emulator, in addition to those based on the TZ PRDO [56,59] and the Mendel-Naor oracle [44]. The new construction is based upon neighborhood covers due to Awerbuch and Peleg [5] and Cohen [15]. Appendix G provides a more detailed table that summarizes our new results. Preliminaries Ultrametrics, HSTs and Branching Events Given an undirected weighted graph G = (V, E), we denote by d G (u, v) the distance between some two vertices u, v ∈ V . When the graph G is clear from the context, we sometimes omit the subscript G and write d(u, v). Given some positive parameter β, we denote by d (β) G (u, v) the weight of the shortest path between u and v, among the paths that has at most β edges. When the graph G is clear from the context, n denotes the number of vertices in G and m denotes the number of edges. A Consistent Choice of Shortest Paths. Given two vertices u, v of a graph G, there can be more than one shortest path in G between u, v. We often want to choose shortest paths in a consistent manner, that is, we assume that we have some fixed consistent rule of how to choose a shortest path between two vertices. For example, when viewing a path as a sequence of edges, choosing shortest paths in a consistent manner can be done by choosing for every u, v ∈ V the shortest path between u, v that is the smallest lexicographically. For Section 5, we will need the following definitions, regarding metric spaces. Definition 1. A metric space is a pair (X, d), where X is a set and d : X → R ≥0 satisfies 1. For every x, y ∈ X, d(x, y) = 0 if and only if x = y. 2. For every x, y ∈ X, d(x, y) = d(y, x). 3. (Triangle Inequality) For every x, y, z ∈ X, d(x, z) ≤ d(x, y) + d(y, z). If d also satisfies that for every x, y, z ∈ X, d(x, z) ≤ max{d(x, y), d(y, z)}, then d is called an ultrametric on X, and (X, d) is called an ultrametric space. Definition 2. A hierarchically (well) separated tree or HST 10 is a rooted tree T = (V, E) with labels 11 ℓ : V → R ≥0 , such that if v ∈ V is a child of u ∈ V , then ℓ(v) ≤ ℓ(u), and if v ∈ V is a leaf in T , then ℓ(v) = 0. Let x, y be two vertices in a rooted tree (e.g., in an HST). Let lca(x, y) denote the lowest common ancestor of x, y, which is a vertex z such that x, y are in its sub-tree, but z does not have a child with the same property. Given an HST, let L be its set of leaves, and define the function d(x, y) = ℓ(lca(x, y)). It is not hard to show that (L, d) is an ultrametric space. In Section 5, we will use the fact, that was proved by Bartal et al. in [7], that every ultrametric can be represented by an HST with this distance function. Another useful notion is that of Branching Events. Given a set of pairs P ⊆ V 2 , for every a = (x, y) ∈ P, let P a be a shortest path between x, y in the graph G (for every a ∈ P we choose a single P a in some consistent manner). For a, b ∈ P, x ∈ V we say that (a, b, x) is a Branching Event if x ∈ P a ∩ P b , and the adjacent edges to x in P a are not the same as the adjacent edges to x in P b . Define Branch(P) as the set of all branching events of P. This notion of Branching Events was introduced in [19] in the analysis of distance preservers. It also plays a key role in the construction by Elkin and Pettie in [30]. In this paper, the authors built a pairwise PRDO with stretch 1. The following theorem is an immediate corollary from Theorem 3.2 in [30] and the discussion that comes after it. Theorem 1. Given an undirected weighted graph G and a set P of pairs of vertices, there is an algorithm that computes an interactive 1-distance preserver with query time O(1) and size O(n + |P| + |Branch(P)|) . Interactive Structures Definition 3 (Interactive Spanner). Given an undirected weighted graph G = (V, E, w), an interactive α-spanner is a pair (S, D) where S is an α-spanner of G, and D is an oracle that answers every query (u, v) ∈ V 2 with a path P u,v ⊆ S such that d G (u, v) ≤ w(P u,v ) ≤ α · d G (u, v) . The size of (S, D) is defined as max{|S|, |D|}, and its query time is the query time of D. Note that in particular, D is a PRDO for G. Definition 4 (Interactive Emulator). Given an undirected weighted graph G = (V, E, w), an in- teractive α-emulator is a pair (H, D), where H = (V, E ′ , w ′ ) is an α-emulator (i.e., an (α, 0)- emulator) of G, and D is an oracle that answers every query (u, v) ∈ V 2 with a path P u,v ⊆ E ′ such that d G (u, v) ≤ w ′ (P u,v ) ≤ α · d G (u, v) . Definition 5 (Interactive Distance Preserver). Given an undirected weighted graph G = (V, E, w) and a set P of pairs of vertices, an interactive α-distance preserver is a pair (S, D), where S is an α-distance preserver of G and P, and D is an oracle that answers every query (u, v) ∈ P with a path P u,v ⊆ S such that d G (u, v) ≤ w(P u,v ) ≤ α · d G (u, v) . Note that D is a pairwise PRDO for G and P. Hierarchy of Sets Let G = (V, E) be an undirected weighted graph. In most of our constructions, we use the notion of a hierarchy of sets V = A 0 ⊇ A 1 ⊇ A 2 ⊇ · · · ⊇ A l = ∅, for an integer parameter l. Given an index i ∈ [0, l − 1] and a vertex v ∈ V , we define 1. p i (v) = the closest vertex to v from A i . The vertex p i (v) is called the i-th pivot of v. 2. B i (v) = {u ∈ A i | d(v, u) < d(v, p i+1 (v)}. The set B i (v) is called the i-th bunch of v. 3.B i (v) = B i (v) ∪ {p i (v)}. The setB i (v) is called the i-th extended bunch of v. 4. B 1/2 i (v) = {u ∈ A i | d(v, u) < 1 2 d(v, p i+1 (v)}. The set B 1/2 i (v) is called the i-th half-bunch of v. 5.B 1/2 i (v) = B 1/2 i (v) ∪ {p i (v)}. The setB 1/2 i (v) is called the i-th extended half-bunch of v. Note that at this point, the sets B l−1 (v),B l−1 (v), B 1/2 l−1 (v),B 1/2 l−1 (v) are not well-defined, since p l (v) is not defined. Thus, we define them simply as the set A l−1 . We also define the following sets, each of them consists of pairs of vertices from V . For every i = 0, 1, ..., l − 1, H i = v∈A i u∈B i (v) {(v, u)}. For every i = 0, 1, ..., l − 1,H i = v∈A i u∈B i (v) {(v, u)}. 3. H = l−1 i=0 H i , andH = l−1 i=0H i . 4. For every i = 0, 1, ..., l − 1, H 1/2 i = v∈A i u∈B 1/2 i (v) {(v, u)}. For every i = 0, 1, ..., l − 1,H 1/2 i = v∈A i u∈B 1/2 i (v) {(v, u)}. 6. H 1/2 = l−1 i=0 H 1/2 i , andH 1/2 = l−1 i=0H 1/2 i . In Appendix A we prove the following theorem, that generalizes a similar result by Elkin and Neiman, Huang and Pettie in [27,34]. Theorem 2. For every choice of a set hierarchy,H 1/2 is a (1 + ϵ, β l )-hopset, simultaneously for all positive ϵ ≤ O(l), where β l = O( l ϵ ) l−1 . Given a parameter t > 1, we sometimes use the notation γ t (ϵ, k) = β ⌈log t k⌉+1 , i.e., γ t (ϵ, k) = O log k ϵ ⌈log t k⌉(2) The following lemma will be useful for building interactive structures. Lemma 1. For every i = 0, 1, ..., l − 1, there is an interactive 1-distance preserver for the set of pairs Proof. Define an oracle D that for every v ∈ V stores a pointer to the next vertex on the shortest path from v to p i (v). Given a query (v, p i (v)), return the path created by following these pointers from v to p i (v). Note that if the vertex x is on the shortest path from v to p i (v), then it is easy to verify that p i (x) = p i (v). Therefore the pointer stored for x keeps us on the shortest path from v to p i (v). Hence, the stretch of D is 1. The query time of D is linear in the size of the returned path. Also, the size of D is O(n). P i = {(v, p i (v)) | v ∈ V } , Lastly, we claim that the union of all the shortest paths that are returned by D is a forest, and therefore contains O(n) edges. This is true because we can orient every edge in these shortest paths in the same direction that the pointers point, and get that from every vertex there is at most one outgoing edge. That means that there are no cycles. We denote this union by S, and conclude that (S, D) is an interactive distance preserver. We now state a useful property of the above notions. A similar claim was proved in [49,30] for one-third -bunches 12 (instead of half-bunches), and later it was extended to half-bunches in [23]. Lemma 2. For every i = 0, 1, ..., l − 1, |Branch(H 1/2 i )| ≤ 4 u∈A i |B i (u)| 3 . 12 The formal definition of a one-third-bunch is analogous to the definition of a half-bunch, but with 1 3 instead of 1 2 : B 1/3 i (v) = {u ∈ Ai | dG(v, u) < 1 3 dG(v, pi+1(v))}. The proof of this lemma can be found in Appendix B. In all our constructions, we employ the following general process of producing the hierarchy of sets. The same process was employed also in [56,30], and in many other works on this subject. We define A 0 to be V , and then for every i = 0, 1, ..., l − 2 sequentially, we construct A i+1 by sampling each vertex from A i independently with some probability q i . In Appendix B we prove the following lemma. Lemma 3. The following inequalities hold: 1. For every i < l − 1, E[|H i |] ≤ n q i i−1 j=0 q j . 2. E[|H l−1 |] ≤ 2 · max{1, (n l−2 j=0 q j ) 2 }. 3. For every i < l − 1, E[|Branch(H 1/2 i )|] ≤ 24 · n q 3 i i−1 j=0 q j . 4. E[|Branch(H 1/2 l−1 )|] ≤ 60 · max{1, (n l−2 j=0 q j ) 4 }. New Interactive Distance Preservers We start by introducing two constructions of interactive (1 + ϵ)-distance preservers. For most choices of the parameters (as long as ϵ > 2 −k c , for a constant c which is approximately 0.58), the second construction has a smaller size than the first one. O(|P| · γ 4/3 (ϵ, k) + n log k + n 1+ 1 k ) , where γ 4/3 (ϵ, k) = O( log k ϵ ) ⌈log 4/3 k⌉ . In addition, there exists a universal 13 (1 + ϵ, γ 4/3 (ϵ, k))-hopset with size O(n 1+ 1 k ) and support size O(n 1+ 1 k + n log k). Proof. We first construct a hierarchy of sets as in Section 3.3, with q i = 1 2 n − (4/3) i 3k and l = ⌈log 4/3 k⌉ + 1 . These choices are similar to the values of sampling probabilities in [49,30,23], which produce near-additive spanners ( [49,23]) and PRDOs ( [30]). Recall that by Theorem 2,H 1/2 is a (1 + ϵ, β l )-hopset, for all ϵ > 0 simultaneously, where β l = O( l ϵ ) l−1 (see Appendix A for a proof). For each i = 0, 1, ..., l − 1, let (S 1 i , D 1 i ) be the interactive 1-distance preserver from Lemma 1 for the set {(v, p i (v)) | v ∈ V }, and let (S 2 i , D 2 i ) be the interactive 1-distance preserver from Theorem 1 for the set of pairs H 1/2 i . Let P be a set of pairs of vertices in G. We define an oracle D for our new interactive distance preserver as follows. 1. For every pair (x, y) ∈ P, the oracle D stores a path P x,y ⊆ G ∪H 1/2 between x, y with length at most β l and weight at most (1 + ϵ)d(x, y). 2. For every i = 0, 1, ..., l − 1, the oracle D stores the oracles D 1 i and D 2 i . For every edge e ∈H 1/2 , the oracle D stores a flag variable f e . If e ∈ {(v, p i (v)) | v ∈ V } for some i, then f e = 1. If e ∈ H 1/2 i for some i, then f e = 2. In both cases, D also store the relevant index i. Next, we describe the algorithm for answering queries. Given a query (x, y) ∈ P, find the stored path P x,y . Now replace each hop-edge e = (a, b) ∈H 1/2 that appears in P x,y ; if f e = 1, replace e with the path returned from D 1 i (a, b); if f e = 2, replace e with the path returned from D 2 i (a, b) . Return the resulting path as an output. Since the query time of D 1 i , D 2 i is linear in the length of the returned path, for every i, the total query time of D is also linear in the size of the output. Thus, the query time of our PRDO is O(1). Also, note that the paths that are returned by the oracles D 1 i , D 2 i have the same weight as the hop-edges they replace. Therefore the resulting path has the same weight as P x,y , which is at most (1 + ϵ)d(x, y). Hence, D has a stretch of 1 + ϵ. The size of D is the sum of its components' sizes. For storing the paths P x,y (item 1 in the description of D) we need O(|P|β l ) space. For storing the flags and the indices (item 3) we need O(|H 1/2 |) space. For each i, the size of D 1 i is O(n) and the size of D 2 i is O(n + |H 1/2 i | + |Branch(H 1/2 i )|) . For estimating these quantities, we use Lemma 3. By inequalities (1) and (3) in Lemma 3, for every i < l − 1, in expectation, |H 1/2 i | ≤ |H i | ≤ n q i i−1 j=0 q j , |Branch(H 1/2 i )| ≤ 24 · n q 3 i i−1 j=0 q j . Summing these two quantities, we get (2) and (4) in Lemma 3, if nq ≥ 1, then |H 1/2 i | + |Branch(H 1/2 i )| ≤ n q i i−1 j=0 q j + 24 · n q 3 i i−1 j=0 q j ≤ 25 · n q 3 i i−1 j=0 q j = 25n · 2 3 n (4/3) i k · 1 2 i n − (4/3) i −1 k = 25 2 i−3 n 1+ 1 k . For i = l − 1, first denote q = l−2 j=0 q j . By inequalities|H 1/2 l−1 | + |Branch(H 1/2 l−1 )| ≤ 2(nq) 2 + 60(nq) 4 ≤ 62(n l−2 j=0 q j ) 4 = 62( 1 2 l−1 n 1− (4/3) l−1 −1 k ) 4 ≤ 62( 1 2 l−1 n 1− k−1 k ) 4 = 62 2 4l−4 n 4 k = O(n 1+ 1 k ) , where the last step is true since k ≥ 3. Otherwise, if nq < 1, then the expected value of |H 1/2 l−1 | + |Branch(H 1/2 l−1 )| is bounded by a constant, which is also O(n 1+ 1 k ). Hence, we get that the total expected size of D is O(|P|β l + |H 1/2 |) + l−1 i=0 O(n + |H 1/2 i | + |Branch(H 1/2 i )|) = O(|P|β l ) + l−1 i=0 O(n + |H 1/2 i | + |Branch(H 1/2 i )|) = O(|P|β l ) + O(n · l) + l−2 i=0 O( 1 2 i−3 n 1+ 1 k ) + O(n 1+ 1 k ) = O(|P|β l + n · l + n 1+ 1 k ) = O(|P|β l + n · log k + n 1+ 1 k ) . Define S =H ∪ (x,y)∈P (P x,y ∩ G) , whereH = l−1 i=0 S 1 i ∪ l−1 i=0 S 2 i . Then, the output paths of D are always contained in S, and S has the same size bound as D. Hence, (S, D) is an interactive (1 + ϵ)-distance preserver with query time O(1) and size O(|P|β l + n · log k + n 1+ 1 k ) . In addition, note thatH is the support set of the hopsetH 1/2 . The analysis above implies that |H| = O(n 1+ 1 k + n · l) = O(n 1+ 1 k + n log k). Since this is true for every ϵ > 0, we conclude that H 1/2 is a universal (1 + ϵ, γ 4/3 (ϵ, k))-hopset with size O(n 1+ 1 k ) and support size O(n 1+ 1 k + n log k). Remark 1. In Theorem 8 in [23], the authors prove thatH (see Section 3.3 for definition) is also a (3 + ϵ, 2(3 + 12 ϵ ) l−1 )-hopset, for ϵ ∈ (0, 12]. By a similar argument, we also prove thatH 1/2 is a (3 + ϵ, (12 + 40 ϵ ) l−1 )-hopset, for ϵ = O(l). For completeness, this proof appears in Appendix A. Using this property ofH 1/2 , we get an interactive (3 + ϵ)-distance preserver with a better size of O(|P|k log 4/3 (12+ 40 ϵ ) + n · log k + n 1+ 1 k ) . Our second construction of an interactive (1 + ϵ)-distance preserver improves the factor of γ 4/3 (ϵ, k) in the size of the preserver from Theorem 3, to almost γ 2 (see Equation (2) for the definition of γ t (ϵ, k)). Since large parts of the proof are very similar to the proof of Theorem 3, we give here only a sketch of the proof, while the full proof appears in Appendix C. Theorem 4. Given an undirected weighted graph G = (V, E), an integer k ∈ [3, log 2 n], a positive parameter ϵ ≤ 1, and a set of pairs P ⊆ V 2 , G has an interactive (1 + ϵ)-distance preserver with query time O(1) and size O(|P| ·γ 2 (ϵ, k) + n · (log k + log log( 1 ϵ )) + n 1+ 1 k ) , whereγ 2 (ϵ, k) = O( log k ϵ ) (1+o(1)) log 2 k+log 4/3 log( 1 ϵ ) . In addition, there exists a universal (1 + ϵ,γ 2 (ϵ, k))-hopset with size O(n 1+ 1 k ) and support size O(n 1+ 1 k + n(log k + log log 1 ϵ )). Sketch of the Proof. We use the same hierarchy of sets as in Theorem 3, up to some level h, such that |A h | ≤ n γ 4/3 (we denote here for short γ 4/3 = γ 4/3 (ϵ, k)). A simple computation shows that h = log 4/3 log k + log 4/3 log( 1 ϵ ) + o(log log k) . For these h levels, we store the same information in our oracle as before. Specifically, for i < h, we store the interactive distance preservers from Theorem 1 on H 1/2 i , and the ones from Lemma 1 on {(v, p i (v)) | v ∈ V }. We actually store the latter preservers for every i = 0, 1, ..., l − 1. For each level i ≥ h in the hierarchy, we change the probability q i to 1 2 ( n γ 4/3 ) − 2 i−h k . We now apply our interactive distance preserver from Theorem 3 on l−1 i=h H 1/2 i . Denote the resulting preserver by (S 3 , D 3 ). We store these interactive distance preservers in our new oracle, together with the same information as before; a path P x,y ⊆ G ∪H 1/2 with low stretch and length, for each (x, y) ∈ P, and flags for the hop-edges. Since the stored interactive distance preservers cover all of the pairs inH 1/2 , we can use the same stretch analysis as before, and now get a stretch of at most (1 + ϵ) 2 . Replacing ϵ by ϵ 3 , we still get a stretch of 1 + ϵ. The query time analysis is the same as before. As for the size analysis, the first difference is that now we need to add the size of (S 3 , D 3 ). Since it is the same preserver from Theorem 3, built on a vertices set of size at most n γ 4/3 , this additional size is O(( n γ 4/3 ) 1+ 1 k ) · γ 4/3 + n · l + n 1+ 1 k ) = O(n · l + n 1+ 1 k ) . Note that in this case we must increase the value of l to approximately log 2 k + h. The second and final difference in the size analysis is that now storing the paths P x,y takes |P| · β l space, with the new l. A simple computation gives that β l = O log k ϵ (1+o(1)) log 2 k+log 4/3 log( 1 ϵ ) =γ 2 (ϵ, k) . In conclusion, while the stretch and the query time stay the same, the size decreases to O(|P|γ 2 (ϵ, k) + n · l + n 1+ 1 k ) = O(|P|γ 2 (ϵ, k) + n(log k + log log( 1 ϵ )) + n 1+ 1 k ) . Accordingly, the support size of the universal hopsetH 1/2 is O(n 1+ 1 k + n(log k + log log( 1 ϵ ))) . A New Interactive Emulator Recall Definition 4 of an interactive emulator. For every positive integer parameter k, the distance oracle of Thorup and Zwick in [56] can be viewed as an interactive (2k − 1)-spanner, and thus, also as an (2k − 1)-emulator, with size O(kn 1+ 1 k ) and query time O(k). Wulff-Nilsen [59] later improved this query time to O(log k). Next, we devise another interactive emulator, based on the (non-path-reporting) distance oracle by Mendel and Naor in [44], with smaller query time and size, albeit with a slightly larger stretch. We will use it later in Section 6, for building our new PRDO. Proof. The graph G = (V, E) induces an n-point metric space, where the metric d(x, y), for x, y ∈ V , is the shortest path metric in G. We apply the following lemma from [44] on this metric space. For definitions of metric spaces, ultrametrics and HSTs, see Section 3. Lemma 4 (Mendel and Naor [44], Lemma 4.2). Given a metric space (V, d) with |V | = n, there is a randomized algorithm that produces a hierarchy of sets V = X 0 ⊇ X 1 ⊇ X 2 ⊇ · · · ⊇ X s = ∅, such that the following properties hold. 1. E[s] ≤ kn 1 k . 2. E[ s−1 j=0 |X j |] ≤ n 1+ 1 k . 3. For every j = 0, 1, ..., s − 1 there is an ultrametric ρ j on V such that for all x, y ∈ V it holds that d(x, y) ≤ ρ j (x, y) and for all x ∈ V, y ∈ X j \ X j+1 , it holds that ρ j (x, y) ≤ O(k) · d(x, y). For the purpose of our proof, we restrict the ultrametric ρ j to the subset X j ⊆ V , i.e., we view (X j , ρ j ) as an ultrametric space, for every j. It is a well-known fact (see Lemma 3.5 in [7] by Bartal et al.) that every ultrametric space (X, ρ) can be represented by an HST, such that its leaves form the set X. Let T j = (N j , E j ) be the HST for the ultrametric space (X j , ρ j ). The leaves of T j are X j and there are labels ℓ j : N j → R ≥0 such that for every x, y ∈ X j , ℓ j (lca(x, y)) = ρ j (x, y), where lca(x, y) is the lowest common ancestor of x, y in T j . For any edge (u, v) ∈ E j , such that u is the parent of v, we assign a weight w j (u, v) = ℓ j (u)−ℓ j (v) 2 . It is easy to check that the weight of the unique path in T j between two leaves x, y is exactly ℓ j (lca(x, y)) = ρ j (x, y). Note that ℓ j (x) = ℓ j (y) = ρ j (x, x) = ρ j (y, y) = 0. We now use the following result by Gupta [33]. Lemma 5 (Gupta [33]). Given a weighted tree T and a subset R of vertices in T , there exists a weighted tree T ′ on R such that for every x, y ∈ R, d T (x, y) ≤ d T ′ (x, y) ≤ 8d T (x, y) . Let T ′ j be the resulting tree from Lemma 5, when applied on the tree T j , where the set R is the leaves of T j , which are exactly X j . Suppose that x, y ∈ X j such that y / ∈ X j+1 . Then, x, y are vertices of T ′ j , and y). We define an oracle D that stores the following: d T ′ j (x, y) ∈ [d T j (x, y), 8d T j (x, y)] = [ρ j (x, y), 8ρ j (x, y)] ⊆ [d(x, y), 8 · O(k) · d(x, y)] , i.e., d(x, y) ≤ d T ′ j (x, y) ≤ O(k) · d(x, 1. For every v ∈ V , the oracle D stores j(v) -the largest index j ∈ [0, , s − 1] such that v ∈ X j . 2. For every v ∈ V , the oracle D stores parent(v), depth(v), which are vectors of length j(v) + 1, such that [parent(v)] j is a pointer to the parent of v in T ′ j , and [depth(v)] j is the number of edges in T ′ j between v and the root of T ′ j (in case v is not a vertex of T ′ j , the entries [parent(v)] j and [depth(v)] j are empty). d(u, v). Using the pointers [parent(·)] j , [depth(·)] j , the oracle D finds the unique path in T ′ j between u, v, and returns it as an output. We can now finally define our interactive emulator (H, D). The sub-graph H is the union graph of the trees T ′ j , for every j = 0, 1, ..., s − 1, and D is the oracle described above. We already saw that for every query (u, v) ∈ V 2 , the oracle D returns a path P u,v ⊆ H between them, such that ) . Hence, the stretch of our emulator is O(k). Given a query (u, v) ∈ V 2 , let j = min{j(u), j(v)}. If, w.l.o.g., j = j(v), then u, v ∈ X j , but v / ∈ X j+1 , and therefore d(u, v) ≤ d T ′ j (u, v) ≤ O(k) ·d(u, v) ≤ w(P u,v ) = d T ′ j (u, v) ≤ O(k) · d(u, v The query time is linear in the length of the returned path. Hence, the query time is O(1). For the sizes of H, D, we have in expectation |H| ≤ s−1 j=0 |T ′ j | ≤ s−1 j=0 |X j | ≤ n 1+ 1 k , |D| ≤ O( v∈V (1 + (j(v) + 1) + (j(v) + 1)) = O(n + v∈V (j(v) + 1)) . For the last sum, note that j(v) + 1 is the number of j's such that v ∈ X j . Hence, if we define an indicator δ v,j ∈ {0, 1} to be 1 iff v ∈ X j , we get v∈V (j(v) + 1) = v∈V s−1 j=0 δ v,j = s−1 j=0 v∈V δ v,j = s−1 j=0 |X j | ≤ n 1+ 1 k . Therefore we still have in expectation that |D| ≤ O(n + n 1+ 1 k ) = O(n 1+ 1 k ). Remark 2. Following the proof of Theorem 5, one can see that the stretch of the resulting interactive emulator is 8c M N , where c M N is the constant coefficient 14 of k in stretch of the ultrametrics ρ j in Lemma 4 (see item 3 in the lemma). We believe that the coefficient 8c M N can be improved. However, in this paper we made no effort to optimize it. A New Path-Reporting Distance Oracle Our new interactive spanner is achieved by combining three main elements: a partial TZ oracle (that we will define below), an interactive emulator, and an interactive distance preserver. Note that we have several possible choices for the interactive emulator and for the interactive distance preserver. For example, we may use either our new interactive emulator from Section 5, or other known constructions of emulators. In addition, the choice of the interactive distance preserver may be from Theorem 3, from Theorem 4 or from Remark 1. By applying different choices of emulator and distance preserver, we get a variety of results, that demonstrate the trade-off between stretch, query time and size. Main Components of the Construction The first ingredient of our new interactive spanner is the partial TZ oracle -an oracle that given a query (u, v) ∈ V 2 , either returns a path between u and v, or returns other vertices u ′ , v ′ which are not far from u, v respectively, and both are in some smaller subset of V . The formal details are in the following lemma. This oracle is actually the same construction as the distance oracle of Thorup and Zwick in [56], but only for h levels. The query algorithm is based on that of Wulff-Nilsen [59]. Lemma 6. Let G = (V, E) be an undirected weighted graph with n vertices, and let 1 ≤ h < k ≤ log 2 n be two integer parameters. There is a set S ⊆ V of size O(n 1− h k ) and an oracle D with size O(h · n 1+ 1 k ), that acts as follows. Given a query (u, v) ∈ V 2 , D either returns a path between u, v with weight at most (2h + 1)d G (u, v), or returns two paths P u,u ′ , P v,v ′ , from u to some u ′ ∈ S and from v to some v ′ ∈ S, such that w(P u,u ′ ), w(P v,v ′ ) ≤ h · d G (u, v) . The query time of the oracle D is O(log h). In addition, there is a set E ′ ⊆ E such that the output paths of D are always contained in E ′ , and |E ′ | = O(h · n 1+ 1 k ) . The proof of Lemma 6 appears in Appendix D. We will call the oracle D a partial TZ oracle. The second and third ingredients of our new interactive spanner are interactive emulator and interactive distance preserver. Table 4 summarizes the results for interactive emulators and distance preservers that will be used. Type Stretch Size Query Time Source Table 4: A summary of results on interactive emulators and distance preservers. The first column indicates whether the construction is of an interactive emulator or interactive distance preserver. The last column specifies the source which the construction is based on. Here, c M N denotes the constant coefficient of k in the stretch of the distance oracle of Mendel and Naor [44]. It might be possible to improve the coefficient 8c M N of the stretch of the interactive emulator from Theorem 5, that appears in the first row of the table (see Remark 2). Note that the construction of [56,59] is actually of an interactive spanner. However, any interactive spanner is also an interactive emulator. Emulator 8c M N · k O(n 1+ 1 k ) O(1) Theorem 5 Emulator 2k − 1 O(k · n 1+ 1 k ) O(log k) [56, 59] Emulator (4 + ϵ)k O(k · log 1+ϵ Λ · n 1+ 1 k ) O(log log k) Theorem 13 Distance Preserver 1 + ϵ O(|P| · γ 4/3 (ϵ, k) + n log k + n 1+ 1 k ) O(1) Theorem 3 Distance Preserver 3 + ϵ O(|P| · k log 4/3 (12+ 40 ϵ ) + n log k + n 1+ 1 k ) O(1) Remark 1 A Variety of Interactive Spanners We now show how to combine the partial TZ oracle, an interactive emulator and an interactive distance preserver, to achieve an interactive spanner. We phrase the following lemma without specifying which of the interactive emulators and distance preservers we use. This way, we can later substitute the emulators and distance preservers from Table 4, and get a variety of results for interactive spanners. Lemma 7. Suppose that for every n-vertex graph, every integer k ∈ [3, log 2 n], and every ϵ ∈ (0, 1], there is an interactive emulator with stretch α E ·k, query time q(k) and size O(k δ n 1+ 1 k ). In addition, suppose that for every such k, ϵ, for every n-vertex graph and every set of pairs P, there is an interactive distance preserver with stretch α P , query time O(1), and size O(k τ ·|P|+n log k +n 1+ 1 k ). Then, for any graph G = (V, E) with n vertices, any integer k ∈ [3, log 2 n], and any 0 ≤ ϵ ≤ 1 2 , there is an integer h = O (δ+τ )k·log log n ϵ log n and an interactive spanner with stretch 2α P · α E ((1 + ϵ)k + O(h)), size O(n log k + h · n 1+ 1 k ) and query time q(k) + O(log h). Proof. Denote σ = δ + τ . Given the graph G = (V, E), denote by D 0 the partial TZ oracle from Lemma 6 with parameter h = σ · k · log log n log n · 1 ϵ . Denote by S ⊆ V the corresponding set from Lemma 6, and recall that the size of S is |S| = O(n 1− h k ). Also, denote by E 0 ⊆ E the "underlying" set of D 0 , which is the set of size O(h · n 1+ 1 k ) that contains every output path of D 0 . Define the graph K = (S, S 2 ) to be the complete graph over the vertices of S, where the weight of an edge (x, y) is defined to be d G (x, y). Let (H E , D E ) be an interactive emulator for K, with parameter k 1 = k(1 + 2ϵ) h , instead of k. That is, (H E , D E ) has stretch α E · k 1 , query time q(k 1 ) and size O(k δ 1 |S| 1+ 1 k 1 ). The set H E can be viewed as a set of pairs of vertices in G. The size of this interactive emulator is O(k δ 1 |S| 1+ 1 k 1 ) = O(k δ (n 1− h k ) 1+ 1 k 1 ) = O(k δ · n 1+ 1 k 1 − h k ) = O(k δ · n 1+ h (1+2ϵ)k − h k ) = O(k δ · n 1− 2ϵ·h (1+2ϵ)k ) = O(k δ · n 1− ϵ·h k ) = O(k δ · n 1− σ·log log n log n ) = O k δ · n log σ n , and therefore the size of this H E is |H E | = O k δ · n log σ n .(3) Now, construct an interactive distance preserver (H P , D P ) on the graph G and the set of pairs P = H E , with stretch α P , query time O(1) and size O(k τ · |H E | + n log k + n 1+ 1 k ) = O(k τ · k δ · n log σ n + n log k + n 1+ 1 k ) = O(k σ · n log σ n + n log k + n 1+ 1 k ) = O(n + n log k + n 1+ 1 k ) = O(n log k + n 1+ 1 k ) . We used the fact that k ≤ log n. Next, define a new oracle D that stores the oracles D 0 , D E and D P . Given a query (u, v) ∈ V 2 , the oracle D uses the oracle D 0 to either obtain a u − v path in E 0 with stretch at most 2h + 1, or obtain two paths, P u,u ′ ⊆ E 0 from u to u ′ ∈ S and P v,v ′ ⊆ E 0 from v to v ′ ∈ S, both with weight at most h · d G (u, v). In the second case, D queries the oracle D E on (u ′ , v ′ ) to find a path P u ′ ,v ′ ⊆ H E between u ′ , v ′ . Then it replaces each edge (x, y) ∈ P u ′ ,v ′ by the path D P (x, y) ⊆ H P returned by the oracle D P . The concatenation of P u,u ′ , these paths, and P v ′ ,v , is a path in E 0 ∪H P ⊆ E between u, v, and we return it as an output. Here, P v ′ ,v is the same path as P v,v ′ , but with the order of the edges reversed. Our interactive spanner is now defined as (E 0 ∪ H P , D). Note that the resulting path described above is of weight at most h · d G (u, v) + e∈P u ′ ,v ′ α P · w(e) + h · d G (u, v) = 2h · d G (u, v) + α P · w(P u ′ ,v ′ ) ≤ 2h · d G (u, v) + α P · α E · k 1 · d G (u ′ , v ′ ) ≤ 2h · d G (u, v) + α P · α E · k 1 · (d G (u ′ , u) + d G (u, v) + d G (v, v ′ )) Lemma 6 ≤ 2h · d G (u, v) + α P · α E · k 1 · (2h + 1)d G (u, v) = (2h + α P · α E · k 1 · (2h + 1))d G (u, v) = 2h + α P · α E · k(1 + 2ϵ) h · (2h + 1) d G (u, v) ≤ α P · α E · 2h + k(1 + 2ϵ) h · (2h + 1) d G (u, v) . ≤ α P · α E · 4h + 1 + k(1 + 2ϵ)(2h + 1) h d G (u, v) ≤ α P · α E · 5h + k(1 + 2ϵ)(2h + 1) h d G (u, v) . Note that since h ≥ 1 ϵ (by its definition), and since 1 + 2ϵ ≤ 2 (because ϵ ≤ 1 2 ), we have (1 + 2ϵ)(2h + 1) h = 2(1 + 2ϵ) + 1 + 2ϵ h ≤ 2(1 + 2ϵ) + 2 1/ϵ = 2(1 + 3ϵ) . Thus, the weight of the resulting path is at most α P · α E · 5h + k(1 + 2ϵ)(2h + 1) h d G (u, v) ≤ α P · α E · (5h + 2(1 + 3ϵ)k) d G (u, v) . = 2α P · α E · 5 2 h + (1 + 3ϵ)k d G (u, v) . Using the same proof, but with ϵ 3 instead of ϵ, we get that our interactive spanner has stretch max{2h + 1, 2α P · α E · (O(h) + (1 + ϵ)k)} = 2α P · α E · (O(h) + (1 + ϵ)k) , as desired. The size of D is the sum of the sizes of D 0 , D E and D P . The size of D 0 , by Lemma 6, is O(h·n 1+ 1 k ). The size of D E has the same bound as of the size of H E , which is O k δ · n log σ n = O(n) (by Equation (3)). Lastly, the size of D P was already shown to be O(n log k + n 1+ 1 k ). We get that the total size of the oracle D is O(n log k + h · n 1+ 1 k ) . In addition, note that the output paths of D are always contained in the set E 0 ∪ H P , which has the same size bound as for the oracle D. We conclude that our interactive spanner (E 0 ∪ H P , D) is of size O(n log k + h · n 1+ 1 k ). Regarding the query time, notice that finding P u,u ′ and P v,v ′ requires O(log h) time (by Lemma 6). Finding the path P u ′ ,v ′ requires q(k 1 ) time, and applying D P on every edge requires linear time in the size of the returned path. Hence, the query time of the resulting oracle is q( k 1 ) + O(log h) ≤ q(k) + O(log h). The following theorems follow from Lemma 7, when using the emulators and distance preservers from Table 4. Theorem 6. For every n-vertex graph and parameters k ∈ [3, log 2 n] and ϵ such that log −O(1) log n ≤ ϵ ≤ 1 2 , there is an interactive spanner with stretch (4 + ϵ)k + O( 1 ϵ · log k · log log k), query time O(log k + log 1 ϵ ), and size O k · log log n · log (3) n ϵ log n · n 1+ 1 k . If also ϵ ≥ log k·log log k k , then we also have an interactive spanner with stretch (4 + ϵ)k, query time O(log k), and the same size as above. Proof. We use the interactive emulator that is based on [56,59] (see Table 4), the interactive distance preserver from Theorem 3, and the notations from Lemma 7, all with ϵ 12 instead of ϵ. Then, we have α E = 2, δ = 1, and q(k) = O(log k), α P = 1 + ϵ 12 , τ = log k (γ 4/3 ( ϵ 12 , k)) = O(log log k + log 1 ϵ ). Thus, σ = δ + τ = O(log log k + log 1 ϵ ) = O(log (3) n) (since k ≤ log n and ϵ ≥ log −O(1) log n). Therefore h = O k · log log n · log (3) n ϵ log n = O 1 ϵ · log k · log log k , where the last step is true since log x·log log x x is a decreasing function for all x ≥ 8. By Lemma 7, there is an interactive spanner with stretch If ϵ ≥ log k·log log k k , then we also have log k · log log k ≤ ϵ 2 k, thus h = O log k·log log k ϵ = O(ϵk) . Hence, the stretch is (4 + ϵ)k + O(h) = (4 + ϵ + O(ϵ))k = (4 + O(ϵ))k. By replacing ϵ with ϵ C , for a sufficiently large constant C, we get a stretch of (4 + ϵ)k. Since in this case we also have 1 ϵ ≤ k, the query time is O(log k). 2 1 + ϵ 12 · 2 1 + ϵ 12 k + O(h) = 4 1 + ϵ 12 2 k + O(h) ≤ (4 + ϵ)k + O(h) , In the following theorem, the precise constant coefficient of the stretch is not specified. Instead it appears in Table 5. O n log k + k · log log n log n · n 1+ 1 k = O(log k · n 1+ 1 k ) . Proof. We use the interactive emulator from Theorem 5, the interactive distance preserver from O(n log k + h · n 1+ 1 k ) = O n log k + k · log log n log n · n 1+ 1 k = O(log k · n 1+ 1 k ) . Similarly to Theorems 6 and 7, more results can be derived by Lemma 7, when using different combinations from Table 4. We specify these results in Table 5 without proof (the proofs are relatively simple and similar to the proofs of Theorems 6 and 7). Note that in this table, we assume that the parameter ϵ is constant. For a precise specification of the dependencies on ϵ, see Table 6 in Appendix G. For general aspect ratio Λ, the size bounds in the second and the fourth rows are O k·log log n h = O (log log k + log k log Λ) · k · log log n log n = O log log n · k · log log n log n = O(1) . Otherwise, if k > √ log n, then log k log Λ = O log log n log log n = O(1), and therefore h = O k · log log k · log log n log n = O k · log log n · log (3) n log n . Similarly, for the fourth row, we have either h = O(1) (for small k) or h = O k·log log n log n (for large k). Note that for constant ϵ, all our interactive spanners have size of Ω k·log log n log n · n 1+ 1 k . We always have k·log log n log n · n 1+ 1 k = n · log log n (log n)/k · 2 log n k ≥ n log log n ≥ n log k, since 2 x > x for every x (in particular for x = (log n)/k). Therefore the term n log k that appears in Lemma 7, in the size of the interactive spanners, is negligible. See also Appendix G for explicit dependencies of these bounds on ϵ. Linear-size Interactive Spanner In order to construct a linear-size oracle (as opposed to size O(n log log n) that can be obtained from Theorem 7), we use a technique that is based on the results of Bezdrighin et al. [9]. In this paper, the authors presented the notion of stretch-friendly partitions, defined as follows. In the following, given a graph G and a subset U of its vertices, G[U ] denotes the sub-graph of G induced by the vertices of U . Definition 6. Let G = (V, E) be an undirected weighted graph, and fix some t > 0. A stretchfriendly t-partition of G is a partition V = q i=1 V i ({V i } q i=1 are called clusters), such that for every i = 1, ..., q, 1. There is a spanning tree T i of G[V i ], rooted at some r i ∈ V i , such that for every v ∈ V i , the unique path in T i between v and r i has at most t edges. 2. If (x, y) ∈ E is such that x ∈ V i and y / ∈ V i , then the weight of every edge on the unique path in T i between x and r i is at most w(x, y). 3. If (x, y) ∈ E is such that x, y ∈ V i , then the weight of every edge on the unique path in T i between x and y is at most w(x, y). In [9], it was proved that for every t ≥ 1, there is a polynomial-time algorithm that computes a stretch-friendly O(t)-partition with at most n t clusters. We use this construction to show that one can reduce the size of an interactive spanner, at the cost of increasing its stretch. A similar proof is presented in [9], where the authors prove that one can reduce the size of a spanner by increasing its stretch. However, in the case of interactive spanners, this reduction that relies on stretch-friendly partitions must be done more carefully, since we also have to prove that the approximate shortest paths in the resulting spanner can be reported efficiently. The proof of the following theorem is deferred to Appendix E. Theorem 8. Suppose that there exists an algorithm A, that given a graph with n vertices, produces an interactive α-spanner with query time q and size h(n). Then, given a positive number t ≥ 1, there exists an algorithm that given a graph G, produces an interactive O(α · t)-spanner with query time O(q) and size O(h( n t ) + n). Given Theorem 8, we apply it to our interactive spanner from Theorem 7, that has stretch O(k), query time O log k·log log n log n and size O k·log log n log n · n 1+ 1 k . We use t = k·log log n log n , and get the following result. For k = O log n log log n , the stretch of this interactive spanner is O(k), and its query time is O(1). Ultra-Sparse Interactive Spanner To produce ultra-sparse interactive spanners, we refine Theorem 8 in the following way (the proof appears in Appendix E). Theorem 10. Suppose that there exists an algorithm A, that given a graph with n vertices, produces an interactive α-spanner with query time q and size h(n). Then, given a positive number t ≥ 1, there exists an algorithm that given a graph G, produces an interactive O(α · t)-spanner with query time O(q + t) and size n + O(h( n t )). Along with Theorem 9 (with k = Θ(log n)), this theorem implies existence of interactive O(log · log log n · t)-spanners with size n + O( n t ) and query time O(log log n + t). By setting t = ω(1), we obtain ultra-sparse PRDOs with stretchÕ(log n) and query time O(log log n) (as long as t = O(log log n)). Appendix A In two separate papers, by Elkin and Neiman and by Huang and Pettie [27,34], the authors proved that the set that is denoted in Section 3.3 byH is a (1 + ϵ, β l )-hopset where β l = O( l ϵ ) l−1 . In this section, we prove thatH 1/2 is also a (1 + ϵ, β l )-hopset, albeit the constant hidden in the O-notation in β l = O( l ϵ ) l−1 becomes larger when we use half-bunches instead of bunches. We use essentially the same proof from [27,34], while rephrasing some of its concepts. SinceH 1/2 ⊆H, this also proves (up to the constant mentioned above) the original result of [27,34] aboutH. We also show thatH 1/2 is a (3 + ϵ, k O(log 1 ϵ ) )-hopset. This proof generalizes a result from [23], where it was shown that this is the case forH 1 , and that in unweighted graphs,H 1/2 is a (3 + ϵ, k O(log 1 ϵ ) )-spanner (more precisely, a version ofH 1/2 , where shortest paths are added instead of virtual edges). First, we restate and prove Theorem 2. Theorem (Theorem 2). Given an undirected weighted graph G = (V, E) and a positive parameter ϵ ≤ O(l),H 1/2 is a (1 + ϵ, O( l ϵ ) l−1 )-hopset. Remark 3. Note that the definition ofH 1/2 is independent of ϵ, i.e.,H 1/2 is a (1 + ϵ, O( l ϵ ) l−1 )- hopset for all ϵ = O(l) simultaneously. Proof. Let h > 1 be some integer. Fix u, v ∈ V and let P be the shortest path between u, v. In the following, given two vertices x, y ∈ P such that x is closer to u than y is, we denote by [x, y] the sub-path of P between x and y. We define a partition tree for P as follows. The partition tree has l levels, and each node in the tree is a sub-path of P . The root of the tree ((l − 1)-th level) is the path P itself. The children of each node Q in level i > 0 are defined by partitioning Q into at most 2h disjoint sub-paths, each of them is either a single edge, or has weight at most w(Q) h . This can be done by setting u 0 to be the first vertex in Q, and then setting u j+1 to be the first vertex after u j such that w([u j , u j+1 ]) > w(Q) h . The sub-path [u j , u j+1 ], without its last edge, has weight at most w(Q) h , and there are at most h such sub-paths. For every node in the partition tree, we will assign a label Good or Bad. For each level i = 0, 1, ..., l − 1 we will later define parameters α i , β i , r i , such that the following invariant is kept for all nodes in the partition tree (here d(x, y) is a shortened notation for d G (x, y), for every x, y ∈ V ). Invariant 1. For every node [x, y] in the i-th level of the partition tree, 1. If [x, y] is Good, d (β i ) G∪H 1/2 (x, y) ≤ α i · d(x, y) . 2. If [x, y] is Bad, d(x, p i+1 (x)) ≤ r i · d(x, y) , d(y, p i+1 (y)) ≤ r i · d(x, y) . To determine suitable values of the parameters α i , β i , r i , we analyse several different scenarios. Let [x, y] be a leaf in the partition tree, i.e., a vertex in level 0. If x, y are connected with an edge or a hopset edge, we assign the label Good to [x, y], and otherwise Bad. If Good was assigned, that means that d (1) G∪H 1/2 (x, y) = d(x, y) . Thus we can choose any α 0 , β 0 ≥ 1 and item 1 of Invariant 1 is satisfied. If [x, y] is Bad, that means that y / ∈ B 1/2 0 (x) and x / ∈ B 1/2 0 (y), which implies d(x, p 1 (x)), d(y, p 1 (y)) ≤ 2d(x, y). Therefore we can choose any r 0 ≥ 2 and item 2 of Invariant 1 is satisfied. Now let [x, y] be in the i-th level of the partition tree, for i > 0. There are three possible cases: 1. If all of the children of [x, y] are labeled as Good, we label [x, y] also as Good. In the first case, we know that for every child [a, b] of [x, y], d d(a, b), and therefore d (2h·β i−1 ) (β i−1 ) G∪H 1/2 (a, b) ≤ α i−1 ·G∪H 1/2 (x, y) ≤ α i−1 · d(x, y). In the second case, we know that for every child [s, t] before [a, b] and after [a ′ , b ′ ] there is a path P s,t ⊆ G ∪H 1/2 , between s, t, with length at most β i−1 and weight at most α i−1 · d(s, t). Also, the following is a path of 3 edges from a to b ′ in G ∪H 1/2 : Q = (a, p i (a)) • (p i (a), p i (b ′ )) • (p i (b ′ ), b ′ ) . Note that in both cases 2 and 3, since [a, b], [a ′ , b ′ ] are Bad, we know in particular that they cannot be just single edges with weight larger than d(x,y) h , because such edges are always labeled as Good. Therefore d(a, b) , d(a ′ , b ′ ) ≤ d(x,y) h , and we get, by Invariant 1, d(a, p i (a)) ≤ r i−1 · d(a, b) ≤ r i−1 h d(x, y), and(4)d(b ′ , p i (b ′ )) ≤ r i−1 · d(a ′ , b ′ ) ≤ r i−1 h d(x, y) .(5) Thus, in case 2, the path Q has weight d(a, p i (a)) + d(p i (a), p i (b ′ )) + d(b ′ , p i (b ′ )) ≤ d(a, p i (a)) + d(p i (a), a) + d(a, b ′ ) + d(b ′ , p i (b ′ )) + d(b ′ , p i (b ′ )) ≤ d(a, b ′ ) + 4 · r i−1 h d(x, y) . We concatenate this path with the paths P s,t , for children [s, t] that appear either before a or after b ′ . We obtain a path from x to y in G ∪H 1/2 , with length at most (2h − 1) · β i−1 + 3 and weight at most [s,t] α i−1 · d(s, t) + d(a, b ′ ) + 4 · r i−1 h d(x, y) ≤ α i−1 · d(x, y) + 4 · r i−1 h d(x, y) = (α i−1 + 4 · r i−1 h )d(x, y) . Summarizing cases 1 and 2, if [x, y] was labeled as Good, we have d (β i ) G∪H 1/2 (x, y) ≤ α i d(x, y), where α i = max{α i−1 , α i−1 + 4 · r i−1 h } = α i−1 + 4 · r i−1 h , and(6)β i = max{2h · β i−1 , (2h − 1) · β i−1 + 3} = 2h · β i−1 .(7) The last equality is true when β i−1 ≥ 3. This does not happen when i = 1; then β 0 = 1 and β 1 = max{2h · 1, (2h − 1) · 1 + 3} = 2h + 2. But then, since h > 1, for i > 1 we do have β i−1 ≥ 3 (note that β i is monotonically increasing). Hence, indeed β i = 2h · β i−1 for i > 1. Finally, in the third case, p i (a), p i (b ′ ) are not connected by an edge ofH 1/2 , and then by inequalities (4), (5) we conclude that d(p i (a), p i+1 (p i (a))) ≤ 2d(p i (a), p i (b ′ )) ≤ 2(d(p i (a), a) + d(a, b ′ ) + d(b ′ , p i (b ′ ))) ≤ 2(d(a, b ′ ) + 2 · r i−1 h d(x, y)) ≤ (2 + 4 · r i−1 h )d(x, y) . The vertex p i+1 (p i (a)) is in A i+1 . Hence, by the definition of p i+1 (x) as the closest A i+1 -vertex to x, we have y) . y) is symmetric. Therefore, we can choose any sequence d(x, p i+1 (x)) ≤ d(x, p i+1 (p i (a))) ≤ d(x, a) + d(a, p i (a)) + d(p i (a), p i+1 (p i (a))) ≤ d(x, y) + r i−1 h d(x, y) + (2 + 4 · r i−1 h )d(x, y) = (3 + 5 · r i−1 h )d(x,The proof of d(y, p i+1 (y)) ≤ (3+5· r i−1 h )d(x,{r i } such that r i ≥ 3 + 5 · r i−1 h(8) holds, and item 2 of Invariant 1 will be satisfied. Assume that h > 5 (this will be ensured in the sequel). We prove by induction that for the sequences choices r i = 3h h−5 , α i = 1 + 72i h and β i = 2(2h) i , inequalities (6), (7), (8) hold (note that the sequence {r i } is independent of i). For i = 0, we have r 0 = 3h h−5 = 3 + 15 h−5 > 2, α 0 = 1 and β 0 = 2 > 1. Thus, as we already saw, Invariant 1 holds for level 0. For i > 0, 3 + 5 · r i−1 h = 3 + 5 h · 3h h − 5 = 3 + 15 h − 5 = r i . Hence, inequality (8) is satisfied. Also, α i−1 + 4 · r i−1 h = 1 + 72(i − 1) h + 12 h − 5 = 1 + 72 h (i − 1 + 1 6 · h h − 5 ) = 1 + 72 h (i − 1 + 1 6 (1 + 5 h − 5 )) ≤ 1 + 72 h (i − 1 + 1 6 (1 + 5)) = 1 + 72i h = α i . This proves inequality (6). Finally, note that β 1 = 2h + 2 ≤ 2 · (2h) 1 , and for i ≥ 2, we have 2h · β i−1 = 2h · 2(2h) i−1 = 2(2h) i = β i . Therefore, inequality (7) is satisfied. We conclude that Invariant 1 holds for every level i. In particular, the invariant holds for the root [u, v]. Since A l = ∅ and p l (u) is not defined, item 2 from the invariant cannot be true for this node. Therefore, item 1 holds, i.e., d (β l−1 ) G∪H 1/2 (u, v) ≤ α l−1 · d(u, v) . Choose h = ⌈ C·l ϵ ⌉, where C is a constant such that C ≥ 72 and ϵ < C·l 5 . There is such constant C since we assumed that ϵ = O(log k) = O(l). Then, h ≥ C·l ϵ > 5 and we get that our hopset has stretch α l−1 = 1 + 72(l − 1) h ≤ 1 + 72(l − 1)ϵ C · l ≤ 1 + (l − 1)ϵ l < 1 + ϵ , and hopbound β l−1 = 2(2h) l−1 = O l ϵ l−1 . The proof of the following theorem is based on ideas from [23]. Proof. We use a similar proof technique as the one of Theorem 2. We fix an integer h > 5 and two vertices u, v in G. We again consider the partition tree of the u − v shortest path. We will assign a label Good or Bad for each node in the tree, and we will define the parameters {α i , β i , r i } such that the same invariant as before is satisfied: if [x, y] is a Good node in the i-th level, then d (β i ) G∪H 1/2 (x, y) ≤ α i · d(x, y) , and if [x, y] is a Bad node in the i-th level, then d(x, p i+1 (x)), d(y, p i+1 (y)) ≤ r i · d(x, y) . Recall that d(·, ·) denotes the distance in G. The definition of Good and Bad nodes in the 0-th level is the same as before: the leaf [x, y] is Good if and only if (x, y) is either an original edge of G or an edge ofH 1/2 . Then we know that if α 0 , β 0 ≥ 1, r 0 ≥ 2, the invariant is satisfied for i = 0. For [x, y] in the i-th level, we define the labels Good and Bad in a slightly different way. We again consider three possible cases: In the first case, similarly to the proof of Theorem 2, we have d y). In the second case, note that the path (2h·β i−1 ) G∪H 1/2 (x, y) ≤ α i−1 · d(x,Q = (x, p i (x)) • (p i (x), p i (y)) • (p i (y), y) is a three-edges path in G ∪H 1/2 between x, y. For bounding its weight, consider some Bad child [a, b] of [x, y]. By the invariant, we know that d(a, p i (a)) ≤ r i−1 · d(a, b) ≤ r i−1 h d(x, y). The last inequality is true since [a, b] must be a sub-path of weight at most d(x,y) h (otherwise, by the partition tree definition, [a, b] must be a single edge in G, but such [a, b] is always Good ). As a result, d(x, p i (x)) ≤ d(x, p i (a)) ≤ d(x, a) + d(a, p i (a)) ≤ d(x, a) + r i−1 h d(x, y) .(9) Similarly, d(y, p i (y)) ≤ d(y, a) + r i−1 h d(x, y). Therefore, we have d(p i (x), p i (y)) ≤ d(p i (x), x) + d(x, y) + d(y, p i (y)) (9) ≤ d(x, a) + r i−1 h · d(x, y) + d(x, y) + d(y, a) + r i−1 h · d(x, y) = 2d(x, y) + 2r i−1 h d(x, y) = (2 + 2r i−1 h )d(x, y) . We conclude that in the second case we have d (3) G∪H 1/2 (x, y) ≤ w(Q) = d(x, p i (x)) + d(p i (x), p i (y)) + d(p i (y), y) ≤ d(x, p i (x)) + (2 + 2r i−1 h )d(x, y) + d(p i (y), y) (9) ≤ d(x, a) + d(a, y) + 2r i−1 h · d(x, y) + (2 + 2r i−1 h )d(x, y) = (3 + 4r i−1 h )d(x, y) . Hence, by the analysis of the first two cases, we conclude that we can choose α i ≥ max{α i−1 , 3 + 4r i−1 h }, β i ≥ max{2h · β i−1 , 3} , and the invariant will hold. In the third case, p i (x), p i (y) are not connected by an edge ofH 1/2 . By the definition of half-bunches, it means that d(p i (x), p i+1 (p i (x))) ≤ 2d(p i (x), p i (y)) (9) ≤ (4 + 4r i−1 h )d(x, y) ,(10) and similarly, d(p i (y), p i+1 (p i (y))) ≤ (4 + 4r i−1 h )d(x, y). We conclude that in the third case, d(x, p i+1 (x)) ≤ d(x, p i+1 (p i (x))) ≤ d(x, p i (x)) + d(p i (x), p i+1 (p i (x))) (9),(10) ≤ d(x, y) + r i−1 h d(x, y) + (4 + 4r i−1 h )d(x, y) ≤ (5 + 5r i−1 h )d(x, y) . Note that in the third case, there exists a child [a, b] of [x, y] labeled as Bad, and thus, Inequality (9) is applicable. Hence, the invariant is satisfied if we choose {α i , β i , r i } such that α i ≥ max{α i−1 , 3 + 4r i−1 h }, β i ≥ max{2h · β i−1 , 3}, r i ≥ 5 + 5r i−1 h ,(11) and also α 0 , β 0 ≥ 1, r 0 ≥ 2. Consider the choices α i = 3 + 20 h−5 , r i = 5h h−5 = 5 + 25 h−5 (for every i) and β i = (2h) i . Note that we have α 0 > 3 > 1, β 0 = 1 and r 0 > 5 > 2 (recall that h > 5). In addition, for all i > 0, 5 + 5r i−1 h = 5 + 5 · 5h h−5 h = 5 + 25 h − 5 = r i , max{α i−1 , 3 + 4r i−1 h } = max{3 + 20 h − 5 , 3 + 20 h − 5 } = 3 + 20 h − 5 = α i , max{2h · β i−1 , 3} = max{(2h) i , 3} = (2h) i = β i . This proves that Condition (11) is satisfied, and therefore the invariant is satisfied. Note that the root [u, v] of the partition tree is in level l − 1. As p l (u), p l (v) are undefined, the condition for [x, y] being Bad never holds. Thus, [u, v] is labeled Good. Hence, we have d (β l−1 ) G∪H 1/2 (u, v) ≤ α l−1 · d(u, v) . Choosing h = 5 + 20 ϵ , we get a stretch of α l−1 = 3 + 20 h−5 ≤ 3 + ϵ and hopbound of β l−1 = (2h) l−1 ≤ (12 + 40 ϵ ) l−1 . Appendix B In this appendix we prove Lemmas 2 and 3 from Section 3.3. Lemma (Lemma 2). For every i = 0, 1, ..., l − 1, |Branch(H 1/2 i )| ≤ 4 u∈A i |B i (u)| 3 . Proof. Let (a, b, x) ∈ Branch(H 1/2 i ) be a branching event, and let a = (u, v), b = (y, z) such that v ∈ B 1/2 i (u) and z ∈ B 1/2 i (y). If d(u, v) ≤ d(y, z), we get d(y, u) ≤ d(y, x) + d(x, u) ≤ d(y, z) + d(u, v) ≤ 2d(y, z) < 2 · 1 2 d(y, p i+1 (y)) = d(y, p i+1 (y)) . See Figure 2 for illustration. Therefore u ∈ B i (y). Similarly, v ∈ B i (y) and also, of course, z ∈ B 1/2 i (y) ⊆ B i (y). If we instead have d(y, z) ≤ d(u, v) , then symmetrically we get y, z, v ∈ B i (u). We proved that Branch(H When we assume that d(u, v) ≤ d(y, z), it follows that the weight of the sub-paths from each of y, z, u, v to x is at most d(y, z) ≤ 1 2 d(y, p i+1 (y)). 1/2 i ) is a subset of the union (u, v), (y, z), x ∈ H 1/2 i × H 1/2 i × V v, y, z ∈ B i (u) and x ∈ P u,v ∩ P y,z ∪ ∪ (u, v), (y, z), x ∈ H 1/2 i × H 1/2 i × V u, v, z ∈ B i (y) and x ∈ P u,v ∩ P y,z . For the first set in this union, its size can be bounded by u∈A i v,y,z∈B i (u) |{x ∈ V | ((u, v), (y, z), x) ∈ Branch(H 1/2 i )}| ≤ ≤ u∈A i v,y,z∈B i (u) 2 = 2 u∈A i |B i (u)| 3 . Here the first inequality follows from the fact that the two shortest paths P u,v , P y,z , that were chosen in a consistent manner, can have at most two branching events between them (see Lemma 7.5 in [19]). We also used the fact that since (u, v), (y, z) ∈ H 1/2 i , then u (as well as v, y, z) must be in A i . Clearly, we can get the same bound for the size of the second set. Hence we conclude that |Branch(H 1/2 i )| ≤ 4 u∈A i |B i (u)| 3 . Next, we prove Lemma 3. Proof of Lemma 3. For proving inequalities (1) and (3), fix i < l − 1. We define a random variable X(v) for every v ∈ V , that equals |B i (v)| if v ∈ A i , and otherwise equals 0. By the definition of H i , we can see that |H i | ≤ v∈A i |B i (v)| = v∈V X(v) . Now fix some v ∈ V and a subset S ⊆ V such that v ∈ S, and suppose that A i = S. We order the vertices of A i by an increasing order of their distances from v. Then B i (v) is a subset of the vertices that come before the first A i+1 -vertex in that list. Every vertex in this list is contained in A i+1 with probability q i . Therefore |B i (v)| ≤ |B i (v)| + 1 is bounded by a geometric random variable with probability q i , when it is conditioned on the event that A i = S. By the law of total expectation, E[X(v) | v ∈ A i ] = S⊆V v∈S E[X(v) | v ∈ A i and A i = S] · Pr[A i = S | v ∈ A i ] = S⊆V v∈S E[|B i (v)| | A i = S] · Pr[A i = S | v ∈ A i ] ≤ S⊆V v∈S 1 q i · Pr[A i = S | v ∈ A i ] = 1 q i . Using the law of total expectation again, we get E[X(v)] = E[X(v) | v ∈ A i ] · Pr[v ∈ A i ] + 0 · Pr[v / ∈ A i ] ≤ 1 q i · Pr[v ∈ A i ] . By the definition of A i , the probability of v to be in A i is the product i−1 j=0 q j . We finally get that E[|H i |] ≤ v∈V E[X(v)] ≤ v∈V 1 q i · i−1 j=0 q j = n q i · i−1 j=0 q j . This proves inequality (1). For inequality (3), we use the fact that for a geometric random variable X with probability p, E[X 3 ] = p 2 − 6p + 6 p 3 ≤ 6 p 3 . Similarly to the computation above, we get E[X(v) 3 | v ∈ A i ] = S⊆V v∈S E[X(v) 3 | v ∈ A i and A i = S] · Pr[A i = S | v ∈ A i ] = S⊆V v∈S E[|B i (v)| 3 | A i = S] · Pr[A i = S | v ∈ A i ] ≤ S⊆V v∈S 6 q 3 i · Pr[A i = S | v ∈ A i ] = 6 q 3 i . Now, by Lemma 2, E[|Branch(H 1/2 i )|] ≤ 4E[ v∈A i |B i (v)| 3 ] = 4E[ v∈V X(v) 3 ] = 4 v∈V E[X(v) 3 ] = 4 v∈V E[X(v) 3 | v ∈ A i ] · Pr[v ∈ A i ] ≤ 4 v∈V 6 q 3 i · Pr[v ∈ A i ] = 24 · n q 3 i · i−1 j=0 q j , which proves inequality (3). For proving inequalities (2) and (4), note that since A l = ∅, thenB l−1 (v) = A l−1 , for any v ∈ A l−1 . Also, notice that |A l−1 | is a binomial random variable with parameter n and probability q = l−2 j=0 q j . For a binomial random variable X ∼ Bin(n, q), E[X 2 ] = nq(1 − q) + (nq) 2 ≤ nq + (nq) 2 ≤ 2 max{1, (nq) 2 } , since either nq ≥ 1, and then nq + (nq) 2 ≤ (nq) 2 + (nq) 2 = 2(nq) 2 , or nq < 1, and then nq + (nq) 2 < 1 + 1 = 2. By definition, |H l−1 | ≤ v∈A l−1 |A l−1 | = |A l−1 | 2 , and therefore, E[|H l−1 |] ≤ E[|A l−1 | 2 ] ≤ 2 max{1, (n l−2 j=0 q j ) 2 } . This proves inequality (2). Another property of a variable X ∼ Bin(n, q) is the following. ≤ nq + 7(nq) 2 + 6(nq) 3 + (nq) 4 ≤ 15 max{1, (nq) 4 } , where the reasoning for the last step is similar to the above. Then, by Lemma 2, |Branch(H l−1 )| ≤ 4 v∈A l−1 |B l−1 (v)| 3 = 4 v∈A l−1 |A l−1 | 3 = 4|A l−1 | 4 . Finally, E[|Branch(H l−1 )|] ≤ 4E[|A l−1 | 4 ] ≤ 4 · 15 max{1, (nq) 4 } = 60 max{1, (n l−2 j=0 q j ) 4 } . This proves inequality (4). Appendix C We bring here the proof of Theorem 4. Theorem (Theorem 4). Given an undirected weighted graph G = (V, E), an integer k ∈ [3, log 2 n], a positive parameter ϵ ≤ 1, and a set of pairs P ⊆ V 2 , G has an interactive (1 + ϵ)-distance preserver with query time O(1) and size O(|P| ·γ 2 (ϵ, k) + n · (log k + log log( 1 ϵ )) + n 1+ 1 k ) , whereγ 2 (ϵ, k) = O( log k ϵ ) (1+o(1)) log 2 k+log 4/3 log( 1 ϵ ) . Proof. For short, we denote γ 4/3 = γ 4/3 (ϵ, k). This time, we choose a different hierarchy of sets than in the proof of Theorem 3, by choosing q i =    1 2 n − (4/3) i 3k i < h 1 2 ( n γ 4/3 ) − 2 i−h k i ≥ h , where l, h will be chosen later. We use Theorem 2, but with an approximation parameter ϵ ′ = ϵ 3 instead of ϵ, to conclude that H 1/2 is a (1 + ϵ 3 , β l )-hopset (β l is still O( 3l ϵ ) l−1 = O( l ϵ ) l−1 ) . We divide this hopset into three disjoint sets. Q 1 = l−1 i=0 {(v, p i (v)) | v ∈ V } , Q 2 = h−1 i=0 H 1/2 i , Q 3 = l−1 i=h H 1/2 i . By the size analysis below, we note that using full bunches instead of half-bunches in the definition of Q 3 also achieves the same results, even with slightly better constant factors hidden by the O-notations. However, for convenience, and since asymptotically the results are similar, we stick with half-bunches. For every i = 0, 1, ..., l − 1, let (S 1 i , D 1 i ) be the interactive distance preserver from Lemma 1 on the set {(v, p i (v)) | v ∈ V }. Let (S 2 i , D 2 i ) be the interactive distance preserver from Theorem 1 on the set H 1/2 i . Lastly, let (S 3 , D 3 ) be the interactive distance preserver from Theorem 3 on the set Q 3 , but with an approximation parameter of ϵ ′ = ϵ 3 instead of ϵ. We now build an interactive distance preserver (S, D), where the oracle D stores the following information. 1. For every (x, y) ∈ P, the oracle D stores a path P x,y ⊆ G ∪H 1/2 between x, y with length at most β l and weight at most (1 + ϵ 3 )d(x, y). Next, we describe the algorithm for answering queries. Given a query (x, y) ∈ P, find the stored path P x,y . Now replace each hop-edge e = (a, b) ∈H 1/2 that appears in P x,y ; if f e = 1, replace e with the path returned from D 1 i (a, b); if f e = 2, replace e with the path returned from D 2 i (a, b); if f e = 3, use D 3 (a, b). Return the resulting path as an output. For every Since the query time of D 1 i , D 2 i , D 3 is linear in the returned path, for every i, we conclude that the total query time of D is also linear in the size of the output. Thus, the query time of this oracle is O(1). Also, note that the paths that are returned from the oracles D 1 i , D 2 i , D 3 have a weight of at most (1 + ϵ 3 ) times the hop-edges they replace (By Lemma 1 and Theorems 1 and 3). Therefore the resulting path has a weight of at most (1 + ϵ 3 ) 2 d(x, y) ≤ (1 + ϵ)d(x, y) , (since ϵ ≤ 1). Hence, the stretch is indeed 1 + ϵ. We now analyse the size of D. The paths P x,y from item 1 of the description of D have a total size of O(|P|β l ). The total size of the oracles D 1 i , D 2 i from items 2, 3 can be bounded by O(n · l + n 1+ 1 k ), as in the proof of Theorem 3. The flags and the indices from item 5 require O(|H 1/2 |) space, which is O(|H 1/2 |) = O(|Q 1 |) + O(|Q 2 |) + O(|Q 3 |) ≤ O(n · l) + O( h−1 i=0 |D 2 i |) + O(|D 3 |) = O(n · l + n 1+ 1 k + |D 3 |) . For estimating the expected size of the oracle D 3 , we need to compute the size of the set of pairs Q 3 . By Lemma 3, for i ∈ [h, l − 1) (using inequality (1)), we have |H i | ≤ n q i i−1 j=0 q j = n · 2( n γ 4/3 ) · 1 2 h n − (4/3) h −1 k · 1 2 i−h ( n γ 4/3 ) − 2 i−h −1 k = 1 2 i−1 n · n − (4/3) h −1 k · ( n γ 4/3 ) 1 k * ≤ 1 2 i−1 n · 1 γ 4/3 · ( n γ 4/3 ) 1 k = 1 2 i−1 ( n γ 4/3 ) 1+ 1 k . For inequality (*) to hold, we need to choose h so that n (4/3) h −1 k ≥ γ 4/3 , i.e., h ≥ log 4/3 ( k log γ 4/3 log n +1). Since k ≤ log 2 n, we can choose h = ⌈log 4/3 (log 2 γ 4/3 + 1)⌉ ≥ log 4/3 (log 2 γ 4/3 + 1) ≥ log 4/3 ( k log 2 γ 4/3 log 2 n + 1) . We substitute γ 4/3 for evaluating h (γ 4/3 = γ 4/3 (ϵ, k) is given by Equation (2)). h = ⌈log 4/3 (log 2 γ 4/3 + 1)⌉ = ⌈log 4/3 ⌈log 4/3 k⌉ log 2 (O( log k ϵ )) ⌉ = = log 4/3 log 4/3 k + log 4/3 log 2 ( 1 ϵ ) + O(log (3) k) . For i = l − 1, denote q = l−2 j=0 q j . By inequality (2), if nq < 1, then the expected value of |H l−1 | is bounded by a constant. Otherwise, |H l−1 | ≤ 2(n l−2 j=0 q j ) 2 = 2n 2 · ( 1 2 h n − (4/3) h −1 k · 1 2 l−1−h ( n γ 4/3 ) − 2 l−1−h −1 k ) 2 ≤ n 2 · 1 2 2l−3 ( 1 γ 4/3 · ( n γ 4/3 ) − 2 l−1−h −1 k ) 2 * * ≤ n 2 · 1 2 2l−3 ( 1 γ 4/3 · ( n γ 4/3 ) − k−1 k ) 2 = 1 2 2l−3 ( n γ 4/3 ) 2−2· k−1 k = 1 2 2l−3 ( n γ 4/3 ) 2 k ≤ 1 2 2l−3 ( n γ 4/3 ) 1+ 1 k . For inequality (**) to hold, we need to choose l so that 2 l−1−h ≥ k. The smallest l that satisfies this is l = ⌈log 2 k⌉ + 1 + h. Substituting h, we get l = ⌈log 2 k⌉ + 1 + h = log 2 k + log 4/3 log 4/3 k + log 4/3 log 2 ( 1 ϵ ) + o(log log k) = = log 2 k · (1 + o(1)) + log 4/3 log( 1 ϵ ) . Using the notation ofγ 2 (ϵ, k) (see Theorem 4 for its definition), we get β l = O( log k ϵ ) (1+o(1)) log 2 k+log 4/3 log( 1 ϵ ) =γ 2 (ϵ, k) . Summing up, the size of Q 3 = l−1 i=h H 1/2 i is l−1 i=h |H 1/2 i | ≤ l−1 i=h 1 2 i−1 ( n γ 4/3 ) 1+ 1 k = O(( n γ 4/3 ) 1+ 1 k ) , so the total size of our new preserver is O(|P|β l ) + O(n · l + n 1+ 1 k ) + O(( n γ 4/3 ) 1+ 1 k · γ 4/3 + n log k + n 1+ 1 k ) = = O(|P|β l + n · (log k + log log( 1 ϵ )) + n 1+ 1 k ) . To complete the proof, we define the distance preserver S to be the union of all the edges P x,y ∩ G, where (x, y) ∈ P, together with l−1 i=0 S 1 i ∪ h−1 i=0 S 2 i ∪ S 3 . Then, the output paths of D are always contained in S, and |S| = O(|D|). Hence, (S, D) is an interactive (1 + ϵ)-distance preserver with query time O(1) and size O(|P|γ 2 (ϵ, k) + n · (log k + log log( 1 ϵ )) + n 1+ 1 k ) . Appendix D Lemma 6 can be derived by a similar analysis as in the paper by Thorup and Zwick [56] and in the paper by Wulff-Nilsen [59]. However, since we apply their construction only for h < k of the levels, we bring here the full proof, for completeness. Lemma (Lemma 6). Let G = (V, E) be an undirected weighted graph with n vertices, and let 1 ≤ h < k ≤ log 2 n be two integer parameters. There is a set S ⊆ V of size O(n 1− h k ) and an oracle D with size O(h · n 1+ 1 k ), that acts as follows. Given a query (u, v) ∈ V 2 , the oracle D either returns a path between u, v with weight at most (2h + 1)d G (u, v), or returns two paths P u,u ′ , P v,v ′ , from u to some u ′ ∈ S and from v to some v ′ ∈ S, such that w(P u,u ′ ), w(P v,v ′ ) ≤ h · d G (u, v) . The query time of the oracle D is O(log h). In addition, there is a set E ′ ⊆ E such that the output paths that D returns are always contained in E ′ , and |E ′ | = O(h · n 1+ 1 k ) . Before presenting the proof, we introduce some notions that will help us to describe the components of the desired oracle. Given two vertices u, v ∈ V and an index i = 0, 1, ..., h, we define: 1. P (u, v, i) = a predicate that indicates whether d G (u, p i (u)) ≤ i · d G (u, v). When considering the sequence of distances d G (u, p 0 (u)), d G (u, p 1 (u)), ..., d G (u, p h (u)), this predicate indicates which elements are considered small. 2. Q(u, v, i) = a predicate that indicates whether (p i (u) ∈ B i (v)) ∨ (p i+1 (v) ∈ B i+1 (u)) (this will be used only for i < h). Since our oracle will store the shortest paths between vertices and the members of their bunches, and the shortest paths between vertices and their pivots, Q actually indicates whether we have enough information to output a path between u, v that passes through either p i (u) or p i+1 (v). 3. ∆ u (i) = d G (u, p i+2 (u)) − d G (u, p i (u) ). This represents the difference between two adjacent elements of even indices in the distance sequence that was mentioned above. This notation will be used only for i + 2 ≤ h. The key observation regarding these notions is provided in the following lemma. Lemma 8. For u, v ∈ V and i ∈ [0, h − 2], if Q(u, v, i) = F ALSE, then ∆ u (i) ≤ 2d G (u, v). Therefore, if P (u, v, i) = T RU E and also Q(u, v, i) = F ALSE, then P (u, v, i + 2) = T RU E. Proof. By the definitions of bunches, assuming Q(u, v, i) = F ALSE, we know that d G (v, p i+1 (v)) ≤ d G (v, p i (u)) ≤ d G (v, u) + d G (u, p i (u)) , and d G (u, p i+2 (u)) ≤ d G (u, p i+1 (v)) ≤ d G (u, v) + d G (v, p i+1 (v)) . Thus, d G (u, p i+2 (u)) ≤ d G (u, v) + d G (v, p i+1 (v)) ≤ d G (u, v) + d G (v, u) + d G (u, p i (u)) , that is, ∆ u (i) = d G (u, p i+2 (u)) − d G (u, p i (u)) ≤ 2d G (u, v). If we also have P (u, v, i) = T RU E, then we know that d G (u, p i (u)) ≤ i · d G (u, v), and so we have d G (u, p i+2 (u)) ≤ 2d G (u, v) + d G (u, p i (u)) ≤ 2d G (u, v) + i · d G (u, v) = (i + 2) · d G (u, v) , i.e., P (u, v, i + 2) = T RU E. In our oracle, we store the shortest paths from every x ∈ V to every y ∈ B i (x), and from every x ∈ V to each of its pivots p i (x), for i ∈ [0, h]. In the query algorithm, given a query (u, v) ∈ V 2 , we would like to find an index i ∈ [0, h) such that Q(u, v, i) = P (u, v, i) = T RU E. For such index i, we either return the shortest path from u to p i (u), concatenated with the shortest path from p i (u) to v (this is in the case where p i (u) ∈ B i (v)), or return the shortest path from v to p i+1 (v), concatenated with the shortest path from p i+1 (v) to u. In the first case, the weight of the returned path is d G (u, p i (u)) + d G (p i (u), v) ≤ 2d G (u, p i (u)) + d G (u, v) ≤ (2i + 1)d G (u, v) ≤ (2h + 1)d G (u, v) . (12) In the second case, since p i (u) / ∈ B i (v), we have d G (v, p i+1 (v)) ≤ d G (v, p i (u)), and thus d G (v, p i+1 (v)) + d G (p i+1 (v), u) ≤ 2d G (v, p i+1 (v)) + d G (u, v) ≤ 2d G (v, p i (u)) + d G (u, v) ≤ 2(d G (v, u) + d G (u, p i (u))) + d G (u, v) ≤ 2d G (u, p i (u)) + 3d G (u, v) ≤ (2i + 3)d G (u, v) ≤ (2(h − 1) + 3)d G (u, v) = (2h + 1)d G (u, v) . Hence, the weight of the returned path is d G (v, p i+1 (v)) + d G (p i+1 (v), u) ≤ (2h + 1)d G (u, v) .(13) In both cases we used the fact that P (u, v, i) = T RU E, and therefore d G (u, p i (u)) ≤ i · d G (u, v). For handling the case that there is no index i such that Q(u, v, i) = P (u, v, i) = T RU E, we have the following lemma. From now on, we assume for simplicity that h is even. Lemma 9. Let (u, v) ∈ V 2 , and let [i 1 , i 2 ] ⊆ [0, h − 2] such that i 1 , i 2 are even, and P (u, v, i 1 ) = T RU E. Then, either there is an even index i ∈ [i 1 , i 2 ] that satisfies Q(u, v, i) = P (u, v, i) = T RU E, or P (u, v, i 2 + 2) = T RU E. Proof. First, if none of the even indices i ∈ [i 1 , i 2 ] satisfy Q(u, v, i) = T RU E, then by Lemma 8 we know that ∆ u (i) ≤ 2d G (u, v) for every even i ∈ [i 1 , i 2 ]. Therefore, when summing ∆ u (i), for all even indices i ∈ [i 1 , i 2 ], we get d G (u, p i 2 +2 (u)) − d G (u, p i 1 (u)) = i ∆ u (i) ≤ i 2 − i 1 + 2 2 · 2d G (u, v) = (i 2 − i 1 + 2)d G (u, v) . Since we know that P (u, v, i 1 ) = T RU E, we conclude that d G (u, p i 2 +2 (u)) ≤ d G (u, p i 1 (u)) + (i 2 − i 1 + 2)d G (u, v) ≤ i 1 · d G (u, v) + (i 2 − i 1 + 2)d G (u, v) = = (i 2 + 2)d G (u, v) . That is, P (u, v, i 2 + 2) = T RU E. Suppose now that there is an even index i ∈ [i 1 , i 2 ] such that Q(u, v, i) = T RU E, and let i be the smallest such index. We now know that in the interval [i 1 , i − 2], we have P (u, v, i 1 ) = T RU E, and there are no indices j ∈ [i 1 , i − 2] with Q(u, v, j) = T RU E. By the first case that we proved above, it follows that P (u, v, i) = T RU E. Recall that by the choice of i, we also know that Q(u, v, i) = T RU E. Therefore i is the desired index. Suppose that there is no even index i ∈ [0, h − 2] such that Q(u, v, i) = P (u, v, i) = T RU E, and there is no even index i ∈ [0, h − 2] such that Q(v, u, i) = P (v, u, i) = T RU E (with the order of u, v switched). In that case, we can simply return the two shortest paths from u to p h (u) and from v to p h (v), because they both have a weight of at most h · d G (u, v), as desired. This is true by Lemma 9, using the the fact that for the index 0, P (u, v, 0) = P (v, u, 0) = T RU E always holds. To find an even index i ∈ [0, h − 2] such that Q(u, v, i) = P (u, v, i) = T RU E, or determine that there is no such index, we perform a sort of binary search. The details of this binary search are described in the proof of the main lemma below. For the sake of this search, we use a designated data structure for every u ∈ V . We now describe this data structure. Definition 7. Given a vertex u ∈ V , we define the max-in-range tree of u to be the following binary tree. Each node in the binary tree represents a sub-interval Note that mid splits the interval [i 1 , i 2 ] into two almost equal parts. Thus the length of an interval in some node is at most half of the interval in the parent of this node. Hence, the depth of the max-in-range tree of every u ∈ V is at most log h, and it contains O(h) nodes. Note also that this tree can be pre-computed, when constructing the oracle. The total size of these trees is O(n · h), and this term is dominated by the oracle's size O(h · n 1+ 1 k ). Proof of Lemma 6. Given k and the n-vertex graph G = (V, E), we start by creating a hierarchy of sets with q i = n − 1 k (see Section 3.3). By the definition of the sets A i , in expectation, |A h | = n h−1 j=0 q j = n · n − h k = n 1− h k . We choose A h to be the desired set S ⊆ V . For every vertex u ∈ V we define the cluster of u as C(u) = {v ∈ V | u ∈ h i=0 B i (v)} . Given v ∈ C(u), denote by P v,u the shortest path from v to u in G. Let q u (v) be a pointer from v to the next vertex after v in P v,u . For every i = 0, 1, ..., h, let (S 1 i , D 1 i ) be the interactive 1-distance preserver from Lemma 1, for the set {(v, p i (v)) | v ∈ V }. Our oracle D stores the following information. Given a query (u, v) ∈ V 2 , we perform a binary search to find an index i such that P (u, v, i) = T RU E and either Q(u, v, i) = T RU E or i = h. This binary search is performed as follows. Suppose that the current search range is [i 1 , i 2 ], and that it appears in the max-in-range tree T (u) ( Finally, when i 1 = i 2 = i, we check whether P (u, v, i) = Q(u, v, i) = T RU E. If this condition does not hold, we say that this binary search failed, and we perform a symmetric binary search for the pair (v, u) -i.e., where the roles of u and v are switched. If this binary search also fails, we return the two shortest paths from u to p h (u) and from v to p h (v), using the oracle D 1 h . Note that indeed p h (u), p h (v) ∈ A h = S. Suppose, however, that one of the two binary searches succeeded. Without loss of generality, assume that P (u, v, i) = Q(u, v, i) = T RU E for the last index i that was considered in the binary search of (u, v). Then in particular p i (u) ∈ B i (v) ∨ p i+1 (v) ∈ B i+1 (u) . We consider two cases. • If p i (u) ∈ B i (v), we use D 1 h to find the shortest path from u to p i (u), and we use the pointers q p i (u) (·), starting at v, to find the shortest path P p i (u),v . The output path is the concatenation of these two paths. • Otherwise, if p i (u) / ∈ B i (v) and p i+1 (v) ∈ B i+1 (u) , we symmetrically output a path that consists of the concatenation between the shortest paths from v to p i+1 (v) and from p i+1 (v) to u. The latter is constructed via the pointers q p i+1 (v) (·), starting at u. If instead the binary search of (v, u) is the one that succeeds, we symmetrically return a path between v and u, that passes through either p i (v) or p i+1 (u). For the query algorithm to be well-defined, first notice that if p i (u) ∈ B i (v), then by definition v ∈ C(p i (u)). Similarly, if p i+1 (v) ∈ B i+1 (u), we have u ∈ C(p i+1 (v)). We claim that for every a, b ∈ V such that b ∈ C(a), we can indeed use only the pointers q a (·), starting at b, to construct the shortest path from a to b. For this purpose, it suffices to prove that x ∈ C(a) for every x on this shortest path, and thus q a (x) is the next vertex after x on this shortest path from b to a. Since b ∈ C(a), we know that there is some i ∈ [0, h] such that a ∈ B i (b), i.e., a ∈ A i and d G (b, a) < d G (b, p i+1 (b)). By the fact that x is on the shortest path between a, b, we can write d G (x, a) = d G (b, a) − d G (b, x) < d G (b, p i+1 (b)) − d G (b, x) ≤ d G (b, p i+1 (x)) − d G (b, x) ≤ d G (b, x) + d G (x, p i+1 (x)) − d G (b, x) = d G (x, p i+1 (x)) . Hence, by definition, a ∈ B i (x), for the same index i ∈ [0, h]. Therefore x ∈ C(a), as desired. For the correctness of the binary search of (u, v), note the following invariant: if the current search range is [i 1 , i 2 ], then P (u, v, i 1 ) = T RU E, and either Q(u, v, i 2 ) = T RU E or i 2 = h. This invariant trivially holds for the first search range [0, h], since d G (u, p 0 (u)) = d G (u, u) = 0 ≤ 0 · d G (u, v) . For smaller search ranges, consider two cases. If the current search range is [i 1 , j], which is the left child of [i 1 , i 2 ] (that satisfies the invariant), then we already have P (u, v, i 1 ) = T RU E. Recall also that the reason of searching in the left child [i 1 , j] is that Q(u, v, j) = T RU E. Hence, in particular, the invariant indeed holds. If the current search range is [mid, i 2 ], which is the right child of [i 1 , i 2 ] (that satisfies the invariant), then it means that for the even index j ∈ [i 1 , mid − 2] that maximizes ∆ u (j), we have Q(u, v, j) = F ALSE. But then, by Lemma 8, we know that ∆ u (j) ≤ 2d G (u, v), and therefore ∆ u (i) ≤ 2d G (u, v) for all i ∈ [i 1 , mid − 2]. Hence, we get d G (u, p mid (u)) = d G (u, p i 1 (u)) + i ∆ u (i) ≤ d G (u, p i 1 (u)) + i 2d G (u, v) ≤ d G (u, p i 1 (u)) + mid − i 1 2 · 2d G (u, v) ≤ d G (u, p i 1 (u)) + (mid − i 1 )d G (u, v) ≤ i 1 · d G (u, v) + (mid − i 1 )d G (u, v) = mid · d G (u, v) . The sum goes over only even indices in [i 1 , mid − 2], and we used the fact that P (u, v, i 1 ) = T RU E. We conclude that here also P (u, v, mid) = T RU E, and also Q(u, v, i 2 ) = T RU E or i 2 = h, since this is a part of the assumption on [i 1 , i 2 ]. As a result, at the end of the algorithm, when we get to a search range of the form [i, i], we either have Q(u, v, i) = T RU E or i = h. If Q(u, v, i) = T RU E, we return a (2h + 1)-stretch path between u, v (See Inequalities (12) and (13)). Otherwise, we conclude that P (u, v, h) = T RU E, and thus the shortest path from u to p h (u) is of weight at most h · d G (u, v). Symmetrically, if the second binary search succeeds, then we return a (2h + 1)-stretch path between u, v, and otherwise we have d G (v, p h (v)) ≤ h · d G (u, v). If both binary searches fail, it implies that the two paths we return are of weight at most h · d G (u, v). Next, we analyse the query time, size and stretch of our oracle D. Its query time consists of performing the two binary searches, then applying D 1 i once or twice for some i ∈ [0, h], and also using the pointers q u (·). The binary searches require O(log h) time (this is the depth of the maxin-range trees T (u), T (v)), while querying D 1 i and using the pointers q u (·) is done within a linear time in the length of the resulting paths. Hence, we get a total query time of O(log h). Regarding the size of D, notice that each oracle D 1 i , for every i = 0, 1, ..., h is of size O(n), therefore storing them requires O(h · n) space. The collection of pointers q u (v), for every u, v ∈ V such that v ∈ C(u), is of expected size u∈V v∈C(u) 1 = v∈V u|v∈C(u) 1 = v∈V u∈ B i (v) 1 ≤ v∈V h i=0 |B i (v)| = h i=0 v∈V |B i (v)| ≤ h i=0 n q i i−1 j=0 q j = h i=0 n 1− i−1 k ≤ (h + 1)n 1+ 1 k , where we used the definition of C(u) and a bound on v∈V |B i (v)| which is achieved similarly to the proof of Lemma 3 in Appendix B. Note that storing p i (v), B i (v) for every v ∈ V and i = 0, 1, ..., h is exactly like storingH i for every i = 0, 1, ..., h. By Lemma 3, the expected total sizes of these sets is h i=0 n q i i−1 j=0 q j = h i=0 n 1− i−1 k ≤ (h + 1)n 1+ 1 k . Lastly, we note that for every u ∈ V , the max-in-range tree T (u) has O(h) nodes, each of them stores one index j. Hence, the space needed to store T (u) for every u ∈ V is O(h · n). We therefore conclude that the total size of D is O(h · n 1+ 1 k ). For the stretch of the oracle D, recall that in the discussions after Lemma 8 and Lemma 9, we already saw that our algorithm either returns a path of weight (2h + 1)d G (u, v) between u, v, or two paths from u to p h (u) and from v to p h (v), with weight at most h · d G (u, v). Finally, notice that the returned paths by the oracle D are contained in the union Z 1 ∪ Z 2 , where Z 1 = h i=0 S 1 i and Z 2 = u∈V v∈C(u) P u,v , and P u,v is the shortest path between u and v. By Lemma 1, we know that each S 1 i is of size O(n), thus |Z 1 | = O(h · n). Regarding Z 2 , notice that each edge of this set is of the from (b, q a (b)) for some a, b ∈ V such that b ∈ C(a). Thus, the total size of this set is bounded by the number of the pointers q u (·), which we already proved is O(h · n 1+ 1 k ). We conclude that there is a set E ′ ⊆ E such that all of the returned paths by the oracle D are in E ′ , and |E ′ | = O(h · n 1+ 1 k ) . The weight of an edge (C, C ′ ) ∈ E H is defined to be w(e(C, C ′ )). We now apply A on the graph H, to get an interactive α-spanner (S, D) with query time q and size h( n t ). We define our oracle for the new interactive spanner to contain the following. 1. The oracle D. 2. For every v ∈ V , its cluster C = C(v), a pointer p(v) to its parent in the tree T C (for r C , we set p(r C ) = r C ) and the number of edges in the path T C [v, r C ]. 3. e(C, C ′ ), for every edge (C, C ′ ) ∈ S. The size of this oracle is then h( n t ) + |V | · O(1) + |S| · O(1) = O(h( n t ) + n). Given a query (u, v) ∈ V 2 , our oracle uses D to find a path Q ⊆ S between C(u) and C(v). Let Q = (C(u) = C 1 , C 2 , ..., C l = C(v)), and for every i ∈ [0, l − 1], denote (x i , y i ) = e(C i , C i+1 ). Also, let y 0 = u and x l = v. Note that for every i = 1, ..., l, the vertices x i , y i−1 are in the same cluster C i , and denote P i = T C i [y i−1 , x i ]. We later show how P i can be found, using the stored data in our oracle. The returned path for the query u, v is then P 1 • (x 1 , y 1 ) • P 2 • (x 2 , y 2 ) • · · · • (x l−1 , y l−1 ) • P l . See Figure 3 for an illustration. Figure 3: The returned path. The dashed lines are the paths P i . Notice that they do not always pass through the root of the respective cluster C i . Next, we describe in more details how to find the paths {P i }. Let x, y be vertices in some cluster C. We want to find the unique path between x, y in T C , i.e., T C [x, y], in time that is proportional to length of this path. If x = y, then the desired path is empty. Otherwise, if |T C [x, r C ]| > |T C [y, r C ]| (the number of edges in T C [x, r C ] is larger than the number of edges in T C [y, r C ]), find T C [p(x), y] recursively and return (x, p(x)) • T C [p(x), y]. Symmetrically, if |T C [y, r C ]| ≥ |T C [x, r C ]|, return (y, p(y)) • T C [p(y), x]. The correctness of this algorithm follows from the fact that if, for example, |T C [y, r C ]| ≥ |T C [x, r C ]|, and x ̸ = y, then x cannot be in the sub-tree of y in T C , hence the path T C [x, y] must pass through p(y). In each recursive call we reduce the sum |T C [x, r C ]| + |T C [y, r C ]| by 1, and therefore the algorithm ends either when x = y = lca(x, y), or when |T C [x, r C ]| = |T C [y, r C ]| = 0 -which also means x = y = r C = lca(x, y). Therefore the time needed to find T C [x, y] is O(1) times the total length of T C [x, lca(x, y)] and T C [y, lca(x, y)], which is O(|T C [x, lca(x, y)] • T C [lca(x, y), y]|) = O(|T C [x, y]|). The running time of the whole query algorithm is proportional to the number of edges in the returned path, plus the query time of D, which is O(q). We now analyse the stretch. For every i, the path P i consists of two paths, P 1 i = T C i [y i−1 , lca(x i , y i−1 )] and P 2 i = T C i [lca(x i , y i−1 ), x i ] . Since these are sub-paths of the paths from y i−1 , x i to the root, both paths have at most O(t) edges. By the stretch-friendly property of C, we know that the weight of each edge in P 1 i is at most w(x i−1 , y i−1 ) (for i > 1), and that the weight of each edge in P 2 i is at most w(x i , y i ) (for i < l). Therefore, the weight of the returned path is at most l i=1 w(P i ) + l−1 i=1 w(x i , y i ) = w(P 1 1 ) + l−1 i=1 w(P 2 i ) + w(x i , y i ) + w(P 1 i+1 ) + w(P 2 l ) ≤ w(P 1 1 ) + l−1 i=1 O(t)w(x i , y i ) + w(P 2 l ) = w(P 1 1 ) + O(t) · w(Q) + w(P 2 l ) , where Q is the returned path from the oracle D in the graph H. Now let P u,v be the shortest path in G between u, v. Let C ′ 1 , C ′ 2 , ..., C ′ f be the list of the clusters of C that P u,v passes through, in the same order that P u,v passes through them. In case it passes through some C ′ j more than once, C ′ j will appear multiple times. Then,P = (C ′ 1 , ..., C ′ f ) is a path in H between C(u) and C(v), and since the stretch of D is α, we get that w(Q) ≤ α · d H (C(u), C(v)) ≤ α · w(P ) . But notice that for each j = 1, ..., f − 1, P u,v contains an edge e j between the clusters C ′ j and C ′ j+1 , and since in H we defined w(C ′ j , C ′ j+1 ) to be the weight of the minimal edge between these clusters, we have w(Q) ≤ α · w(P ) = α f −1 j=1 w(C ′ j , C ′ j+1 ) ≤ α f −1 j=1 w(e j ) ≤ α · w(P u,v ) . It is left to bound the terms w(P 1 1 ), w(P 2 l ). We show that w(P 1 1 ) ≤ O(t) · w(P u,v ), and the proof that w(P 2 l ) ≤ O(t) · w(P u,v ) is symmetric. Recall that Q = (C 1 , ..., C l ) is the path returned by D for the query (C(u), C(v)). If the entire shortest path P u,v between u, v in G is contained in the cluster C(u), then in particular C(u) = C(v) = C 1 . Thus, Q is of length 0, and the returned path from our oracle is just T C 1 [u, v], which is the concatenation of two paths with O(t) edges. By the stretch-friendly property, for every edge (x, y) ∈ P u,v , the edges of T C 1 [x, y] have weight at most w(x, y) ≤ w(P u,v ). Notice that the union (x,y)∈Pu,v T C 1 [x, y] is a (not necessarily simple) path in T C 1 between u and v, and so it contains T C 1 [u, v]. Therefore each one of the O(t) edges in T C 1 [u, v] has weight of at most w(P u,v ). Thus, the returned path has weight of at most O(t) · w(P u,v ). Otherwise, let v ′ be the first vertex in P u,v which is not in C 1 , and let u ′ be the vertex before v ′ in P u,v . Similarly to the first case, we can prove that w(T C 1 [u, u ′ ]) ≤ O(t) · w(P u,v ). In addition, again by the stretch-friendly property, each of the O(t) edges in T C 1 [r C 1 , u ′ ] has weight of at most w(u ′ , v ′ ) ≤ w(P u,v ). Since P 1 1 ⊆ T C 1 [u, r C 1 ] ⊆ T C 1 [u, u ′ ] ∪ T C 1 [r C 1 , u ′ ], we get w(P 1 1 ) ≤ w(T C 1 [u, u ′ ]) + w(T C 1 [r C 1 , u ′ ]) ≤ O(t) · w(P u,v ) + O(t) · w(P u,v ) = O(t) · w(P u,v ) . We proved that the weight of the returned path is at most w(P 1 1 ) + w(P 2 l ) + O(t) · w(Q) ≤ O(t) · w(P u,v ) + O(t) · w(P u,v ) + O(t)α · w(P u,v ) = O(α · t) · w(P u,v ) . Thus the stretch of our new interactive spanner is O(α · t). For completing the proof, notice that the output paths from our oracle are always a concatenation of paths T C [x, y] for some C ∈ C and x, y ∈ C, and edges e(C, C ′ ), where (C, C ′ ) ∈ S. These paths are always contained is the set C∈C T C ∪ {e(C, C ′ ) | (C, C ′ ) ∈ S}. The set C∈C T C is a forest, and thus has size at most n − 1. The set {e(C, C ′ ) | (C, C ′ ) ∈ S} is of size |S| ≤ h( n t ). Therefore, the size of this entire set is also O(h( n t ) + n). To prove Theorem 10 (i.e., to build ultra-sparse interactive spanners), we slightly change the construction from the proof of Theorem 8. the following notations regarding graph covers. Given an undirected weighted graph G = (V, E), a cover of G is a collection S = {S 1 , S 2 , ..., S t } of subsets of V (called clusters), such that V = t i=1 S i . Note that the clusters are not necessarily disjoint. Given a cluster S ∈ S, let G[S] be the induced sub-graph of G, by the vertices of S. We define the radius of S, denoted by Rad(S), to be the minimal number r ≥ 0 such that in every S ∈ S, there is a vertex v S ∈ S that satisfies d G[S] (v S , u) ≤ r, for all u ∈ S. If such r does not exist, we denote Rad(S) = ∞. Lastly, denote by |S| the number of clusters in S. The following theorem is from [5]. Theorem 12 (Theorem 3.1 in [5]). Let k ≥ 1 be an integer, let G = (V, E) be an undirected weighted n-vertex graph, and let S be a cover of G. There exists another cover T of G that satisfies the following properties. 1. For every S ∈ S there is a cluster T ∈ T such that S ⊆ T . 2. Rad(T ) ≤ (2k − 1)Rad(S). 3. For every v ∈ V , the number of clusters T ∈ T such that v ∈ T , is at most 2k|S| 1 k . Using this theorem, we now construct the desired interactive emulator. Theorem 13. Given an undirected weighted n-vertex graph G = (V, E), an integer parameter k ≥ 1, and a parameter ϵ > 0, there is an interactive (4 + ϵ)k-emulator with size O(k · log Λ ϵ · n 1+ 1 k ) and query time O(log log k + log 1 ϵ ), where Λ = maxu,v d G (u,v) min u̸ =v d G (u,v) is the aspect ratio of G. Proof. Throughout the proof we assume that the minimal weight of an edge in G is 1, as otherwise the weights can be re-scaled by dividing each weight by the minimal weight of an edge. This way, the aspect ratio is Λ = max u,v d G (u, v), i.e., the diameter of G. Note that for every two distinct vertices u, v ∈ V , the distance d G (u, v) is between 1 and Λ. Fix some positive number W > 0. Given the graph G, we define a cover S W by taking the W -neighborhood of each vertex in G: for every v ∈ V , denote S(v, W ) = {u ∈ V | d G (v, u) ≤ W } . The collection S W is then defined as S W = {S(v, W ) | v ∈ V }. This is a cover of G with radius Rad(S W ) ≤ W . By Theorem 12, there is a cover T W such that 1. For every v ∈ V , there is a cluster T (v, W ) ∈ T W such that S(v, W ) ⊆ T (v, W ). We call T (v, W ) the home-cluster of v. 2. Rad(T W ) ≤ (2k − 1)Rad(S W ) ≤ (2k − 1)W . That is, for every T ∈ T W there is a vertex v T ∈ T such that every vertex in the induced graph G[T ] has distance at most (2k − 1)W from v T . Hence, there is a sub-tree of G that is rooted at v T , consists of the vertices of T , and every vertex of this tree has distance at most (2k − 1)W from the root v T . Let q T (u), for every u ∈ T , be a pointer to the next vertex on the unique path in this tree from u to v T . Similarly, h T (u) denotes the number of edges in this unique path, i.e., the depth of u in this tree. 3. For every u ∈ V , the number of clusters T ∈ T W such that u ∈ T is at most 2k|S W | 1 k = 2k · n 1 k(14) We call the cover T W a neighborhood cover for the parameter W . For a given ϵ > 0, we define a hierarchy of neighborhood covers: we construct the neighborhood cover T W , for every W ∈ {(1 + ϵ) i } λ i=0 , where λ = ⌈log 1+ϵ Λ⌉. For every v ∈ V and i ∈ [0, λ], we define a pointer HC i (v) to the home-cluster of v in the neighborhood cover T (1+ϵ) i . Recall also that every cluster T ∈ T (1+ϵ) i has a spanning tree rooted at a vertex v T . For every vertex u ∈ T , the pointer q T (u) points at the next vertex on the unique path from u to the vertex v T , and the variable h T (u) denotes the number of edges in this path. We now describe the construction of our interactive emulator. We define an oracle D that stores the following information. 1. For every v ∈ V and i ∈ [0, λ], the oracle D stores HC i (v), the home-cluster of v in T (1+ϵ) i . 2. For every i ∈ [0, λ], every cluster T ∈ T (1+ϵ) i , and every vertex u ∈ T , the oracle D stores the pointer q T (u) and the variable h T (u). 3. The oracle D stores the distance oracle of Mendel and Naor [44] for the graph G. Next, we describe the algorithm for answering queries. Given a query (u, v) ∈ V 2 , our oracle D first queries the Mendel-Naor distance oracle on (u, v). Recall that this distance oracle is not pathreporting. Thus, the result is only an estimated =d(u, v) such that d G (u, v) ≤d ≤ c M N ·k·d G (u, v), or equivalentlyd c M N · k ≤ d G (u, v) ≤d .(15) Here c M N is the constant coefficient of the stretch in the Mendel-Naor distance oracle. Ifd > Λ, we can safely replaced by Λ, and Inequality (15) will still be satisfied. Then, our oracle D finds the smallest i such that (1 + ϵ) i ∈ [d c M N ·k ,d], and also u is in the home-cluster of v, in the cover T (1+ϵ) i . It finds this i by performing a binary search in the range I = log 1+ϵ d c M N ·k , ⌈log 1+ϵd ⌉ . During this binary search, if the current search range is [i 1 , i 2 ], and the index i is in the middle of this range, then the oracle D refers to the home-cluster T = HC i (v), and checks whether q T (u) is defined (that is, if u ∈ T ). If it is defined, then the binary search continues in the range [i 1 , i], and otherwise in the range [i + 1, i 2 ]. Lastly, the oracle D uses the pointers q T (·) and the variables h T (·), to find a path in the spanning tree of T between u and v. This is done in the same way as in the proof of Theorem 8 in Appendix E. The resulting path is the output path of D for the query (u, v). For the query algorithm to be well defined, note that I ⊆ [0, λ], and thus we did construct the neighborhood cover T (1+ϵ) i for every i ∈ I. This is true since ⌈log 1+ϵd ⌉ ≤ ⌈log 1+ϵ Λ⌉ = λ. We also show that there must be an index i in the range I such that u is in the home-cluster of v in T (1+ϵ) i . Indeed, since we know that d G (u, v) ≤d (by Inequality (15)), then for i ′ = ⌈log 1+ϵd ⌉ ∈ I, we have (1 + ϵ) i ′ ≥d ≥ d G (u, v) . Since the home-cluster of v in the cover T (1+ϵ) i ′ contains all of the vertices within distance (1 + ϵ) i ′ from v, then in particular, it must contain u. To analyse the stretch of the oracle D, we consider two cases regarding the index i, which is the smallest in I such that u is in the home-cluster of v in T (1+ϵ) i . In the first case, if i − 1 is also in I, then we know that u is not in the home-cluster of v in T (1+ϵ) i−1 . Since this home-cluster contains all of the vertices within distance (1 + ϵ) i−1 from v, it must be that (1 + ϵ) i−1 < d G (u, v) . In the second case, where i = ⌊log 1+ϵ d c M N ·k ⌋, then we also know by Inequality (15) that (1 + ϵ) i−1 < (1 + ϵ) log 1+ϵ d c M N ·k =d c M N · k ≤ d G (u, v) . In both cases we saw that (1 + ϵ) i−1 < d G (u, v). Now, note that the output path is the concatenation of two paths, from u to x and from x to v, where x is the lowest common ancestor of u, v in the spanning tree of the home-cluster of v in T (1+ϵ) i . Therefore, the weight of this path is at most twice the radius of this home-cluster, which by Theorem 12 is Rad(T (1+ϵ) i ) ≤ (2k − 1)(1 + ϵ) i . We conclude that the weight of the resulting output path is at most 2(2k − 1)(1 + ϵ) i < 4k · (1 + ϵ) i < 4(1 + ϵ)k · d G (u, v) . That is, our oracle D has stretch 4(1 + ϵ)k. Lastly, finding the unique path between u, v in the spanning tree of the home-cluster of v takes linear time in the length of the output path. We conclude that the query time of D is O(log log k + log 1 ϵ ). Regarding the size of the oracle D, note that storing the Mendel-Naor distance oracle (item 3 in the description of D) requires O(n 1+ 1 k ) space. The variables that store the home-cluster for every vertex v and cover T (1+ϵ) i (item 1) requires O(n · λ) = O(n log 1+ϵ Λ) space. Lastly, to store the pointers q T (·) and the variables h T (·) (item 2), we need O   λ i=0 T ∈T (1+ϵ) i |T |   = O λ i=0 v∈V |{T ∈ T (1+ϵ) i | v ∈ T }| (14) = O λ i=0 v∈V 2k · n 1 k = O(k · log 1+ϵ Λ · n 1+ 1 k ) space. We get a total size of O(k · log 1+ϵ Λ · n 1+ 1 k ) = O(k · log Λ ϵ · n 1+ 1 k ) for the oracle D. Define H as the sub-graph of G, which is the union of all the output paths from D. Note that H is contained in the union of the spanning trees of every cluster of every neighborhood cover T (1+ϵ) i . The total size of H is therefore |H| ≤ λ i=0 T ∈T (1+ϵ) i |T | = O k · log Λ ϵ · n 1+ 1 k , where the last step is justified by the same argument as in the size bound above. We finally proved that (H, D) is an interactive emulator with stretch 4(1 + ϵ)k, size O(k · log Λ ϵ · n 1+ 1 k ), and query time O(log log k +log 1 ϵ ). By replacing ϵ by ϵ 4 , we get the assertion of the theorem. Appendix G In this appendix we specify the dependencies on the parameter ϵ, that were omitted from Table 5 (where we assumed that ϵ was constant). Stretch Size Coefficient Query Time Emulator Distance Preserver (4 + ϵ)k Table 6: A variety of results for interactive spanners by Lemma 7, when using the interactive emulators and distance preservers from Table 4. The O-notations are omitted. The column Size Coefficient specifies the factor that multiplies n 1+ 1 k in the size of each interactive spanner. More accurately, if the value of this column is X, then the size of the respective interactive spanner is O(⌈X⌉ · n 1+ 1 k ). and Hopsets with Small Support Size . . . . . . . . . . . . . . . . . . . . 7 2.2 PRDOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 New Combinatorial and Algorithmic Notions . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Preliminaries 13 3.1 Ultrametrics, HSTs and Branching Events . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Interactive Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Hierarchy of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Components of the Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 A Variety of Interactive Spanners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2.1 Linear-size Interactive Spanner . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.2.2 Ultra-Sparse Interactive Spanner . . . . . . . . . . . . . . . . . . . . . . . . the edges it uses are still virtual (not necessarily belong to G). with query time O(1) and size O(n). Theorem 3 . 3Given an undirected weighted graph G = (V, E), an integer k ≥ 3, a positive parameter ϵ ≤ O(log k), and a set of pairs P ⊆ V 2 , there exists an interactive (1 + ϵ)-distance preserver with query time O(1) and size Theorem 5 . 5Given an undirected weighted n-vertex graph G = (V, E) and an integer k ≥ 1, there is an interactive O(k)-emulator with size O(n 1+ 1 k ) and query time O(1). query time q(k) + O(log h) = O(log k + log 1 ϵ ), and sizeO(n log k + h · n 1+ 1 k ) = O k · log log n · log (3) n ϵ log n · n 1+ 1 k ,as desired (for the last equation, note that for k ≤ log n log (3) n we have n log k ≤ n 1+ 1 k , and for k > log n log (3) n we have log k ≤ k·log log n·log ( Theorem 7 . 7For every n-vertex graph and an integer parameter k ∈ [3, log 2 n], there is an interactive spanner with stretch O(k), query time O log k·log log n log n = O(log log k), and size Remark 1 , 1and the notations from Lemma 7, all with some constant ϵ. Then, we haveα E = O(1), δ = 0, q(k) = O(1), α P = 3 + ϵ = O(1), and τ = log 4/3 (12 + 40 ϵ ) = O(1). Thus, σ = O(1) and h = O k · log log n log n = O (log k) ,where the last step is true since log x x is a decreasing function for all x ≥ 3. By Lemma 7, there is an interactive spanner with stretch2 · O(1) · ((1 + ϵ)k + O(h)) = O(k) , query time q(k) + O(log h) = O log k·log log n log n = O(log log k), and size O k log log Λ·log log n log n·log k · n 1+ 1 k , respectively. The query times are both at most O(log log k + log (3) Λ). These more general bounds imply the Theorem 9 . 9For every n-vertex graph and an integer parameter k ∈ [3, log 2 n], there is an interactive spanner with stretch O k · k·log log n log n = O(k log k), query time O log k·log log n log n = O(log log k) and size O(n 1+ 1 k ). 2 . 2If there are children of [x, y] that are labeled as Bad, let [a, b] and [a ′ , b ′ ] be the first and the last children of [x, y] which are labeled as Bad. If p i (a) and p i (b ′ ) are connected by a hop-edge, we still label [x, y] as Good. 3. If p i (a), p i (b ′ ) are not connected by a hop-edge, we label [x, y] as Bad. Figure 1 : 1If the dashed line between p i (a), p i (b ′ ) represents a hop-edge in H, then [x, y] is labeled as Good. Otherwise, it is labeled as Bad. Theorem 11 . 11Given an undirected weighted graph G = (V, E) and a positive parameter ϵ ≤ O(l), H 1/2 is a (3 + ϵ, (12 + 40 ϵ ) l−1 )-hopset, for any ϵ = O(l) simultaneously. Figure 2 : 2A branching event. i = 0, 1, ..., l − 1, the oracle D stores the oracles D 1 i . 3. For every i = 0, 1, ..., h − 1, the oracle D stores the oracles D 2 i . 4. The oracle D stores the oracle D 3 .5.For every e ∈H 1/2 , the oracle D stores a flag variable f e . If e ∈ {(v, p i (v)) | v ∈ V } for some i, then f e = 1. If e ∈ H i ≥ h, f e = 3. In all cases, D also stores the relevant index i. [i 1 , i 2 ] ⊆ [0, h], such that i 1 , i 2 are even. The root represents the full interval [0, h] and the leaves are intervals of the form [i, i], for some i ∈ [0, h]. The left and right children of a node [i 1 , i 2 ] (where i 1 < i 2 ) are [i 1 , j] and [mid, i 2 ] respectively, where mid is the even index among i 1 +i 2 2 − 1, i 1 +i 2 2 , and j is the even index in [i 1 , mid − 2] that maximizes ∆ u (j). The query time of the oracle D consists of applying the Mendel-Naor distance oracle, which requires O(1) time, then performing a binary search in the range I = [⌊log 1+ϵ d c M N ·k ⌋, ⌈log 1+ϵd ⌉]. Each step in this binary search is done in a constant time (checking whether u is in the home-cluster of v). Thus, the running time of this binary search is O(log |I|) = O log log 1+ϵd − log 1+ϵ d c M N · k = O(log log 1+ϵ (c M N · k)) = O(log log k + log 1 ϵ ) . Distance Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Ultra-Sparse PRDOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Distance Preservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Introduction 1 1.1 2 A Technical Overview Table 2 : 2A summary of results on path-reporting distance oracles. We mostly omit O-notations, except for the stretch parameter in one of our results. In fact, our query time in this result is O log k·log log nlog n Table 5 : 5A variety of results for interactive spanners by Lemma 7, when using the interactive emulators and distance preservers fromTable 4. The O-notations are omitted from the size and query time columns. For simplicity, in all of the above, the parameter ϵ > 0 is considered to be constant. In the second and the fourth rows we assume Λ = n O(1) . In rows 5 and 6 the query times are in fact at most O log k log log k·log log nlog n and O log k·log log n log n , respectively. appear in the table (for the case where Λ = n O(1) ), since if k ≤ √ log n, then for the second row we have 1 . 1If all of the children of [x, y] are labeled as Good, [x, y] is also labeled as Good. 2. If there are children of [x, y] that are labeled as Bad, and p i (x), p i (y) are connected by a hop-edge, we still say that [x, y] is Good.3. If p i (x), p i (y) are not connected by a hop-edge (and [x, y] has Bad children), we label [x, y] as Bad. at the beginning, [i 1 , i 2 ] = [0, h]). Let [i 1 , j] and [mid, i 2 ] be the left and right children of [i 1 , i 2 ], respectively (see Definition 7). If Q(u, v, j) = T RU E, continue to search in the range [i 1 , j]. Otherwise, if Q(u, v, j) = F ALSE, continue to [mid, i 2 ]. Although many works also study the construction time of distance oracles, in this work we focus on the query time-stretch-size trade-offs of our distance oracles, and do not try to optimize their construction time. However, all our distance oracles can be constructed inÕ(mn) time.2 The query time is not explicated in[5,15]. In Appendix F we argue that their construction of neighborhood covers gives rise to PRDOs with these parameters. By "purely additive stretch" we mean that the multiplicative stretch is 1. A branching event[19] between two paths P1, P2 is a triple (P1, P2, x) such that x is a vertex on P1, P2, and the adjacent edges to x in P1 are not exactly the same as in P2. See Section 3.1. A related notion of path-reporting low-hop spanners for metric spaces was recently introduced and studied in[35,32]. In the context of graphs, these objects are stronger than interactive emulators (because of the low-hop guarantee), but weaker than interactive spanners (as they do not provide paths in the input graph). Also, the constructions of[35,32] for general metrics have size Ω(n log n) for all choices of parameters. We actually give here the definition of a 1-HST. The definition of a k-HST, for a general k, is given in[7].11 The original definition by Bartal in[6] uses weights on the edges instead of labels on the vertices. We use a different equivalent notion, that was given by Bartal et al. in another paper[7]. By universal (α(ϵ), β(ϵ))-hopset, we mean that this hopset is a (α(ϵ), β(ϵ))-hopset for every ϵ simultaneously. Naor and Tao[45] showed that cMN can be 33, and argued that it can hardly be smaller than 16. i−h k Acknowledgements We wish to thank Arnold Filtser for reminding us that spanning metric Ramsey trees[1]can be viewed as a PRDO, and for helpful discussions about spanning clan embeddings.Appendix EIn this appendix we prove Theorem 8.Theorem (Theorem 8). Suppose we have an algorithm A that given a graph with n vertices, produces an interactive α-spanner with query time q and size h(n). Given a positive number t > 0, there exists an algorithm that given a graph G, produces an interactive O(α · t)-spanner with query time O(q) and size O(h( n t ) + n). Proof. Given G = (V, E) and t, let C be a stretch-friendly O(t)-partition of G with |C| ≤ n t (See definition 6). Denote by C(v) the unique cluster in C that contains the vertex v. Recall that every C ∈ C has a spanning tree T C , rooted at some vertex r C , such that for every v ∈ C, the unique path between v and r C in T C has O(t) edges. For x, y ∈ C, we denote by T C [x, y] the unique path in T C between the vertices x, y. Generally, T C [x, y] is the concatenation of T C [x, lca(x, y)] and T C [lca(x, y), y], where lca(x, y) is the lowest common ancestor of x, y in T C . Now, the cluster graph H = (V H , E H ) is defined as follows. The vertices set V H is the set of clusters C. For every C, C ′ ∈ C, denote by e(C, C ′ ) the edge in E with the minimal weight among the edges between C and C ′ . If there is no such edge, denote e(C, C ′ ) = ⊥. Then,Proof of Theorem 10. As in the proof of Theorem 8, we construct a stretch-friendly O(t)-partition C of G with |C| ≤ n t . For every C ∈ C, let T C and r C denote the corresponding spanning tree and root, respectively, and let T C [x, y] denote the unique path in T C between the vertices x, y ∈ C. The cluster graph H = (V H , E H ) and the edges e(C, C ′ ) are defined in the same way as in the proof of Theorem 8. Now apply the algorithm A on H, to obtain an interactive α-spanner (S, D) with query time q and size h( n t ). Our oracle for the new interactive spanner will still contain the oracle D and the edges {e(C, C ′ ) | (C, C ′ ) ∈ S}. Also, for every v ∈ V , we will store the pointer p(v) to its parent in the tree T C , of the cluster C such that v ∈ C. We will no longer store C(v) (see definition in the proof of Theorem 8) and the number of edges in the path T C [v, r C ].Given a query (u, v) ∈ V 2 , we will follow the pointers from u and from v until we get to the roots r u , r v of the respective clusters that contains u and v. As a result, in time O(t) we discover the identities of the two respective clusters (i.e., we find C(u), C(v)), and the number of edges in the path from u, v to r u , r v in the spanning trees of their respective clusters. From this point on, the query algorithm proceeds as described in the proof of Theorem 8.The stretch of the resulting oracle remains O(α · t). 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AlgorithmsM. Thorup and U. Zwick. Spanners and emulators with sublinear distance errors. In Proc. of Symp. on Discr. Algorithms, pages 802-809, 2006. Compact oracles for reachability and approximate distances in planar digraphs. Mikkel Thorup, Journal of the ACM (JACM). 516Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM (JACM), 51(6):993-1024, 2004. Approximate distance oracles with improved query time. Christian Wulff-Nilsen, 10.1137/1.9781611973105.39Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013. Sanjeev Khannathe Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013New Orleans, Louisiana, USAChristian Wulff-Nilsen. Approximate distance oracles with improved query time. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 539-549. SIAM, 2013. doi:10.1137/1.9781611973105.39. All pairs lightest shortest paths. Uri Zwick, Proceedings of the thirty-first annual ACM symposium on Theory of Computing. the thirty-first annual ACM symposium on Theory of ComputingUri Zwick. All pairs lightest shortest paths. In Proceedings of the thirty-first annual ACM symposium on Theory of Computing, pages 61-69, 1999. Exact and approximate distances in graphs-a survey. Uri Zwick, Algorithms-ESA 2001: 9th Annual European SymposiumÅrhus. DenmarkSpringerProceedingsUri Zwick. Exact and approximate distances in graphs-a survey. In Algorithms-ESA 2001: 9th Annual European SymposiumÅrhus, Denmark, August 28-31, 2001 Proceedings, pages 33-48. Springer, 2001. All pairs shortest paths using bridging sets and rectangular matrix multiplication. Uri Zwick, Journal of the ACM (JACM). 493Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM (JACM), 49(3):289-317, 2002. A slightly improved sub-cubic algorithm for the all pairs shortest paths problem with real edge lengths. Uri Zwick, Algorithmica. 462Uri Zwick. A slightly improved sub-cubic algorithm for the all pairs shortest paths problem with real edge lengths. Algorithmica, 46(2):181-192, 2006. . For Every U ∈ V And V ∈ C, u), the oracle D stores the pointer q u (v)For every u ∈ V and v ∈ C(u), the oracle D stores the pointer q u (v). For every i = 0, 1, ..., h and v ∈ V , the oracle D stores the pivot p i (v) and the bunch B i (v). For every i = 0, 1, ..., h and v ∈ V , the oracle D stores the pivot p i (v) and the bunch B i (v).
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[ "Optimal binning: mathematical programming formulation", "Optimal binning: mathematical programming formulation" ]
[ "Guillermo Navas-Palencia [email protected] " ]
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The optimal binning is the optimal discretization of a variable into bins given a discrete or continuous numeric target. We present a rigorous and extensible mathematical programming formulation to solve the optimal binning problem for a binary, continuous and multi-class target type, incorporating constraints not previously addressed. For all three target types, we introduce a convex mixed-integer programming formulation. Several algorithmic enhancements, such as automatic determination of the most suitable monotonic trend via a Machine-Learning-based classifier, and implementation aspects are thoughtfully discussed. The new mathematical programming formulations are carefully implemented in the open-source python library OptBinning. * This version contains a new objective function and the maximum p-value constraint for the continuous target.
null
[ "https://export.arxiv.org/pdf/2001.08025v3.pdf" ]
210,861,134
2001.08025
dbb4b8771f8ef4d311aea137ef1d0a01d6a198b8
Optimal binning: mathematical programming formulation December 12, 2022 * Guillermo Navas-Palencia [email protected] Optimal binning: mathematical programming formulation December 12, 2022 * The optimal binning is the optimal discretization of a variable into bins given a discrete or continuous numeric target. We present a rigorous and extensible mathematical programming formulation to solve the optimal binning problem for a binary, continuous and multi-class target type, incorporating constraints not previously addressed. For all three target types, we introduce a convex mixed-integer programming formulation. Several algorithmic enhancements, such as automatic determination of the most suitable monotonic trend via a Machine-Learning-based classifier, and implementation aspects are thoughtfully discussed. The new mathematical programming formulations are carefully implemented in the open-source python library OptBinning. * This version contains a new objective function and the maximum p-value constraint for the continuous target. Introduction Binning (grouping or bucketing) is a technique to discretize the values of a continuous variable into bins (groups or buckets). From a modeling perspective, the binning technique may address prevalent data issues such as the handling of missing values, the presence of outliers and statistical noise, and data scaling. Furthermore, the binning process is a valuable interpretable tool to enhance the understanding of the nonlinear dependence between a variable and a given target while reducing the model complexity. Ultimately, resulting bins can be used to perform data transformations. Binning techniques are extensively used in machine learning applications, exploratory data analysis and as an algorithm to speed up learning tasks; recently, binning has been applied to accelerate learning in gradient boosting decision tree [13]. In particular, binning is widely used in credit risk modeling, being an essential tool for credit scorecard modeling to maximize differentiation between high-risk and low-risk observations. There are several unsupervised and supervised binning techniques. Common unsupervised techniques are equal-width and equal-size or equal-frequency interval binning. On the other hand, well-known supervised techniques based on merging are Monotone Adjacent Pooling Algorithm (MAPA), also known as Maximum Likelihood Monotone Coarse Classifier (MLMCC) [22] and ChiMerge [14], whereas other techniques based on decision trees are CART [2], Minimum Description Length Principle (MDLP) [3] and, more recently, condition inference trees (CTREE) [10]. The binning process might require to satisfy certain constraints. These constraints might range from requiring a minimum number of records per bin to monotonicity constraints. This variant of the binning process is known as the optimal binning process. The optimal binning is generally solved by iteratively merging an initial granular discretization until imposed constraints are satisfied. Performing this fine-tuning manually is likely to be unsatisfactory as the number of constraints increases, leading to suboptimal or even infeasible solutions. However, we note that this manual adjustment has been encouraged by some authors [21], legitimating the existing interplay of "art and science" in the binning process. There are various commercial software tools for solving the optimal binning problem 1 . Software IBM SPSS and the MATLAB Financial Toolbox, use MDLP and MAPA as default algorithm, respectively. The most advanced tool to solve the optimal binning problem is available in the SAS Enterprise Miner software. A limited description of the proprietary algorithm can be found in [18], where two mixed-integer programming (MIP) formulations are sketched: a mixed-integer linear programming (MILP) formulation to obtain a fast probably suboptimal solution, and a mixed-integer nonlinear programming (MINLP) formulation to obtain an optimal solution. The suboptimal formulation is the default method due to computational time limitations (MILP techniques are considerably more mature). We note that the SAS implementation allows most of the constraints required in credit risk modeling, becoming an industry standard. Besides, there exist a few open-source solutions, but the existing gap comparing to the commercial options in terms of capabilities is still significant. Among the available alternatives, we mention the MATLAB implementation of the monotone optimal binning in [17], and the R specialized packages smbinning [8], relying on CTREE, and MOB [23], which merely include basic functionalities. In this paper, we develop a rigorous and extensible mathematical programming formulation for solving the optimal binning problem. This general formulation can efficiently handle binary, continuous, and multi-class target type. The presented formulations incorporate the constraints generally required to produce a good binning [21], and new constraints not previously addressed. For all three target types, we introduce a convex mixed-integer programming formulation, ranging from a integer linear programming (ILP) formulation for the simplest cases to a mixed-integer quadratic programming (MIQP) formulation for those cases adding more involved constraints. The remainder of the paper is organized as follows. Section 2 introduces our general problem formulation and the corresponding mixed-integer programming formulation for each supported target. We focus on the formulation for binary target, investigating various formulation variants. Then, in Section 3 we discuss in detail several algorithmic aspects such as the automatic determination of the optimal monotonic trend and the development of presolving algorithms to efficiently solve large size instances. Section 4 includes experiments with real-world datasets and compares the performance of supported solvers for large size instances. Finally, in Section 5, we present our conclusions and discuss possible research directions. Mathematical programming formulation The optimal binning process comprises two steps: A pre-binning process that generates an initial granular discretization, and a subsequent refinement or optimization to satisfy imposed constraints. The pre-binning process uses, for example, a decision tree algorithm to calculate the initial split points. The resulting m split points are sorted in strictly ascending order, s 1 < s 2 < . . . < s m to create n = m + 1 pre-bins. These pre-bins are defined by the intervals (−∞, s 1 ), [s 1 , s 2 ), . . . , [s m , ∞). Given n pre-bins, the decision variables consist of a binary lower triangular matrix (indicator variables) of size n, X ij ∈ {0, 1}, ∀(i, j) ∈ {1, . . . , n : i ≥ j}. The starting point is a diagonal matrix, meaning that initially all pre-bins are selected. A basic feasible solution must satisfy the following constraints: • All pre-bins are either isolated or merged to create a larger bin interval, but cannot be erased. Each column must contain exactly one 1. n i=1 X ij = 1, j = 1, . . . , n.(1) • Only consecutive pre-bins can be merged. Continuity by rows, no 0 − 1 gaps are allowed. X ij ≤ X ij+1 , i = 1, . . . , n; j = 1, . . . , i − 1. (2) • A solution has a last bin interval of the form [s k , ∞), for k ≤ n. The binary decision variable X nn = 1. To clarify, Figure 1 shows an example of a feasible solution. In this example, pre-bins corresponding to split points (s 2 , s 3 , s 4 ) and (s 5 , s 6 ) are merged, thus having an optimal binning with bin intervals (−∞, s 1 ), [s 1 , s 4 ), [s 4 , s 6 ), [s 6 , ∞). The described problem can be seen as a generalized assignment problem. A direct formulation of metrics involving ratios such as the mean or most of the divergence measures on merged bins leads to a non-convex MINLP formulation, due to the ratio of sums of binary variables. Solving non-convex MINLP problems to optimality is a challenging task requiring the use of global MINLP solvers, especially for large size instances. Investigating the binary lower triangular matrix in Figure 1, it can be observed, by analyzing the constraints in Equations (1) and (2) imposing continuity by rows, that a feasible solution is entirely characterized by the position of the first 1 for each row. This observation permits the pre-computation of the set of possible solutions by rows, obtaining an aggregated matrix with the shape of X for each involved metric. Consequently, the nonconvex objective function and constraints are linearized, resulting in a convex formulation by exploiting problem information. Using this reformulation, we shall see that the definition of constraints for binary, continuous and multi-class target are almost analogous. Mixed-integer programming formulation for binary target Given a binary target y used to discretize a variable x into n bins, we define the normalized count of non-events (NE) p i , case y = 0, and events (E) q i , case y = 1, for each bin i as p i = r N E i r N E T , q i = r E i r E T , where r N E i , r E i , r N E T and r E T are the number of non-event and event records per bin, and the total number of non-event records and event records, respectively. Next, we define the Weight of Evidence (WoE) and event rate (D) for each bin, WoE i = log r N E i /r N E T r E i /r E T , D i = r E i r E i + r N E i . The Weight of Evidence WoE i and event rate D i for each bin are related by means of the functional equations WoE i = log 1 − D i D i + log r E T r N E T = log r E T r N E T − logit(D i ) D i = 1 + r N E T r E T e WoEi −1 = 1 + e WoEi−log r E T r N E T −1 , where D i can be characterized as a logistic function of WoE i , and WoE i can be expressed in terms of the logit function of D i . This shows that WoE is inversely related to the event rate. The constant term log(r E T /r N E T ) is the log ratio of the total number of event and the total number of non-events. Divergence measures serve to assess the discriminant power of a binning solution. The Jeffreys' divergence [11], also known as Information Value (IV) within the credit risk industry, is a symmetric measure expressible in terms of the Kullback-Leibler divergence D KL (P ||Q) [15] defined by J(P ||Q) = IV = D KL (P ||Q) + D KL (Q||P ) = n i=1 (p i − q i ) log p i q i . The IV statistic is unbounded, but some authors have proposed rules of thumb to settings quality thresholds [21]. Alternatively, the Jensen-Shannon divergence is a bounded symmetric measure also expressible in terms of the Kullback-Leibler divergence JSD(P ||Q) = 1 2 (D(P ||M ) + D(Q||M )) , M = 1 2 (P + Q), and bounded by JSD(P ||Q) ∈ [0, log (2)]. Note that these divergence measures cannot be computed when r N E i = 0 and/or r E i = 0. Other divergences measures without this limitation are described in [24]. A good binning algorithm for binary target should be characterized by the following properties [21]: 1. Missing values are binned separately. 2. Each bin should contain at least 5% observations. No bins should have 0 non-events or events records. Property 1 is adequately addressed in many implementations, where missing and special values are incorporated as additional bins after the optimal binning terminates. Property 2 is a usual constraint to enforce representativeness. Property 3 is required to compute the above divergence measures. Let us define the parameters of the mathematical programming formulation: n ∈ N number of pre-bins. r N E T ∈ N total number of non-event records. r E T ∈ N total number of event records. r N E i ∈ N number of non-event records per pre-bin. r E i ∈ N number of event records per pre-bin. r i = r N E i + r E i number of records per pre-bin. r N E min ∈ N minimum number of non-event records per bins. r N E max ∈ N maximum number of non-event records per bins. r E min ∈ N minimum number of event records per bins. r E max ∈ N maximum number of event records per bins. b min ∈ N minimum number of bins. b max ∈ N maximum number of bins. The objective function is to maximize the discriminant power among bins, therefore, maximize a divergence measure. The IV can be computed using the described parameters and the decision variables X ij , yielding IV = n i=1   i j=1 r N E z r N E T − r E z r E T X ij   log i j=1 r N E j /r N E T X ij i j=1 r E j /r E T X ij , The IV is the sum of the IV contributions per bin, i.e., the sum by rows. As previously stated, given the constraints in Equations (1) and (2), an aggregated low triangular matrix V ij ∈ R + 0 , ∀(i, j) ∈ {1, . . . , n : i ≥ j} with all possible IV values from bin merges can be pre-computed as follows V ij =   i z=j r N E z r N E T − r E z r E T   log i z=j r N E z /r N E T i z=j r E z /r E T , i = 1, . . . , n; j = 1, . . . , i.(3) The optimal IV for each bin is determined by using the remarked observation that a solution is characterized by the position of the first 1 for each row, thus, using the continuity constraint in (2), we obtain V i· = V i1 X i1 + i j=2 V ij (X ij − X ij−1 ) ⇐⇒ V ii X ii + i−1 j=1 (V ij − V ij+1 )X ij .(4) for i = 1, . . . , n. The latter formulation is preferred to reduce the fill-in of the matrix of constraints. Similarly, a lower triangular matrix of event rates D ij ∈ [0, 1], ∀(i, j) ∈ {1, . . . , n : i ≥ j} can be pre-computed as follows D ij = i z=j r E z i z=j r z , i = 1, . . . , n; j = 1, . . . , i. The ILP formulation, with no additional constraints such as monotonicity constraints, can be stated as follows max X n i=1 V ii X ii + i−1 j=1 (V ij − V ij+1 )X ij (5a) s.t. n i=j X ij = 1, j = 1, . . . , n (5b) X ij − X ij+1 ≤ 0, i = 1, . . . , n; j = 1, . . . , i − 1 (5c) b min ≤ n i=1 X ii ≤ b max (5d) r min X ii ≤ i j=1 r j X ij ≤ r max X ii , i = 1, . . . , n (5e) r N E min X ii ≤ i j=1 r N E j X ij ≤ r N E max X ii , i = 1, . . . , n (5f) r E min X ii ≤ i j=1 r E j X ij ≤ r E max X ii , i = 1, . . . , n (5g) X ij ∈ {0, 1}, ∀(i, j) ∈ {1, . . . , n : i ≥ j} (5h) Apart from the constraints (5b) and (5c) already described, constraint (5d) imposes a lower and upper bound on the number of bins. Other range constraints (5e-5g) limit the number of total, non-event and event records per bin. Note that to increase sparsity, the range constraints are not implemented following the standard formulation to avoid having the data twice in the model. For example, constraint (5d) is replaced by d + n i=1 X ii − b max = 0, 0 ≤ d ≤ b max − b min . Monotonicity constraints Monotonicity constraints between the event rates of consecutive bins can be imposed to ensure legal compliance and business constraints. Three types of monotonic trends are considered: the usual ascending/descending, and two types of unimodal forms, concave/convex and peak/valley. This modeling flexibility can help to capture overlooked or unexpected patterns, providing new insights to enrich models. Note that work in [18] uses the WoE approach instead. Applying Equation (4), the optimal event rate for each bin is given by D i· = D ii X ii + i−1 j=1 (D ij − D ij+1 )X ij , i = 1, . . . , n.(6) Monotonic trend: ascending and descending. The formulation for monotonic ascending trend can be stated as follows, D zz X zz + z−1 j=1 (D zj − D zj+1 )X zj + β(X ii + X zz − 1) ≤ 1 + (D ii − 1)X ii + i−1 j=1 (D ij − D ij+1 )X ij , i = 2, . . . , n; z = 1, . . . i − 1. The term 1 + (D ii − 1)X ii or simply 1 − X ii , is used to ensure that event rates are in [0, 1], and the ascending constraint is satisfied even if bin i is not selected. Note that this is a big-M formulation M + (D ii − M )X ii using M = 1, which suffices given D ∈ [0, 1], however, a tighter (non integer) M = max({D ij : i = 1, . . . , n; i ≥ j}), can be used instead. The parameter β is the minimum event rate difference between consecutive bins. The term β(X ii + X zz − 1) is required to ensure that the difference between two selected bins i and z is greater or equal than β. Similarly, for the descending constraint, D ii X ii + i−1 j=1 (D ij − D ij+1 )X ij + β(X ii + X zz − 1) ≤ 1 + (D zz − 1)X zz + z−1 j=1 (D zj − D zj+1 )X zj , i = 2, . . . , n; z = 1, . . . i − 1. Monotonic trend: concave and convex. The concave and convex trend can be achieved by taking the definition of concavity/convexity on equally spaced points: −x i+1 + 2x i − x i−1 ≥ 0 concave x i+1 − 2x i + x i−1 ≥ 0 convex Thus, replacing Equation (6) in the previous definition of concavity we obtain the concave trend constraints, − D ii X ii + i−1 z=1 (D iz − D iz+1 )X iz + 2 D jj X jj + j−1 z=1 (D jz − D jz+1 )X jz − D kk X kk + k−1 z=1 (D kz − D kz+1 )X kz ≥ X ii + X jj + X kk − 3, for i = 3, . . . n; j = 2, . . . , i − 1 and k = 1, . . . , j − 1. Similarly, for convex trend we get D ii X ii + i−1 z=1 (D iz − D iz+1 )X iz − 2 D jj X jj + j−1 z=1 (D jz − D jz+1 )X jz D kk X kk + k−1 z=1 (D kz − D kz+1 )X kz ≥ X ii + X jj + X kk − 3, for i = 3, . . . n; j = 2, . . . , i − 1 and k = 1, . . . , j − 1. Note that term X ii + X jj + X kk − 3 is used the preserve redundancy of constraints when not all bins i, j and k are selected, given that D ∈ [0, 1]. Monotonic trend: peak and valley. The peak and valley trend 2 define an event rate function exhibiting a single trend change or reversal. The optimal trend change position is determined by using disjoint constraints, which can be linearized using auxiliary binary variables. The resulting additional constraints are as follows, i − n(1 − y i ) ≤ t ≤ i + ny i , i = 1, . . . , n (7a) t ∈ [0, n] (7b) y i ∈ {0, 1}, i = 1, . . . , n(7c) where t is the position of the optimal trend change bin, y i are auxiliary binary variables and n in (7a) is the smallest big-M value for this formulation while preserving the redundancy of constraints. Furthermore, for the peak trend we incorporate the following constraints, y i + y z + 1 + (D zz − 1)X zz + z−1 j=1 (D zj − D zj+1 )X zj ≥ D ii X ii + i−1 j=1 (D ij − D ij+1 )X ij , i = 2, . . . , n; z = 1, . . . , i − 1, 2 − y i − y z + 1 + (D ii − 1)X ii + i−1 j=1 (D ij − D ij+1 )X ij ≥ D zz X zz + z−1 j=1 (D zj − D zj+1 )X zj , i = 2, . . . , n; z = 1, . . . , i − 1. Similarly, for the valley trend we include, y i + y z + 1 + (D ii − 1)X ii + i−1 j=1 (D ij − D ij+1 )X ij ≥ D zz X zz + z−1 j=1 (D zj − D zj+1 )X zj , i = 2, . . . , n; z = 1, . . . , i − 1, 2 − y i − y z + 1 + (D zz − 1)X zz + z−1 j=1 (D zj − D zj+1 )X zj ≥ D ii X ii + i−1 j=1 (D ij − D ij+1 )X ij , i = 2, . . . , n; z = 1, . . . , i − 1. Note that none of these constraints are necessary if the position of the change bin t is fixed in advance. For example, given t, the valley trend constraints are replaced by two sets of constraints; one to guarantee a descending monotonic trend before t and another to guarantee an ascending monotonic trend after t. Devising an effective heuristic to determine the optimal t can yield probably optimal solutions while reducing the problem size significantly. Additional constraints Reduction of dominating bins. To prevent any particular bin from dominating the results, it might also be required that bins have at most a certain number of (total/nonevent/event) records using constraints (5e -5g). Furthermore, we might produce more homogeneous solutions by reducing a concentration metric such as the standard deviation of the number of total/non-event/event records among bins. Three concentration metrics are considered: standard deviation, Herfindahl-Hirschman Index (HHI) [17] and the difference between the largest and smallest bin. The standard deviation among the number of records for each bin is given by std =    1 m − 1 n i=1   i j=1 r j X ij − X ii m n i=1 i j=1 r j X ij   2    1/2 , where m = n i=1 X ii is the optimal number of bins. Let us define the following auxiliary variables µ = 1 m n i=1 i j=1 r j X ij , w i = i j=1 r j X ij − µX ii , i = 1, . . . n. Taking w = (w 1 , . . . , w n ) T , the standard deviation t can be incorporated to the formulation with a different representation. Since std = (w T w/(m − 1)) 1/2 then ||w|| 2 (m − 1) 1/2 ≤ t ⇐⇒ ||w|| 2 2 ≤ (m − 1)t 2 . The non-convex MINLP formulation using the parameter γ to control the importance of the term t, max X,µ,w n i=1 V ii X ii + i−1 j=1 (V ij − V ij+1 )X ij − γt (10a) s.t. (5b -5h) (10b) n i=1 w 2 i ≤ (m − 1)t 2 (10c) µ = 1 m n i=1 i j=1 r j X ij (10d) w i = i j=1 r j X ij − µX ii , i = 1, . . . n (10e) m = n i=1 X ii (10f) m ≥ 0 (10g) µ ≥ 0 (10h) w i ∈ R, i = 1, . . . , n.(10i) A widely used metric to quantify concentration is HHI, which can be employed to asses the quality of a binning solution. Lower values of HHI correspond to more homogeneous bins. The HHI of the number of records for each bin is given by HHI = 1 r 2 T n i=1   i j=1 r j X ij   2 , where r T = n i=1 r i is the total number of records. The MIQP formulation using the parameter γ to control the importance of HHI is stated as max X n i=1 V ii X ii + i−1 j=1 (V ij − V ij+1 )X ij − γ r 2 T n i=1   i j=1 r j X ij   2 (11a) s.t. (5b -5h)(11b) An effective MILP formulation can be devised using a simplification of the standard deviation approach based on reducing the difference between the largest and smallest bin. The MILP formulation is given by max X,pmin,pmax n i=1 V ii X ii + i−1 j=1 (V ij − V ij+1 )X ij − γ(p max − p min ) (12a) s.t. (5b -5h) (12b) p min ≤ r T (1 − X ii ) + i j=1 r j X ij , i = 1, . . . , n (12c) p max ≥ i j=1 r j X ij , i = 1, . . . , n (12d) p min ≤ p max (12e) p min ≥ 0. (12f) p max ≥ 0. (12g) Maximum p-value constraint. A necessary constraint to guarantee that event rates between consecutive bins are statistically different is to impose a maximum p-value constraint setting a significance level α. Suitable statistical tests are the Z-test, Pearson's Chi-square test or Fisher 's exact test. To perform these statistical tests we require an aggregated matrix of non-event and event records per bin, R N E ij = i z=j r N E z , R E ij = i z=j r E z , i = 1, . . . , n; j = 1, . . . , i. The preprocessing procedure to detect pairs of pre-bins that do not satisfy the p-value constraints using the Z-test is shown in Algorithm 1. if Z-test(x, y, w, z) < zscore then These constraints are added to the formulation by imposing that, at most, one of the bins violating the maximum p-value constraint can be selected. Two cases are considered depending on j: X ij + X kl ≤ 1 + X kl−1 if j = 1, X ij + X kl ≤ 1 + X ij−1 + X kl−1 if j > 1 , ∀(i, j, k, l) ∈ I. These two cases are represented in Figure 2. Finally, note that previous constraints can be rewritten such that they resemble continuity constraints (5c), respectively Case j > 1 (right). X ij + (X kl − X kl−1 ) ≤ 1, and (X ij − X ij−1 ) + (X kl − X kl−1 ) ≤ 1. Mixed-integer programming reformulation for local and heuristic search The number of binary decision variables X is n(n + 1)/2. For large n the N P-hardness of the combinatorial optimization problem might limit the success of tree-search techniques. A first approach to tackle this limitation is reformulating the problem to reduce the number of decision variables. First, observe in Figure 1 that a solution is fully characterized by the diagonal of X. Thus, having the diagonal we can place the ones on the positions satisfying unique assignment (1) and continuity (2) constraints. On the other hand, to return indexed elements in any aggregated matrix in the original formulation, we require the position of the first one by row. We note that this information can be retrieved by counting the number of consecutive zeros between ones (selected bins) of the diagonal. To perform this operation we use two auxiliary decision variables: an accumulator of preceding zeros a i and the preceding run-length of zeros z i . A similar approach to counting consecutive ones is introduced in [12]. The described approach is illustrated in Figure 3. The positions in z are zero-based indexes of the reversed rows of the aggregated matrices. For example, the aggregated lower triangular matrix R E is now computed backward: R E ij = j z=i r E z for i = 1, . . . , n; j = 1, . . . , i. Same for the aggregated matrices V , D, R and R N E . Let us define the parameters of the mathematical programming formulation: n ∈ N number of pre-bins. b min ∈ N minimum number of bins. V [i,zi] ∈ R + 0 Information value. D [i, b max ∈ N maximum number of bins. and the decision variables: x i ∈ {0, 1} binary indicator variable. a i ∈ N 0 accumulator of preceding zeros. z i ∈ N 0 preceding run-length of zeros. The new formulation with 3n decision variables is stated as follows max X n i=1 V [i,zi] x i (13a) s.t. x n = 1 (13b) a i = (a i−1 + 1)(1 − x i ), i = 1, . . . , n (13c) z i = a i−1 (1 − x i−1 )x i , i = 1, . . . , n (13d) b min ≤ n i=1 x i ≤ b max (13e) r min ≤ n i=1 R [i,zi] x i ≤ r max , i = 1, . . . , n (13f) r N E min ≤ n i=1 R N E [i,zi] x i ≤ r N E max , i = 1, . . . , n (13g) r E min ≤ n i=1 R E [i,zi] x i ≤ r E max , i = 1, . . . , n (13h) x i ∈ {0, 1}, i = 1, . . . , n (13i) a i ∈ N 0 , i = 1, . . . , n (13j) z i ∈ N 0 , i = 1, . . . , n(13k) This MINLP formulation is particularly suitable for Local Search (LS) and heuristic techniques, where decision variable z i can be used as an index, for example, V [i,zi] . The nonlinear constraints (13d) and (13e) are needed for counting consecutive zeros. After the linearization of these constraints via big-M inequalities or indicator constraints [12], the formulation is adequate for Constraint Programming (CP). Additional constraints such as monotonicity constraints can be incorporated to (13b -13k) in a relatively simple manner: Monotonic trend ascending D [i,zi] x i + 1 − x i ≥ D [j,zj ] x j + β(x i + x j − 1), i = 2, . . . , n; z = 1, . . . , i − 1. Monotonic trend descending zj ] x j , i = 2, . . . , n; z = 1, . . . , i − 1. D [i,zi] x i + β(x i + x j − 1) ≤ 1 − x j + D [j, Monotonic trend concave −D [i,zi] x i + 2D [j,zj ] x j − D [k,z k ] x k ≥ x i + x j + x k − 3, for i = 3, . . . , n; j = 2, . . . , i − 1; k = 1, . . . , j − 1. Monotonic trend convex D [i,zi] x i − 2D [j,zj ] x j + D [k,z k ] x k ≥ x i + x j + x k − 3, for i = 3, . . . , n; j = 2, . . . , i − 1; k = 1, . . . , j − 1. Monotonic trend peak: constraints (7a -7c) and y i + y j + 1 + (D [j,zj ] − 1)x j − D [i,zi] x i ≥ 0 2 − y i − y j + 1 + (D [i,zi] − 1)x i − D [j,zj ] x j ≥ 0, for i = 2, . . . , n; z = 1, . . . , i − 1. Monotonic trend valley: constraints (7a -7c) and y i + y j + 1 + (D [i,zi] − 1)x i − D [j,zj ] x j ≥ 0 2 − y i − y j + 1 + (D [j,zj ] − 1)x j − D [i,zi] x i ≥ 0 for i = 2, . . . , n; z = 1, . . . , i − 1. In Section 4, we compare the initial CP/MIP formulation to the presented LS formulation for large size instances. Mixed-integer programming formulation for continuous target The presented optimal binning formulation given a binary target can be seamlessly extended to a continuous target. Following the methodology developed for Equations (3) and (4), we could adapt the IV statistic for a continuous target as described in [5] IV = n i=1 |µ − u i | r i r T , where µ ∈ R is the target mean for all records (global target mean), r T is total number of records, and u i and r i are the target mean and number of records for each bin, respectively. The goal of this metric is to obtain bins with a target mean as different as possible from the global target mean. However, we note that the usage of this metric as objective function might tend to reduce granularity by selecting only a few bins with large differences. Therefore, we use only the p-norm distance (L 1 -norm of L 2 -norm) term and exclude the relative number of records term. An aggregated lower triangular matrix L ij ∈ R + 0 , ∀(i, j) ∈ {1, . . . , n : i ≥ j} can be pre-computed as follows, L ij = µ − U ij p , U ij = j z=i s z j z=i r z , where U ij ∈ R is the aggregated matrix of target mean values, and s i is the sum of target values for each pre-bin. A more robust approach might replace the mean by an order statistic, but unfortunately, the aggregated matrix computed from the pre-binning data would require approximation methods. Finally, replacing L ij in the objective function (5a), the resulting formulation is given by max X n i=1 L ii X ii + i−1 j=1 (L ij − L ij+1 )X ij (16a) s.t. (5b -5h) (16b) Monotonicity constraints As for the binary target case, we can impose monotonicity constraints between the mean value of consecutive bins. Since establishing tight bounds for U ij ∈ R is not trivial, we discard a big-M formulation. Another traditional technique such as the use of SOS1 sets is also discarded to avoid extra variables and constraints. Instead, we state the ascending monotonic trend in double implication form as follows X ii = 1 and X zz = 1 =⇒ U zz X zz + z−1 j=1 (U zj − U zj+1 )X zj + β ≤ U ii X ii + i−1 j=1 (U ij − U ij+1 )X ij , i = 2, . . . , n; z = 1, . . . i − 1. The enforced constraint must be satisfied iff the two literals X ii and X zz are true, otherwise the constraint is ignored. This is half-reified linear constraint [4]. The parameter β is the minimum mean difference between consecutive bins. Similarly, for the descending constraint, X ii = 1 and X zz = 1 =⇒ U ii X ii + i−1 j=1 (U ij − U ij+1 )X ij + β ≤ U zz X zz + z−1 j=1 (U zj − U zj+1 )X zj , i = 2, . . . , n; z = 1, . . . i − 1. Furthermore, the concave and convex trend can be written in triple implication form using literals X ii , X jj and X kk . The concave trend constraints are X ii = 1, X zz = 1 and X kk = 1 =⇒ − U ii X ii + i−1 z=1 (U iz − U iz+1 )X iz + 2 U jj X jj + j−1 z=1 (U jz − U jz+1 )X jz − U kk X kk + k−1 z=1 (U kz − U kz+1 )X kz ≥ 0, for i = 3, . . . n; j = 2, . . . , i − 1 and k = 1, . . . , j − 1. Similarly, for convex trend we get X ii = 1, X zz = 1 and X kk = 1 =⇒ U ii X ii + i−1 z=1 (U iz − U iz+1 )X iz − 2 U jj X jj + j−1 z=1 (U jz − U jz+1 )X jz U kk X kk + k−1 z=1 (U kz − U kz+1 )X kz ≥ 0, for i = 3, . . . n; j = 2, . . . , i − 1 and k = 1, . . . , j − 1. The formulation can be extended to support valley and peak trend. The peak trend requires constraints (7a -7c) and X ii = 1 and X zz = 1 =⇒M (y i + y z ) + U zz X zz + z−1 j=1 (U zj − U zj+1 )X zj ≥ U ii X ii + i−1 j=1 (U ij − U ij+1 )X ij , M (2 − y i − y z ) + U ii X ii + i−1 j=1 (U ij − U ij+1 )X ij ≥ U zz X zz + z−1 j=1 (U zj − U zj+1 )X zj , for i = 2, . . . , n; z = 1, . . . , i − 1. The big-M formulation to handle disjoint constraints in (7a -7c) requires an effective bound, we suggest M = max({|U ij | : i = 1, . . . , n : i ≥ j}). Similarly, for the valley trend we include constraints and X ii = 1 and X zz = 1 =⇒M (y i + y z ) + U ii X ii + i−1 j=1 (U ij − U ij+1 )X ij ≥ U zz X zz + z−1 j=1 (U zj − U zj+1 )X zj , M (2 − y i − y z ) + U zz X zz + z−1 j=1 (U zj − U zj+1 )X zj , ≥ U ii X ii + i−1 j=1 (U ij − U ij+1 )X ij , for i = 2, . . . , n; z = 1, . . . , i − 1. Additional constraints Maximum p-value constraint. Similar to the binary target case, we can impose a maximum p-value constraint between consecutive bins to ensure that their means are statistically different. In this case, the T-test for means is appropriate, although it requires the standard deviation. To compute an aggregate matrix of standard deviations, SD ij , we could use SD ij = j z=i ss z j z=i r z − j z=i s z j z=i r z 2 , where ss i is the sum of squared target values for each pre-bin. Finally, the algorithm and procedure described in Section 2.1.2 to incorporate these constraints can be readily reused. Mixed-integer programming formulation for multi-class target A simple approach to support a multi-class target is to use the one-vs-rest scheme with n C distinct classes. This scheme consists of building a binary target for each class. The resulting mathematical formulation closely follows the formulation for binary target, max X n C c=1 n i=1 V c ii X ii + i−1 j=1 (V c ij − V c ij+1 )X ij (19a) s.t. (5b -5e) (19b) Note that for this formulation we need an aggregated matrix V and D for each class c. It is important to emphasize that the monotonicity constraints in Section 2.1.1 act as linking constraints among classes, otherwise, n C optimal binning problems with binary target could be solved separately. Again, additional constraints described in Section 2.1.2 can be naturally incorporated with minor changes. 3 Algorithmic details and implementation Automatic monotonic trend algorithm Our approach to automate the monotonic trend decision employs a Machine Learning (ML) classifier that predicts, given the pre-binning data, the most suitable monotonic trend to maximize discriminatory power. In particular, we aim to integrate an off-line classifier, hence we are merely interested in ML classification algorithms easily embeddable. Recently, a similar approach was implemented in the commercial solver CPLEX to make automatic decisions over some algorithmic choices [19]. For this study, a dataset is generated with 417 instances collected from public datasets. We design a set of 16 numerical features, describing the pre-binning instances in terms of number of pre-bins, distribution of records per pre-bin and trend features. The most relevant trend features are: number of trend change points, linear regression coefficient sense, area of the convex hull, and area comprised among extreme trend points. The labeling procedure consists of solving all instances selecting the ascending (A), descending (D), peak (P) and valley (V) monotonic trend. Concave and convex trends are discarded due to being a special case of peak and valley, respectively. In what follows, without loss of generality, we state the procedure for the binary target case: if the relative difference between IV with ascending/descending monotonic trend and IV with peak/valley trend is less than 10%, the ascending/descending monotonic trend is selected, due to the lesser resolution times. Table 1 summarizes the composition of the dataset with respect to assigned labels. We note that the dataset is slightly unbalanced, being predominant the descending label (D). To perform experiments, the dataset is split into train and test subsets in a stratified manner to treat unbalanced data. The proportion of the test set is 30%. Three interpretable multi-class classification algorithms are tested, namely, logistic regression, decision trees (CART) and Support vector Machine (SVM) using the Python library Scikit-learn [20]. All three algorithms are trained using option class weight="balanced". Throughout the learning process, we discard 8 features and perform hyperparameter optimization for all three algorithms. These experiments show that SVM and CART have similar classification measures, and we decide to choose CART (max depth 5) to ease implementation. On the test set, the trained CART has a weighted average accuracy, precision and recall of 88%. See classification measures and the confusion matrix in Table 2. We observe that various instances of the minority class (V) are misclassified, indicating that more instances or new features might be required to improve classification measures. Improving this classifier is part of ongoing research. Presolving algorithm The mathematical programming formulation is a hard combinatorial optimization problem that does not scale well as the number of pre-bins increases. To reduce solution times we need to reduce the search space to avoid deep tree searches during branching. The idea is to develop a presolving algorithm to fix bins not satisfying monotonicity constraints, after that the default presolver may be able to reduce the problem size significantly. The presolving algorithm applies to the binary target case and was developed after several observations about the aggregated matrix of event rates D. Algorithm 2 shows the implemented approach for the ascending monotonicity trend. Presolving algorithm for the descending monotonicity is analogous, only requiring inequalities change. Algorithm 2 Preprocessing ascending monotonicity Binning quality score To assess the quality of binning for binary target, we develop a binning quality score considering the following aspects: • Predictive power: IV rule of thumb [21] in Table 3. • Statistical significance: bin event rates must be statistically different, therefore large p-values penalize the quality score. • Homogeneity: binning with homogeneous bin sizes or uniform representativeness, increases reliability. To account for all these aspects, we propose a rigorous binning quality score function Proposition 3.1 Given a binning with Information Value ν, p-values between consecutive bins p i , i = 1, . . . , n − 1 and normalized bins size s i , i = 1, . . . , n, the binning quality score function is defined as Q(ν, p, s) = ν c exp −ν 2 /(2c 2 ) + 1/2 n−1 i=1 (1 − p i ) 1 − n i=1 s 2 i 1 − 1/n ,(20) where Q(ν, p, s) ∈ [0, 1] and c = 1 5 2 log(5/3) is the best a priori IV value in [0.3, 0.5). Proof: Given the rule of thumb in Table 3, let us consider the set of statistical distributions with positive skewness, positive fat-tail, and support on the semi-infinite interval [0, ∞). The function should penalize large values of Information Value ν, and fast decay is expected after a certain threshold indicating over-prediction. This fast decay is a required property that must be accompanied by the following statement: lim ν→0 f (ν) = lim ν→∞ f (ν) = 0 Among the available distributions satisfying aforementioned properties, we select the Rayleigh distribution, which probability density function is given by f (ν; c) = ν c 2 e −ν 2 /(2c 2 ) , ν ≥ 0. This is a statistical distribution, not a function, hence we need a scaling factor so that max ν∈[0,∞) f (ν; c) = 1: the maximum value of a unimodal probability distribution is the mode c, thus γ = f (c, c) = 1 c √ e =⇒ f (ν, c) γ = ν exp −ν 2 /(2c 2 ) + 1/2 c . The optimal c such that f (a) = f (b) for b > a can be obtained by solving f (b; c)−f (a; c) = 0 for c, which yields c * = √ b 2 − a 2 2 log(b/a) . Term n−1 i=1 (1 − p i ) assesses the statistical significance of the bins. Furthermore, term 1− n i=1 s 2 i 1−1/n = 1−HHI * , where HHI * is the normalized Herfindahl Hirschman Index, assesses the homogeneity/uniformity of the bin sizes. For example, if we consider that the boundaries of the interval with strong IV predictive power in Table 3, a = 0.3 and b = 0.5, should produce the same quality score, c * (a, b) = Table 4 shows the value of f (ν, c * ) for various IV values ν; note the fast decay of f (ν, c * ) when ν > 0.5. Implementation The presented mathematical programming formulations are implemented using Google OR-Tools [16] with the open-source MILP solver CBC [7], and Google's BOP and CP-SAT solvers. Besides, the specialized formulation in Section 2.1.3 is implemented using the commercial solver LocalSolver [1]. The python library OptBinning 3 has been developed throughout this work to ease usability and reproducibility. Much of the implementation effort focuses on the careful implementation of constraints and the development of fast algorithms for preprocessing and generating the model data. A key preprocessing algorithm is a pre-binning refinement developed to guarantee that no bins have 0 non-events and events records in the binary target case. Categorical variables require special treatment: pre-bins are ordered in ascending order with respect to a given metric; the event rate for binary target and the target mean for a continuous target. The original data is replaced by the ordered indexes and is then used as a numerical (ordinal) variable. Furthermore, during preprocessing, the non-representative categories may be binned into an "others" bin. Similarly, missing values and special values are incorporated naturally as additional bins after the optimal binning is terminated. Experiments The experiments were run on an Intel(R) Core(TM) i5-3317 CPU at 1.70GHz, using a single core, running Linux. Two binning examples are shown in Tables 5 and 6, using Fair Isaac (FICO) credit risk dataset [6] (N = 10459) and Home Credit Default Risk Kaggle competition dataset [9] (N = 307511), respectively. Example in Table 5 uses the variable AverageMInFile (Average Months in File) as an risk driver. FICO dataset imposes monotonicity constraints to some variables, in particular, for this variable, the event rate must be monotonically decreasing. Moreover, the dataset includes three special values/codes defined as follows: For the sake of completeness, we also include a few random missing values on the dataset. As shown in Table 5, these values are separately treated by incorporating a Special and Missing bin. Regarding computation time, this optimal binning instance is solved in 0.08 seconds. The optimization time accounts for 91% of the total time, followed by the pre-binning time representing about 6%. The remaining 3% is spent in pre-processing and post-processing operations. Example in Table 6 uses the categorical variable ORGANIZATION TYPE from the Kaggle dataset. This variable has 58 categories, and we set the non-representative categories cut-off to 0.01. Note that the bin just before the Special bin corresponds to the bin with non-representative categories, which is excluded from the optimization problem, hence monotonicity constraint does not apply. This optimal binning instance is solved in 0.25 seconds. For categorical variables, most of the time is spent on pre-processing, 71% in this particular case, whereas the optimization problem is solved generally faster. Benchmark CP/MIP vs local search heuristic For large instances, we compare the performance of Google OR-Tools' solvers BOP (MIP) and CP-SAT against LocalSolver. For these tests we select two variables from Home Credit Default Risk Kaggle competition dataset [9] (N = 307511). We aim to perform a far finer binning than typical in many applications to stress the performance of classical solvers for large combinatorial optimization problems. Tables 7 and 8 show results for varying number of pre-bins n and monotonic trends. In test 1 from Table 7 , LocalSolver does not improve after 10 seconds, not being able to reduce the optimality gap. In test 2, LocalSolver outperforms CP-SAT, finding the optimal solution after 5 seconds, 28x faster. In test 3, solution times are comparable. Results reported in Table 8 are also interesting; in test 1, BOP and CP-SAT solvers cannot find an optimal solution after 1000 seconds. LocalSolver finds the best found feasible solution after 30 seconds. Nevertheless, we recall that the described heuristic for peak/valley trend introduced in Section 2.1.1 could reduce resolution times substantially, obtaining times comparable to those when choosing ascending/descending monotonic trend. Conclusions We propose a rigorous and flexible mathematical programming formulation to compute the optimal binning. This is the first optimal binning algorithm to achieve solutions for nontrivial constraints, supporting binary, continuous and multi-class target, and handling several monotonic trends rigorously. Importantly, the size of the decision variables and constraints used in the presented formulations is independent of the size of the datasets; they are entirely controlled by the starting solution computed during the pre-binning process. In the future, we plan to extend our methodology to piecewise-linear binning and multivariate binning. Lastly, the code is available at https://github.com/guillermo-navas-palencia/ optbinning to ease reproducibility. Figure 1 : 1Lower triangular matrix X. Initial solution after pre-binning (left). Optimal solution with 4 bins after merging pre-bins (right). Algorithm 1 1Maximum p-value constraint using Z-test 1: procedure p-value violation indices(n, R N E , R E , α) Figure 2 : 2General violation constraint between two feasible solutions. Case j = 1 (left). Figure 3 : 3New decision variables suitable for counting consecutive zeros. j = 1, . . . , n − i − 1 do 7: if D i+1+j,i − D i+1+j,i+1+j > 0 then (ν, c * ) 0 0.083 0.404 0.938 0.938 0.610 0.282 0.171 0.005 • - 9 : 9No Bureau Record or No Investigation • -8: No Usable/Valid Trades or Inquiries • -7: Condition not Met (e.g. No Inquiries, No Delinquencies) Table 1 : 1Number of instances and percentage for each label. Table 2 : 2Classification measures (left) and confusion matrix (right) for CART on the test set. Table 3 : 3Information Value rule of thumb. Table 4 : 4Function values for various ν values. Bin Count Count (%) Non-event Event Event rateWoE IV JS (−∞, 30.5) 544 0.052013 99 445 0.818015 -1.41513 0.087337 0.010089 [30.5, 48.5) 1060 0.101348 286 774 0.730189 -0.907752 0.076782 0.009281 [48.5, 54.5) 528 0.050483 184 344 0.651515 -0.537878 0.014101 0.001742 [54.5, 64.5) 1099 0.105077 450 649 0.590537 -0.278357 0.008041 0.001002 [64.5, 70.5) 791 0.075629 369 422 0.533502 -0.046381 0.000162 0.000020 [70.5, 74.5) 536 0.051248 262 274 0.511194 0.0430441 0.000095 0.000012 [74.5, 81.5) 912 0.087198 475 437 0.479167 0.171209 0.002559 0.000320 [81.5, 101.5) 2009 0.192083 1141 868 0.432056 0.361296 0.025000 0.003108 [101.5, 116.5) 848 0.081078 532 316 0.372642 0.608729 0.029532 0.003636 [116.5, ∞) 1084 0.103643 702 382 0.352399 0.696341 0.049039 0.006009 Special 558 0.053351 252 306 0.548387 -0.106328 0.000601 0.000075 Missing 490 0.046850 248 242 0.493878 0.112319 0.000592 0.000074 Table 5 : 5Example optimal binning using variable AverageMInFile from FICO dataset.Bin Count Count (%) Non-event Event Event rate WoE IV JS [XNA, School] 64267 0.208991 60751 3516 0.054709 0.416974 0.030554 0.003792 [Medicine, ...] 31845 0.103557 29673 2172 0.068205 0.182104 0.003182 0.000397 [Other] 16683 0.054252 15408 1275 0.076425 0.0594551 0.000187 0.000023 [Business ...] 16537 0.053777 15150 1387 0.083873 -0.0416281 0.000095 0.000012 [Transport: ...] 81221 0.264124 73657 7564 0.093129 -0.156466 0.006905 0.000862 [Security, ...] 55150 0.179343 49424 5726 0.103826 -0.277067 0.015465 0.001927 [Housing, ...] 41808 0.135956 38623 3185 0.076182 0.0629101 0.000524 0.000065 Special 0 0 0 0 0 0 0 0 Missing 0 0 0 0 0 0 0 0 Table 6 : 6Example optimal binning using categorical variable ORGANIZATION TYPE from Home Credit Default Risk Kaggle competition dataset. Table 7 : 7Variable REGION POPULATION RELATIVE. Performance comparison Google OR-Tools' CP-SAT vs LocalSolver. Time in seconds.n monotonic trend solver variables constraints time solution gap 100 peak cp 5151 15150 t 0.11721972 - 100 peak mip 5151 15150 t 0.11786335 * - 100 peak ls 401 10101 1 0.11666556 1.0% 100 peak ls 401 10101 5 0.11735812 0.4% 100 peak ls 401 10101 10 0.11771822 0.1% 100 peak ls 401 10101 30 0.11786335 0.0% 100 ascending cp 5050 10933 2.3 0.05175782 - 100 ascending ls 300 5000 1 0.05175782 0.0% Table 8 : 8Variable DAYS EMPLOYED. Performance comparison Google OR-Tools' CP-SAT/BOP vs LocalSolver. Time in seconds. *: Best feasible solution. t: 1000 seconds exceeded. To the author's knowledge, at the time of writing, these tools are restricted to the problem of discretizing a variable with respect to a binary target. 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Pro- ceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 186(1007):453-461, 1946. A difficult MIP construct: counting consecutive 1's. E Kalvelagen, E. Kalvelagen. A difficult MIP construct: counting consecutive 1's. LightGBM: A Highly Efficient Gradient Boosting Decision Tree. G Ke, Q Meng, T Finley, T Wang, W Chen, W Ma, Q Ye, T Liu, Advances in Neural Information Processing Systems. I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. GarnettCurran Associates, Inc30G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, and T. Liu. LightGBM: A Highly Efficient Gradient Boosting Decision Tree. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 3146-3154. Curran Associates, Inc., 2017. R Kerber, Chimerge, Discretization of Numeric Attributes. AAAI-92 Proceedings. R. Kerber. ChiMerge: Discretization of Numeric Attributes. AAAI-92 Proceedings, 1992. On Information and Sufficiency. S Kullback, R A Leibler, Ann. Math. Statist. 221S. Kullback and R. A. Leibler. On Information and Sufficiency. Ann. Math. Statist., 22(1):79-86, 1951. P Laurent, F Vincent, Google OR-Tools. P. Laurent and F. Vincent. Google OR-Tools 7.7. https://developers.google.com/ optimization/, 2020. Monotone optimal binning algorithm for credit risk modeling. P Mironchyk, V Tchistiakov, P. Mironchyk and V. Tchistiakov. Monotone optimal binning algorithm for credit risk modeling. 2017. Rigorous Constrained Optimization Binning for Credit Scoring. I Oliveira, M Chari, S Haller, SAS Global Forum 2008 -Data Mining and Predictive Modelling. I. Oliveira, M. Chari, and S. Haller. Rigorous Constrained Optimization Binning for Credit Scoring. SAS Global Forum 2008 -Data Mining and Predictive Modelling, 2008. Learning a Classification of Mixed-Integer Quadratic Programming Problems. A Lodi, P Bonami, G Zarpellon, Integration of Constraint Programming. W.-J. van HoeveSpringer-VerlagLecture Notes in Computer ScienceA. Lodi P. Bonami and G. Zarpellon. Learning a Classification of Mixed-Integer Quadratic Programming Problems. In W.-J. van Hoeve, editor, Integration of Con- straint Programming, Artificial Intelligence, and Operations Research -CPAIOR 2018, Lecture Notes in Computer Science, pages 595-604. Springer-Verlag, 2018. Scikit-learn: Machine Learning in Python. F Pedregosa, G Varoquaux, A Gramfort, V Michel, B Thirion, O Grisel, M Blondel, P Prettenhofer, R Weiss, V Dubourg, J Vanderplas, A Passos, D Cournapeau, M Brucher, M Perrot, E Duchesnay, Journal of Machine Learning Research. 12F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine Learning in Python . Journal of Machine Learning Research, 12:2825-2830, 2011. Credit Risk Scorecards: Developing And Implementing Intelligent Credit Scoring. N Siddiqi, Wiley and SAS Business Series. WileyN. Siddiqi. Credit Risk Scorecards: Developing And Implementing Intelligent Credit Scoring. Wiley and SAS Business Series. Wiley, 2005. Credit Scoring and Its Applications. L C Thomas, D B Edelman, J N Crook, Society for Industrial and Applied Mathematics. L. C. Thomas, D. B. Edelman, and J. N. Crook. Credit Scoring and Its Applications. Society for Industrial and Applied Mathematics, 2002. Monotonic Optimal Binning (MOB) for Risk Scorecard Development. L Wensui, L. WenSui. Monotonic Optimal Binning (MOB) for Risk Scorecard Development, 2020. Metric Divergence Measures and Information Value in Credit Scoring. G Zeng, Journal of Mathematics. G. Zeng. Metric Divergence Measures and Information Value in Credit Scoring. Journal of Mathematics, 2013:1-10, 2013.
[ "https://github.com/guillermo-navas-palencia/", "https://github.com/guillermo-navas-palencia/optbinning" ]
[ "Axion Star Explosions: A New Source for Axion Indirect Detection", "Axion Star Explosions: A New Source for Axion Indirect Detection" ]
[ "Miguel Escudero \nTheoretical Physics Department\nCERN\n1211Geneva 23Switzerland\n", "Charis Kaur Pooni \nPhysics\nKing's College London\nWC2R 2LSStrand, LondonUK\n", "Malcolm Fairbairn \nPhysics\nKing's College London\nWC2R 2LSStrand, LondonUK\n", "Diego Blas \nGrup de Física Teòrica\nDepartament de Física\nUniversitat Autònoma de Barcelona\n08193BellaterraSpain\n\nInstitut de Fisica d'Altes Energies (IFAE)\nThe Barcelona Institute of Science and Technology\nCampus UAB08193Bellaterra (Barcelona)Spain\n", "Xiaolong Du \nCarnegie Observatories\n813 Santa Barbara Street91101PasadenaCAUSA\n\nDepartment of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA\n", "David J E Marsh \nPhysics\nKing's College London\nWC2R 2LSStrand, LondonUK\n" ]
[ "Theoretical Physics Department\nCERN\n1211Geneva 23Switzerland", "Physics\nKing's College London\nWC2R 2LSStrand, LondonUK", "Physics\nKing's College London\nWC2R 2LSStrand, LondonUK", "Grup de Física Teòrica\nDepartament de Física\nUniversitat Autònoma de Barcelona\n08193BellaterraSpain", "Institut de Fisica d'Altes Energies (IFAE)\nThe Barcelona Institute of Science and Technology\nCampus UAB08193Bellaterra (Barcelona)Spain", "Carnegie Observatories\n813 Santa Barbara Street91101PasadenaCAUSA", "Department of Physics and Astronomy\nUniversity of California\n90095Los AngelesCAUSA", "Physics\nKing's College London\nWC2R 2LSStrand, LondonUK" ]
[]
If dark matter is composed of axions, then axion stars form in the cores of dark matter halos. These stars are unstable above a critical mass, decaying to radio photons that heat the intergalactic medium, offering a new channel for axion indirect detection. We recently provided the first accurate calculation of the axion decay rate due to axion star mergers. In this work we show how existing data concerning the CMB optical depth leads to strong constraints on the axion photon coupling in the mass range 10 −14 eV ma 10 −8 eV. Axion star decays lead to efficient reionization of the intergalactic medium during the dark ages. By comparing this non-standard reionization with Planck legacy measurements of the Thompson optical width, we show that couplings in the range 10 −14 GeV −1 gaγγ 10 −10 GeV −1 are excluded for our benchmark model of axion star abundance. Future measurements of the 21cm emission of neutral hydrogen at high redshift could improve this limit by an order of magnitude or more, providing complementary indirect constraints on axion dark matter in parameter space also targeted by direct detection haloscopes.Numerical simulations of axion-like dark matter cosmologies have shown that a solitonic core forms in the center of every dark matter halo, see[1,2]for the first simulations and[3][4][5][6][7]for further studies. In Ref.[8] we have explicitly calculated the number and energy density evolution of axion stars from hierarchical structure formation using semi-analytic models. Ref.[2] demonstrated a power law relation between the mass of an axion star and its host halo:where z is the redshift, M h is the virial halo mass, α is a power law exponent, and [2]:where m a the axion-like dark matter mass and M min is the smallest halo mass that could host a soliton of mass M S at a given redshift (given by the Jeans scale) [9, 10].Ref.[2] found a core-halo mass relation of the form Refs. [7,11]suggest that there is an intrinsic diversity in the core-halo mass relation. Importantly, all numerical simulations of axion-like dark matter cosmologies show that the core-halo mass relation is bounded between α = 1/3 and α = 3/5. The Monte Carlo merger tree model we employed in Ref.[8]shows that an average slope α = 2/5 captures this diversity well for both the soliton mass function and, more importantly, the merger rate. InFigure 1we show isocontours of constant M h for various values of M S and m a for the case α = 2/5. For the other cases we refer the reader to Appendix A. FIG. 1. In blue dashed: isocontours of halo masses that host a given axion star mass at redshift z = 20 depending upon the value of the axion mass for the benchmark core halo mass relation MS ∝ M 2/5 h . In red we show isocontour lines of critical axion stars (M decay S ) as a function of several gaγγ values.Soliton Explosions. Solitonic cores, also referred to as axion stars have ultrahigh phase-space occupation numbers that can trigger collective processes which cannot occur in vacuum. Parametric resonance can lead to exponentially fast decays of axion stars into photons[12][13][14][15]or into relativistic axions[16][17][18], meaning axion stars become unstable above a certain mass[12,15](vector dark matter solitons have a similar instability [19]): M decay S 8.4 × 10 −5 M 10 −11 GeV −1 g aγγ 10 −13 eV m a , (3) arXiv:2302.10206v1 [hep-ph] 20 Feb 2023 2 FIG. 2. Schematic of reionization caused by axion star explosions. Appreciable formation of dark matter halos hosting sufficiently massive axion stars occurs for z 100, see [8]. When an axion star decays, it releases a huge number of low-energy photons, which are absorbed by inverse Bremsstrahlung, leading to heating of the IGM. If the IGM becomes hot enough, reionization occurs. Energetically, reionization requires a fraction of around f decay DM≈ 10 −9 of the dark matter to decay. For the lowest mass axion-like particles, ma 5 × 10 −13 eV, axion star decay is kinematically blocked at early times, leading to a population of super critical stars which decay all at once in a burst once the plasma frequency falls low enough to allow the decay. This leads to patchy reionization. At higher axion masses, plasma blocking is not efficient at the relevant redshifts, and instead super-critical stars formed by major mergers decay immediately as they form, leading to a more uniform and continuous reionization history.where g aγγ is the axion-photon coupling. Super-critical solitons explode into photons with E γ = m a /2 in a timescale given by the light crossing time of the soliton (in units c = 1):The energy released in the explosion of a supercritical soliton is E = M S − M decay S . There are, however, two factors that could prevent this explosion: 1) If the axion-like particle possesses self interactions, then axion stars can decay into axions leading a "Bosenova" above a critical mass M B−Nova set by the quartic self coupling λ[16][17][18][20][21][22]. For a quartic coupling typical of a cosine instanton potential, λ = m 2 a /f 2 a , and axion-photon coupling g aγγ α EM /f a , one finds that M decay S > M B−Nova . This means that for nominal values of g aγγ the axion stars will not actually decay into photons but rather into relativistic axions, with possible limits from such a decay explored in Ref.[23]. However, there are many models that feature significantly enhanced g aγγ interactions, see e.g.[24], or suppressed quartic interactions, see e.g.[25,26]. In what follows, we will consider that the relevant effect is decay into photons and thus our results will apply to scenarios such as those previously mentioned. It is also possible that the initial contraction phase of a Bosenova may itself trigger parametric resonance decay to photons[12].2) In the parameter space of interest for our study, axion stars are always hosted by halos of mass M h < 10 5 M which means that they lie below the baryonic Jeans scale and are therefore surrounded by an intergalactic medium (IGM) with average properties. The latter contains ionized particles, which could kinematically block the axion decay by plasma effects. More concretely, the photon gets an effective mass-squared that is proportional to the number of free electrons in the plasma[27]:Axion star decay is blocked until the plasma frequency drops to ω p (z decay ) < m a /2, resulting in an explosion of all the axion stars with M S > M decay S at specific redshift:z decay 32 m a 10 −13 eV 2/3 − 1 .Once the plasma frequency has dropped such that parametric resonance decay is no longer blocked, then supercritical axion stars will explode as soon as they are formed. In Ref.[8] we calculated the cosmological evolution of the number/energy density of critical axion stars using the extended Press-Schechter formalism and Monte Carlo merger trees. We considered the case that an axion star will only explode either if it is plasma blocked until it is super critical, or if it is formed by a major merger defined by the merger of two axion stars of comparable mass, such that mass increase happens rapidly.Soliton explosions lead to axion dark matter decay, and can inject energy into the IGM comparable to or greater than energy emission from astrophysical processes, such as core collapse supernovae[8], and thus offer a new method of indirect detection of axions.3 Heating the IGM.Figure 2outlines qualitatively the mechanism just described: axion stars start to form and grow when halos form and merge, which happens appreciably only at z 100. Once an axion star becomes massive enough and the plasma frequency is low enough, parametric resonance can take place and the star rapidly explodes into low energy photons of E γ = m a /2. These photons are absorbed by the IGM, heat it, and in turn via collisional ionizations lead to an increased ionization fraction of the Universe.Axion star decay results in the release of a huge number of low energy photons with E γ = m a /2. We will be interested in m a < 10 −8 eV, whose associated low energy photons are efficiently absorbed via inverse Bremsstrahlung on ionized particles in the intergalactic medium, namely via γ e − p → e − p processes[28]. The net result of this absorption is to heat the IGM. In turn, once the temperature of the IGM goes above T ∼ 1 eV, collisional ionization processes e − H → 2e − p ionize the Universe. As a result, the decay of axion stars may result into a period of early reionization, which is strongly constrained by Planck legacy data[29,30]. This forms the basis of the constraints derived in this paper, see also[31][32][33][34][35][36]for constraints of this type for other dark matter candidates.Importantly, depending upon the absorption length scale of the photons produced by axion stars the effect on the IGM can be different. For m a 5 × 10 −13 eV the photons are absorbed within very small volumes and this generates a shockwave very much like the one formed by supernova explosions as the amount of energy released is very similar[37]. On the other hand, for m a 5 × 10 −13 eV the absorption length-scale is larger than the inter-separation of axion stars results in a homogeneous cosmological increase in the global IGM temperature. It is important to highlight that in principle a fraction as small as f DM ∼ 3 × 10 −9 13.6 eV n b /ρ DM of dark matter converted into heat in the IGM would be enough to fully reionize the Universe. Methodology. We model the IGM temperature by including the heating generated by axion star decays and accounting for Compton and adiabatic cooling. Collisional ionization dominates in our scenario which we implement using the fitted rates in[38]. We then track the free electron density x e using the effective 3-level atom approximation[39,40], including free protons and He + . We use the same rates and expressions as the recombination code RECFAST including their fudge factor[41,42]-modulo that we change T b → T γ in the terms in Eqs.(1) and (2) of [42] that account for photon ionization as the approximation T b T γ breaks down for z < 200. We have explicitly checked our numerical code against the results of [34] that studies decaying dark matter into very low energy photons such as the ones we consider. Finally, in the regime m a 5 × 10 −13 eV the photons from axion star explosions are absorbed at very small scales and generate shockwaves which ionize small patches of the Universe. We track such explosions until they are unable to ionize more IGM. We follow standard methods used for supernova remnants[37,[43][44][45], and find results very close to those expected from the Sedov-Taylor solutions[46,47].Constraints from the CMB Optical Depth. We derive conservative constraints on reionization driven by axion star explosions by using the 95% upper limit on the integrated Thompson optical depth to reionization from Planck CMB observations: τ reio < 0.068[29,30]. The optical depth is measured by joint analysis of CMB temperature, polarisation, and baryon acoustic oscillations in the galaxy correlation function. A prior on the redshift of reionization z reio ≥ 6.5 is adopted to enforce consistency with the Gunn-Peterson optical depth[48].The axion parameter space excluded at 2σ confidence as a result of this constraint is shown in the two regions in dark red inFigure 3, and the resulting evolution of the ionization fraction and IGM temperature for representative models is shown inFigure 4. Constraints in the region m a 5×10 −13 eV are a result of patchy reionization generated by the decay of all critical axion stars in the Universe once the axion decay is kinematically allowed, namely when ω p (z decay ) = m a /2. For m a 5 × 10 −13 eV the photon plasma mass is small compared to the axion mass, and the constraints arise instead from supercritical axion stars formed via mergers providing continuous heating and uniform reionization. The intermediate region of parameter space is constrained only at 1σ. The intermediate region leads to a smaller increase in τ because when some reionization takes place the photon plasma mass grows and blocks further axion star decays. This can explicitly be seen in the green curve of the upper panel ofFigure 4. The intermediate region of parameter space will be further tested in the near future. For example, LiteBIRD will provide a cosmic variance limited measurement of the large scale EE polarisation spectrum, thereby significantly improving the determination of τ reio[49].We note that the Planck constraints do not extend to arbitrarily large g aγγ -the larger the value of g aγγ , the smaller the mass of the halos that host critical soliton stars. In axion-like dark matter cosmologies structure formation is suppressed at M h < M min , see Eq.(2), and couplings g aγγ 10 −10 correspond to axion stars hosted in halos whose abundance is suppressed. In addition, we notice that the bounds change their shape for m a 5 × 10 −11 eV. For this region of parameter space the photon absorption efficiency in the IGM is smaller than one and this weakens the constraints.Lyman-α and Spectral Distortions. We considered two other existing measurements that can limit this scenario for IGM heating. Since axion star explosions occur when Compton scattering is inefficient (z 10 4 ), they generate y-type distortions of the CMB energy spectrum. The change to the photon energy density from such an effect must satisfy δρ γ /ρ γ 5 × 10 −5 based on COBE/FIRAS measurements[34,55], which trans-4 FIG. 3. Parameter space excluded by Planck measurements of the Thompson optical depth. We highlight also reach of future 21cm surveys and constraints from X/γ-Ray observations [50-52] and CAST[53]. The evolution of the free electron fraction and the baryon temperature is shown for the points highlighted by colored symbols inFigure 4.lates to f decay DM 2 × 10 −7 for decaying DM: significantly weaker than the Planck optical depth constraint. At redshifts 2 z 7 the Lyman-α forest can measure the IGM temperature see e.g.[56][57][58]. The IGM temperature at these redshifts satisfies T b 1.5 × 10 4 K and since Compton cooling is the main cooling process (which we include in our calculations), this means that temperature histories where axion star decay causes T b to exceed this threshold would be ruled out by these data. This is shown as the upper contour inFigure 3, with an accompanying T b (z) inFigure 4(bottom panel). Lyman-α constraints are not competitive with the Planck optical depth constraint.21cm Cosmology. Measurements of the hyperfine 21cm transition are expected to revoluzionize our understanding of the cosmic dawn and the epoch of reionization, see[59,60]for reviews. There have already been several interesting bounds on the emission power of the 21cm line at various redshifts [61, 62], and even a putative detection [63] (which is disputed see, e.g. [62]). The observational status of the cosmological 21cm line is still in its infancy but it is expected that ongoing and upcoming experiments such as HERA [64] and SKA [65] among others should be able to robustly detect the 21cm line and provide a view of the thermal state of the Universe at redshifts 4 z 30 [66], see [67] for a compilation of experiments targeting this line.As highlighted inFigure 4, axion star explosions greatly enhance the IGM temperature at the redshifts where the 21cm line will be targeted z 35. In particular, the rise of T b at large redshifts z 20 is a feature that is not easy to achieve astrophysically, see e.g.[68]. The sky-averaged 21 cm brightness temperature can be 10 −4
null
[ "https://export.arxiv.org/pdf/2302.10206v1.pdf" ]
257,050,478
2302.10206
33bb344360496d0353535f74fb25aa282824714d
Axion Star Explosions: A New Source for Axion Indirect Detection Miguel Escudero Theoretical Physics Department CERN 1211Geneva 23Switzerland Charis Kaur Pooni Physics King's College London WC2R 2LSStrand, LondonUK Malcolm Fairbairn Physics King's College London WC2R 2LSStrand, LondonUK Diego Blas Grup de Física Teòrica Departament de Física Universitat Autònoma de Barcelona 08193BellaterraSpain Institut de Fisica d'Altes Energies (IFAE) The Barcelona Institute of Science and Technology Campus UAB08193Bellaterra (Barcelona)Spain Xiaolong Du Carnegie Observatories 813 Santa Barbara Street91101PasadenaCAUSA Department of Physics and Astronomy University of California 90095Los AngelesCAUSA David J E Marsh Physics King's College London WC2R 2LSStrand, LondonUK Axion Star Explosions: A New Source for Axion Indirect Detection If dark matter is composed of axions, then axion stars form in the cores of dark matter halos. These stars are unstable above a critical mass, decaying to radio photons that heat the intergalactic medium, offering a new channel for axion indirect detection. We recently provided the first accurate calculation of the axion decay rate due to axion star mergers. In this work we show how existing data concerning the CMB optical depth leads to strong constraints on the axion photon coupling in the mass range 10 −14 eV ma 10 −8 eV. Axion star decays lead to efficient reionization of the intergalactic medium during the dark ages. By comparing this non-standard reionization with Planck legacy measurements of the Thompson optical width, we show that couplings in the range 10 −14 GeV −1 gaγγ 10 −10 GeV −1 are excluded for our benchmark model of axion star abundance. Future measurements of the 21cm emission of neutral hydrogen at high redshift could improve this limit by an order of magnitude or more, providing complementary indirect constraints on axion dark matter in parameter space also targeted by direct detection haloscopes.Numerical simulations of axion-like dark matter cosmologies have shown that a solitonic core forms in the center of every dark matter halo, see[1,2]for the first simulations and[3][4][5][6][7]for further studies. In Ref.[8] we have explicitly calculated the number and energy density evolution of axion stars from hierarchical structure formation using semi-analytic models. Ref.[2] demonstrated a power law relation between the mass of an axion star and its host halo:where z is the redshift, M h is the virial halo mass, α is a power law exponent, and [2]:where m a the axion-like dark matter mass and M min is the smallest halo mass that could host a soliton of mass M S at a given redshift (given by the Jeans scale) [9, 10].Ref.[2] found a core-halo mass relation of the form Refs. [7,11]suggest that there is an intrinsic diversity in the core-halo mass relation. Importantly, all numerical simulations of axion-like dark matter cosmologies show that the core-halo mass relation is bounded between α = 1/3 and α = 3/5. The Monte Carlo merger tree model we employed in Ref.[8]shows that an average slope α = 2/5 captures this diversity well for both the soliton mass function and, more importantly, the merger rate. InFigure 1we show isocontours of constant M h for various values of M S and m a for the case α = 2/5. For the other cases we refer the reader to Appendix A. FIG. 1. In blue dashed: isocontours of halo masses that host a given axion star mass at redshift z = 20 depending upon the value of the axion mass for the benchmark core halo mass relation MS ∝ M 2/5 h . In red we show isocontour lines of critical axion stars (M decay S ) as a function of several gaγγ values.Soliton Explosions. Solitonic cores, also referred to as axion stars have ultrahigh phase-space occupation numbers that can trigger collective processes which cannot occur in vacuum. Parametric resonance can lead to exponentially fast decays of axion stars into photons[12][13][14][15]or into relativistic axions[16][17][18], meaning axion stars become unstable above a certain mass[12,15](vector dark matter solitons have a similar instability [19]): M decay S 8.4 × 10 −5 M 10 −11 GeV −1 g aγγ 10 −13 eV m a , (3) arXiv:2302.10206v1 [hep-ph] 20 Feb 2023 2 FIG. 2. Schematic of reionization caused by axion star explosions. Appreciable formation of dark matter halos hosting sufficiently massive axion stars occurs for z 100, see [8]. When an axion star decays, it releases a huge number of low-energy photons, which are absorbed by inverse Bremsstrahlung, leading to heating of the IGM. If the IGM becomes hot enough, reionization occurs. Energetically, reionization requires a fraction of around f decay DM≈ 10 −9 of the dark matter to decay. For the lowest mass axion-like particles, ma 5 × 10 −13 eV, axion star decay is kinematically blocked at early times, leading to a population of super critical stars which decay all at once in a burst once the plasma frequency falls low enough to allow the decay. This leads to patchy reionization. At higher axion masses, plasma blocking is not efficient at the relevant redshifts, and instead super-critical stars formed by major mergers decay immediately as they form, leading to a more uniform and continuous reionization history.where g aγγ is the axion-photon coupling. Super-critical solitons explode into photons with E γ = m a /2 in a timescale given by the light crossing time of the soliton (in units c = 1):The energy released in the explosion of a supercritical soliton is E = M S − M decay S . There are, however, two factors that could prevent this explosion: 1) If the axion-like particle possesses self interactions, then axion stars can decay into axions leading a "Bosenova" above a critical mass M B−Nova set by the quartic self coupling λ[16][17][18][20][21][22]. For a quartic coupling typical of a cosine instanton potential, λ = m 2 a /f 2 a , and axion-photon coupling g aγγ α EM /f a , one finds that M decay S > M B−Nova . This means that for nominal values of g aγγ the axion stars will not actually decay into photons but rather into relativistic axions, with possible limits from such a decay explored in Ref.[23]. However, there are many models that feature significantly enhanced g aγγ interactions, see e.g.[24], or suppressed quartic interactions, see e.g.[25,26]. In what follows, we will consider that the relevant effect is decay into photons and thus our results will apply to scenarios such as those previously mentioned. It is also possible that the initial contraction phase of a Bosenova may itself trigger parametric resonance decay to photons[12].2) In the parameter space of interest for our study, axion stars are always hosted by halos of mass M h < 10 5 M which means that they lie below the baryonic Jeans scale and are therefore surrounded by an intergalactic medium (IGM) with average properties. The latter contains ionized particles, which could kinematically block the axion decay by plasma effects. More concretely, the photon gets an effective mass-squared that is proportional to the number of free electrons in the plasma[27]:Axion star decay is blocked until the plasma frequency drops to ω p (z decay ) < m a /2, resulting in an explosion of all the axion stars with M S > M decay S at specific redshift:z decay 32 m a 10 −13 eV 2/3 − 1 .Once the plasma frequency has dropped such that parametric resonance decay is no longer blocked, then supercritical axion stars will explode as soon as they are formed. In Ref.[8] we calculated the cosmological evolution of the number/energy density of critical axion stars using the extended Press-Schechter formalism and Monte Carlo merger trees. We considered the case that an axion star will only explode either if it is plasma blocked until it is super critical, or if it is formed by a major merger defined by the merger of two axion stars of comparable mass, such that mass increase happens rapidly.Soliton explosions lead to axion dark matter decay, and can inject energy into the IGM comparable to or greater than energy emission from astrophysical processes, such as core collapse supernovae[8], and thus offer a new method of indirect detection of axions.3 Heating the IGM.Figure 2outlines qualitatively the mechanism just described: axion stars start to form and grow when halos form and merge, which happens appreciably only at z 100. Once an axion star becomes massive enough and the plasma frequency is low enough, parametric resonance can take place and the star rapidly explodes into low energy photons of E γ = m a /2. These photons are absorbed by the IGM, heat it, and in turn via collisional ionizations lead to an increased ionization fraction of the Universe.Axion star decay results in the release of a huge number of low energy photons with E γ = m a /2. We will be interested in m a < 10 −8 eV, whose associated low energy photons are efficiently absorbed via inverse Bremsstrahlung on ionized particles in the intergalactic medium, namely via γ e − p → e − p processes[28]. The net result of this absorption is to heat the IGM. In turn, once the temperature of the IGM goes above T ∼ 1 eV, collisional ionization processes e − H → 2e − p ionize the Universe. As a result, the decay of axion stars may result into a period of early reionization, which is strongly constrained by Planck legacy data[29,30]. This forms the basis of the constraints derived in this paper, see also[31][32][33][34][35][36]for constraints of this type for other dark matter candidates.Importantly, depending upon the absorption length scale of the photons produced by axion stars the effect on the IGM can be different. For m a 5 × 10 −13 eV the photons are absorbed within very small volumes and this generates a shockwave very much like the one formed by supernova explosions as the amount of energy released is very similar[37]. On the other hand, for m a 5 × 10 −13 eV the absorption length-scale is larger than the inter-separation of axion stars results in a homogeneous cosmological increase in the global IGM temperature. It is important to highlight that in principle a fraction as small as f DM ∼ 3 × 10 −9 13.6 eV n b /ρ DM of dark matter converted into heat in the IGM would be enough to fully reionize the Universe. Methodology. We model the IGM temperature by including the heating generated by axion star decays and accounting for Compton and adiabatic cooling. Collisional ionization dominates in our scenario which we implement using the fitted rates in[38]. We then track the free electron density x e using the effective 3-level atom approximation[39,40], including free protons and He + . We use the same rates and expressions as the recombination code RECFAST including their fudge factor[41,42]-modulo that we change T b → T γ in the terms in Eqs.(1) and (2) of [42] that account for photon ionization as the approximation T b T γ breaks down for z < 200. We have explicitly checked our numerical code against the results of [34] that studies decaying dark matter into very low energy photons such as the ones we consider. Finally, in the regime m a 5 × 10 −13 eV the photons from axion star explosions are absorbed at very small scales and generate shockwaves which ionize small patches of the Universe. We track such explosions until they are unable to ionize more IGM. We follow standard methods used for supernova remnants[37,[43][44][45], and find results very close to those expected from the Sedov-Taylor solutions[46,47].Constraints from the CMB Optical Depth. We derive conservative constraints on reionization driven by axion star explosions by using the 95% upper limit on the integrated Thompson optical depth to reionization from Planck CMB observations: τ reio < 0.068[29,30]. The optical depth is measured by joint analysis of CMB temperature, polarisation, and baryon acoustic oscillations in the galaxy correlation function. A prior on the redshift of reionization z reio ≥ 6.5 is adopted to enforce consistency with the Gunn-Peterson optical depth[48].The axion parameter space excluded at 2σ confidence as a result of this constraint is shown in the two regions in dark red inFigure 3, and the resulting evolution of the ionization fraction and IGM temperature for representative models is shown inFigure 4. Constraints in the region m a 5×10 −13 eV are a result of patchy reionization generated by the decay of all critical axion stars in the Universe once the axion decay is kinematically allowed, namely when ω p (z decay ) = m a /2. For m a 5 × 10 −13 eV the photon plasma mass is small compared to the axion mass, and the constraints arise instead from supercritical axion stars formed via mergers providing continuous heating and uniform reionization. The intermediate region of parameter space is constrained only at 1σ. The intermediate region leads to a smaller increase in τ because when some reionization takes place the photon plasma mass grows and blocks further axion star decays. This can explicitly be seen in the green curve of the upper panel ofFigure 4. The intermediate region of parameter space will be further tested in the near future. For example, LiteBIRD will provide a cosmic variance limited measurement of the large scale EE polarisation spectrum, thereby significantly improving the determination of τ reio[49].We note that the Planck constraints do not extend to arbitrarily large g aγγ -the larger the value of g aγγ , the smaller the mass of the halos that host critical soliton stars. In axion-like dark matter cosmologies structure formation is suppressed at M h < M min , see Eq.(2), and couplings g aγγ 10 −10 correspond to axion stars hosted in halos whose abundance is suppressed. In addition, we notice that the bounds change their shape for m a 5 × 10 −11 eV. For this region of parameter space the photon absorption efficiency in the IGM is smaller than one and this weakens the constraints.Lyman-α and Spectral Distortions. We considered two other existing measurements that can limit this scenario for IGM heating. Since axion star explosions occur when Compton scattering is inefficient (z 10 4 ), they generate y-type distortions of the CMB energy spectrum. The change to the photon energy density from such an effect must satisfy δρ γ /ρ γ 5 × 10 −5 based on COBE/FIRAS measurements[34,55], which trans-4 FIG. 3. Parameter space excluded by Planck measurements of the Thompson optical depth. We highlight also reach of future 21cm surveys and constraints from X/γ-Ray observations [50-52] and CAST[53]. The evolution of the free electron fraction and the baryon temperature is shown for the points highlighted by colored symbols inFigure 4.lates to f decay DM 2 × 10 −7 for decaying DM: significantly weaker than the Planck optical depth constraint. At redshifts 2 z 7 the Lyman-α forest can measure the IGM temperature see e.g.[56][57][58]. The IGM temperature at these redshifts satisfies T b 1.5 × 10 4 K and since Compton cooling is the main cooling process (which we include in our calculations), this means that temperature histories where axion star decay causes T b to exceed this threshold would be ruled out by these data. This is shown as the upper contour inFigure 3, with an accompanying T b (z) inFigure 4(bottom panel). Lyman-α constraints are not competitive with the Planck optical depth constraint.21cm Cosmology. Measurements of the hyperfine 21cm transition are expected to revoluzionize our understanding of the cosmic dawn and the epoch of reionization, see[59,60]for reviews. There have already been several interesting bounds on the emission power of the 21cm line at various redshifts [61, 62], and even a putative detection [63] (which is disputed see, e.g. [62]). The observational status of the cosmological 21cm line is still in its infancy but it is expected that ongoing and upcoming experiments such as HERA [64] and SKA [65] among others should be able to robustly detect the 21cm line and provide a view of the thermal state of the Universe at redshifts 4 z 30 [66], see [67] for a compilation of experiments targeting this line.As highlighted inFigure 4, axion star explosions greatly enhance the IGM temperature at the redshifts where the 21cm line will be targeted z 35. In particular, the rise of T b at large redshifts z 20 is a feature that is not easy to achieve astrophysically, see e.g.[68]. The sky-averaged 21 cm brightness temperature can be 10 −4 If dark matter is composed of axions, then axion stars form in the cores of dark matter halos. These stars are unstable above a critical mass, decaying to radio photons that heat the intergalactic medium, offering a new channel for axion indirect detection. We recently provided the first accurate calculation of the axion decay rate due to axion star mergers. In this work we show how existing data concerning the CMB optical depth leads to strong constraints on the axion photon coupling in the mass range 10 −14 eV ma 10 −8 eV. Axion star decays lead to efficient reionization of the intergalactic medium during the dark ages. By comparing this non-standard reionization with Planck legacy measurements of the Thompson optical width, we show that couplings in the range 10 −14 GeV −1 gaγγ 10 −10 GeV −1 are excluded for our benchmark model of axion star abundance. Future measurements of the 21cm emission of neutral hydrogen at high redshift could improve this limit by an order of magnitude or more, providing complementary indirect constraints on axion dark matter in parameter space also targeted by direct detection haloscopes. Numerical simulations of axion-like dark matter cosmologies have shown that a solitonic core forms in the center of every dark matter halo, see [1,2] for the first simulations and [3][4][5][6][7] for further studies. In Ref. [8] we have explicitly calculated the number and energy density evolution of axion stars from hierarchical structure formation using semi-analytic models. Ref. [2] demonstrated a power law relation between the mass of an axion star and its host halo: M S = M h M min (z) α M min (z),(1) where z is the redshift, M h is the virial halo mass, α is a power law exponent, and [2]: M where m a the axion-like dark matter mass and M min is the smallest halo mass that could host a soliton of mass M S at a given redshift (given by the Jeans scale) [9,10]. Ref. [2] found a core-halo mass relation of the form M S ∝ M 1/3 h while Refs. [4][5][6] have found M S ∝ M 3/5 h . Refs. [7,11] suggest that there is an intrinsic diversity in the core-halo mass relation. Importantly, all numerical simulations of axion-like dark matter cosmologies show that the core-halo mass relation is bounded between α = 1/3 and α = 3/5. The Monte Carlo merger tree model we employed in Ref. [8] shows that an average slope α = 2/5 captures this diversity well for both the soliton mass function and, more importantly, the merger rate. In Figure 1 we show isocontours of constant M h for various values of M S and m a for the case α = 2/5. For the other cases we refer the reader to Appendix A. Soliton Explosions. Solitonic cores, also referred to as axion stars have ultrahigh phase-space occupation numbers that can trigger collective processes which cannot occur in vacuum. Parametric resonance can lead to exponentially fast decays of axion stars into photons [12][13][14][15] or into relativistic axions [16][17][18], meaning axion stars become unstable above a certain mass [12,15] (vector dark matter solitons have a similar instability [19]): [8]. When an axion star decays, it releases a huge number of low-energy photons, which are absorbed by inverse Bremsstrahlung, leading to heating of the IGM. If the IGM becomes hot enough, reionization occurs. Energetically, reionization requires a fraction of around f decay DM ≈ 10 −9 of the dark matter to decay. For the lowest mass axion-like particles, ma 5 × 10 −13 eV, axion star decay is kinematically blocked at early times, leading to a population of super critical stars which decay all at once in a burst once the plasma frequency falls low enough to allow the decay. This leads to patchy reionization. At higher axion masses, plasma blocking is not efficient at the relevant redshifts, and instead super-critical stars formed by major mergers decay immediately as they form, leading to a more uniform and continuous reionization history. M decay S 8.4 × 10 −5 M 10 −11 GeV −1 g aγγ 10 −13 eV m a ,(3) where g aγγ is the axion-photon coupling. Super-critical solitons explode into photons with E γ = m a /2 in a timescale given by the light crossing time of the soliton (in units c = 1): τ decay S r c day 8.4 × 10 −5 M M S 10 −13 eV m a 2 .(4) The energy released in the explosion of a supercritical soliton is E = M S − M decay S . There are, however, two factors that could prevent this explosion: 1) If the axion-like particle possesses self interactions, then axion stars can decay into axions leading a "Bosenova" above a critical mass M B−Nova set by the quartic self coupling λ [16][17][18][20][21][22]. For a quartic coupling typical of a cosine instanton potential, λ = m 2 a /f 2 a , and axion-photon coupling g aγγ α EM /f a , one finds that M decay S > M B−Nova . This means that for nominal values of g aγγ the axion stars will not actually decay into photons but rather into relativistic axions, with possible limits from such a decay explored in Ref. [23]. However, there are many models that feature significantly enhanced g aγγ interactions, see e.g. [24], or suppressed quartic interactions, see e.g. [25,26]. In what follows, we will consider that the relevant effect is decay into photons and thus our results will apply to scenarios such as those previously mentioned. It is also possible that the initial contraction phase of a Bosenova may itself trigger parametric resonance decay to photons [12]. 2) In the parameter space of interest for our study, axion stars are always hosted by halos of mass M h < 10 5 M which means that they lie below the baryonic Jeans scale and are therefore surrounded by an intergalactic medium (IGM) with average properties. The latter contains ionized particles, which could kinematically block the axion decay by plasma effects. More concretely, the photon gets an effective mass-squared that is proportional to the number of free electrons in the plasma [27]: ω 2 p (z) = 4πα EM n e (z) m e .(5) Axion star decay is blocked until the plasma frequency drops to ω p (z decay ) < m a /2, resulting in an explosion of all the axion stars with M S > M decay S at specific redshift: z decay 32 m a 10 −13 eV 2/3 − 1 .(6) Once the plasma frequency has dropped such that parametric resonance decay is no longer blocked, then supercritical axion stars will explode as soon as they are formed. In Ref. [8] we calculated the cosmological evolution of the number/energy density of critical axion stars using the extended Press-Schechter formalism and Monte Carlo merger trees. We considered the case that an axion star will only explode either if it is plasma blocked until it is super critical, or if it is formed by a major merger defined by the merger of two axion stars of comparable mass, such that mass increase happens rapidly. Soliton explosions lead to axion dark matter decay, and can inject energy into the IGM comparable to or greater than energy emission from astrophysical processes, such as core collapse supernovae [8], and thus offer a new method of indirect detection of axions. Heating the IGM. Figure 2 outlines qualitatively the mechanism just described: axion stars start to form and grow when halos form and merge, which happens appreciably only at z 100. Once an axion star becomes massive enough and the plasma frequency is low enough, parametric resonance can take place and the star rapidly explodes into low energy photons of E γ = m a /2. These photons are absorbed by the IGM, heat it, and in turn via collisional ionizations lead to an increased ionization fraction of the Universe. Axion star decay results in the release of a huge number of low energy photons with E γ = m a /2. We will be interested in m a < 10 −8 eV, whose associated low energy photons are efficiently absorbed via inverse Bremsstrahlung on ionized particles in the intergalactic medium, namely via γ e − p → e − p processes [28]. The net result of this absorption is to heat the IGM. In turn, once the temperature of the IGM goes above T ∼ 1 eV, collisional ionization processes e − H → 2e − p ionize the Universe. As a result, the decay of axion stars may result into a period of early reionization, which is strongly constrained by Planck legacy data [29,30]. This forms the basis of the constraints derived in this paper, see also [31][32][33][34][35][36] for constraints of this type for other dark matter candidates. Importantly, depending upon the absorption length scale of the photons produced by axion stars the effect on the IGM can be different. For m a 5 × 10 −13 eV the photons are absorbed within very small volumes and this generates a shockwave very much like the one formed by supernova explosions as the amount of energy released is very similar [37]. On the other hand, for m a 5 × 10 −13 eV the absorption length-scale is larger than the inter-separation of axion stars results in a homogeneous cosmological increase in the global IGM temperature. It is important to highlight that in principle a fraction as small as f DM ∼ 3 × 10 −9 13.6 eV n b /ρ DM of dark matter converted into heat in the IGM would be enough to fully reionize the Universe. Methodology. We model the IGM temperature by including the heating generated by axion star decays and accounting for Compton and adiabatic cooling. Collisional ionization dominates in our scenario which we implement using the fitted rates in [38]. We then track the free electron density x e using the effective 3-level atom approximation [39,40], including free protons and He + . We use the same rates and expressions as the recombination code RECFAST including their fudge factor [41,42] -modulo that we change T b → T γ in the terms in Eqs. (1) and (2) of [42] that account for photon ionization as the approximation T b T γ breaks down for z < 200. We have explicitly checked our numerical code against the results of [34] that studies decaying dark matter into very low energy photons such as the ones we consider. Finally, in the regime m a 5 × 10 −13 eV the photons from axion star explosions are absorbed at very small scales and generate shockwaves which ionize small patches of the Universe. We track such explosions until they are unable to ionize more IGM. We follow standard methods used for supernova remnants [37,[43][44][45], and find results very close to those expected from the Sedov-Taylor solutions [46,47]. Constraints from the CMB Optical Depth. We derive conservative constraints on reionization driven by axion star explosions by using the 95% upper limit on the integrated Thompson optical depth to reionization from Planck CMB observations: τ reio < 0.068 [29,30]. The optical depth is measured by joint analysis of CMB temperature, polarisation, and baryon acoustic oscillations in the galaxy correlation function. A prior on the redshift of reionization z reio ≥ 6.5 is adopted to enforce consistency with the Gunn-Peterson optical depth [48]. The axion parameter space excluded at 2σ confidence as a result of this constraint is shown in the two regions in dark red in Figure 3, and the resulting evolution of the ionization fraction and IGM temperature for representative models is shown in Figure 4. Constraints in the region m a 5×10 −13 eV are a result of patchy reionization generated by the decay of all critical axion stars in the Universe once the axion decay is kinematically allowed, namely when ω p (z decay ) = m a /2. For m a 5 × 10 −13 eV the photon plasma mass is small compared to the axion mass, and the constraints arise instead from supercritical axion stars formed via mergers providing continuous heating and uniform reionization. The intermediate region of parameter space is constrained only at 1σ. The intermediate region leads to a smaller increase in τ because when some reionization takes place the photon plasma mass grows and blocks further axion star decays. This can explicitly be seen in the green curve of the upper panel of Figure 4. The intermediate region of parameter space will be further tested in the near future. For example, LiteBIRD will provide a cosmic variance limited measurement of the large scale EE polarisation spectrum, thereby significantly improving the determination of τ reio [49]. We note that the Planck constraints do not extend to arbitrarily large g aγγ -the larger the value of g aγγ , the smaller the mass of the halos that host critical soliton stars. In axion-like dark matter cosmologies structure formation is suppressed at M h < M min , see Eq. (2), and couplings g aγγ 10 −10 correspond to axion stars hosted in halos whose abundance is suppressed. In addition, we notice that the bounds change their shape for m a 5 × 10 −11 eV. For this region of parameter space the photon absorption efficiency in the IGM is smaller than one and this weakens the constraints. Lyman-α and Spectral Distortions. We considered two other existing measurements that can limit this scenario for IGM heating. Since axion star explosions occur when Compton scattering is inefficient (z 10 4 ), they generate y-type distortions of the CMB energy spectrum. The change to the photon energy density from such an effect must satisfy δρ γ /ρ γ 5 × 10 −5 based on COBE/FIRAS measurements [34,55], which trans- We highlight also reach of future 21cm surveys and constraints from X/γ-Ray observations [50][51][52] and CAST [53]. The evolution of the free electron fraction and the baryon temperature is shown for the points highlighted by colored symbols in Figure 4. lates to f decay DM 2 × 10 −7 for decaying DM: significantly weaker than the Planck optical depth constraint. At redshifts 2 z 7 the Lyman-α forest can measure the IGM temperature see e.g. [56][57][58]. The IGM temperature at these redshifts satisfies T b 1.5 × 10 4 K and since Compton cooling is the main cooling process (which we include in our calculations), this means that temperature histories where axion star decay causes T b to exceed this threshold would be ruled out by these data. This is shown as the upper contour in Figure 3, with an accompanying T b (z) in Figure 4 (bottom panel). Lyman-α constraints are not competitive with the Planck optical depth constraint. 21cm Cosmology. Measurements of the hyperfine 21cm transition are expected to revoluzionize our understanding of the cosmic dawn and the epoch of reionization, see [59,60] for reviews. There have already been several interesting bounds on the emission power of the 21cm line at various redshifts [61,62], and even a putative detection [63] (which is disputed see, e.g. [62]). The observational status of the cosmological 21cm line is still in its infancy but it is expected that ongoing and upcoming experiments such as HERA [64] and SKA [65] among others should be able to robustly detect the 21cm line and provide a view of the thermal state of the Universe at redshifts 4 z 30 [66], see [67] for a compilation of experiments targeting this line. As highlighted in Figure 4, axion star explosions greatly enhance the IGM temperature at the redshifts where the 21cm line will be targeted z 35. In particular, the rise of T b at large redshifts z 20 is a feature that is not easy to achieve astrophysically, see e.g. [68]. The sky-averaged 21 cm brightness temperature can be Figure 3 with symbols. For the scenarios we show there are no astrophysical sources of heating or reionization. We also show in black the ΛCDM evolution for xe from the Planck best fit cosmology and T b from the fiducial model of [54]. In the lower panel we highlight scenarios that can be tested with future 21cm observations, conservatively (long dashed) and optimistically (dashed). written as [69]: T 21 = 27 mK x HI Ω b h 2 0.023 0.15 Ω m h 2 1 + z 10 1/2 1 − T γ T S ,(7) where x HI is the fraction of neutral hydrogen in the Universe and T S is the spin temperature. At z 25 the spin temperature is T S T b [70] and we then see that for T b > T γ the 21cm signal could reach a maximum in emission of ∼ 35 mK at z 20 provided T b T γ . This is in strong contrast with what is expected in most cold IGM models where T b (z = 20) ∼ 7 K and which would lead to a strong absorption feature with T 21 ∼ −200 mK. The sensitivity of SKA in this redshift range is expected to be ∆T 21 ∼ 10 mK, which means that SKA should be able to clearly differentiate between these two cases and thus future 21cm observations will test scenarios where axion star explosions heat the IGM. In Figure 3 we highlight in light red the region of parameter space that could be tested by an experiment such as SKA by demanding that T 21 < 30 mK at z 20 (equivalent to T b 500 K by z 20). A tighter limit can be arrived at if one could differentiate any model from ΛCDM that causes T b > T γ at z 10. This is shown by the bottom light line in Figure 3. 21cm measurements not only improve the sensitivity to much smaller g aγγ couplings but also expand the sensitivity to axion mass. This is because for m a 10 −7 eV the photons generated can still be absorbed by inverse Bremsstrahlung and increase T b observably while not achieving ionization necessary to be probed by τ alone. Conclusions. Parametric-resonance instability of axion stars leads to an enhanced decay rate of axion dark matter into low energy photons. We have shown how the production of such photons heats the IGM, and alters the optical depth of CMB photons. Planck legacy measurements of the optical depth form the basis of our new constraints on g aγγ highlighted in Figure 3, which is the strongest limit on axion dark matter in the relevant mass range by more than an order of magnitude. Future 21cm measurements of the IGM during the dark ages could improve this limit by more than an order of magnitude. The overlap of these indirect probes with the target regions of the DM-Radio haloscope program [71] invites further careful consideration of axion star explosions as a new tool in the hunt for dark matter. Outlook. Aspects of this model that require further exploration include primarily a detailed study of soliton merger rates in simulations displaying a core-halo diversity in the redshift and particle mass ranges of interest. While our Monte Carlo merger trees in [8] suggest that M S ∝ M 2/5 h accounts well for diversity in the soliton merger rate only simulations in the redshifts of inter-ests can fully support or modify this conclusion. On the side of particle physics, the strongest assumption of our analysis is that axion quartic couplings are suppressed, while g aγγ remains large. Construction of explicit axionlike dark matter models above the QCD line is required to address how natural such couplings are. Nevertheless, simulations of axion star explosions including both quartic and photon couplings may yet find that Bosenova triggered photon parametric resonance can occur for more standard coupling ratios. Furthermore, determining the true reach of 21cm measurements requires a full simulation of the 21cm anisotropies including anomalous injection from axion stars. We suspect that anisotropies in this model will differ strongly from ΛCDM, and thus bounds may even be more powerful than our estimates. Acknowledgements. ME would like to thank Andrea Caputo, Josh Eby, and Sam Witte for very useful discussions. MF gratefully received funding via the STFC particle theory grant STFC-ST/T000759/1. CKP is funded via an STFC quota studentship. DB is supported by a 'Ayuda Beatriz Galindo Senior' from the Spanish 'Ministerio de Universidades', grant BG20/00228. The research leading to these results has received funding from the Spanish Ministry of Science and Innovation (PID2020-115845GB-I00/AEI/10.13039/501100011033). IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. DJEM is supported by an Ernest Rutherford fellowship from STFC. In the main text we have shown all results for the benchmark core-halo mass relation in Eq. (1) with α = 2/5. Here we show the results for other two benchmark core-halo mass relations: α = 1/3 and α = 3/5. The former was the first one found in the literature [1,2]. The case α = 3/5 was found in Refs. [4][5][6]. In addition, Refs. [7,11] have shown evidence not for a strict core-halo mass relation but rather for diversity. Importantly, all numerical simulations show that axion stars lie somewhere in between α = 1/3 and α = 3/5. In addition, our Monte Carlo merger tree [8] shows that α = 2/5 captures well the effect of diversity in the merger rate of axion stars and in consequence represents our nominal choice for the core-halo mass relation. For completeness, in this appendix we show the results for these other two scenarios. In particular, Figure S1 shows the M S -M h relation for the two cases while Figure S2 shows the resulting constraints for each of them. Supplementary Material Appendix B: Decaying dark matter fraction and axion star merger evolution Planck constraints on the axion parameter space are strongly related to the amount of decaying dark matter that is injected as heat in the IGM. There are two distinct regions of parameter space. For m a 5 × 10 −13 eV there are three key effects: 1) axion stars cannot decay until the photon plasma mass is low enough, 2) the explosion thus occurs as a burst of energy when ω p (z) < m a /2, and 3) the photons generated from axion star explosions are absorbed on very small volumes and this generates shockwaves and bubbles of ionized IGM. This leads to patchy ionization. In this case, the constraints on the parameter space closely resemble regions of constant decaying dark matter density evaluated at the redshift of decay, z decay . This is explicitly shown in the left panel of Figure S3. There we can see that the region of parameter space excluded by Planck closely follows a region of parameter space where f decay DM 10 −6 . It is important to note that this is almost three orders of magnitude larger than what would energetically be needed h . We can clearly see that for a fixed axion star mass the mass of a halo hosting it is much smaller for the case α = 3/5 than for the one with α = 1/3. This in turn means that significantly smaller gaγγ couplings can be probed. We highlight in grey regions of parameter space for which such axion stars cannot have possibly formed by z = 20 as their masses would lie below the effective Jeans mass generated by quantum pressure, see Eq. (2). if all the energy were to be injected homogeneously as heat in the IGM. However, since the observable τ is strongly sensitive to the volume of ionized IGM in the Universe this patchy reionization is much less efficient in fully ionizing the entire Universe. This is analogous to what happens in standard reionization where the most efficient sources of the global ionization of the Universe are UV and X-ray photons with mean free paths similar to the size of the observable Universe. In the region of parameter space with m a 5 × 10 −13 eV the absorption of photons from axion star explosions takes place over distances that are larger than the typical inter-separation of axion stars. That means that the process can effectively be seen as homogeneous. In addition, in this case the emission is continuous because the photon plasma mass is small compared to m a /2. In this regime what matters is how much energy is released into heat and on which timescale does the IGM cool. The former we explicitly calculated in Ref. [8]. The latter is simply given by Compton cooling and reads (see Eq. (E1c)): τ Compton = 45 4π 2 1 T 4 γ m e σ T 2.3 × 10 12 years 1 (1 + z) 4 . (B1) Thus, given energy arguments we expect regions of parameter space where df decay DM /dt/τ Compton 3 × 10 −9 can lead to reionization of the Universe. This is explicitly shown in the right panel of Figure S3 where we see that the Planck exclusion region is parallel and very close to this line. We notice that 21 cm observations will be sensitive to rates that are ∼ 1 order of magnitude smaller than those currently tested by Planck measurements of τ reio . FIG. S3. Left: isocontours of f decay DM compared with the constraints obtained from Planck. Right: isocontours of df decay DM /dt/τCompton. We can clearly appreciate how the constraints follow the same shape and follow closely the case 3 × 10 −9 as expected from energy arguments. The region ma 10 −10 eV deviates from this expectations because the probability of photon absorption in the IGM is smaller than 1. Appendix C: Cosmological Photon Plasma Mass There are always some charged particles in the early Universe and this changes the global propagation properties of photons in cosmology. In particular, the photon gets a mass-squared that is proportional to the number of free electrons in the plasma [27]: ω 2 p (z) = 4πα EM n e (z) m e ,(C1) where α EM is the electromagnetic fine structure constant, n e is the (free) electron number density and m e is the electron mass. Photons with energy E γ < ω p (z) cannot propagate. This in turn has important implications for the axion decay into photons because by pure kinematics the stars cannot explode into photons unless m a > 2ω p (z). In Figure S4 we show the plasma frequency as a function of redshift for the Planck best fit ΛCDM cosmology (as well as for considering a fully reionized Universe). We see that axions with masses m a < 2 × 10 −14 eV cannot decay and thus our constraints are restricted to sufficiently massive axion-like particles 1 . The main cosmological implication of the plasma mass is that all the axion stars will not be able to explode until ω p (z) ≤ m a /2. For the range of redshifts z reio z 800 the relation between the redshift of decay and the axion mass is approximately given by: This means that for axion masses in the range 2 × 10 −14 m a 10 −12 eV all axion stars that are massive enough will explode and convert all their energy into very low energy photons that can subsequently heat the intergalactic medium. For m a 10 −12 eV we will instead expect that axion stars to explode as soon as they form. In this regime, we expect a continuous emission of energy across redshift. S5. Typical length scale over which photons from axion star decays are absorbed in the IGM via inverse Bremsstrahlung on free electron-proton pairs with a plasma at two different temperatures, T b = 9 K, 1 eV. We show it for the reference value of z = 20 and with a pre-reionization free electron fraction xe = 2 × 10 −4 . We can appreciate three regimes: at ma 10 −8 eV the photons have mean free paths larger than the size of the Universe and therefore there are no cosmological signatures, for ma 5×10 −11 eV the length-scale of absorption is always smaller than 1/H which mean that all of the photons will be absorbed even if T b ∼ 1 eV which is approximately the temperature of an ionized IGM. Finally, for ma 5 × 10 −13 eV the photons from axion stars are absorbed on very small lengthscales and thus generates an intense shock-wave which in turn generates a sort of patchy reionization. 1. For m a 10 −8 eV the photons have mean free paths larger than the size of the observable Universe and thus axion star explosions will not lead to relevant cosmological implications. 2. For 5 × 10 −13 eV m a 10 −8 eV the photons produced from axion star explosions will be absorbed by the IGM on length scales larger than the one it will take for this region to reach T b ∼ 1 eV, which is: L T b = 3 M S /(n b 3T b ) 0.4 kpc M S 8 × 10 −5 M 1/3 T b eV 1/3 21 1 + z . (D5) That means that the absorption of photons in this region of parameter space can be seen as homogeneous, so it will take several nearby axion star explosions to heat up the IGM. This scale is labelled as typical bubble size in Figure S5. 3. For m a 5 × 10 −13 eV the photons are absorbed on very small lengthscales. Since a large amount of energy is released per axion star decay E ∼ M S ∼ 10 −4 M the injection of these photons will lead to shockwaves in the IGM very similar to supernova remnants. The lower limit on m a of 2 × 10 −14 eV arises due to the photon plasma mass in the Universe. It is important to note that these considerations have been made with x e ∼ 2 × 10 −4 as expected pre-reionization in ΛCDM. As soon as x e grows photons from axion star decays will be absorbed faster in the IGM, see Eq. (D3). In what follows, in Appendix E we outline our modeling of energy injections for m a 5×10 −13 eV and in Appendix F we consider the case of m a 5 × 10 −13 eV. Appendix E: Continuous Axion Star Mergers and Explosions: Homogeneous IGM Heating and Reionization For m a 5 × 10 −13 eV axion stars explode into photons that are absorbed across lengthscales which are larger than the typical interseparation of axion stars. In this regime, the injection of energy can be seen as homogeneous. As discussed in the main text, we track the evolution of the baryon temperature including the new heating source from axion star explosions and also account for Compton and adiabatic cooling. We also consider for the net baryon cooling generated by collisional ionizations e H → 2e p. We track the free electron fraction using the effective 3-level atom and follow the evolution of protons (p) and HeII ≡ He + . x i ≡ n i /n H so that x e ≡ n e /n H = x p + x HeII . The evolution equations that describe them are: S6. Here we compare the free electron fraction evolution in a scenario of very light decaying dark matter that decays homogeneously and following a typical exponential decay law [34]. For the comparison we have used the very same values as in Figure 15 of [34]. We can appreciate the excellent agreement. FIG. Finally, we account for collisional ionizations, i.e. e H → 2e p type processes. The change on the number of free ions is given by dx p dt coll = σv H x e n H (1 − x p ) ,(E6a) dx HeII dt coll = σv HeII x e n H (f He − x HeII ) ,(E6b) where f He = n He /n H , and where σv H and Helium, σv HeII , are the collisional excitation rates for Hydrogen and Helium taken from [38]. The net cooling generated by collisional ionizations is simply given by dE dV dz coll = − n H (1 + z)H E H × dx p dt coll + E HeII × dx HeII dt coll ,(E7) where E H = 13.6 eV and E HeII = 24.6 eV To cross check our code we considered also the decaying dark matter scenario studied in Ref. [34]. There the authors consider a scenario with dark matter decaying into very low energy photons as we do. In their case the decay is modelled by a simple exponential law and in Figure S6 we show the excellent agreement for the evolution of the free electron fraction when we use their model of dark matter. Thompson CMB Optical Depth: Finally, after solving Equations (E1) we then calculate several integrated quantities. In particular, the Thompson optical depth to recombination which reads: τ = σ T n free e,0 zmax 0 x e (z) (1 + z) 2 H(z) dz.(E8) In practice we choose z max = 50 and since we know the Universe was reionized by z = 6 we use x e = 1 for z ≤ 6 irrespective of the reionization evolution that axion star explosions lead to. Solutions for this reionization evolution are shown in Figure 4. Spectral Distortions: Additional heating of the IGM due to axion star explosions can generate a y-type distortion of the CMB at the redshifts of interest. The maximum amount of energy that could go into the CMB from axion star explosions is given by, y 1 4 δρ γ ρ γ = 1 4 zmax zmin dQ/dz ρ γ dz = 1 4 zmax zmin 1 ρ γ dE dV dz dep,h dz,(E9) where dQ/dz is the heating rate and ρ γ is the energy density of CMB photons. In Eq. (E9), we have chosen to integrate from z min = 0 to z max = 400, given z = 400 is a high enough redshift before axion stars have started From the initial conditions in Eq. (F3) we can then solve Eqs. (F1) with L Explosion = 0. The last thing needed is the other luminosities which can be written as [43]: L Compton = 2π 3 45 σ T m e T 4 γ pR 3 ,(F4) where T γ is the CMB temperature at a given redshift, and L Ionization = f m n b E H 4π 2 R 2 (Ṙ − HR) ,(F5) where here n b is the background baryon number density, E H 13.6 eV is the energy it takes to ionize a hydrogen atom, and f m is the fraction of the baryonic mass kept in the interior of the bubble which by construction is f m 1. In our calculations, for concreteness we take f m = 0.1 but we have checked that smaller values yield very similar results. Finally, we stop the integration when the pressure of the interior of the bubble is p 2 T crit n b with T crit 15000 K which roughly corresponds to the temperature at which the IGM becomes is fully ionized. For smaller pressures the bubble will not be able to ionize the swept IGM and the radius at this point will tell us the region that has been ionized by the axion star explosion. Figure S7 shows the numerical solution to Eqs. (F1) for one energy injection example. For concreteness, we show the result for z decay = 20 and for two different axion star masses. There we can clearly see the steep decrease of the pressure as a result of the increase in the bubble radius. We note that we do not find any significant dependence of the results with respect to the redshift of injection. By numerically solving for all the axion-star masses of interest we find that the comoving size that the bubbles of ionized material reach is: (F6) We have also found that the characteristic time it takes the bubble to reach such a radius is much shorter than the age of the Universe and is approximately given by: t bubble 0.2 Myr M S 8 × 10 −5 M 1/3 .(F7) Eq. (F6) is very useful as it tells us the characteristic size of ionized bubbles in the Universe. This, together with the number density of such axions stars will allow us to understand the global reionization driven by this process. Once the bubble pressure reaches the IGM one, the bubbles will stuck. From this point onward the bubble will simply experience Hubble expansion and we can treat the free electron fraction and its temperature following the standard equations for recombination/ionization. In particular, we use the very same equations as in Eq. (E1) but without a heating term, namely, with dE/(dV dz)| dep,h = 0. We then solve the set of equations (E1) starting with x e = 1 and T b = 1 eV as expected from the final state of the bubble. What happens from then onwards is essentially that the bubbles cool due to Compton cooling and this leads to recombination which in turn reduces x e . We simply then calculate the volume average free electron density by taking this result and multiplying it by n crit S × R 3 where we obtain n crit S from [8] and R is given by Eq. (F6). An example of the evolution of this volume averaged ionization fraction can be seen in Figure 4 in dashed blue. We can clearly appreciate the downward trend of x e as generated by Compton cooling. FIG. 1 . 1In blue dashed: isocontours of halo masses that host a given axion star mass at redshift z = 20 depending upon the value of the axion mass for the benchmark core halo mass relation MS ∝ M2/5 h . In red we show isocontour lines of critical axion stars (M decay S ) as a function of several gaγγ values. FIG. 3 . 3Parameter space excluded by Planck measurements of the Thompson optical depth. FIG. 4 . 4Evolution of xe (upper) and T b (lower) for the six scenarios highlighted in ( Miguel Escudero, Charis Kaur Pooni, Malcolm Fairbairn, Diego Blas, Xiaolong Du, and David J. E. Marsh) This supplementary material contains: Appendix A: Constraints for other core-halo mass relations in the parameter space g aγγ and m a . Appendix B: Decaying dark matter fraction from axion star explosions as a function of g aγγ and m a . Appendix C: Evolution of the photon plasma mass and its effect on axion star decays. Appendix D: Absorption rate of photons generated from axion star explosions in the IGM. Appendix E: Evolution equations for x e and T b in the presence of homogeneous and continuous energy releases from axion star mergers and explosions. Relevant for m a 5 × 10 −13 eV. Appendix F: Patchy reionization evolution generated by axion star explosions where m a 5 × 10 −13 eV. Evolution of shockwaves and ionized bubbles and its impact on the global x e . Appendix A: Results for other Core-halo mass relations FIG. S1 . S1Isocontours of halo masses (dashed blue) that host an axion star mass (y axis) as a function of axion mass (x axis) at a redshift z = 20, see Eq. (1). In red we show the value of a critical axion star above which the star can explode into photons given an axion-photon coupling in red, see Eq. (3). The left panel corresponds to the Schive relation, with MS ∝ M FIG. S2 . S2Equivalent to Figure 3 but for two other core-halo mass relations MS ∝ M ). In light red dashed lines we show the benchmark case with MS ∝ M . S4. Minimum mass of an axion that can decay into photons as a function of redshift, ma > 2 ωp(z). In red we show the evolution according to the Planck ΛCDM best fit cosmology and in purple we show the evolution in a fully ionized Universe. 10 −14 10 −13 10 −12 10 −11 10 −10 10 −9 10 −8 10 −7 FIG. S7. Time evolution of the shockwave resulting from an axion star explosion into photons that are readily absorbed in the IGM. Top to bottom we show the comoving radius, the velocity normalised to the Hubble rate, and the pressure inside the bubble as a function of time. FIG. 2. Schematic of reionization caused by axion star explosions. Appreciable formation of dark matter halos hosting sufficiently massive axion stars occurs for z 100, seearXiv:2302.10206v1 [hep-ph] 20 Feb 2023 We note that there are under dense regions in the Universe where the plasma frequency could be smaller and lead to further con-straints at ma < 2 × 10 −14 eV, see[32,33,72,73] for considerations of this type in the context of dark photon dark matter. Appendix D: Absorption of photons from axion star explosions in the IGMAxion star explosions generate a huge number of very low energy photons with E γ = m a /2. For m a 10 −8 eV the most efficient absorption process of these photons in the IGM is free-free absorption also known as inverse Bremsstrahlung, γ e − p → e − p. The rate at which these photons are absorbed by the plasma is[28]:where σ T is the Thompson cross section, T b is the temperature of the electron-baryon fluid and where Λ BR (E γ , z) = g BR α EM n p m 3g BR is the Gaunt factor that for E γ T b can be approximated by g BR = √ 3 π log(2.25T b /E γ ). Plugging in numerical values we find:where here we have normalized the rate to the thermodynamic values of T b and x e as expected pre-reionization in ΛCDM at z = 20. We note that the rate has a strong redshift dependence and that it scales as m −2 a and T. This means that as the temperature of the baryons raises the free free absorption becomes less effective. Importantly, it also scales as x 2 e which means that as the Universe becomes ionized the absorption becomes highly efficient. This rate should be compared with the Hubble expansion H(z) to see if these photons will be absorbed, which is well described around the redshifts of interest byFigure S5explicitly shows the absorption lengthscale of photons produced from axion star decays for two characteristic baryon temperatures, T b = 9 K, and T b = 1 eV, which correspond to a cool IGM and one where the IGM is would be fully ionized. From this figure we notice three distinctive regions of parameter space:which represent the rate equations for the free electron fraction contribution due to Hydrogen, Eq. (E1a) Helium, Eq. (E1b), recombination (α), photoionization (β) and collisional processes (coll). Both recombination and collisional excitations depend on the baryon temperature, T b , which is solved via the rate equation, Eq. (E1c), where the first term is simply adiabatic cooling, the second corresponds to Compton cooling, the third to the net gas cooling generated by possible collisional ionizations, and the last term is a result of heating by axion star explosions. The energy deposited as heat in the IGM by axion star explosions is explicitly given by:where here the Θ function ensures that no emission is generated if the photon plasma mass blocks the decay, the last factor is an efficiency factor that takes into account that photons may not be absorbed if their mean free path is large, and finally the fraction of dark matter decaying into photons from axion star mergers is obtained from[8]and approximately reads as:The relevant energy transitions in the effective-three level model correspond to the ground state (1s), the second level with two quantum states (2s and 2p), and the third level (c), which denotes the continuum. Direct recombination and photoionization are prohibited, and the Lyman-α decay channel is heavily dampened by the optically thick plasma during the recombination epoch. Therefore, the only possible route to recombination (the ground state, 1s) is to go via the 2s− →1s γγ-photon decay channel. This occurs at a much slower rate compared to Lyman-α, Λ H,2s1s = 8.22458 s −1 which has a transitional energy, E H,2s1s = E H,2p1s = 10.2 eV, for Hydrogen, and Λ He,2s1s = 51.3 s −1 with an energy transition of E He,2s1s = 20.62 eV, for Helium. This creates an overall effective decay rate for energy transitions between 2− →1, where γγ-decay channel dominates at earlier times. This effective decay rate is encapsulated by the Peebles C-factor, which scales the rate equations in Eq. (E1a) and (E1b) due to these physical effects. The C-factor for Hydrogen, C H , and Helium, C He , are given by, For m a 5 × 10 −13 eV photons generated by axion star explosions in the early Universe are absorbed on very small lengthscales (seeFigure S5). Axion star explosions release a huge amount of energy M S ∼ 10 −4 M (seeFigure 1) and will generate a shockwave very similar to those generated by supernova explosions that would be very violent and fully reionize the IGM around it. In this appendix we describe how we track the evolution of the bubbles generated by these explosions and calculate the volume average ionization fraction of the Universe. This scenario leads to patchy ionization.The expansion of post explosionary shock-waves in a dense medium was first studied in the context of nuclear bombs[46,47]. It was later realised that this could also be applied to astrophysical environments such as around supernova explosions[77]and the equations were subsequently developed to include the internal and external pressure changes, the heating of the interior of the bubble[78], energy lost to the shockwave due to the emission of radiation[79], the self gravity of the shell and the expansion of the Universe[37,43,80].In order to model the behaviour of the expanding bubble with radius R and enclosed mass M , we use the following two coupled equations (see, e.g.,[43][44][45]):where a dot represents derivative w.r.t time, Ω m is the total density parameter, Ω b the baryon density parameter, and H is the Hubble expansion rate. L tot represents the total luminosity and p is the bubble pressure resulting from this luminosity. The first term in equation (F1a) represents the driving pressure of the outflow, the second is drag due to accelerating the IGM from velocity HR to velocityṘ, the third is the self gravity of the expanding shell the fourth is the self gravity of the entire halo. The first term in equation (F1b) represents the increase in pressure caused by injection of energy while the second term is the drop in pressure caused by adiabatic expansion. In particular, the total luminosity is given by:where here L Explosion is the luminosity that is generated by the explosion of the axion star into photons, L Compton is the luminosity lost via Compton cooling against the CMB, and L Ionization takes into account the energy lost in ionising the swept IGM by the shockwave. Since the timescale for the explosion is very short compared to any other time scale, see Eq.(3), and the one for absorption as well, see Eq. 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White, Galaxy For- mation and Evolution (2010). With this expression, we have found that for the parameter space that is currently excluded by Planck constraints on τ the y distortion is much smaller than the one currently tested by COBE/FIRAS. In particular, for the red circle benchmark point in Figure 3 we find |y| = 3 × 10 −7 . This is two orders of magnitude smaller than the current sensitivity. For the diamond point in purple that can be tested by 21cm observations, we find |y| = 1 × 10 −8. which could be within the sensitivity of future CMB observationsto decay. With this expression, we have found that for the parameter space that is currently excluded by Planck constraints on τ the y distortion is much smaller than the one currently tested by COBE/FIRAS. In particular, for the red circle benchmark point in Figure 3 we find |y| = 3 × 10 −7 . This is two orders of magnitude smaller than the current sensitivity. For the diamond point in purple that can be tested by 21cm observations, we find |y| = 1 × 10 −8 , which could be within the sensitivity of future CMB observations.
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[ "REAL-TIME SEMANTIC SEGMENTATION ON FPGAS FOR AUTONOMOUS VEHICLES WITH HLS4ML", "REAL-TIME SEMANTIC SEGMENTATION ON FPGAS FOR AUTONOMOUS VEHICLES WITH HLS4ML" ]
[ "Nicolò Ghielmetti ", "Vladimir Loncar ", "Maurizio Pierini ", "Marcel Roed ", "Sioni Summers ", "Thea Aarrestad ", "Christoffer Petersson ", "Zenseact Gothenburg ", "Sweden Hampus Linander ", "Jennifer Ngadiuba ", "Kelvin Lin ", "Philip Harris ", "\nInstitute for Particle Physics and Astrophysics\nEuropean Organization for Nuclear Research (CERN)\nETH Zürich\nCH-1211, 8093Geneva 23, ZürichSwitzerland, Switzerland\n", "\nFermi National Accelerator Laboratory Batavia\nUniversity of Gothenburg Gothenburg\n40530, 60510ILSweden, USA\n", "\nUniversity of Washington Seattle\n98195WAUSA\n", "\nMassachusetts Institute of Technology Cambridge\n02139MAUSA\n" ]
[ "Institute for Particle Physics and Astrophysics\nEuropean Organization for Nuclear Research (CERN)\nETH Zürich\nCH-1211, 8093Geneva 23, ZürichSwitzerland, Switzerland", "Fermi National Accelerator Laboratory Batavia\nUniversity of Gothenburg Gothenburg\n40530, 60510ILSweden, USA", "University of Washington Seattle\n98195WAUSA", "Massachusetts Institute of Technology Cambridge\n02139MAUSA" ]
[]
In this paper, we investigate how field programmable gate arrays can serve as hardware accelerators for real-time semantic segmentation tasks relevant for autonomous driving. Considering compressed versions of the ENet convolutional neural network architecture, we demonstrate a fully-on-chip deployment with a latency of 4.9 ms per image, using less than 30% of the available resources on a Xilinx ZCU102 evaluation board. The latency is reduced to 3 ms per image when increasing the batch size to ten, corresponding to the use case where the autonomous vehicle receives inputs from multiple cameras simultaneously. We show, through aggressive filter reduction and heterogeneous quantization-aware training, and an optimized implementation of convolutional layers, that the power consumption and resource utilization can be significantly reduced while maintaining accuracy on the Cityscapes dataset.
10.1088/2632-2153/ac9cb5
[ "https://arxiv.org/pdf/2205.07690v1.pdf" ]
248,811,025
2205.07690
b4900ad55d1c167101914976fd25d20f491d7627
REAL-TIME SEMANTIC SEGMENTATION ON FPGAS FOR AUTONOMOUS VEHICLES WITH HLS4ML May 17, 2022 Nicolò Ghielmetti Vladimir Loncar Maurizio Pierini Marcel Roed Sioni Summers Thea Aarrestad Christoffer Petersson Zenseact Gothenburg Sweden Hampus Linander Jennifer Ngadiuba Kelvin Lin Philip Harris Institute for Particle Physics and Astrophysics European Organization for Nuclear Research (CERN) ETH Zürich CH-1211, 8093Geneva 23, ZürichSwitzerland, Switzerland Fermi National Accelerator Laboratory Batavia University of Gothenburg Gothenburg 40530, 60510ILSweden, USA University of Washington Seattle 98195WAUSA Massachusetts Institute of Technology Cambridge 02139MAUSA REAL-TIME SEMANTIC SEGMENTATION ON FPGAS FOR AUTONOMOUS VEHICLES WITH HLS4ML May 17, 2022deep learning · FPGA · convolutional neural network In this paper, we investigate how field programmable gate arrays can serve as hardware accelerators for real-time semantic segmentation tasks relevant for autonomous driving. Considering compressed versions of the ENet convolutional neural network architecture, we demonstrate a fully-on-chip deployment with a latency of 4.9 ms per image, using less than 30% of the available resources on a Xilinx ZCU102 evaluation board. The latency is reduced to 3 ms per image when increasing the batch size to ten, corresponding to the use case where the autonomous vehicle receives inputs from multiple cameras simultaneously. We show, through aggressive filter reduction and heterogeneous quantization-aware training, and an optimized implementation of convolutional layers, that the power consumption and resource utilization can be significantly reduced while maintaining accuracy on the Cityscapes dataset. Introduction Deep Learning has strongly reshaped computer vision in the last decade, bringing the accuracy of image recognition applications to unprecedented levels. Improved pattern recognition capabilities have had a significant impact on the advancement of research in science and technology. Many of the challenges faced by future scientific experiments, such as the CERN High Luminosity LHC [1] or the Square Kilometer Array observatory [2], and technological challenges faced by, for example, the automotive industry, will require the capability of processing large amounts of data in real-time, often through edge computing devices with strict latency and power-consumption constraints. This requirement has generated interest in the development of energy-effective neural networks, resulting in efforts like tinyML [3], which aims to reduce power consumption as much as possible without negatively affecting the model accuracy. Advances in Deep Learning for computer vision have had a crucial impact on the development of autonomous vehicles, enabling the vehicles to perceive their environment at ever-increasing levels of accuracy and detail. Deep Neural Networks are used for finding patterns and extracting relevant information from camera images, such as the precise location of the surrounding vehicles and pedestrians. In order for an autonomous vehicle to drive safely and efficiently, it must be able to react fast and make quick decisions. This imposes strict latency requirements on the neural networks that are deployed to run inference on resource-limited embedded hardware in the vehicle. In addition to algorithmic development, computer vision for autonomous vehicles has benefited from technological advances in parallel computing architecture [4]. The possibility of performing network training and inference on graphics processing units (GPUs) has made large and complex networks computationally affordable and testable on real-life problems. Due to their high efficiency, GPUs have become a common hardware choice in the automotive industry for on-vehicle Deep Learning inference. In this paper, we investigate the possibility of exploiting field-programmable gate arrays (FPGAs) as a low-power, inference-optimized, highly parallelisable alternative to GPUs. By applying aggressive filter-reduction and quantization of the model bit precision at training time, and by introducing a highly optimized firmware implementation of convolutional layers, we achieve the compression required to fit semantic segmentation models on FPGAs. We do so by exploiting and improving the hls4ml library, which provides an automatic conversion of a given Deep Neural Network into C++ code, which is given as input to a high level synthesis (HLS) library. The HLS library then translates this into FPGA firmware, to be deployed on hardware. Originally developed for scientific applications in particle physics that require sub-microsecond latency [5][6][7][8][9][10][11], hls4ml has been successfully applied outside the domain of scientific research [12,13], specifically in the context of tinyML applications [14]. Applying model compression at training time is crucial in order to minimize resource-consumption and maximize the model accuracy. To do so, we rely on quantization-aware training through the QKeras [15] library, which has been interfaced to hls4ml in order to guarantee an end-to-end optimal training-to-inference workflow [12]. As a baseline, we start from the ENet [16] architecture, designed specifically to perform pixel-wise semantic segmentation for tasks requiring low latency operations. We modify the architecture, removing resource-consuming asymmetric convolutions, and dilated or strided convolutions. In addition, we apply filter ablation and quantization at training time. Finally, we optimize the implementation of convolutional layers in hls4ml in order to significantly reduce the resource consumption. With these steps, we obtain a good balance between resource utilization and accuracy, enabling us to deploy the whole network on a Xilinx ZCU102 evaluation board [17]. This paper is organized as follows: The baseline dataset and model are described in sections 2 and 3, respectively. The model compression and the specific optimization necessary to port the compressed model to the FPGA are described in section 4 and 5. Conclusions are given in section 6. Dataset Our experiments are performed using the Cityscapes dataset [18], which involves 5,000 traffic scene images collected in 50 different cities with varying road types and seasons. These images have fine-grained semantic segmentation annotations with pixel-level classification labels. We have limited ourselves to the four semantic classes Road, Car, Person and Background. According to the standard Cityscapes split, 2975 images are used for training, 500 for validation and 1525 for testing. We crop and resize the original images to have an input resolution of 240×152 pixels. As a pre-processing step, we normalize all pixel values (integer values in the range [0, 255]) to be in the [0, 1] range by dividing each one by 256. In this way all inputs are smaller than one and can be represented by a fixed-point datatype using only 8 bits (log 2 (256)) (see section 4). An example image from the dataset is shown in Fig. 1, together with a visualization of its semantic segmentation mask. Baseline model The architecture we use is inspired by a fully convolutional residual network called Efficient Neural Network (ENet) [16]. This network was designed for low latency and minimal resource usage. It is designed as a sequence of blocks, summarized in Table 1. The initial block, shown in the left figure in Fig. 2, encodes the input into a 32 × 120 × 76 tensor, which is then processed by a set of sequential blocks of bottlenecks. The first three blocks constitute the downsampling encoder, where each block consists of a series of layers as summarized in the left diagram in Fig. 3. The final two blocks provide an upsampling decoder, as illustrated in the right diagram in Fig. 3 Some differences from the original architecture in [16] is that we do not use asymmetric, dilated, or strided convolutions. To further reduce the resource usage, we use three bottlenecks per block instead of five, and we merge convolutional layers with batch normalization layers by rescaling convolutional filter weights with batch normalization parameters (implemented through a QConv2DBatchnorm layer [19]). When we use quantization-aware training, this allows us to directly quantize the merged weights during the forward pass, rather than quantizing the batch normalization parameters and the convolutional filters separately. This merging of layers saves resources on the FPGA, since only the merged weights are used. Performing the merging already during training, ensures that the weights used during training and during inference are quantized the same way. The baseline ENet model is obtained fixing the six f hyperparameters of Table 1 to (32, 64, 64, 64, 128, 48). This choice results in an architecture with 1.1 · 10 6 parameters, yielding a mIoU=63.2% and an accuracy of 91.5%. Model compression We consider two compression techniques for the model at hand: filter-wise homogeneous pruning, obtained by reducing the number of filters on all the convolutional layers; and quantization, i.e., reducing the number of bits allocated for the numerical representation of the network components and the output of each layer computation. In addition, we use the AutoQKeras library [12], distributed with QKeras, to optimize the numerical representation of each component at training time as a hyperparameter. This is done using a mathematical model of the inference power consumption as a constraint in the loss function. Maxpool(2) Pad (2) Concat Skip connection Conv (3,29) BatchNorm ReLU Upsample (2) Pad (1) Conv(2,4) Figure 2: Initial (left) and final (right) block architecture. In the two diagrams, conv(k, f ) represents a convolutional layer with f k × k filters; maxpool(k) and upsample(k) represent a k × k max pooling or upsample layer, respectively; and pad(p) represents padding by p pixels in the lower and right directions. Table 2: Architecture reduction through internal filter ablation and corresponding performance. As a reference, the baseline architecture is reported on the first row. Highlighted in bold the three models considered further in this work. Filter multiplicity reduction Model name f i f 1 f 2 f 3 f 4 f 5 Parameters Normally, network pruning consists of zeroing specific network parameters that have little impact on the model performance. This could be done at training time or after training. In the case of convolutional layers, a generic pruning of the filter kernels would result in sparse kernels. It would then be difficult to take advantage of pruning during inference. To deal with this, filter ablation (i.e., the removal of an entire kernel) was introduced [20]. When filter ablation is applied, one usually applies a restructuring algorithm (e.g., Keras Surgeon [21]) to rebuild the model into the smaller-architecture model that one would use at inference. In this work, we take a simpler (and more drastic) approach: we treat the number of filters in the convolutional layers as a single hyperparameter, fixed across the entire network. We then reduce its value and repeat the training, looking for a good compromise between accuracy and resource requirements. We repeat the procedure with different target filter multiplicities. The result of this procedure is summarized in Table 2, where different pruning configurations are compared to the baseline Enet model. Out of these models, we select two configurations that would be affordable on the FPGA at hand: a four-filters (Enet4) and an eight-filter (Enet8) configuration. As a reference for comparison, we also consider one version with 16 filters, Enet16, despite it being too large to be deployed on the FPGA in question. We then proceed by quantizing these models through quantization-aware training to further reduce the resource consumption. Skip branch Main branch Maxpool (2,2) Pad (1) Conv (1,f o ) Pad(2) Conv(3,f i ) Conv(2,f i ) Conv(1,f o ) BatchNorm BatchNorm BatchNorm BatchNorm ReLU ReLU Add ReLU Skip branch Main branch Pad(2) Conv(3,f i ) Conv(2,f i ) BatchNorm BatchNorm Conv(1,f o ) BatchNorm Conv(1,f o ) BatchNorm ReLU ReLU Add Upsample (2) ReLU Upsample(2) Figure 3: Downsample encoder (left) and upsample decoder (right) blocks. In the figures, Conv(k, f ) represents a convolutional layer with f k × k filters, Maxpool(k) represents a k × k max pooling layer, Upsample(k) represents a k × k upsampling layer, and Pad(p) represents padding by p pixels in the lower and right directions. Blue boxes represent convolutional and batch normalization layers that in the model used are single merged layers. Homogeneous quantization-aware training Homogeneous quantization-aware training (QAT) consists of repeating the model training while forcing the numerical representation of its weight and activation functions to a fixed T, I precision, where T is the total number of bits and I is the number of integer bits. Doing so, the model training converges to a minimum that might not be the absolute minimum of the full-precision training, but that would minimize the performance loss once quantization is applied. In practice, we perform a homogeneous QAT replacing each layer of the model with its QKeras equivalent and exploiting the existing QKeras-to-hls4ml interface for FPGA deployment. We study the impact of QAT for T ∈ 2, 4, 8 with I = 0, on the pruned models desribed above (Enet4, Enet8, and Enet16). The resulting performance is shown in Table 3, where we label the three quantization configurations as Q2, Q4, and Q8, respectively. The resulting resource utilization for Enet4 and Enet8 falls within the regime of algorithms that we could deploy on the target FPGA. We observe similar drops in accuracy when going from full precision to Q8 and from Q4 to Q2, but little differences between the Q4 and Q8 models. In this respect, Q4 would offer a better compromise between accuracy and resources than Q8. Out of these, the models with the highest accuracy and mIoU that would be feasible to fit on the FPGA, is the 8 filter model quantized to 8 bits (Enet8Q8) and the 8 filter model quantized to 4 bits (Enet8Q4). Model name Quantization f i f 1 f 2 f 3 f 4 f 5 mIoU (%) Accuracy (%) Enet16 -32 16 16 Table 3: Homogeneously and heterogeneously quantized models with indicated bitwidth and filter architecture together with their validation mean IOU trained with quantization aware training using QKeras. The corresponding values before quantization (from Table 2) are also reported in the three first rows. The quantization of the model does not have to be homogeneous across layers. In fact, it has been demonstrated that a heterogeneous quantization is the best way to maintain high accuracy at low resource-cost [22]. We therefore define one final model with an optimized combination of quantizers. Heterogeneous quantization aware training Heterogeneous QAT consists in applying different quantization to different network components. For deep networks, one typically deals with the large number of possible configurations by using an optimization library. In our case, we use AUTOQKERAS [12]. In AUTOQKERAS, a hyperparameter search over individual layer quantization conditions and filter counts is performed. Since the model contains skip connections, the scan over number of filters needs to be handled with care. In particular, we use the block features of AUTOQKERAS to ensure that the filter count matches throughout a bottleneck, so that the tensor addition of the skip connection will have valid dimensions. The search for best hyperparameters, including the choice of indivdual quantizers for kernels and activations, is carried out using a Bayesian strategy where the balance between accuracy and resource usage is controlled by targeting a metric derived from them both [12]. In our search we permit e.g. a 4% decrease in accuracy if the resource usage also is halved at the same time. FPGA implementation, deployment and results Resource-efficient convolution algorithm The hls4ml library has an implementation of convolutional layers that is aimed at low-latency designs [13]. However, this implementation comes at the expense of high resource utilization. This is due to the number of times pixels of the input image are replicated to maintain the state of a sliding input window. For convolutional layers operating on wider images, like in our case, this overhead can be prohibitively large. In order to reduce the resource consumption of the convolutional layers of the model, we introduce a new algorithm that is more resource efficient. The new implementation, dubbed "line buffer", uses shift registers to keep track of previously seen pixels. The primary advantage of the line buffer implementation over the previous one is the reduction of the size of the buffer needed to store the replicated pixels. For an image of size H ×W , with a convolution kernel of size K ×L, the line buffer allocates K − 1 buffers (chain of shift registers) of depth W for the rows of the image, while the previous implementation allocates K 2 buffers of depth K × (W − K + 1) for the elements in the sliding input window. The algorithm is illustrated on Fig. 5. Initially, each new pixel read from the input image stream is pushed into the shift register chain. If the shift register is full, the first element will be popped and it will be pushed into the next shift register in chain. The process is repeated for all K − 1 shift registers in the chain. The popped pixels are stacked with the input pixel into a column vector and are pushed as the rightmost column of the input window. The pixels popped from the leftmost column of the input window are not used further. In our implementation, the propagation of new pixels through the shift register chain and the insertion into the sliding input window are completed in a single clock cycle, making the implementation as efficient as the existing hls4ml implementation. To compute the output from the populated sliding input window, we rely on the existing routines of hls4ml. We rely on a set of counters to keep track of input window state to know when to produce an output. The algorithm for maintaining the chain of shift registers and populating the sliding input window can be adapted for use in the pooling layers as well. To compare the two implementations, we consider the resource utilization of an ENet bottleneck block consisting of 8 filters, implemented using either method. The results are summarized in Table 4. We observe a substantial reduction in BRAM usage, at the price of a small increase in LUT utilization. Implementation BRAM DSP FF LUT Encoded [13] 4,752 5,632 195,344 291,919 Line buffer 4,064 5,632 176,620 305,494 Improvement -15% 0% -1% +5% Table 4: Comparison of previous and proposed hls4ml implementation of the convolutional layer, in terms of relative reduction of resource utilization. The estimates are made targeting an xczu9eg-2ffvb1156 MPSoC device on a ZCU102 development kit. FIFO depth optimization With the dataflow compute architecture of hls4ml, layer compute units are connected with FIFOs, implemented as memories in the FPGA. These FIFOs contribute to the overall resource utilisation of the design. The read and write pattern of these FIFOs depends on the dataflow through the model, which is not predictable before the design has been scheduled by the HLS compiler, and is generally complex. With previous hls4ml releases, these memories have therefore been assigned a depth corresponding to the dimensions of the tensor in the model graph as a safety precaution. To optimize this depth and thereby reduce resource consumption, we implemented an additional step in the compilation of the model to hardware. By using the clock-cycle accurate RTL simulation of the scheduled design, we can monitor the actual occupancy of each FIFO in the model when running the simulation over example images. This enables us to extract and set the correct size of the FIFOs, reducing memory usage compared to the baseline. By applying this procedure, we observe a memory efficiency l O l l F l = 19.5%, where the index l runs across the layers, O l is the observed occupancy for the l-th layer, and F l is the corresponding FIFO depth. The corresponding mean occupancy is found to be l O l F l = 4.7%. We then resize every FIFO to its observed maximum occupancy and rerun the C-Synthesis, thereby saving FPGA resources and allowing larger models to fit on the FPGA. Table 5 shows the impact of such an optimization on the FPGA resources for one example model, Enet8Q8, demonstrating a significant reduction of resources. Results The hardware we target is a Zynq UltraScale+ MPSoC device (xczu9eg-2ffvb1156) on a ZCU102 development kit, which targets automotive applications. After reducing the FPGA resource consumption through the methods described above, the highest accuracy models highlighted in Table 3 are synthesized. These are the homogeneously quantized Enet8Q8 and Enet8Q4 models, as well as the heterogeneously quantized EnetHQ model. To find the lowest latency implementation, we run several attempts varying the reuse factor (RF) and the clock period. The RF indicates how many times a multiplier can be reused (zero for a fully parallel implementation). Lower RF leads to lower latency, but higher resource usage. We targeted reuse factors of 10, 20, 50, 100, and clock periods of 5, 7, 10 ns. For each model, we then chose the configuration yielding the lowest latency. For Enet8Q8, this is a target clock period of 7 ns and RF=10. For Enet8Q4 and EnetHQ we use a clock period of 7 ns and RF = 6. Inference performance of this model was measured on the ZCU102 target device. The final latency and resource utilization report is shown in Table 5.3. Table 6: Accuracy, mIoU, latency and resource utilization for the EnetHQ, Enet8Q4 and Enet8Q8 models. The latency is quoted for a batch size b = 1 and b = 10. Resources are expressed as a percentage of those available on the xczu9eg-2ffvb1156 MPSoC device on the ZCU102 development kit. The last row is a comparison to work presented in Ref. [23]. Model Acc. a The former is without considering data transfer, pre-and post-processing. The number in parenthesis includes these additional overheads, averaged over 58 images, and is more comparable to the numbers we present. We measured the time taken by the accelerator to produce a prediction on batches of images, with batch sizes of b = 1 and b = 10. The same predictions have been executed 10 5 times, and the time average is taken as the latency. The single image latency (batch size of 1) is 4.8-4.9 ms for all three models. Exploiting the data flow architecture, the latency to process images in a batch size of 10 is less than 10 times the the latency observed for a batch size of 1. While in a real-world deployment of this model the latency to return the predictions of a single image is the most important metric, a system comprised of multiple cameras may be able to benefit from the speedup of batched processing by batching over the images captured simultaneously from different cameras. The model with the highest accuracy and lowest resource consumption is the heterogeneously quantized EnetHQ model. This model has an mIoU of 36.8% and uses less than 30% of the total resources. Similar work on performing semantic segmentation on FPGAs include Ref. [23] and a comparison is given in Table 5.3. Here, the original ENet model [16] is trained and evaluated on the Cityscapes dataset, and then deployed on a Xilinx Zynq 7035 FPGA using the Xilinx Vitis AI Deep Learning Processor Unit (DPU). There are some crucial differences between the approach taken here and that of Ref. [23]. In order to achieve the lowest possible latency, we implement a fully on-chip design with high layer parallelism. We optimize for latency, rather than frame rate, such that in a real-life application the vehicle response time could be minimized. Keeping up with the camera frame rate is a minimal requirement, but a latency lower than the frame interval can be utilized. In our approach, each layer is implemented as a separate module and data is streamed through the architecture layer by layer. Dedicated per-layer buffers ensure that just enough data is buffered in order to feed the next layer. This is highly efficient, but limits the number of layers that can be implemented on the FPGA. Consequently, in order to fit onto the FPGA in question, our model is smaller and achieves a lower mIoU. Ref. [23] does not quote a latency, but a frame rate. A best-case latency is then computed as the inverse of this frame rate, which corresponds to 30.38 ms. However, this does not include any overhead latency like data transfer, pre-and post-processing. Including these, the average time per image increases to 720 ms. Conclusions In this paper, we demonstrate that we can perform semantic segmentation on a single FPGA on a Zynq MPSoC device using a compressed version of ENet. The network is compressed using automatic heterogeneous quantization at training time and a filter ablation procedure, and is then evaluated on the Cityscapes dataset. Inference is executed on hardware with a latency of 4.9 ms per image, utilizing 18% of the DSPs, 30% of the LUTs, 16% of the FFs and 25 % of the BRAMs. Processing the images in batches of ten results in a latency of 30 ms per batch, which is significantly faster than ten times the single-image batch inference latency. This is relevant when batching over images captured from different cameras simultaneously. By introducing an improved implementation of convolutional layers in hls4ml, we significantly reduce resource consumption, allowing for a fully-on-chip deployment of larger convolutional neural networks. This avoids latency overhead caused by data transfers between off-chip memory and FPGA processing elements, or between multiple devices. Also taking into account the favorable power-efficiency of FPGAs, we conclude that FPGAs offer highly interesting, low-power alternatives to GPUs for on-vehicle deep learning inference and other computer vision tasks requiring low-power and low-latency. For evaluation metrics we use two typical figures of merit for semantic segmentation:• The model accuracy (Acc), defined as Acc = T P +T N T P +T N +F P +F N , where T P , T N , F P , and F N are the fraction of true positives, true negatives, false positives, and false negatives, respectively. • The mean of the class-wise Intersection over Union (mIoU), i.e., the average across classes of the Intersection-Over-Union (defined as IOU = T P T P +F P +F N ). Figure 1 : 1An downsampled image from the Cityscapes dataset (left) and the corresponding semantic segmentation target (right), in which the pixels belong to one of the classes {background (blue), road (teal), car (yellow), person (red)}. Figure 4 : 4Validation accuracy versus number of bits for the models tested during heterogeneous QAT with AUTO-QKERAS, for the six blocks in the network. The circle size represents the number of filters, while the color refers to the median bits for the convolutions in the corresponding block. Details on the grid of options considered in the optimization are given in the text. Figure 5 : 5Schematic representation of the new hls4ml implementation of Convolutional layers, as described in the text.The hyperparameter scan is done sequentially over the blocks, i.e. the Bayesian search over quantization and filter count of the initial layer is performed first and is then frozen for the hyperparameter scan of the first bottleneck and so on. The rest of the model is kept in floating point until everything in the end is quantized. Figure 4 4shows the outcome of the heterogeneous QAT, in terms of validation accuracy and total number of bits for the six blocks in the network. The optimal configuration search is performed taking as a baseline the Enet4 model, scanning the kernel bits in {4, 8} and fixing the number of kernels to four times a by-layer multiplicative chosen in {0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0}. The optimal configuration (EnetHQ) is obtained for f i = 8, f 1 = 2, f 2 = 4, f 3 = 8, f 4 = 4, and f 5 = 3, resulting in 4.7 · 10 3 parameters, a mIoU=36.8% and an accuracy of 81.1%. Out of all the quantized models, both homogeneous and heterogeneous, this is the one which performs the best. . The final block is shown in the right diagram of Fig. 2. Model architecture parametrized by the number of filters in the bottlenecks f i , with i = 1, ..., 5.Layer Type Output resolution Initial downsample f 0 × 120 × 76 3× bottleneck 1 downsample f 1 × 60 × 38 3× bottleneck 2 downsample f 2 × 30 × 19 3× bottleneck 3 f 3 × 30 × 19 3× bottleneck 4 upsample f 4 × 60 × 38 3× bottleneck 5 upsample f 5 × 120 × 76 Final upsample 4 × 240 × 152 Table 1: Effect of FIFO depth optimization on FPGA resource usage and model latency. The values in the table are taken from Vivado HLS estimates of resource usage. A comparison using physical resource usage is unfeasible since the model without optimization can not be synthesized. The estimates are made targeting an xczu9eg-2ffvb1156 MPSoC device on a ZCU102 development kit.Optimisation BRAM LUT FF DSP Latency No 7,270 676,760 230,913 228 3.577 ms Yes 1,398 437,559 146,392 228 3.577 ms Improvement -81% -35% -37% 0% 0% Table 5: A G , 10.23731/CYRM-2017-004High-Luminosity Large Hadron Collider. 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[ "https://github.com/google/qkeras.", "https://github.com/google/qkeras/blob/master/qkeras/qconv2d_batchnorm.py.", "https://github.com/BenWhetton/keras-surgeon." ]
[ "Automated Expected Amortised Cost Analysis of Probabilistic Data Structures", "Automated Expected Amortised Cost Analysis of Probabilistic Data Structures", "Automated Expected Amortised Cost Analysis of Probabilistic Data Structures", "Automated Expected Amortised Cost Analysis of Probabilistic Data Structures" ]
[ "Lorenz Leutgeb \nMax Planck Institute for Informatics and Graduate School of Computer Science\nSaarland Informatics Campus\nSaarbrückenGermany\n", "Georg Moser \nDepartment of Computer Science\nUniversität Innsbruck\nAustria\n", "Florian Zuleger \nInstitute of Logic and Computation 192\n\n", "\nTechnische Universität Wien\nAustria\n", "Lorenz Leutgeb \nMax Planck Institute for Informatics and Graduate School of Computer Science\nSaarland Informatics Campus\nSaarbrückenGermany\n", "Georg Moser \nDepartment of Computer Science\nUniversität Innsbruck\nAustria\n", "Florian Zuleger \nInstitute of Logic and Computation 192\n\n", "\nTechnische Universität Wien\nAustria\n" ]
[ "Max Planck Institute for Informatics and Graduate School of Computer Science\nSaarland Informatics Campus\nSaarbrückenGermany", "Department of Computer Science\nUniversität Innsbruck\nAustria", "Institute of Logic and Computation 192\n", "Technische Universität Wien\nAustria", "Max Planck Institute for Informatics and Graduate School of Computer Science\nSaarland Informatics Campus\nSaarbrückenGermany", "Department of Computer Science\nUniversität Innsbruck\nAustria", "Institute of Logic and Computation 192\n", "Technische Universität Wien\nAustria" ]
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In this paper, we present the first fully-automated expected amortised cost analysis of self-adjusting data structures, that is, of randomised splay trees, randomised splay heaps and randomised meldable heaps, which so far have only (semi-) manually been analysed in the literature. Our analysis is stated as a type-and-effect system for a first-order functional programming language with support for sampling over discrete distributions, non-deterministic choice and a ticking operator. The latter allows for the specification of fine-grained cost models. We state two soundness theorems based on two different-but strongly related-typing rules of ticking, which account differently for the cost of non-terminating computations. Finally we provide a prototype implementation able to fully automatically analyse the aforementioned case studies.
10.1007/978-3-031-13188-2_4
[ "https://export.arxiv.org/pdf/2206.03537v1.pdf" ]
249,461,538
2206.03537
fb73c7d16aae9510392264b6bd909ca4525aa766
Automated Expected Amortised Cost Analysis of Probabilistic Data Structures Lorenz Leutgeb Max Planck Institute for Informatics and Graduate School of Computer Science Saarland Informatics Campus SaarbrückenGermany Georg Moser Department of Computer Science Universität Innsbruck Austria Florian Zuleger Institute of Logic and Computation 192 Technische Universität Wien Austria Automated Expected Amortised Cost Analysis of Probabilistic Data Structures amortised cost analysis · functional programming · proba- bilistic data structures · automation · constraint solving In this paper, we present the first fully-automated expected amortised cost analysis of self-adjusting data structures, that is, of randomised splay trees, randomised splay heaps and randomised meldable heaps, which so far have only (semi-) manually been analysed in the literature. Our analysis is stated as a type-and-effect system for a first-order functional programming language with support for sampling over discrete distributions, non-deterministic choice and a ticking operator. The latter allows for the specification of fine-grained cost models. We state two soundness theorems based on two different-but strongly related-typing rules of ticking, which account differently for the cost of non-terminating computations. Finally we provide a prototype implementation able to fully automatically analyse the aforementioned case studies. Introduction Probabilistic variants of well-known computational models such as automata, Turing machines or the λ-calculus have been studied since the early days of computer science (see [15,16,23] for early references). One of the main reasons for considering probabilistic models is that they often allow for the design of more efficient algorithms than their deterministic counterparts (see e.g. [6,21,23]). Another avenue for the design of efficient algorithms has been opened up by Sleator and Tarjan [31,33] with their introduction of the notion of amortised complexity. Here, the cost of a single data structure operation is not analysed in isolation but as part of a sequence of data structure operations. This allows for the design of algorithms where the cost of an expensive operation is averaged out over multiple operations and results in a good overall worst-case cost. Both methodologiesprobabilistic programming and amortised complexity-can be combined for the design of even more efficient algorithms, as for example in randomized splay trees [1], where a rotation in the splaying operation is only performed with some probability (which improves the overall performance by skipping some rotations while still guaranteeing that enough rotations are performed). In this paper, we present the first fully-automated expected amortised cost analysis of probabilistic data structures, that is, of randomised splay trees, randomised splay heaps, randomised meldable heaps and a randomised analysis of a binary search tree. These data structures have so far only (semi-)manually been analysed in the literature. Our analysis is based on a novel type-and-effect system, which constitutes a generalisation of the type system studied in [13,17] to the non-deterministic and probabilistic setting, as well as an extension of the type system introduced in [34] to sublinear bounds and non-determinism. We provide a prototype implementation that is able to fully automatically analyse the case studies mentioned above. We summarize here the main contributions of our article: (i) We consider a first-order functional programming language with support for sampling over discrete distributions, non-deterministic choice and a ticking operator, which allows for the specification of fine-grained cost models. (ii) We introduce compact small-step as well as big-step semantics for our programming language. These semantics are equivalent wrt. the obtained normal forms (ie., the resulting probability distributions) but differ wrt. the cost assigned to non-terminating computations. (iii) Based on [13,17], we develop a novel type-and-effect system that strictly generalises the prior approaches from the literature. (iv) We state two soundness theorems (see Section 5.3) based on two different-but strongly related-typing rules of ticking. The two soundness theorems are stated wrt. the small-step resp. big-step semantics because these semantics precisely correspond to the respective ticking rule. The more restrictive ticking rule can be used to establish (positive) almost sure termination (AST) while the more permissive ticking rule supports the analysis of a larger set of programs (which can be very useful in case termination is not required or can be established by other means); in fact, the more permissive ticking rule is essential for the precise cost analysis of randomised splay trees. We note that the two ticking rules and corresponding soundness theorems do not depend on the details of the type-and-effect system, and we believe that they will be of independent interest (e.g., when adapting the framework of this paper to other benchmarks and cost functions). (v) Our prototype implementation ATLAS strictly extends and earlier vrsion discussed in [17] and all our earlier evaluation results can be replicated (and sometime improved). With our implementation and the obtained experimental results we make two contributions to the complexity analysis of data structures: 1. We automatically infer bounds on the expected amortised cost, which could previously only be obtained by sophisticated pen-and-paper proofs. In particular, we verify that the amortised costs of randomised variants of self-adjusting data structures improve upon their non-randomised variants. In Table 1 we state the expected cost of the randomised data structures and their deterministic counterparts; the benchmarks are detailed in Section 2. probabilistic deterministic [17] Splay Tree insert 3 /4 log 2 (|t|) + 3 /4 log 2 (|t| + 1) 2 log 2 (|t|) + 3 /2 delete 9 /8 log 2 (|t|) 5 /2 log 2 (|t|) + 3 splay 9 /8 log 2 (|t|) 3 /2 log 2 (|t|) Splay Heap insert 3 /4 log 2 (|h|) + 3 /4 log 2 (|h| + 1) 1 /2 log 2 (|h|) + log 2 (|h| + 1) + 3 /2 delete min 3 /4 log 2 (|h|) log 2 (|h|) Meldable Heap insert log 2 (|h|) + 1 delete min 2 log 2 (|h|) not applicable meld log 2 (|h1|) + log 2 (|h2|) Coin Search Tree insert 3 /2 log 2 (|t|) + 1 /2 delete log 2 (|t|) not applicable delete max 3 /2 log 2 (|t|) + 1 /2 Table 1: Expected Amortised Cost of Randomised Data Structures. We also state the deterministic counterparts considered in [17] for comparison. 2. We establish a novel approach to the expected cost analysis of data structures. While the detailed study of Albers et al. in [1] requires a sophisticated penand-paper analysis, our approach allows us to fully-automatically compare the effect of different rotation probabilities on the expected cost (see Table 2 of Section 6). Related Work. The generalisation of the model of computation and the study of the expected resource usage of probabilistic programs has recently received increased attention (see e.g. [2,4,5,7,9,10,14,19,20,22,25,34,35]). We focus on related work concerned with automations of expected cost analysis of deterministic or non-deterministic, probabilistic programs-imperative or functional. (A probabilistic program is called non-deterministic, if it additionally makes use of non-deterministic choice.) In recent years the automation of expected cost analysis of probabilistic data structures or programs has gained momentum, cf. [2-5, 20, 22, 25, 34, 35]. Notably, the Absynth prototype by [25], implement Kaminski's ert-calculus, cf. [14] for reasoning about expected costs. Avanzini et al. [5] introduce the tool ecoimp, which generalises the Absynth prototype and provides a modular and thus a more efficient and scalable alternative for non-deterministic, probabilistic programs. In comparison to these works, we base our analysis on a dedicated type system finetuned to express sublinear bounds; further our prototype implementation ATLAS derives bounds on the expected amortised costs. Neither is supported by Absynth or eco-imp. Martingale based techniques have been implemented, e.g., by Peixin Wang et al. [35]. Related results have been reported by Moosbrugger et al. [22]. Meyer et al. [20] provide an extension of the KoAT tool, generalising the concept of alternating size and runtime analysis to probabilistic programs. Again, these innovative tools are not suited to the benchmarks considered in our work. With respect to probabilistic functional programs, Di Wang et al. [34] provided the only prior expected cost analysis of (deterministic) probabilistic programs; this work is most closely related to our contributions. Indeed, our typing rule (ite : coin) stems from [34] and the soundness proof wrt. the big-step semantics is conceptually similar. Nevertheless, our contributions strictly generalise their results. First, our core language is based on a simpler semantics, giving rise to cleaner formulations of our soundness theorems. Second, our type-andeffect provides two different typing rules for ticking, a fact we can capitalise on in additional strength of our prototype implementation. Finally, our amortised analysis allows for logarithmic potential functions. A bulk of research concentrates on specific forms of martingales or Lyapunov ranking functions. All these works, however, are somewhat orthogonal to our contributions, as foremostly termination (ie. AST or PAST) is studied, rather than computational complexity. Still these approaches can be partially suited to a variety of quantitative program properties, see [32] for an overview, but are incomparable in strength to the results established here. Structure. In the next section, we provide a bird's eye view on our approach. Sections 3 and 4 detail the core probabilistic language employed, as well as its small-and big-step semantics. In Section 5 we we introduce the novel type-andeffect system formalising and state soundness of the system wrt. the respective semantics. In Section 6 we present evaluation results of our prototype implementation ATLAS. Finally, we conclude in Section 7. Overview of Our Approach and Results In this section, we first sketch our approach on an introductory example and then detail the benchmarks and results depicted in Table 1 in the Introduction. Introductory Example Consider the definition of the function descend, depicted in Figure 1. The expected amortised complexity of descend is log 2 (|t|), where |t| denotes the size of a tree (defined as the number of leaves of the tree). 4 Our analysis is set up in terms of template potential functions with unknown coefficients, which will be instantiated by our analysis. Following [13,17], our potential functions are composed of two types of resource functions, which can express logarithmic amortised cost: For a sequence of n trees t 1 , . . . , t n and coefficients a i ∈ N, b ∈ Z, with n i=1 a i + b 0, the resource function p (a1,...,an,b) (t 1 , . . . , t n ) := log 2 (a 1 · |t 1 | + · · · + a n · |t n | + b) denotes the logarithm of a linear combination of 1 descend t = match t with 2 | leaf → leaf 3 | node l a r → if coin 1/2 Denotes p = 1 /2, which is default and could be omitted. 4 then let xl = (descend l) in node xl a r The symbol denotes a tick. 5 else let xr = (descend r) in node l a xr Fig. 1: descend function the sizes of the trees. The resource function rk(t), which is a variant of Schoenmakers' potential, cf. [26,29,30], is inductively defined as (i) rk(leaf) := 1; (ii) rk(node l d r) := rk(l) + log 2 (|l|) + log 2 (|r|) + rk(r), where l, r are the left resp. right child of the tree node l d r, and d is some data element that is ignored by the resource function. (We note that rk(t) is not needed for the analysis of descend but is needed for more involved benchmarks, e.g. randomised splay trees.) With these resource functions at hand, our analysis introduces the coefficients q * , q (1,0) , q (0,2) , q * , q (1,0) , q (0,2) and employs the following Ansatz : 5 q * · rk(t) + q (1,0) · p (1,0) (t) + q (0,2) · p (0,2) (t) c descend (t) + + q * rk(descend t) + q (1,0) · p (1,0) (descend t) + q (0,2) · p (0,2) (descend t) . Here, c descend (t) denotes the expected cost of executing descend on tree t, where the cost is given by the ticks as indicated in the source code (each tick accounts for a recursive call). The result of our analysis will be an instantiation of the coefficients, returning q (1,0) = 1 and zero for all other coefficients, which allows to directly read off the logarithmic bound log 2 (|t|) of descend. Our analysis is formulated as a type-and-effect system, introducing the above template potential functions for every subexpression of the program under analysis. The typing rules of our system give rise to a constraint system over the unknown coefficients that capture the relationship between the potential functions of the subexpressions of the program. Solving the constraint system then gives a valid instantiation of the potential function coefficients. Our type-andeffect system constitutes a generalisation of the type system studied in [13,17] to the non-deterministic and probabilistic setting, as well as an extension of the type system introduced in [34] to sublinear bounds and non-determinism. In the following, we survey our type-and-effect system by means of example descend. A partial type derivation is given in Figure 2. For brevity, type judgements and the type rules are presented in a simplified form. In particular, we restrict our attention to tree types, denoted as T. This omission is inessential to the actual complexity analysis. For the full set of rules see the Appendix. We now discuss this type derivation step by step. Let e denote the body of the function definition of descend, cf. Figure 1. Our automated analysis infers an annotated type by verifying that the type judgement t : T|Q e : T|Q is derivable. Types are decorated with annotations Q := [q * , q (1,0) , q (0,2) ] and Q := [q * , q (1,0) , q (0,2) ]-employed to express the potential carried by the arguments to descend and its results. Annotations fix = q * · rk(t) + q (1,0) · p (1,0) (t) + q (0,2) · p (0,2) (t) and (ii) Φ(e : T|Q ) := q * · rk(e) + q (1,0) · p (1,0) (e) + q (0,2) · p (0,2) (e). By our soundness theorems (see Section 5.3), such a typing guarantees that the expected amortised cost of descend is bounded by the expectation (wrt. the value distribution in the limit) of the difference between Φ(t : T|Q) and Φ(descend t : T|Q ). Because e is a match expression, the following rule is applied (we only state a restricted rule here, the general rule can be found in the Appendix): ε|∅ leaf : T|Q l : T, r : T|Q 1 e 1 : T|Q t : T|Q match t with|leaf → leaf |node l a r → e 1 : T|Q (match) Here e 1 denotes the subexpression of e that corresponds to the node case of match. Apart from the annotations Q, Q 1 and Q , the rule (match) constitutes a standard type rule for pattern matching. With regard to the annotations Q and Q 1 , (match) ensures the correct distribution of potential by inducing the constraints q 1 1 = q 1 2 = q * q 1 (1,1,0) = q (1,0) q 1 (1,0,0) = q 1 (0,1,0) = q * q 1 (0,0,2) = q (0,2) , where the constraints are immediately justified by recalling the definitions of the resource functions p (a1,...,an,b) (t 1 , . . . , t n ) := log 2 (a 1 · |t 1 | + · · · + a n · |t n | + b) and rk(t) = rk(l) + log 2 (|l|) + log 2 (|r|) + rk(r). The next rule is a structural rule, representing a weakening step that rewrites the annotations of the variable context. The rule (w) allows a suitable adaptation of the coefficients based on the following inequality, which holds for any substitution σ of variables by values, Φ(σ; l : T, r : T|Q 1 ) Φ(σ; l : T, r : T|Q 2 ). l : T, r : T|Q 2 e 1 : T|Q l : T, r : T|Q 1 e 1 : T|Q (w) In our prototype implementation this comparison is performed symbolically. We use Farkas' Lemma in conjunction with two facts about the logarithm to then ( node (meld h2l (node h1l h1x h1r)) h2x h2r ) 7 else ( node h2l h2x (meld h2r (node h1l h1x h1r)) ) 8 else Omitted for brevity, symmetric to the the depicted case. linearise this symbolic comparison, namely the monotonicity of the logarithm and the fact that 2 + log 2 (x) + log 2 (y) 2 log 2 (x + y) for all x, y 1. For example, Farkas' Lemma in conjunction with the latter fact gives rise to q 1 (0,0,2) + 2f q 2 (0,0,2) q 1 (1,1,0) − 2f q 2 (1,1,0) q 1 (1,0,0) + f q 2 (1,0,0) q 1 (0,1,0) + f q 2 (0,1,0) , for some fresh rational coefficient f 0 introduced by Farkas' Lemma. After having generated the constraint system for descend, the solver is free to instantiate f as needed. In fact in order to discover the bound log 2 (|t|) for descend, the solver will need to instantiate f = 1 /2, corresponding to the inequality log 2 (|l| + |r|) 1 /2 log 2 (|l|) + 1 /2 log 2 (|r|) + 1. So far, the rules did not refer to sampling and are unchanged from their (non-probabilistic) counterpart introduced in [13,17]. The next rule, however, formalises a coin toss, biased with probability p. Our general rule (ite : coin) is depicted in Figure 12 and is inspired by a similar rule for coin tosses that has been recently been proposed in the literature, cf. [34]. This rule specialises as follows to our introductory example: Here e 2 and e 3 respectively, denote the subexpressions of the conditional and in addition the crucial condition Q 2 = 1 /2 · Q 3 + 1 /2 · Q 4 holds. This condition, expressing that the corresponding annotations are subject to the probability of the coin toss, gives rise to the following constraints (among others) q 2 (0,0,2) = 1 /2 · q 3 (0,0,2) + 1 /2 · q 4 (0,0,2) q 2 (0,1,0) = 1 /2 · q 3 (0,1,0) + 1 /2 · q 4 (0,1,0) q 2 (1,0,0) = 1 /2 · q 3 (1,0,0) + 1 /2 · q 4 (1,0,0) . In the following, we will only consider one alternative of the coin toss and proceed as in the partial type derivation depicted in Figure 1 (ie. we state the thenbranch and omit the symmetric else-branch). Thus next, we apply the rule for the let expression. This rule is the most involved typing rule in the system 1 splay a t = match t with 2 | node cl c cr → match cl with 3 | node bl b br → match (splay a bl) 1/2 with Recursive call costs 1 /2. 4 | node al a1 ar → if coin 5 then (node al a1 (node ar b (node br c cr))) 1/2 Rotation costs 1 /2. proposed in [13,17]. However, for our leading example it suffices to consider the following simplified variant: l : T|Q 4 (descend l) : T|Q 6 l : T|Q 7 node x l a r : T|Q l : T, r : T|Q 3 let x l = (descend l) in node x l a r : T|Q (let) Focusing on the annotations, the rule (let : tree) suitably distributes potential assigned to the variable context, embodied in the annotation Q 3 , to the recursive call within the let expression (via annotation Q 4 ) and the construction of the resulting tree (via annotation Q 7 ). The distribution of potential is facilitated by generating constraints that can roughly be stated as two "equalities", that is, (i) "Q 3 = Q 4 + D" and (ii) "Q 7 = D + Q 6 ". Equality (i) states that the input potential is split into some potential Q 4 used for typing (descend l) and some remainder potential D (which however is not constructed explicitly and only serves as a placeholder for potential that will be passed on). Equality (ii) states that the potential Q 7 used for typing node x l a r equals the remainder potential D plus the leftover potential Q 6 from the typing of (descend l) . The (tick : now) rule then ensures that costs are properly accounted for by generating constraints for Q 4 = Q 5 + 1. Finally, the type derivation ends by the application rule, denoted as (app), that verifies that the recursive call is well-typed wrt. the (annotated) signature of the function descend : T|Q → T|Q , ie. the rule enforces that Q 5 = Q and Q 6 = Q . We illustrate (a subset of) the constraints induced by (let), (tick : now) and (app): q 3 (1,0,0) = q 4 (1,0) q 3 (0,1,0) = q 7 (0,1,0) q 1 = q 6 1 q 4 (0,2) = q 5 (0,2) + 1 q 3 (0,0,2) = q 4 (0,2) q 3 2 = q 7 2 q (1,0) = q 6 (1,0) q 4 (1,0) = q 5 (1,0) q 3 1 = q 4 1 q (0,2) = q 6 (0,2) q 6 1 = q 7 1 q 5 (1,0) = q (1,0) , where (i) the constraints in the first three columns-involving the annotations Q 3 , Q 4 , Q 6 and Q 7 -stem from the constraints of the rule (let : tree); (ii) the constraints in the last column-involving Q 4 , Q 5 , Q and Q -stem from the constraints of the rule (tick : now) and (app). For example, q 3 (1,0,0) = q 4 (1,0) and q 3 (0,1,0) = q 7 (0,1,0) distributes the part of the logarithmic potential represented by Q 3 to Q 4 and Q 7 ; q 6 1 = q 7 1 expresses that the rank of the result of evaluating the recursive call can be employed in the construction of the resulting tree node x l a r; q 4 (1,0) = q 5 (1,0) and q 4 (0,2) = q 5 (0,2) + 1 relate the logarithmic resp. constant potential according to the tick rule, where the addition of one accounts for 1 insert d t = match t with 2 | leaf → node leaf d leaf 3 | node l a r → if coin 1/2 Assuming probability 1 /2 for a < d. 4 then node (insert d l) a r 5 else node l a (insert d r) Fig. 5: insert function of a Binary Search Tree with randomized comparison the cost embodied by the tick rule; q 5 (1,0) = q (1,0) stipulates that the potential at the recursive call site must match the function type. Our prototype implementation ATLAS collects all these constraints and solves them fully automatically. Following [13,17], our implementation in fact searches for a solution that minimises the resulting complexity bound. For the descend function, our implementation finds a solution that sets q (1,0) to 1, and all other coefficients to zero. Thus, the logarithmic bound log 2 (|t|) follows. Overview of Benchmarks and Results Randomised Meldable Heaps. Gambin et al. [12] proposed meldable heaps as a simple priority-queue data structure that is guaranteed to have expected logarithmic cost for all operations. All operations can be implemented in terms of the meld function, which takes two heaps and returns a single heap as a result. The partial source code of meld is given in Figure 3 (the full source code of all examples can be found in the Appendix). Our tool ATLAS fully-automatically infers the bound log 2 (|h1|) + log 2 (|h2|) on the expected cost of meld. Randomised Splay Trees. Albers et al. in [1] proposed these splay trees as a variation of deterministic splay trees [31], which have better expected runtime complexity (the same computational complexity in the O-notation but with smaller constants). Related results have been obtained by Fürer [11]. The proposal is based on the observation that it is not necessary to rotate the tree in every (recursive) splaying operation but that it suffices to perform rotations with some fixed positive probability in order to reap the asymptotic benefits of self-adjusting search trees. The theoretical analysis of randomised splay trees [1] starts by refining the cost model of [31], which simply counts the number of rotations, into one that accounts for recursive calls with a cost of c and for rotations with a cost of d. We present a snippet of a functional implementation of randomised splay trees in Figure 4. We note that in this code snippet we have set c = d = 1 /2; this choice is arbitrary; we have chosen these costs in order to be able to compare the resulting amortised costs to the deterministic setting of [17], where the combined cost of the recursive call and rotation is set to 1; we note that our analysis requires fixed costs c and d but these constants can be chosen by the user; for example one can set c = 1 and d = 2.75 corresponding to the costs observed during the experiments in [1]. Likewise the probability of the coin toss has been arbitrarily set to p = 1 /2 but could be set differently by the user. (We remark that to the best of our knowledge no theoretical analysis has been conducted on how to chose the best value of p for given costs c and d.) Our analysis is able to 6: delete_max function of a Coin Search Tree with one rotation fully automatically infer an amortised complexity bound of 9 /8 log 2 (|t|) for splay (with c, d and p fixed as above), which improves on the complexity bound of 3 /2 log 2 (|t|) for the deterministic version of splay as reported in [17], confirming that randomisation indeed improves the expected runtime. We remark on how the amortised complexity bound of 9 /8 log 2 (|t|) for splay is computed by our analysis. Our tool ATLAS computes an annotated type for splay that corresponds to the inequality 3 /4 rk(t)+ 9 /8 log 2 (|t|)+ 3 /4 c splay (t)+ 3 /4 rk(splay t)+ 3 /4. By setting φ(t) := rk(t)+ 3 /4 as potential function in the sense of Tarjan and Sleator [31,33], the above inequality allows us to directly read out an upper bound on the amortised complexity a splay (t) of splay (we recall that the amortised complexity in the sense of Tarjan and Sleator is defined as the sum of the actual costs plus the output potential minus the input potential): 1 pre−condition: t is not a leaf 2 delete_max t = match t with 3 | node l b r → match r with 4 | leaf → ( l,b ) 5 | node rl c rr → match rr with 6 | leaf → (( node l b rl ) ,c ) 7 | rr → let ( t',max ) = (delete max rr) in match t' with 8 | node rrl1 x xa → ( node ( node ( node l b rl ) c rrl1 ) x xa,max ) Fig.a splay (t) = c splay (t) + φ(splay t) − φ(t) 9 /8 · log 2 (|t|). Probabilistic Analysis of Binary Search Trees. We present a probabilistic analysis of a deterministic binary search tree, which offers the usual contains, insert, and delete operations, where delete uses delete_max given in Figure 6, as a subroutine (the source code of the missing operations is given in the Appendix). We assume that the elements inserted, deleted and searched for are equally distributed; hence, we conduct a probabilistic analysis by replacing every comparison with a coin toss of probability one half. We will refer to the resulting data structure as Coin Search Tree in our benchmarks. The source code of insert is given in Figure 5. Our tool ATLAS infers an logarithmic expected amortised cost for all operations, ie., for insert and delete_max we ob- tain (i) 3 /2 rk(t) + 1 /2 log 2 (|t|) c insert (t) + 3 /2 rk(insert t); and (ii) 3 /2 rk(t) + 1 /2 log 2 (|t|) c delete_max (t) + 3 /2 rk(delete_max t) , from which we obtain an expected amortised cost of 1 /2 log 2 (|t|) for both functions. Probabilistic Functional Language Preliminaries. Let R + 0 denote the non-negative reals and R +∞ 0 their extension by ∞. We are only concerned with discrete distributions and drop "discrete" in the following. Let A be a countable set and let D(A) denote the set of (sub)distributions d over A, whose support supp(µ) := {a ∈ A | µ(a) = 0} is countable. Distributions are denoted by Greek letters. For µ ∈ D(A), we may write µ = {a pi i } i∈I , assigning probabilities p i to a i ∈ A for every i ∈ I, where I is a suitable chosen index set. We set |µ| := i∈I p i . If the support is finite, we • ::= < | > | = e ::= f x1 . . . xn | e a/b | false | true | e1 • e2 | if x then e1 else e2 | if nondet then e1 else e2 | if coin a/b then e1 else e2 | leaf | node x1 x2 x3 | match x with | leaf → e1 | node x1 x2 x3 → e2 | ( x1 , x2 ) | match x with | ( x1 , x2 ) → e | let x = e1 in e2 | xsimply write µ = {a p1 1 , . . . , a pn n } The expected value of a function f : A → R + 0 on µ ∈ D(A) is defined as E µ (f ) := a∈supp(µ) µ(a) · f (a). Further, we denote by i∈I p i · µ i the convex combination of distributions µ i , where i∈I p i 1. As by assumption i∈I p i 1, i∈I p i · µ i is always a (sub-)distribution. Syntax. In Figure 7, we detail the syntax of our core probabilistic (first-order) programming language. With the exception of ticks, expressions are given in let-normal form to simplify the presentation of the operational semantics and the typing rules. In order to ease the readability, we make use of mild syntactic sugaring in the presentation of actual code (as we already did above). To make the presentation more succinct, we assume only the following types: a set of base types B such as Booleans Bool = {true, false}, integers Int, or rationals Rat, product types, and binary trees T, whose internal nodes are labelled with elements b : B, where B denotes an arbitrary base type. Values are either of base types, trees or pairs of values. We use lower-case Greek letters (from the beginning of the alphabet) for the denotation of types. Elements t : T are defined by the following grammar which fixes notation. t ::= leaf | node t 1 b t 2 . The size of a tree is the number of leaves: |leaf| := 1, |node t a u| := |t| + |u|. We skip the standard definition of integer constants n ∈ Z as well as variable declarations, cf. [27]. Furthermore, we omit binary operators with the exception of essential comparisons. As mentioned, to represent sampling we make use of a dedicated if-then-else expression, whose guard evaluates to true depending on a coin toss with fixed probability. Further, non-deterministic choice is similarly rendered via an if-then-else expression. Moreover, we make use of ticking, denoted by an operator · a/b to annotate costs, where a, b are optional and default to one. Following Avanzini et al. [2], we represent ticking · as an operation, rather than in let-normal form, as in [34]. This allows us to suit a big-step semantics that only accumulates the cost of terminating expressions. The set of all expressions is denoted E. A typing context is a mapping from variables V to types. Type contexts are denoted by upper-case Greek letters, and the empty context is denoted ε. A program P consists of a signature F together with a set of function definitions of the form f x 1 . . . x n = e f , where the x i are variables and e f an expression. let x = w in e2 → e2[x → w] if true then e1 else e2 → e1 if coin a/b then e1 else e2 → {e a/b 1 , e 1−a/b 2 } if false then e1 else e2 → e2 if nondet then e1 else e2 → e1 if nondet then e1 else e2 → e2 match leaf with |leaf->e1 |node x0 x1 x2->e2 → e1 f x1σ . . . x k σ → eσ match node t a u with |leaf->e1 |node x0 x1 x2->e2 → e2 match (t,u) with |(t,u)->e → e e a/b a/b → e Here we assume f x1 . . . x k = e ∈ P, σ a substitution respecting the signature of f and w is a value. When considering some expression e that includes function calls we will always assume that these function calls are defined by some program P. A substitution or (environment) σ is a mapping from variables to values that respects types. Substitutions are denoted as sets of assignments: σ = {x 1 → t 1 , . . . , x n → t n }. We write dom(σ) to denote the domain of σ. Operational Semantics Small-Step Semantics. The small-step semantics is formalised as a (weighted) non-deterministic, probabilistic abstract reduction system [4,8] over M(E). In this way (expected) cost, non-determinism and probabilistic sampling are taken care of. Informally, a probabilistic abstract reduction system is a transition systems where reducts are chosen from a probability distribution. A reduction wrt. such a system is then given by a stochastic process [8], or equivalently, as a reduction relation over multidistributions [4], which arise naturally in the context of non-determinism (we refer the reader to [4] for an example that illustrates the advantage of multidistributions in the presence of non-determinism). More precisely, multidistributions are countable multisets {a pi i } i∈I over pairs p i : a i of probabilities 0 < p i 1 and objects a i ∈ A with i∈I p i 1. (For ease of presentation, we do not distinguish notationally between sets and multisets.) Multidistributions over objects A are denoted by M(A). For a multidistribution µ ∈ M(A) the induced distribution µ ∈ D(A) is defined in the obvious way by summing up the probabilities of equal objects. Following [5], we equip transitions with (positive) weights, amounting to the cost of the transition. Formally, a (weighted) Probabilistic Abstract Reduction System (PARS) on a countable set A is a ternary relation · · → · ⊆ A×R + 0 ×D(A). For a ∈ A, a rule a c → {b µ(b) } b∈A indicates that a reduces to b with probability µ(b) and cost c ∈ R + 0 . Note that any right-hand-side of a PARS is supposed to be a full distribution, ie. the probabilities in µ sum up to 1. Given two objects a and b, a c → {b 1 } will be written a We suit the one-step reduction relation → given in Figure 8 as a (nondeterministic) PARS over multidistributions. As above, we sometimes identify Dirac distributions {e 1 } with e. Evaluation contexts are formed by let expressions, as in the following grammar: C ::= | let x = C in e. We denote with C[e] the result of substitution the empty context with expression e. Contexts are exploited to lift the one-step reduction to a ternary weighted reduction relation The relation · −→ constitutes the operational (small-step) semantics of our simple probabilistic function language. Thus µ c −→ ν states that the submultidistribution of objects µ evolves to a submultidistribution of reducts ν in one step, with an expected cost of c. Note that since · → is non-deterministic, so is the reduction relation · −→. We now define the evaluation of an expression e ∈ E wrt. to the small-step relation c → b for brevity. An object a ∈ A is called terminal if there is no rule a c → µ, denoted a →. v → (NF) {v 1 } 0 −→ {v 1 } e c → {e p i i }i∈I (Step) {C[e 1 ]} c −→ {C[ei] p i }i∈I µi c i −→ νi i pi 1 (Conv) i pi · µi i p i c i −→ i pi · νi· −→ ⊆ M(E) × R +∞ 0 × M(E), cf.· −→: We set e c −→ ∞ µ, if there is a (pos- sibly infinite) sequence {e 1 } c1 −→ µ 1 c2 −→ µ 2 c3 −→ . . . with c = n c n and µ = lim n→∞ µ n V , where µ n V denotes the restriction of the distribution µ n (induced by the multidistribution µ n ) to a (sub-)distribution over values. Note that the µ n V form a CPO wrt. the pointwise ordering, cf. [36]. Hence, the fixed point µ = lim n→∞ µ n V exists. We also write e −→ ∞ µ in case the cost of the evaluation is not important. (Positive) Almost Sure Termination. A program P is almost surely terminating (AST ) if for any substitution σ, and any evaluation eσ −→ ∞ µ, we have that µ forms a full distribution. For the definition of positive almost sure termination we assume that every statement of P is enclosed in an ticking operation with cost one; we note that such a cost models the length of the computation. We say P is positively almost surely terminating (PAST ), if for any substitution σ, and any evaluation eσ c −→ ∞ µ, we have c < ∞. It is well known that PAST implies AST, cf. [8]. Big-Step Semantics. We now define the aforementioned big-step semantics. We first define approximate judgments σ n c e ⇒ µ, see Figure 10, which say that in derivation trees with depth up to n the expression e evaluates to a subdistribution µ over values with cost c. We now consider the cost c n and subdistribution µ n in σ n cn e ⇒ µ n for n → ∞. Note that the subdistributions µ n in σ n cn e ⇒ µ n form a CPO wrt. the pointwise ordering, cf. [36]. Hence, there exists a fixed point µ = lim n→∞ µ n . Moreover, we set c = lim n→∞ c n (note that either c n converges to some real c ∈ R +∞ 0 or we have c = ∞). We now define the big-step judgments e is not a value σ 0 0 e ⇒ {} σ 0 0 leaf ⇒ {leaf 1 } x1σ = t x2σ = b x3σ = u σ 0 0 node x1 x2 x3 ⇒ {(node t b u) 1 } xσ = v σ 0 0 x ⇒ {v 1 } x1σ = t x2σ = u σ 0 0 (x1,x2) ⇒ {(t,u) 1 } f y1 . . . y k = e ∈ P σ n c e ⇒ µ σ n+1 c f x1 . . . x k ⇒ µ σ n c 1 e1 ⇒ ν for all w ∈ supp(ν): σ[x → w] n cw e2 ⇒ µw σ n+1 c 1 + w∈supp(ν) ν(w)·cw let x = e1 in e2 ⇒ w∈supp(ν) ν(w) · µw xσ = leaf σ n c e1 ⇒ µ σ n+1 c match x with | leaf -> e1 | node x0 x1 x2 -> e2 ⇒ µ σ n c e ⇒ µ σ n+1 c+|µ|·a/b e a/b ⇒ µ xσ = node t a u σ n c e2 ⇒ µ σ n+1 c match x with | leaf -> e1 | node x0 x1 x2 -> e2 ⇒ µ xσ = false σ n c e2 ⇒ µ σ n+1 c if x then e1 else e2 ⇒ µ xσ = (t,u) σ n c e ⇒ µ σ n+1 c match x with | (x1,x2) -> e ⇒ µ xσ = true σ n c e1 ⇒ µ σ n+1 c if x then e1 else e2 ⇒ µ σ n c e1 ⇒ µ σ n+1 c if nondet then e1 else e2 ⇒ µ σ n c e2 ⇒ µ σ n+1 c if nondet then e1 else e2 ⇒ µ σ n c 1 e1 ⇒ µ1 σ n c 2 e2 ⇒ µ2 p = a /b σ n+1 pc 1 +(1−p)c 2 if coin a/b then e1 else e2 ⇒ pµ1 + (1 − p)µ2 Here σ[x → w] denotes the update of the environment σ such that σ[x → w](x) = w and the value of all other variables remains unchanged. For function application we set σ := {y1 → x1σ, . . . , y k → x k σ}. In the rules covering match we set σ := σ {x0 → t, x1 → a, x2 → u} and σ := σ {x0 → t, x2 → u} for trees and tuples respectively. σ c e ⇒ µ by setting µ = lim n→∞ µ n and c = lim n→∞ c n for σ n cn e ⇒ µ n . We want to emphasise that the cost c in σ c e ⇒ µ only counts the ticks on terminating computations. plies that σ c e ⇒ µ for some c c. Moreover, if eσ almost-surely terminates, we can choose c = c in both cases. Type-and-Effect System for Expected Cost Analysis Resource Functions In Section 2, we introduced a variant of Schoenmakers' potential function, denoted as rk(t), and the additional potential functions p (a1,...,an,b) (t 1 , . . . , t n ) = log 2 (a 1 · |t 1 | + · · · + a n · |t n | + b), denoting the log 2 of a linear combination of tree sizes. We demand n i=1 a i + b 0 (a i ∈ N, b ∈ Z) for well-definedness of the latter; log 2 denotes the logarithm to the base 2. Throughout the paper we stipulate log 2 (0) := 0 in order to avoid case distinctions. Note that the constant function 1 is representable: 1 = λt. log 2 (0 · |t| + 2) = p (0,2) . We are now ready to state the resource annotation of a sequence of trees. In case of an annotation of length 1, we sometimes write q * instead of q 1 . We may also write Φ(v : α|Q) for the potential of a value of type α annotated with Q. Both notations were already used above. Note that only values of tree type are assigned a potential. We use the convention that the sequence elements of resource annotations are denoted by the lower-case letter of the annotation, potentially with corresponding sub-or superscripts. Example 1. Let t be a tree. To model its potential as log 2 (|t|) in according to Definition 1, we simply set q (1,0) := 1 and thus obtain Φ(t|Q) = log 2 (|t|), which describes the potential associated to the input tree t of our leading example descend above. Let σ be a substitution, let Γ denote a typing context and let x 1 : T, . . . , x n : T denote all tree types in Γ . A resource annotation for Γ or simply annotation is an annotation for the sequence of trees x 1 σ, . . . , x n σ. We define the potential of the annotated context Γ |Q wrt. a substitution σ as Φ(σ; Γ |Q) := Φ(x 1 σ, . . . , x n σ|Q). An annotated signature F maps functions f to sets of pairs of annotated types for the arguments and the annotated type of the result: F(f ) := α 1 × · · · × α n |Q → β 1 × · · · × β k |Q m = |Q|, 1 = |Q | . We suppose f takes n arguments of which m are trees; m n by definition. Similarly, the return type may be the product β 1 × · · · × β k . In this case, we demand that at most one β i is a tree type. 6 Instead of α 1 × · · · × α n |Q → β 1 × · · · × β k |Q ∈ F(f ), we sometimes succinctly write f : α|Q → β|Q where α, β denote the product types α 1 × · · · × α n , β 1 × · · · × β k , respectively. It is tacitly understood that the above syntactic restrictions on the length of the annotations Q, Q are fulfilled. For every function f , we also consider its cost-free variant from which all ticks have been removed. We collect the cost-free signatures of all functions in the set F cf . q (a,b) ) a,b∈Z ]. We leave it to the reader to specify the coefficients in Q, Q so that the rule (app) as depicted in Section 2 can indeed be employed to type the recursive call of descend. a,b) ) a,b∈N ] be an annotation and let K be a rational such that q (0,2) + K 0. Then, Q := Q + K is defined as follows: a,b) ) a,b∈N ], where q (0,2) := q (0,2) + K and for all (a, b) = (0, 2) q (a,b) := q (a,b) . Recall that q (0,2) is the coefficient of function p (0,2) (t) = log 2 (0|t| + 2) = 1, so the annotation Q+K increments or decrements cost from the potential induced by Q by |K|, respectively. Further, we define the multiplication of an annotation Q by a constant K, denoted as K · Q pointwise. Moreover, let P = [p * ] ∪ [(p (a,b) ) a,b∈N ] be another annotation. Then the addition P + Q of annotations P, Q is similarly defined pointwise. Let Q = [q * ] ∪ [(q (Q = [q * ] ∪ [(q ( Typing Rules The non-probabilistic part of the type system is given in Figs. B.1 and B.2. In contrast to the type system employed in [13,17], the cost model is not fixed but controlled by the ticking operator. Hence, the corresponding application rule (app) has been adapted. Costing of evaluation is now handled by a dedicated ticking operator, cf. Figure 11. In Figure 12, we give the rule (ite : coin) responsible for typing probabilistic conditionals. We remark that the core type system, that is, the type system given in Fig. 12, as well as Figs. B.1 and B.2, ignoring annotations, enjoys subject reduction and progress in the following sense, which is straightforward to verify. → iff e is a value. Soundness Theorems A program P is called well-typed if for any definition f (x 1 , . . . , x n ) = e ∈ P and any annotated signature f : α 1 × · · · × α n |Q → β|Q , we have a corresponding typing x 1 : α 1 , . . . , x k : α k |Q e : β|Q . A program P is called cost-free well-typed, if the cost-free typing relation is used (which employs the cost-free signatures of all functions). Theorem 2 (Soundness Theorem for (tick : now)). Let P be well-typed. Suppose Γ |Q e : α|Q and eσ c −→ ∞ µ. Then Φ(σ; Γ |Q) c + E µ (λv.Φ(v|Q )). Further, if Γ |Q cf e : α|Q , then Φ(σ; Γ |Q) E µ (λv.Φ(v|Q )). Corollary 1. Let P be a well-typed program such that ticking accounts for all evaluation steps. Suppose Γ |Q e : α|Q . Then e is positive almost surely terminating (and thus in particular almost surely terminating). Theorem 3 (Soundness Theorem for (tick : defer)). Let P be well-typed. Suppose Γ |Q e : α|Q and σ c e ⇒ µ. Then, we have Φ(σ; Γ |Q) c + E µ (λv.Φ(v|Q )). Further, if Γ |Q cf e : α|Q , then Φ(σ; Γ |Q) E µ (λv.Φ(v|Q )). We comment on the trade-offs between Theorems 2 and 3. As stated in Corollary 1 the benefit of Theorem 2 is that when every recursive call is accounted for by a tick, then a type derivation implies the termination of the program under analysis. The same does not hold for Theorem 3. However, Theorem 3 allows to type more programs than Theorem 2, which is due to the fact that (tick : defer) rule is more permissive than (tick : now). This holds in particular when termination is assumed (or can be established by other means). We exemplify this difference on the foo function, see Figure 13. Theorem 3 supports the derivation of the type rk(t) + log 2 (|t|) + 1 rk(foo t) + 1, while Theorem 2 does not. This is due to the fact that potential can be "borrowed" with Theorem 3. To wit, from the potential rk(t) + log 2 (|t|) + 1 for foo one can derive the potential rk(l ) + rk(r ) for the intermediate context after both letexpression (note there is no +1 in this context, because the +1 has been used to pay for the ticks around the recursive calls). Afterwards one can restore the +1 by weakening rk(l ) + rk(r ) to rk(foo t) + 1 (using in addition that rk(t) 1 for Table 2: Coefficients q such q · log 2 (|t|) is a bound on the expected amortized complexity of splay depending on the probability p of a rotation and the cost c of a recursive call, where the cost of a rotation is 1 − c. Coefficients are additionally presented in decimal representation to ease comparison. all trees t). On the other hand, we cannot "borrow" with Theorem 2 because the rule (tick : now) forces to pay the +1 for the recursive call immediately (but there is not enough potential to pay for this). In the same way, the application of rule (tick : defer) and Theorem 3 is essential to establish the logarithmic amortised costs of randomised splay trees. (We note that the termination of foo as well as of splay is easy to establish by other means: it suffices to observe that recursive calls are on sub-trees of the input tree). Implementation and Evaluation Implementation. Our prototype ATLAS is an extension of the tool described in [17]. In particular, we rely on the preprocessing steps and the implementation of the weakening rule as reported in [17] (which makes use of Farkas' Lemma in conjunction with selected mathematical facts about the logarithm). We only use the fully-automated mode reported in [17]. We have adapted the generation of the constraint system to the rules presented in this paper. We rely on Z3 [24] for solving the generated constraints. We use the optimisation heuristics of [17] for steering the solver towards solutions that minimize the resulting expected amortised complexity of the function under analysis. Evaluation. We present results for the benchmarks described in Section 2 (plus a randomised version of splay heaps, the source code can be found in the Appendix) in Table 1. Table 3 details the computation time for type checking our results. Note that type inference takes considerably longer (tens of hours). To the best of our knowledge this is the first time that an expected amortised cost could be inferred for these data structures. By comparing the costs of the operations of randomised splay trees and heaps to the costs of their deterministic versions (see Table 1), one can see the randomised variants have equal or lower complexity in all cases (as noted in Section 2 we have set the costs of the recursive call and the rotation to 1 /2, such that in the deterministic case, which corresponds to a coin toss with p = 1, these costs will always add up to one). Clearly, setting the costs of the recursion to the same value as the cost of the rotation does not need to reflect the relation of the actual costs. A more accurate estimation of the relation of these two costs will likely require careful experimentation with data Table 3: Number of assertions, solving time for type checking, and maximum memory usage (in mebibytes) for the combined analysis of functions per-module. The number of functions and lines of code is given for comparison. structure implementations, which we consider orthogonal to our work. Instead, we report that our analysis is readily adapted to different costs and different coin toss probabilities. We present an evaluation for different values of p, recursion cost c and rotation cost 1 − c in Table 2. In preparing Table 2 the template q * · rk(t) + q (1,0) · log 2 (|t|) + q (0,2) was used for performance reasons. The memory usage according to Z3's "max memory" statistic was 7129MiB per instance. The total runtime was 1H45M, with an average of 11M39S and a median of 2M33S. Two instances took longer time (36M and 49M). Deterministic benchmarks. For comparison we have also evaluated our tool ATLAS on the benchmarks of [17]. All results could be reproduced by our implementation. In fact, for the function SplayHeap.insert it yields an improvement of 1 /4 log 2 (|h|), ie. 1 /2 log 2 (|h|) + log 2 (|h| + 1) + 3 /2 compared to 3 /4 log 2 (|h|) + log 2 (|h| + 1) + 3 /2. We note that we are able to report better results because we have generalised the resource functions p (a1,...,am,b) (t 1 , . . . , t m ) := log 2 (a 1 · |t 1 | + · · · + a m · |t m | + b) to also allow negative values for b (under the condition that i a i + b ≥ 1) and our generalised (let : tree) rule can take advantage of these generalized resource functions (see Fig. B.1 for a statement of the rule and the proof of its soundness as part of the proof of Theorem 3). Conclusion In this paper, we present the first fully-automated expected amortised cost analysis of self-adjusting data structures, that is, of randomised splay trees, randomised splay heaps and randomised meldable heaps, which so far have only (semi-) manually been analysed in the literature. In future work, we envision to extend our analysis to related probabilistic settings such as skip lists [28] and randomised binary search trees [18]. We note that adaptation of the framework developed in this paper to new benchmarks will likely require to identify new potential functions and the extension of the type-effect-system with typing rules for these potential functions. Further, on more theoretical grounds we want to clarify the connection of the here proposed expected amortised cost analysis with Kaminski's ert-calculus, cf. [14], and study whether the expected cost transformer is conceivable as a potential function. A Benchmark: Probabilistic Analysis of Binary Search Trees We present a probabilistic analysis of a deterministic binary search tree, which offers the usual contains, insert, and delete operations, where delete uses delete_max as a subroutine (the source code of all operations is given in Fig. C.4). We assume that the elements inserted, deleted and searched for are equally distributed; hence, we conduct a probabilistic analysis by replacing every comparison with a coin toss of probability one half. We will refer to the resulting data structure as Coin Search Tree in our benchmarks. Our tool ATLAS infers an logarithmic expected amortised cost for all operations, e.g., for insert and delete_max we obtain 3 /2 rk(t) + 1 /2 log 2 (|t|) c insert (t) + 3 /2 rk(insert t) 3 /2 rk(t) + 1 /2 log 2 (|t|) c delete_max (t) + 3 /2 rk(delete_max t) , from which we obtain an expected amortised cost of 1 /2 log 2 (|t|) for both functions. B Omitted Definitions B.1 Type System: Non-Probabilistic Part The non-probabilistic and structural typing rules are given in Figure B.1 and B.2 respectively. B.2 Soundness Theorems The proof of the soundness theorems makes use of the following lemma, whose proof can be found in [13]. Lemma 2. Assume i q i log 2 a i q log 2 b for some rational numbers a i , b > 0 and q i q. Then, i q i log 2 (a i + c) q log 2 (b + c) for all c 1. Theorem 2 (Soundness Theorem for (tick : now)). Let P be well-typed. Suppose Γ |Q e : α|Q and eσ c −→ ∞ µ. Then Φ(σ; Γ |Q) c + E µ (λv.Φ(v|Q )). Further, if Γ |Q cf e : α|Q , then Φ(σ; Γ |Q) E µ (λv.Φ(v|Q )). eσ c −→ n µ ⇒ Φ(σ; Γ |Q) c + v∈supp(µ V ) µ(v) · Φ(v|Q ) . This is sufficient to prove the claim because by the definition of eσ c −→ ∞ µ there is some (possibly infinite) sequence {eσ 1 } c1 −→ µ 1 c2 −→ µ 2 c3 −→ . . . with c = n c n and µ = lim n→∞ µ n V . In order to prove the above equation, we proceed by main induction on n -which we will call main induction hypothesis (MIH)-and side-induction on the length of the type derivation Π of Γ |Q e : α|Q -which we will call side induction hypothesis (SIH). We recall that the type system considered in this proof only uses the ticking rule (tick : now) (and not the rule (tick : defer)). In the following, we will only consider selected interesting cases and refer the reader to [13] for the cases that are very similar or equal to the non-probabilistic setting. We now consider eσ c −→ n µ for some n 0 and the type derivation Π of Γ |Q e : α|Q . We first deal with the case that Π ends in a structural rule, cf. Figure B. 2: Case. Suppose the last rule in Π be of the following form: Γ |Q e : α|Q Γ |Q + K e : α|Q + K , where K 0. By SIH, we have that Φ(σ; Γ |Q) c + v∈supp(µ V ) µ(v) · Φ(v|Q ), from which we obtain Φ(σ; Γ |Q + K) = Φ(σ; Γ |Q) + K c + v∈supp(µ V ) µ(v) · Φ(v|Q ) + K c + v∈supp(µ V ) µ(v) · Φ(v|Q + K) , as v∈supp(µ V ) µ(v) 1 and Φ(v|Q + K) = Φ(v|Q ) + K. Case. Let Π end in the following weakening rule applicaton Γ |P e : α|P Φ(Γ |P ) Φ(Γ |Q) Φ(Γ |P ) Φ(Γ |Q ) Γ |Q e : α|Q (w) . By SIH, we have Φ(σ; Γ |P ) c + v∈supp(µ V ) µ(p) · Φ(v|P ). Due to the assumption of the (w) rule, we have Φ(σ; Γ |Q) Φ(σ; Γ |P ) c + v∈supp(µ V ) µ(p) · Φ(v|P ) c + v∈supp(µ V ) µ(p) · Φ(v|Q ) . Case. (share) and (w : var) can be dealt with in the same way, we refer the reader to [13] for the details. We now assume that Π ends in a syntax-directed rule, cf. Figure B.1, and proceed by a case distinction on eσ, respectively the first step of eσ c −→ n µ: Case. First we assume eσ is a value. By definition of · −→ we have µ = {v} and c = 0. There are several subcases to consider, eg. eσ = node t b u, eσ = leaf, eσ = ( e 1 , e 2 ), etc. For these cases we can essentially proceed as in the nonprobabilistic setting, cf. [13]. Exemplarily, we consider the subcase where eσ = ( e 1 , e 2 ). for at most one i, α i = T q i = q * q (a,c) = q (a,c) x 1 : α 1 , x 2 : α 2 |Q (x 1 ,x 2 ) : α 1 × α 2 |Q (pair) where by assumptions in (pair), Q and Q are annotations of length one, respectively. Note that if α 1 = T and α 2 = T, then Q = Q = ∅ and the theorem holds trivially. Thus, we assume that α 1 = T and α 2 = T. By definition and the constraints incorporated in (pair), we obtain Φ(σ; x 1 : α 1 , x 2 : α 2 |Q) Φ(( e 1 , e 2 )|Q ) = v∈{( e1 , e2 ) 1 } 1 · Φ(v|Q ) , from which the claim follows. In order to prove the claim for eσ c −→ n µ we need to split the n-step derivation into n 1 -step and n 2 -step derivations for e 1 and e 2 with n 1 + n 2 + 1 = n, where the one step accounts for substituting the value to which e 1 has evaluated into e 2 . However, we cannot only consider one such split because evaluating e 1 to a normal form will in general need a different number of steps according to the probabilistic choices encountered in the derivation. Hence, we will consider all possible splits. For this, we consider e 1 σ ci −→ i ν i for all 0 i n. We recall that the ν i are pointwise ordered on values, ie. we have ν i V ν j V for i j. Hence, we can define ξ i = ν i V − ν i−1 V for all 0 < i n. Note that for w pi i ∈ ξ i we have that the probability that e 1 σ evaluates to the value w i in i steps is exactly p i . Let w be some value to which e 1 σ has evaluated to in i steps. We then note that let x = w in e 2 0 −→ 1 {e 2 [x → w]}. Thus, we can apply the MIH to e 2 σ[x → w] and obtain that e 2 σ[x → w] cw,i −→ n−i−1 µ w,i ⇒ Φ(σ[x → w]; ∆, x : α|R) c w,i + v∈supp(µw,i V ) µ w,i (v) · Φ(v|Q ) , ( †) for suitably defined distributions µ w,i and costs c w,i . We now consider the SIH applied to e 1 σ c1 −→ n ν, ie. we have that e 1 σ c1 −→ n ν ⇒ Φ(σ; Γ |P ) c 1 + w∈supp(ν V ) ν(w) · Φ(w|P ) . ( ‡) By the definition of the ξ i we have that ν V = n i=1 ξ i . We then consider eσ c −→ n µ. We observe that µ V = n i=1 w p i ∈ξi p i · µ w,i V and c = c 1 + n i=1 w p i ∈ξi p i · c w,i for distributions µ w,i and costs c w,i defined as above. Further, we will establish below that Φ(σ; Γ, ∆|Q) + w∈supp(ν V ) ν(w) · Φ(w|P ) ≥ Φ(σ; Γ |P ) + w∈supp(ν V ) ν(w) · Φ(σ[x → w]; ∆, x : α|R) . ( ) We finally calculate using ( †), ( ‡) and ( ) that Φ(σ; Γ, ∆|Q) c 1 + w∈supp(ν V ) ν(w) · Φ(σ[x → w]; ∆, x : α|R) = c 1 + n i=1 w∈supp(ξi) ξ i (w) · Φ(σ[x → w]; ∆, x : α|R) c 1 + n i=1 w p i ∈ξi p i · (c w,i + v∈supp(µw,i V ) µ w,i (v)Φ(v|Q )) = c 1 + n i=1 v p i i ∈ξi p i · c w,i + n i=1 w p i ∈ξi p i · v∈supp(µw,i V ) µ w,i (v)Φ(v|Q ) = c + v∈supp(µ V ) µ(v) · Φ(v|Q )) . It remains to establish ( ). For this we proceed by a case distinction on whether e 1 is of tree type, i.e., whether the rule Π ends in an application of the (let : tree)or of the (let : base)-rule. We treat the simpler case first and consider that e 1 is not of tree type. Then, Π ends in an application of the (let : base)-rule, ie. p i = q i p ( a,c) = q ( a, 0,c) Γ |P e 1 : α|P ∀ b = 0 (q ( 0, b,c) = r ( b,c) ) ∆, x : α|R e 2 : β|Q r ( 0,c) = p (c) r j = q m+j α = T Γ, ∆|Q let x = e 1 in e 2 : β|Q (let : base) . We note that ( ) follows directly from the constraints in the (let : base) together with the fact that ν is a (sub-)distribution. Finally, we now suppose that e 1 is of type tree. Then, the type derivation Π ends in an application of the (let : tree)-rule. Γ |P e 1 : T|P ∀ b = 0, d = 0 Γ |P ( b,d,e) cf e 1 : T|P ( b,d,e) ∆, x : T|R e 2 : β|Q Γ, ∆|Q let x = e 1 in e 2 : β|Q (let : tree) , where we have elided all arithmetic constraints for readability. By definition and due to the constraints expressed in the typing rule, we have that Φ(σ; Γ, ∆|Q) = i q i rk(t i ) + j q j rk(u j ) + a, b,c q ( a, b,c) log 2 ( a| t| + b| u| + c) Φ(σ; Γ |P ) = i q i rk(t i ) + a,c q ( a, 0,c) log 2 ( a| t| + c) Φ(w|P ) = r k+1 rk(w) + d,e r ( 0,d,e) log 2 (d|w| + c) Φ(σ; Γ |P ( b,d,e) ) = a,c p ( b,d,e) ( a,c) log 2 ( a| t| + c) Φ(w|P ( b,d,e) ) = p ( b,d,e) (d,max (e,0)) log 2 (d|w| + max {e, 0}) Φ(σ[x → w]; ∆, x : T|R) = j q j rk(u j ) + r k+1 rk(w) + b,d,e r ( b,d,e) log 2 ( b| u| + d|w| + e) , where we set t := t 1 , . . . , t m and u := u 1 , . . . , u k , denoting the substitution instances of the variables in Γ , ∆, respectively. (We recall that the well-definedness of log 2 is implicitly assumed: i a i + j b j + c 1 resp. a + b + c 1 as well as j b j + d + e 1 are satisfied. ) We now consider some w ∈ supp(ν V ). By definition of non-deterministic (cost-free) version of e 1 there is a small-step derivation e 1 σ q ( a, b,c) log 2 ( a| t| + b| u| + c) b = 0,d =0 r ( b,d,e) log 2 ( b| u| + d|w| + e) . Because the above equation holds for any w ∈ supp(ν V ) we can deduce that b = 0, a = 0 q ( a, b,c) log 2 ( a| t| + b| u| + c) w∈supp(ν V ) ν(w) ·   b = 0, a = 0 q ( a, b,c) log 2 ( a| t| + b| u| + c)   w∈supp(ν V ) ν(w) · b = 0,d =0 r ( b,d,e) log 2 ( b| u| + d|w| + e) , using that ν is a (sub-)distribution, i.e., that w∈supp(ν V ) ν(w) ≤ 1. We now note that ( ) follows directly from the above inequality and the constraints in the (let : tree) together with the fact that ν is a (sub-)distribution. Case. Let e be a conditional and assume the last rule in Π is of the following form: Γ |Q e 1 : α|Q Γ |Q e 2 : α|Q Γ, x : Bool|Q if x then e 1 else e 2 : α|Q (ite) . (The case where the condition is performed non-deterministically, is treated analogousloy.) By assumption, we either have ( In both case MIH yields that Φ(σ; Γ |Q) c + v∈supp(µ V ) µ(p) · Φ(v|Q ), from which the theorem follows as Φ(σ; Γ, x : Bool|Q) = Φ(σ; Γ |Q) by definition. Case. Let e be ticking statement and let the last rule in Π be of the following form: Γ |Q e : α|Q Γ |Q + a /b e a/b : α|Q (tick : now) . By definition, we have {eσ } a /b → {eσ} c− a /b −→ n−1 µ and by MIH, we have Φ(σ; Γ |Q) (c − a /b) + v∈supp(µ V ) µ(p) · Φ(v|Q ) . Hence Φ(σ; Γ |Q + a /b) c + v∈supp(µ V ) µ(p) · Φ(v|Q ). Case. Let e be a probabilistic branching statement, that is eσ = if coin(p) then e 1 else e 2 , and let the last rule in Π be of the following form Γ |Q 1 e 1 : α|Q Γ |Q 2 e 2 : α|Q p = a /b Q = p · Q 1 + (1 − p) · Q 2 Γ |Q if coin a/b then e 1 else e 2 : α|Q (ite : coin) . By definition, we have {eσ} 0 → {e p 1 , e 1−p 2 } c −→ n−1 µ. By definition of · −→ there exists (sub)distribution µ 1 , µ 2 s.t. {e 1 } c1 −→ m1 µ 1 and {e 2 } c2 −→ m2 µ 2 , where m 1 , m 2 < n, µ = p · µ 1 + (1 − p) · µ 2 and c = p · c 1 + (1 − p) · c 2 . Further by SIH, we conclude (i) Φ(σ; Γ |Q 1 ) c 1 + v∈supp(µ1 V ) µ 1 (p) · Φ(v|Q ) and (ii) Φ(σ; Γ |Q 2 ) c 2 + v∈supp(µ2 V ) µ 2 (p) · Φ(v|Q ). Hence, we obtain Φ(σ; Γ |Q) = Φ(σ; Γ |p · Q 1 + (1 − p) · Q 2 ) = p · Φ(σ; Γ |Q 1 ) + (1 − p) · Φ(σ; Γ |Q 2 ) c 1 + c 2 + (p · v q ∈µ1 V q · Φ(v|Q )) + ((1 − p) · v q ∈µ2 V q · Φ(v|Q )) = c + v q ∈µ1 V p · q · Φ(v|Q ) + v q ∈µ2 V (1 − p) · q · Φ(v|Q ) = c + v q ∈p·µ1 V +(1−p)·µ2 V q · Φ(v|Q ) = c + v q ∈µ V q · Φ(v|Q ) , from which we conclude the case. Case. We consider the application rules (app) and (app : cf) and restrict our argument to the former, as the proof for the cost-free variant is similar, but simpler. We consider the costed typing α 1 × · · · × α n |P → β|P ∈ F(f ) α 1 × · · · × α n |Q → β|Q ∈ F cf (f ) K ∈ Q + 0 x 1 : α 1 , . . . , x n : α n |(P + K · Q) f (x 1 , . . . , x n ) : β|(P + K · Q ) (app) . Let f (x 1 , . . . , x k ) = e ∈ P, as P is well-typed, we have Γ |P e : β|P and Γ |Q cf e : β|Q by assumption. Further, by definition {eσ} 0 → e c −→ n−1 µ. We conclude by MIH that Φ(σ; Γ |P ) c + v∈supp(µ V ) µ(p) · Φ(v|P ) and Φ(σ; Γ |Q) v∈supp(µ V ) µ(p) · Φ(v|Q ). Hence Φ(σ; Γ |P + K · Q) = Φ(σ; Γ |P ) + K · Φ(σ; Γ |Q) c + v∈supp(µ V ) µ(p) · Φ(v|P ) + K · v∈supp(µ V ) µ(p) · Φ(v|Q ) = c + v∈supp(µ V ) µ(p) · (Φ(v|P ) + K · Φ(v|Q )) = c + v∈supp(µ V ) µ(p) · Φ(v|P + K · Q ) . Theorem 3 (Soundness Theorem for (tick : defer)). Let P be well-typed. Suppose Γ |Q e : α|Q and σ c e ⇒ µ. Then, we have Φ(σ; Γ |Q) c + E µ (λv.Φ(v|Q )). Further, if Γ |Q cf e : α|Q , then Φ(σ; Γ |Q) E µ (λv.Φ(v|Q )). Proof. The setup (and most of the cases) of this proof follow the proof of Theorem 2: It suffices to prove for every n 0 that σ n c e ⇒ µµ ⇒ Φ(σ; Γ |Q) c + E µ (λv.Φ(v|Q )) We proceed by main induction on n -which we will call main induction hypothesis (MIH)-and side-induction on the length of the type derivation Π of Γ |Q e : α|Q -which we will call side induction hypothesis (SIH). For the majority of the cases, the arguments can be easily suited from those employed in proof of Theorem 2. Thus, we only consider a restricted set of cases that may be of independent interest. We now consider σ n+1 c e ⇒ µ for some n 0 and the type derivation Π of Γ |Q e : α|Q . The cases where Π ends in a structural rule, cf. Figure B.2, can be dealt with in the same way as in the proof of Theorem 2. We now assume that Π ends in a syntax-directed rule, cf. Figure B.1, and proceed by a case distinction on eσ, respectively the first step of σ n+1 c e ⇒ µ: Case. Consider e = let x = e 1 in e 2 . Let w be some value. We apply the MIH to σ[x → w] n cw e 2 ⇒ µ w and obtain that σ[x → w] n cw e 2 ⇒ µ w ⇒ Φ(σ[x → w]; ∆, x : α|R) c w + v∈supp(µw V ) µ w (v) · Φ(v|Q ) ,( †) for suitably defined distributions µ w and costs c w . We further apply the MIH to σ n c1 e 1 ⇒ ν, ie. we obtain that σ n c1 e 1 ⇒ ν ⇒ Φ(σ; Γ |P ) c 1 + w∈supp(ν V ) ν(w) · Φ(w|P ) .( ‡) Further, we will establish below that c 1 + w∈supp(ν V ) ν(w) · (c w + v∈supp(µw V ) µ w (v) · Φ(v|Q )) = c 1 + w∈supp(ν V ) ν(w) · c w + + w∈supp(ν V ) v∈supp(µw V ) ν(w) · µ w (v) · Φ(v|Q ) = c + v∈supp(µ V ) µ(v) · Φ(v|Q )) . where we have used for the last equality that µ = w∈supp(ν) ν(w) · µ w and c = c 2 + w∈supp(ν) ν(w)·c w according to the definition of the big-step semantics. We finally note that ( ) can be established in the same way as in the proof of Theorem 2 (for both the (let : base)and the (let : tree)-rule case). Case. Let e be a probabilistic branching statement, that is eσ = if coin(p) then e 1 else e 2 , and let the last rule in Π be of the following form Γ |Q 1 e 1 : α|Q Γ |Q 2 e 2 : α|Q p = a /b Q = p · Q 1 + (1 − p) · Q 2 Γ |Q if coin a/b then e 1 else e 2 : α|Q (ite : coin) . By definition, there exists distributions µ 1 and µ 2 such that σ n c1 e 1 ⇒ µ 1 , σ n c2 e 2 ⇒ µ 2 , µ = p · µ 1 + (1 − p) · µ 2 and c = p · c 1 + (1 − p) · c 2 . By MIH, we conclude Φ(σ; Γ |Q) c 1 + E µ1 (λv.Φ(v|Q )) Φ(σ; Γ |Q 2 ) c 2 + E µ2 (λv.Φ(v|Q )) Hence, we obtain Φ(σ; Γ |Q) = Φ(σ; Γ |p · Q 1 + (1 − p) · Q 2 ) = p · Φ(σ; Γ |Q 1 ) + (1 − p) · Φ(σ; Γ |Q 2 ) p · c 1 + (1 − p) · c 2 + (p · v q ∈µ1 q · Φ(v|Q )) + ((1 − p) · v q ∈µ2 q · Φ(v|Q )) = c + v q ∈µ1 p · q · Φ(v|Q ) + v q ∈µ2 (1 − p) · q · Φ(v|Q ) = c + v q ∈p·µ1+(1−p)·µ2 q · Φ(v|Q ) = c + v q ∈µ q · Φ(v|Q ) , from which we conclude the case. By MIH, we obtain Φ(σ; Γ |Q) c + E µ (λv.Φ(v|Q )), from which we conclude that Φ(σ; Γ |Q) c + v∈supp(µ) µ(v) · Φ(v|Q ) = c + v∈supp(µ) µ(v) · Φ(v|Q ) − a /b + a /b = c + v∈supp(µ) µ(v) a /b + v∈supp(µ) µ(v) · Φ(v|Q ) − a /b = c + |µ| · a /b + v∈supp(µ) µ(v) · Φ(v|Q − a /b) = c + |µ| · a /b + E µ (λv.Φ(v|Q − a /b)) . Here, we exploit the definition of Q − a /b and the definition of |µ| in the secondto-last line and the definition of expectations in the last line. C Function Definitions Below, we use a notation for ticks that is easier to type with standard keyboard layouts, i.e. the tilde symbol followed by cost and the subexpression, ∼ a/b e, instead of a tick mark and cost in the superscript, e a/b . Randomised Splay Trees. The benchmark comprises the functions splay, splay max, insert, and delete, see α1 × · · · × αn|P → β|P ∈ F (f ) α1 × · · · × αn|Q → β|Q ∈ F cf (f ) K ∈ Q + 0 x1 : α1, . . . , xn : αn|(P + K · Q) f (x1, . . . , xn) : β|(P + K · Q ) (app) To ease notation, we set a := a1, . . . , am, b := b1, . . . , b k for vectors of indices ai, bj ∈ N. Further, i ∈ {1, . . . , m}, j ∈ {1, . . . , k}, and a, b, d ∈ N and c, e ∈ Z, where we recall that c, e must be chosen such that i ai + j bj + c 1 resp. a + b + c 1 as well as j bj + d + e 1 are satisfied. Sequence elements of annotations, which are not constrained are set to zero. Note that the conditions in (pair) on coefficients are vacuously true, if α1 = T and α2 = T. Fig. 2 : 2descend:T|Q → T|Q l : T|Q5 descend l : T|Q6 (app) l : T|Q4 (descend l) : T|Q6 (tick : now) x l : T, r : T|Q7 node x l a r : T|Q l : T, r : T|Q3 let x l = (descend l) in node x l a r : T|Q (let) l : T, r : T|Q2 if coin 1/2 then e2 else e3 : T|Q (ite : coin) l : T, r : T|Q1 if coin 1/2 then e2 else e3 : T|Q (w) t : T|Q match t with|leaf → leaf |node l a r → e1 : T|Q (match) Partial Type Derivation for Function descend the coefficients of the resource functions in the corresponding potential functions, e.g., (i) Φ(t : T|Q) : Fig. 3 : 3Partial meld function of Randomised Meldable Heaps l : T, r : T|Q 4 e 3 : T|Q l : T, r : T|Q 3 let x l = (descend l) in node x l a r : T|Q l : T, r : T|Q 2 if coin 1/2 then e 2 else e 3 : T|Q (ite : coin) node ( node al a1 ar ) b br ) c cr No rotation. Fig. 4 : 4Partial splay function of Randomised Splay Trees (zigzig-case) Fig. 7 : 7A Core Probabilistic (First-Order) Programming Language Fig. 8 : 8One-Step Reduction Rules Fig. 9 : 9Probabilistic Reduction Rules of Distributions of Expressions Figure 9 . 9(In (Conv), refers to the usual notion of multiset union.) Fig. 10 : 10Big-Step Semantics. Theorem 1 ( 1Equivalence). Let P be a program and σ a substitution. Then,(i) σ c e ⇒ µ implies that eσ c −→ ∞ µ for some c ≥ c, and (ii) eσ c −→ ∞ µ im-Γ |Q e : α|Q Γ |Q + a /b e a/b : α|Q (tick : now) Γ |Q e : α|Q Γ |Q e a/b : α|Q − a /b (tick : defer)Fig. 11: Ticking Operator. Note that a, b are not variables but literal numbers.Γ |Q1 e1 : α|Q Γ |Q2 e2 : α|Q p = a /b Q = p · Q1 + (1 − p) · Q2Γ |Q if coin a/b then e1 else e2 : α|Q (ite : coin)Fig. 12: Conditional expression that models tossing a coin. Definition 1 . 1A resource annotation or simply annotation of length m is a sequence Q = [q 1 , . . . , q m ] ∪ [(q (a1,...,am,b) ) ai,b∈N ], vanishing almost everywhere. The length of Q is denoted |Q|. The empty annotation, that is, the annotation where all coefficients are set to zero, is denoted as ∅. Let t 1 , . . . , t m be a sequence of trees. Then, the potential of t m , . . . , t n wrt. Q is given byΦ(t 1 , . . . , t m |Q) := m i=1 q i ·rk(t i )+ a1,...,am∈N,b∈Z q (a1,...,am,b) ·p (a1,...,am,b) (t 1 , . . . , t m ) . Example 2 . 2Consider the function descend depicted in Figure 2. Its signature is formally represented as T|Q → T|Q , where Q := [q * ] ∪ [(q (a,b) ) a,b∈Z ] and Q := [q * ] ∪ [( Fig. 13 : 13Function foo illustrates the difference between (tick : now) and (tick : defer). Lemma 1. Let e be such that e : α holds. Then: (i) If e c → {e pi i } i∈I , then e i : α holds for all i ∈ I. (ii) The expression e is in normal form wrt. i . We will establish for every n 0 that Case. Consider e = match x with| leaf → e 1 | node x 1 x 2 x 3 → e 2 , and suppose further xσ= node lu b v, that is, {eσ} 0 −→ {e 2 } c −→ n−1 µ.Because Π ends with a syntax-directed rule, Π must in fact end with an application of the (match) rule, ie.r ( a,a,a,b) = q ( a,a,b) p ( a,c) = a+b=c q ( a,a,b) Γ |P + q m+1 e 1 : α|Q r m+1 = r m+2 = q m+1 q i = r i = p i r ( 0,1,0,0) = r ( 0,0,1,0) = q m+1Γ, x 1 : T, x 2 : B, x 3 : T|R e 2 : α|Q Γ, x : T|Q match x with | leaf -> e 1 | node x 1 x 2 x 3 -> e 2 : α|Q (match) .Note that q m+1 denotes the coefficient of rk(xσ) in the definition of Φ(σ; Γ, x : T|Q). By definition and the constraints given in the rule, we obtain:Φ(σ; Γ, x : T|Q) = Φ(σ; Γ, x 1 : T, x 2 : B, x 3 : T|R) .By MIH we have Φ(σ; Γ, x 1 : T,x 2 : B, x 3 : T|R) c + v∈supp(µ V ) µ(v)Φ(v|Q ), from which the case follows directly.Case. Consider e = let x = e 1 in e 2 . Note that if e 0 then − max {−e, 0} = 0 and if e < 0 then − max {−e, 0} = e. Thus in the former we add the sum b| u| to both sides of the inequality (B.2), while in the second case we add b| u| + e. Note the conditions (i) q ( a, b,c) = (d,e) p ( b,d,e) ( a,c+max{−e,0}) and (ii) r ( b,d,e) = p ( b,d,e) (d,max {e,0}) for all b = 0, a = 0, d = 0. Thus, we can sum up those equations for all b = 0, a = 0, d = 0 and obtain that b = 0, a = 0 i) {eσ 1 } = {if true then e 1 else e n−1 µ or (ii) {eσ 1 } = {if false then e 1 else e w) · Φ(σ[x → w]; ∆, x : α|R) Π ends with the rule Γ |Q e : α|Q Γ |Q e a/b : α|Q − a /b (tick : defer) . Figure C. 1 . 1Randomized Splay Heaps. The benchmark comprises the functions insert, and delete min, see Figure C.2. Meldable Heaps. The benchmark comprises the functions meld, insert, and delete min, see Figure C.3.Coin Search Tree. The benchmark comprises the functions insert, contains, delete, and delete max, seeFigure C . 4 . 4Tree. The benchmark comprises the function descend, seeFigure C.5. ∀c 2 q (c) = a+b=c q (a,b) K = q * ∅|Q + K leaf : T|Q (leaf)q1 = q2 = q * q (1,0,0) = q (0,1,0) = q * q (a,a,c) = q (a,c) x1 : T, x2 : B, x3 : T|Q node x1 x2 x3 : T|Q (node) • ∈ {<, >, =} x1 : α, x2 : α|Q x1 • x2 : Bool|Q (cmp) x a variable x : α|Q x : α|Q (var) for at most one i, αi = T qi = q * q (a,c) = q (a,c) x1 : α1, x2 : α2|Q (x1,x2) : α1 × α2|Q (pair) Γ |Q e1 : α|Q Γ |Q e2 : α|Q Γ, x : Bool|Q if x then e1 else e2 : α|Q (ite) r ( a,a,a,b) = q ( a,a,b) p ( a,c) = a+b=c q ( a,a,b) Γ |P + qm+1 e1 : α|Q rm+1 = rm+2 = qm+1 qi = ri = pi r ( 0,1,0,0) = r ( 0,0,1,0) = qm+1 Γ, x1 : T, x2 : B, x3 : T|R e2 : α|Q Γ, x : T|Q match x with | leaf -> e1| node x1 x2 x3 -> e2 : α|Q (match) Γ, x1 : α1, x2 : α2|Q e : β|Q for at most one i, αi = T Γ, x : α1 × α2|Q match x with | (x1,x2) -> e : β|Q (match : pair) pi = qi p ( a,c) = q ( a, 0,c) Γ |P e1 : α|P ∀ b = 0 (q ( 0, b,c) = r ( b,c) ) ∆, x : α|R e2 : β|Q r ( 0,c) = p (c) rj = qm+j α = T Γ, ∆|Q let x = e1 in e2 : β|Q (let : base) pi = qi p ( a,c) = q ( a, 0,c) rj = qm+j r k+1 = p * r ( 0,d,e) = p (d,e) ∀ b = 0 r ( b,0,c) = q ( 0, b,c) ∀ b = 0, a = 0, d = 0 q ( a, b,c) = (d,e) p ( b,d,e) ( a,c+max {−e,0}) ∀ b = 0, d = 0 r ( b,d,e) = p ( b,d,e) (d,max {e,0}) ∧ ∀(d , e ) = (d, max {e, 0}) p e1 : T|P ∀ b = 0, d = 0 Γ |P ( b,d,e) cf ndt e1 : T|P ( b,d,e) ∆, x : T|R e2 : β|Q Γ, ∆|Q let x = e1 in e2 : β|Q (let : tree) Fig . B.1: Syntax-Directed Type Rules: Non-Probabilisitc Part. Γ |R e : β|Q ri = qi r ( a,b) = q ( a,0,b) Γ, x : α|Q e : β|Q (w : var) Γ, x : α, y : α|Q e[x, y] : β|Q Γ, z : α| (Q) e[z, z] : β|Q (share) Γ |P e : α|P Φ(Γ |P ) Φ(Γ |Q) Φ(Γ |P ) Φ(Γ |Q ) Γ |Q e : α|Q (w) Γ |Q e : α|Q K 0 Γ |Q + K e : α|Q + K (shift) Fig. B.2: Structural Type Rules: Non-Probabilisitc Part. An amortised analysis may always default to a wort-case analysis. In particular the analysis of descend in this section can be considered as a worst-case analysis. However, we use the example to illustrate the general setup of our amortised analysis. For ease of presentation, we elide the underlying semantics for now and simply write "descend t" for the resulting tree t , obtained after evaluating descend t. The restriction to at most one tree type in the resulting type is non-essential and could be lifted. However, as our benchmark functions do not require this extension, we have elided it for ease of presentation. 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[ "LINEAR MAXIMUM RANK DISTANCE CODES OF EXCEPTIONAL TYPE", "LINEAR MAXIMUM RANK DISTANCE CODES OF EXCEPTIONAL TYPE" ]
[ "Daniele Bartoli ", "Giovanni Zini ", "Ferdinando Zullo " ]
[]
[]
Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of F q n -linear maximum rank distance codes, introducing the notion of exceptional linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial MRD context. We move towards the classification of exceptional linear MRD codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index.
10.1109/tit.2023.3243682
[ "https://arxiv.org/pdf/2110.08102v1.pdf" ]
239,009,505
2110.08102
dd31514c5bddcda8855e7920bf39bda34ccaf0ca
LINEAR MAXIMUM RANK DISTANCE CODES OF EXCEPTIONAL TYPE 15 Oct 2021 Daniele Bartoli Giovanni Zini Ferdinando Zullo LINEAR MAXIMUM RANK DISTANCE CODES OF EXCEPTIONAL TYPE 15 Oct 2021arXiv:2110.08102v1 [cs.IT] Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of F q n -linear maximum rank distance codes, introducing the notion of exceptional linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial MRD context. We move towards the classification of exceptional linear MRD codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index. Introduction Let q be a prime power, n be a positive integer, and denote by F q n the finite field with q n elements. Rank metric codes can be seen as sets of F q -linear endomorphisms of F q n equipped with the rank distance, that is the distance between two elements is defined as the (linear algebraic) rank of their difference. Since the F q -algebra of the F q -linear endomorphisms of F q n and the F qalgebra L n,q = n−1 i=0 a i x q i : a i ∈ F q n of F q -linearized polynomials of q-degree smaller than n are isomorphic, each rank metric code can be also seen as a subset of L n,q . For any f (x) = n−1 i=0 a i x q i , we define deg q (f (x)) = max{i : a i = 0} and mindeg q (f (x)) = min{i : a i = 0}. In this context, a rank metric code C ⊆ L n,q with minimum distance d achieving the equality in the Singleton-like bound (1) | C | ≤ q n(n−d+1) is called maximum rank distance (MRD) code. Rank metric codes and in particular MRD codes have been introduced several times [14,17] and have been widely investigated in the last few years, due to applications in network coding [37] and cryptography [23]. Two rank metric codes C 1 , C 2 ⊆ L n,q are said to be equivalent if there exist two invertible F q -linearized polynomials g(x), h(x) ∈ L n,q and a field automorphism ρ ∈ Aut(F q n ) such that C 1 = g • C ρ 2 •h = {g • f ρ • h : f ∈ C 2 }, where f ρ (x) := n−1 i=0 ρ(a i )x σ i if f (x) = n−1 i=0 a i x σ i and the composition has to be considered modulo x q n − x. In order to study the equivalence between two rank metric codes, one can make use of the idealisers. They have been introduced in [22], where the left and right idealisers of a rank metric code C ⊆ L n,q are defined respectively as L(C) = {ϕ(x) ∈ L n,q : ϕ • f ∈ C for all f ∈ C}, R(C) = {ϕ(x) ∈ L n,q : f • ϕ ∈ C for all f ∈ C}. Such objects have also been investigated in [27], where they have been called respectively middle and right nuclei. In this paper we are interested in F q n -linear MRD codes, that is MRD codes C such that L(C) is equivalent to F n = {αx : α ∈ F q n }; see [36,Definition 12]. Thus, every F q n -linear MRD code is equivalent to an F q n -subspace of L n,q and we will always consider MRD codes which are F q nsubspaces of L n,q . The MRD condition for F q n -linear rank metric codes reads as follows. By (1), an F q n -linear rank metric code C ⊆ L n,q , with dim F q n (C) = k and minimum distance d, is an MRD code if and only if d = n − k + 1, or equivalently, dim Fq (ker(f )) ≤ k − 1, for all f ∈ C \{0}. This paper is devoted to the investigation of F q n -linear MRD codes which are exceptional. An F q n -linear MRD code C ⊆ L n,q is an exceptional MRD code if the rank metric code C m = C F q nm ⊆ L nm,q is an MRD code for infinitely many m. Only two families of exceptional F q n -linear MRD codes are known: (G) G k,s = x, x q s , . . . , x q s(k−1) F q n , with gcd(s, n) = 1, see [14,17,20]; (T) H k,s (δ) = x q s , . . . , x q s(k−1) , x + δx q sk F q n , with gcd(s, n) = 1 and N q n /q (δ) = (−1) nk , see [28,35]. The first family is known as generalized Gabidulin codes and the second one as generalized twisted Gabidulin codes. Although the definition of exceptional F q n -linear MRD codes appears in this paper for the first time, it has been already studied in particular subcases in different contexts. In the case k = 2, exceptional MRD codes have been considered via so-called exceptional scattered polynomials. Let f (x) ∈ L n,q and t be a nonnegative integer t ≤ n − 1. Then f is said to be scattered of index t if for every x, y ∈ F * q n f (x) x q t = f (y) y q t ⇔ y x ∈ F q , or equivalently, (2) dim Fq (ker(f (x) − mx q t )) ≤ 1, for every m ∈ F q n . The term scattered arises from a geometric framework; see [10]. Indeed, f is scattered of index t if and only the F q -subspace [35], taking into account (2), pointed out the following connection between scattered polynomials and F q n -linear MRD codes: f is scattered of index t if and only if C f,t = x q t , f (x) F q n is an MRD code with dim F q n (C) = 2. The polynomial f is said to be exceptional scattered of index t if it is scattered of index t as a polynomial in L nm,q , for infinitely many m; see [8]. Taking into account (2), a polynomial f is exceptional scattered of index t if and only if the corresponding MRD code C f,t is exceptional. While several families of scattered polynomials have been constructed in recent years [7, 10, 11, 13, 24-26, 28, 29, 31, 35, 39, 40], only two families of exceptional ones are known: U t,f = {(x q t , f (x)) : x ∈ F q n } has the property that dim Fq (U t,f ∩ v F q n ) ≤ 1 for every nonzero vector v ∈ F 2 q n , that is U t,f is scattered with respect to the Desarguesian spread { v F q n : v ∈ F 2 q n \ {(0, 0)}}. Sheekey in (Ps) f (x) = x q s of index 0, with gcd(s, n) = 1 (polynomials of so-called pseudoregulus type); (LP) f (x) = x + δx q 2s of index s, with gcd(s, n) = 1 and N q n /q (δ) = 1 (so-called LP polynomials). Such two families correspond to the known exceptional F q n -linear MRD codes (G) and (T). Several tools have already been proposed in the study of exceptional scattered polynomials, related to algebraic curves or Galois extensions of function fields; see [2,4,6,8,15]. However, their classification is still unknown when the index is greater than 1. For k > 2, the only known families of F q n -subspaces of L q,n corresponding to MRD codes are (G) and (T) as described above and Delsarte dual codes of the MRD codes associated with scattered polynomials. In [9] it has been shown that the only exceptional F q n -linear MRD codes spanned by monomials are the codes (G), in connection with so-called Moore exponent sets. It is therefore natural to investigate exceptional F q n -linear MRD codes not generated by monomials. To this aim, we generalize the notion of Moore exponent set; see Section 3. Using a connection between the generators of F q n -linear rank metric codes C and certain algebraic hypersurfaces X C , we obtain a partial classification of exceptional F q n -linear MRD codes. Tools from intersection theory (see Section 2) yield sufficient conditions on the generators for C to be MRD, via the existence of F q n -rational absolutely irreducible components of X C . Our main results can be summarized as follows. Main Theorem. Let C ⊆ L n,q be an exceptional k-dimensional F q n -linear MRD code containing at least a separable polynomial f (x) and a monomial. If k > 3, assume also that q > 5. Let t be the minimum integer such that x q t ∈ C. • If t = 0 and C = x q t , g 2 (x), g 3 (x), . . . , g k (x) F q n , with deg q (g 2 (x)) < · · · < deg q (g k (x)) and (q, deg q (g 2 (x))) / ∈ {(2, 2), (2,4), (3,2), (4, 2), (5, 2)}, then C is a generalized Gabidulin code. • If t > 0 and C = x q t , f (x), g 3 (x), . . . , g k (x) F q n , with deg(g i (x)) > max{q t , deg(f (x))} for each i = 3, . . . , k, then f (x) is exceptional scattered of index t. When C contains a separable polynomial and a monomial, we call the non-negative integer t of Main Theorem the index of C. Preliminaries on algebraic varieties An algebraic hypersurface is an algebraic variety that can be defined by a single polynomial equation. An algebraic hypersurface defined over a field K is absolutely irreducible if the associated polynomial is irreducible over every algebraic extension of K. An absolutely irreducible K-rational component of a hypersurface V, defined by the polynomial F , is simply an absolutely irreducible hypersurface which is associated to a factor of F defined over K. For a finite field F q , let F q denote its algebraic closure. Also, P m (K) (resp. A m (K)) denotes the m-dimensional projective (resp. affine) space over the field K. We recall some known results on algebraic hypersurfaces of which our approach will make use. Lemma 2.1. [1, Lemma 2.1] Let H be an absolutely irreducible hypersurface and X be an F qrational hypersurface of P m (F q ). If X ∩ H has a non-repeated F q -rational absolutely irreducible component, then X has a non-repeated F q -rational absolutely irreducible component. With the symbol I(P, A ∩ B) we denote the intersection multiplicity of two plane curves in A 2 (K) at a point P ∈ A 2 (K). Classical results on such an integer can be found in most of the textbooks on algebraic curves. For other concepts related to algebraic varieties we refer to [18]. For the special case of curves, a good reference is [16]. Lemma 2.2. [19, Proposition 2] Let F ∈ F q [X, Y ] be such that F = AB for some A, B ∈ F q [X, Y ]. Let P = (u, v) be a point in the affine plane A 2 (F q ) and write F (X + u, Y + v) = F m (X, Y ) + F m+1 (X, Y ) + · · · , where F i is zero or homogeneous of degree i and F m = 0. Suppose that F m = L m for some linear polynomial L ∈ F q [X, Y ] such that L ∤ F m+1 . Then I(P, A ∩ B) = 0, where A and B are the curves defined by A and B respectively. 5] Let F ∈ F q [X, Y ] be such that F = AB for some A, B ∈ F q [X, Y ]. Let P = (u, v) be a point in the affine plane A 2 (F q ) and write F (X + u, Y + v) = F m (X, Y ) + F m+1 (X, Y ) + · · · , where F i is zero or homogeneous of degree i and F m = 0. Suppose that F m = L m for some linear polynomial L ∈ F q [X, Y ] such that L | F m+1 and L 2 ∤ F m+1 . Then I(P, A ∩ B) = 0 or m, where A and B are the curves defined by A and B respectively. Lemma 2.4. [16, Section 3.3] Let F ∈ F q [X, Y ] be such that F = AB for some A, B ∈ F q [X, Y ]. Let P = (u, v) be a point in the affine plane A 2 (F q ) and write F (X + u, Y + v) = F m (X, Y ) + F m+1 (X, Y ) + · · · , where F i is zero or homogeneous of degree i and F m = 0. Suppose that F m factors into m distinct linear factors in F q [X, Y ]. Then I(P, A ∩ B) ≤ ⌊m 2 /2⌋. Lemma 2.5. Let F (X 1 , . . . , X n ) ∈ F q [X 1 , . . . , X n ] with F (X 1 , . . . , X n ) = F m (X 1 , . . . , X n ) + F m+1 (X 1 , . . . , X n ) + · · · + F M (X 1 , . . . , X n ), where F i is zero or homogeneous of degree i and F m F M = 0. If F m or F M contains a non-repeated absolutely irreducible F q -rational factor, then F (X 1 , . . . , X n ) contains a non-repeated absolutely irreducible F q -rational factor. Proof. Let G be the non-repeated absolutely irreducible F q -rational factor in F m (resp. F M ). Consider the unique absolutely irreducible factor F ′ of F such that G | F ′ m (resp. G | F ′ M ). If F ′ were not defined over F q , then there would exist another absolutely irreducible factor F ′′ = σ(F ′ ) = F ′ of F satisfying G | F ′′ m (resp. G | F ′′ M ), where σ is the q-Frobenius automorphism of F q [X 1 , . . . , X n ], whence G 2 | F m (resp. G 2 | F M ), a contradiction. In the sequel we will investigate hypersurfaces connected with Moore polynomial sets; see Definition 3.3. In particular, we are interested in getting information on the existence of F q -rational absolutely irreducible components of curves contained in such hypersurfaces. The approach that we follow has been used for the first time by Janwa, McGuire and Wilson [19] to classify functions on F p n that are almost perfect nonlinear for infinitely many n, in particular for monomial functions. It can be summarized by the following theorem. Theorem 2.6. [21, Lemma 2] Let C ⊂ P 2 (F q ) be a curve of degree d and let S be the set of its singular points. Also, let i(P ) denote the maximum possible intersection multiplicity of two putative components of C at P ∈ C. If P ∈S i(P ) < 2d 2 9 , then C possesses at least one absolutely irreducible component defined over F q . Moore polynomial sets and MRD codes Let q be a prime power and n be a positive integer. Consider k F q n -linearly independent polynomials f 1 (x), f 2 (x), . . . , f k (x) ∈ L n,q and denote by f the k-tuple (f 1 (x), . . . , f k (x)). Define M f (x 1 , . . . , x k ) =      f 1 (x 1 ) f 2 (x 1 ) · · · f k (x 1 ) f 1 (x 2 ) f 2 (x 2 ) · · · f k (x 2 ) . . . . . . . . . . . . f 1 (x k ) f 2 (x k ) · · · f k (x k )      . For any A = {α 1 , . . . , α k } ⊆ F q n , define M f ,A = M f (α 1 , . . . , α k ). Lemma 3.1. If α 1 , . . . , α k are F q -linearly dependent, then det(M f ,A ) = 0. Proof. Without loss of generality, suppose that k ≥ 2 and α 1 = k i=2 b i α i with a i ∈ F q . Then M f,A =      k i=2 b i f 1 (α i ) k i=2 b i f 2 (α i ) · · · k i=2 b i f k (α i ) f 1 (α 2 ) f 2 (α 2 ) · · · f k (α 2 ) . . . . . . . . . . . . f 1 (α k ) f 2 (α k ) · · · f k (α k )      , so that the first row of M f ,A is a linear combination of the remaining rows. Then det(M f,A ) = 0. The converse of Lemma 3.1 is not true in general, the following being a counterexample. Example 3.2. Let k and n be positive integers with n even and k ≤ n. Consider f = (x, x q 2 , . . . , x q 2(k−1) ). Let A = {α 1 , . . . , α k } be a subset of F q n . Then M f,A is the Moore matrix M f ,A =       α 1 α q 2 1 · · · α q 2(k−1) 1 α 2 α q 2 2 · · · α q 2(k−1) 2 . . . . . . . . . . . . α k α q 2 k · · · α q 2(k−1) k       , and det(M f ,A ) = 0 if and only if the elements α 1 , . . . , α k are F q 2 -linearly independent; see [30, Corollary 2.1.95]. Therefore, if α 1 , . . . , α k−1 are F q -linearly independent elements in F q n and α k ∈ α 1 , . . . , α k−1 F q 2 \ α 1 , . . . , α k−1 Fq , then det(M f,A ) = 0 even though {α 1 , . . . , α k } are F q -linearly independent. The following definition identifies the tuples f for which the converse of Lemma 3.1 holds and it will be crucial in our investigation for exceptional MRD codes. Definition 3.3. Let f = (f 1 (x), . . . , f k (x)), where k is a positive integer and f 1 (x), . . . , f k (x) ∈ L n,q . We say that f is a Moore polynomial set for q and n if, for any α 1 , . . . , α k ∈ F q n , det      f 1 (α 1 ) f 2 (α 1 ) · · · f k (α 1 ) f 1 (α 2 ) f 2 (α 2 ) · · · f k (α 2 ) . . . . . . . . . . . . f 1 (α k ) f 2 (α k ) · · · f k (α k )      = 0 =⇒ dim Fq α 1 , . . . , α k Fq < k. If f is a Moore polynomial set for q and nm for infinitely many m, we say that f is an exceptional Moore polynomial set for q and n. Moore polynomial sets can be characterized in terms of MRD codes as follows. Theorem 3.4. Let k and n be positive integers with k ≤ n, and denote by f the k-tuple (f 1 (x), . . . , f k (x)), where f 1 (k), . . . , f k (x) ∈ L n,q are F q n -linearly independent. The F q n -linear rank metric code C f = f 1 (x), . . . , f k (x) F q n is an MRD code if and only if f is a Moore polynomial set for q and n. Proof. Suppose that M f,A is singular for some A = {α 1 , . . . , α k } ⊆ F q n , that is, there exist b 1 , . . . , b k ∈ F q n such that k i=1 b i f i (α j ) = 0, for every j ∈ {1, . . . , k}. This means that A is contained in the kernel of F (x) = k i=1 b i f i (x) ∈ C f . Since C f is an MRD code, it follows that dim Fq (ker(F )) ≤ k − 1, and hence α 1 , . . . , α k are F q -linearly dependent. Conversely, suppose that f is a Moore polynomial set for q and n. Assume by contradiction that there exists g(x) ∈ C f with dim Fq (ker(g(x))) ≥ k and write g( x) = k i=1 b i f i (x) with b i ∈ F q n . Let A = {α 1 , . . . , α k } ⊆ ker(g(x)) where α 1 , . . . , α k are F q -linearly independent. Then M f ,A is singular because its columns are F q n -linearly dependent through k i=1 b i f i (α j ) = 0 for all j = 1, . . . , k. Therefore, f is not a Moore polynomial set for q and n. As a natural consequence, a characterization of the exceptionality property is obtained. Corollary 3.5. Let C ⊆ L n,q be an F q n -linear rank metric code. The following are equivalent: • C is an exceptional MRD code. • Every F q n -basis {f 1 (x), . . . , f k (x)} of C defines an exceptional Moore polynomial set f = (f 1 (x), . . . , f k (x)) for q and n. • There exists an F q n -basis {f 1 (x), . . . , f k (x)} of C for which f = (f 1 (x), . . . , f k (x)) is an exceptional Moore polynomial set for q and n. We will investigate exceptional MRD codes by means of exceptional Moore polynomial sets. Moore polynomial sets and varieties over finite fields In this section we study exceptional F q n -linear MRD codes C ⊆ L n,q of dimension k under the assumption that C contains a monomial. Up to equivalence, we can assume that C contains a separable polynomial. We denote by t the smallest non-negative integer such that x q t ∈ C. Remark 4.1. If C is an F q n -linear MRD code in L n,q , then C contains an invertible map f (x) (see [27, Lemma 2.1] and [33, Lemma 52]), and hence f −1 • C contains the identity x. If C is exceptional, then max{deg q (g(x)) : g(x) ∈ C F q nm } does not depend on the infinitely many m's for which C F q nm is MRD. On the contrary, max{deg q (g(x)) : g(x) ∈ f −1 • C F q nm } may depend on m, so that f −1 • C may not be exceptional. On the other hand, the assumption that C contains a separable polynomial does not affect the exceptionality of C, since max{deg q (g(x)) : g(x) ∈ C} decreases by min{mindeg q (g(x)) : g(x) ∈ C}. Assumptions 4.2. Note that there exist f 1 (x), . . . , f k (x) ∈ C such that the following hold: (1) f 1 (x), . . . , f k (x) are monic and F q n -linearly independent; (2) M 1 := deg q (f 1 (x)), . . . , M k := deg q (f k (x)) are all distinct; (3) m 1 := mindeg q (f 1 (x)), . . . , m k := mindeg q (f k (x) ) are all distinct, and m i = 0 for some i; (4) f 1 (x) = x q t ; (5) for any i, if f i (x) is a monomial then m i = M i ≥ t. Therefore, by Corollary 3.5, we investigate Moore polynomial sets as in the following definition. . . , f k (x)) ⊆ L k n,q satisfying Assumptions 4.2 is said to be a Moore polynomial set for q and n of index t. A key tool in our approach is a link between Moore polynomial sets ans algebraic hypersurface. To this aim, we introduce the following F q n -rational hypersurfaces: U := U f ⊂ P k (F q n ) is the hypersurface defined by the affine equation F f (X 1 , . . . , X k ) := det(M f (X 1 , . . . , X k )) = 0, and V ⊂ P k (F q n ) is the hypersurface U (x,x q ,...,x q k−1 ) . Note that F (x,x q ,...,x q k−1 ) (X 1 , . . . , X k ) = (a1:...:a k )∈P k−1 (Fq) (a 1 X 1 + · · · + a k X k ), with a suitable choice of the representative for the points (a 1 : . . . : a k ). Since f 1 (x), . . . , f k (x) are F q -linearized, the polynomial F (x,x q ,...,x q k−1 ) (X 1 , . . . , X k ) divides F f (X 1 , . . . , X k ), so that V is a component of U. Therefore we can define the F q n -rational variety W ⊂ P k (F q n ) with affine equation W : F f (X 1 , . . . , X k )/F (x,x q ,...,x q k−1 ) (X 1 , . . . , X k ) = 0. The link between Moore polynomial sets and algebraic hypersurfaces W is straightforward. Proof. For any A = {α 1 , . . . , α k } ⊆ F q n , the condition det(M f ,A ) = 0 is equivalent to (α 1 , . . . , α k ) being an affine F q n -rational point of W, while the condition dim Fq ( α 1 , . . . , α k Fq ) < k is equivalent to (α 1 , . . . , α k ) being a point of V. The claim follows. In the case when f 1 (x), . . . , f k (x) are monomials, Theorem 4.4 was already noticed (using a slightly different terminology) and used in [9] to prove the following result. . . , i k } be a set of non-negative integers with 0 < i 2 < . . . < i k such that I is not in arithmetic progression. Suppose that one of the following holds: • |I| = 3 and n > 4i k + 2; • |I| > 3, q > 5 and n > 13 3 i k + log q (13 · 2 10/3 ). Then (x, x q i 2 , . . . , x q i k ) is not a Moore polynomial set for q and n. In the sequel, we will use the following notation: for any i = 1, . . . , k, write f i (x) = Mi j=mi a ij x q j and f i (x, z) := Mi j=mi a ij x q j z q M j −q j . 4.1. Moore polynomial sets of index 0. In this section we investigate Moore polynomial sets of index 0, so that f 1 (x) = x. Without loss of restriction, we assume M 1 = 0 < M 2 < · · · < M k . Theorem 4.6. Suppose that one of the following holds: • k = 3 and n > 4M 3 + 2; • k > 3, q > 5 and n > 13 3 M k + log q (13 · 2 10/3 ). If f is a Moore polynomial set for q and n of index 0, then (M 1 = 0, M 2 , . . . , M k ) is in arithmetic progression and (m σ(1) = 0, m σ(2) , . . . , m σ(k) ) is in arithmetic progression for some σ ∈ S k with σ(1) = 1. Proof. In order to prove the claim on the M i 's, consider the intersection W ∞ = W ∩ H ∞ between W and the hyperplane at infinity H ∞ ⊂ P k (F q n ). Note that W ∞ ⊂ P k−1 (F q n ) is defined by W ∞ : F (x,x q M 2 ,...,x q M k ) (X 1 , . . . , X k )/F (x,x q ,...,x q k−1 ) (X 1 , . . . , X k ) = 0. Suppose that (M 1 , . . . , M k ) is not in arithmetic progression. Then it has been shown in [9, Theorems 3.1 and 4.2] that U ∞ contains an F q n -rational non-repeated absolutely irreducible component X . It follows by Lemma 2.5 that W has an F q n -rational non-repeated absolutely irreducible component. Then, as shown in [9] (in page 9 for k = 3, and in page 17 for k > 3), there exists an affine F q n -rational point in W \ V. Thus, f is not a Moore polynomial set for q and n by Theorem 4.4. Now suppose that (m σ(1) , . . . , m σ(k) ) is not in arithmetic progression for any σ ∈ S k . Consider the tangent variety T of W at the origin O. Then T : F (x q m σ(1) ,x q mσ (2) ,...,x q mσ (k) ) (X 1 , . . . , X k )/F (x,x q ,...,x q k−1 ) (X 1 , . . . , X k ) = 0. Now the same arguments as above show that T has an F q n -rational non-repeated absolutely irreducible component. Then W has an F q n -rational non-repeated absolutely irreducible component by Lemma 2.5. Therefore, as in [9], W has an affine F q n -rational point not in V, so that f is not a Moore polynomial set for q and n. In the rest of this subsection, f is a Moore polynomial set for q and n of index 0, satisfying the assumptions of Theorem 4.6, so that M 1 = 0, M 2 = M, . . . , M k = (k − 1)M . Let λ 3 , . . . , λ k be F q -linearly independent elements of F q n and define H f (X 1 , X 2 ) = F f (X 1 , X 2 , λ 3 , . . . , λ k ) ∈ F q n [X 1 , X 2 ]. Since f is a Moore polynomial set, H f (X 1 , X 2 ) ≡ 0. Let D f ⊂ P 2 (F q n ) be the curve defined by H f (X 1 , X 2 ) = 0. We denote by H f (X 1 , X 2 , T ) the homogeneization of H f (X 1 , X 2 ), i.e. H f (X 1 , X 2 , T ) := det        X 1 f 2 (X 1 , T ) · · · f k (X 1 , T ) X 2 f 2 (X 2 , T ) · · · f k (X 2 , T ) λ 3 T f 2 (λ 3 )T q M · · · f k (λ 3 )T q (k−1)M . . . . . . . . . . . . λ k T f 2 (λ k )T q M · · · f k (λ k )T q (k−1)M        T q (k−2)M −1 q M −1 . Lemma 4.7. If D f has a non-repeated F q n -rational absolutely irreducible component not contained in D x,x q ,...,x q k−1 , then W has a non-repeated F q n -rational absolutely irreducible component not contained in V. Proof. Consider the variety W 3 ⊂ P 3 (F q n ) defined by W 3 : F f (X 1 , X 2 , X 3 , λ 4 , . . . , λ k )/F (x,x q ,...,x q k−1 ) (X 1 , X 2 , X 3 , λ 4 , . . . , λ k ) = 0. Let Π 3 ⊂ P 3 (F q n ) be the hyperplane with affine equation X 3 = λ 3 . By the assumptions, W 3 ∩Π 3 has a non-repeated F q n -rational absolutely irreducible component. Hence, by Lemma 2.1, W 3 has a nonrepeated F q n -rational absolutely irreducible component. The claim follows by repeatedly applying this argument to W 3 , . . . , W k = W. Remark 4.8. It is readily seen that (x, f (x)) is a Moore polynomial set for q and n if and only if f (x) is scattered of index 0 over F q n . By the results in [6,8], if n > 4 deg q (f (x)) and (x, f (x)) is a Moore polynomial set for q and n, then f (x) is a monomial with gcd(n, deg q (f (x))) = 1. Next results deal with the case k > 2. Theorem 4.9. Let f = (f 1 (x) = x, f 2 (x), f 3 (x)) be a Moore polynomial set for q and n of index 0 with 0 < M 2 < M 3 . If n > 4M 3 + 2, then f 2 (x) ∈ L n,q is scattered of index 0. Proof. By Theorem 4.6, M 3 = 2M where M = M 2 . Suppose that f 2 (x) ∈ L n,q is not scattered of index 0, so that there exist λ, µ ∈ F * q n such that λ/µ / ∈ F q and f 2 (λ)/λ = f 2 (µ)/µ. By [5, Corollary 3.4], we can assume that µ / ∈ F q and f 2 (λ) = 0. Let λ 3 = λ and define D f as above. Let D ′ f be the image of D f under the F q n -rational projectivity ϕ : (X 1 : X 2 : T ) → (T : X 2 − X 1 : X 1 ). Note that the point P = (1 : 1 : 0) ∈ D f is mapped by ϕ to O = (0 : 0 : 1). The curve D ′ f has affine equation H ′ f (X 1 , X 2 ) = 0, where H ′ f (X 1 , X 2 ) = H f (1, X 2 +1, X 1 ) = det   1 f 2 (1, X 1 ) f 3 (1, X 1 ) X 2 + 1 f 2 (1, X 1 ) + f 2 (X 2 , X 1 ) f 3 (1, X 1 ) + f 3 (X 2 , X 1 ) λ f 2 (λ)X q M −1 1 f 3 (λ)X q 2M −1 1   = det   1 f 2 (1, X 1 ) f 3 (1, X 1 ) X 2 f 2 (X 2 , X 1 ) f 3 (X 2 , X 1 ) λ f 2 (λ)X q M −1 1 f 3 (λ)X q 2M −1 1   = −λf 2 (X 2 , X 1 ) + f 2 (λ)X 2 X q M −1 1 + G(X 1 , X 2 ), for some G(X 1 , X 2 ) ∈ F q n [X 1 , X 2 ] of degree bigger than q M . The homogeneous polynomial L(X 1 , X 2 ) = −λf 2 (X 2 , X 1 ) + f 2 (λ)X 2 X q M −1 1 has X 2 − µX 1 as a non-repeated factor in F q n [X 1 , X 2 ], since µ is a root of the separable polynomial L(1, X 2 ) ∈ F q n [X 2 ]. Therefore D ′ f has a non repeated F q n -rational absolutely irreducible component, and the same holds for D f . Since µ / ∈ F q , such a component of D f is not contained in D (x,x q ,x q 2 ) . By Lemma 4.7, W has a non-repeated F q n -rational absolutely irreducible component Z not contained in V. The degree of V is q 2 + q + 1, and the degree of Z is at most q 2M + q M − q 2 − q. Thus, by [38,Corollary 7], Z has an affine F q n -rational point not on V, a contradiction to Theorem 4.4. From m 2 = M it follows that m 3 ∈ {2M, M/2}. Suppose by contradiction that f 3 (x) = x q 2M , so that m 3 = M/2 < M , and in particular M is even. Choose λ 3 = λ ∈ F * q n such that f 2 (λ)f 3 (λ) = 0. Via Theorem 2.6, we will prove that the variety W 3 with affine equation W 3 : F f (X 1 , X 2 , λ)/F (x,x q ,x q 2 ) (X 1 , X 2 , λ) = 0 has a non-repeated F q n -rational absolutely irreducible component not contained in V. Suppose that W 3 splits into two components A and B sharing no common absolutely irreducible component. Letà andB be two components of D f sharing no common absolutely irreducible components and such that A ⊆Ã, B ⊆B. Singular points of W 3 are also singular points of D f , and the intersection multiplicity of A and B at a point is at most the intersection multiplicity of A andB at that point. We start the inspection of singular points of W 3 from affine ones. Let P = (α, β) ∈ P 2 (F q n ) be an affine point of D f . The point P is singular for D f if and only if f 3 (λ)f 2 (α) − f 2 (λ)f 3 (α) = 0 and f 3 (λ)f 2 (β) − f 2 (λ)f 3 (β) = 0, that is f 3 (λ)α q M − λ q M f 3 (α) = f 3 (λ)β q M − λ q M f 3 (β) = 0. Also, the intersection multiplicity ofà andB at P equals the intersection multiplicity of τ (Ã) and τ (B) at τ (P ) = O = (0, 0), where τ is the translation (X 1 , X 2 ) → (X 1 − α, X 2 − α). The image D ′ f of D f under τ has affine equation H ′ f (X 1 , X 2 ) = 0, with H ′ f (X 1 , X 2 ) = H f (X 1 + α, X 2 + β) = det    X 1 + α X q M 1 + α q M f 3 (X 1 ) + f 3 (α) X 2 + β X q M 2 + β q M f 3 (X 2 ) + g(β) λ λ q M f 3 (λ)    = a (λα q M − αλ q M )X q m 3 2 − (λβ q M − βλ q M )X q m 3 1 − λ q M (X 1 X q m 3 2 − X 2 X q m 3 1 ) + G(X 1 , X 2 ), where a = 0 is the coefficient of x q m 3 in f 3 (x), and G(X 1 , X 2 ) ∈ F q n [X 1 , X 2 ] has degree bigger than q m3 + 1. We denote respectively by ( H ′ f ) q m 3 and (H ′ f ) q m 3 +1 the homogeneous polynomials (λα q M − αλ q M )X q m 3 2 − (λβ q M − βλ q M )X q m 3 1 and X 1 X q m 3 2 − X 2 X q m 3 1 . If non-vanishing, they are, up to a scalar multiple, the homogeneous parts of smallest degrees in H ′ f (X 1 , X 2 ). Note that the q m3 + 1 linear factors of (H ′ f ) q m 3 +1 are all distinct. • There are at most q 2M singular points (α, β) of D f which satisfy λα q M − αλ q M = λβ q M − βλ q M = 0. In this case, (H ′ f ) q m 3 +1 is the non-zero homogeneous part of smallest degree in H ′ f (X 1 , X 2 ). Thus O is an ordinary (q m3 + 1)-fold point for D ′ f , and by Lemma 2.4 the intersection multiplicity of τ (Ã) and τ (B) at O is at most (q m3 + 1) 2 /4. • There are at most 2(q 2M−m3 − 1) · q M singular points (α, β) of D f which satisfy either λα q M − αλ q M = 0 = λβ q M − βλ q M or λα q M − αλ q M = 0 = λβ q M − βλ q M . In this case, (H ′ f ) q m 3 = X q m 3 2 or (H ′ f ) q m 3 = X q m 3 1 up to a non-zero scalar multiple, and hence gcd(H ′ f ) q m 3 , (H ′ f ) q m 3 +1 = X 2 or X 1 . By Lemma 2.3, the intersection multiplicity of τ (Ã) and τ (B) at O is at most q m3 . • There are at most (q 2M−m3 − 1) · (q m3 − 1) · q M singular points (α, β) of D f which satisfy λα q M − αλ q M = 0, λβ q M − βλ q M = 0, and η(λα q M − αλ q M ) = ξ(λβ q M − βλ q M ) for some (ξ : η) ∈ P 1 (F q mq m3 . • If a singular point (α, β) of D f satisfies η(λα q M − αλ q M ) = ξ(λβ q M − βλ q M ) for some (ξ : η) / ∈ P 1 (F q m 3 ), then (H ′ f ) q m 3 and (H ′ f ) q m 3 +1 are coprime. In this case, by Lemma 2.2, the intersection multiplicity of τ (Ã) and τ (B) at O is 0. Since the homogeneus part of largest degree in H f (X 1 , X 2 ) is λ · X q M 1 · γ∈F q M (X 2 − γX 1 ) q M , the points at infinity of D f are (0 : 1 : 0) and (1 : γ : 0) with γ ∈ F q M . As the map (X 1 : X 2 : T ) → (X 2 : X 1 : T ) maps (0 : 1 : 0) to (1 : 0 : 0) and leaves invariant the curves D f and D (x,x q ,x q 2 ) , it is enough to consider the points P γ = (1 : γ : 0) with γ ∈ F q M . The intersection multiplicity of A andB at P γ equals the intersection multiplicity of σ(Ã) and σ γ (B) at σ γ (P ) = O = (0, 0), where σ γ : (X 1 : X 2 : T ) → (T : X 2 − γT : X 1 ). The image D ′′ f of D f under σ γ has affine equation H ′′ f (X 1 , X 2 ) = 0, where H ′′ f (X1, X2) = H f (1, X2 + α, X1) = det   1 f2(1, X1) f3(1, X1) X2 + γ f2(X2 + γ, X1) f3(X2 + γ, X1) λ f2(λ)X q M −1 1 f3(λ)X q 2M −1 1   = det    1 1 f3(1, X1) X2 X q M 2 f3(X2, X1) + f3(γ, X1) − γf3(1, X1) λ λ q M X q M −1 1 f3(λ)X q 2M −1 1    = λ q M X2X q M −1 1 − λX q M 2 + G(X1, X2), for some G(X 1 , X 2 ) of degree greater than q M (here, we used that the constant term in f 3 (γ, X 1 ) is the same as in γf 3 (1, X 1 )). Since λ q M X 2 X q M −1 1 − λX q M 2 is homogeneous and separable in each variable, 0 is an ordinary q M -fold point for D ′′ f , and by Lemma 2.4 the intersection multiplicity of A andB at P α is at most q 2M /4. The same holds at (0 : 1 : 0). Summing up, the number of intersection points of two components of H f (X 1 , X 2 , T ) = 0, counted with multiplicity, satisfies P I(P, A∩B) ≤ q 2M (q m3 + 1) 2 4 +2(q 2M−m3 −1)q M+m3 +(q 2M−m3 −1)(q m3 −1)q M+m3 +(q M +1) q 2M 4 . Since (q, M ) / ∈ {(2, 2), (2,4), (3,2), (4,2), (5, 2)}, the above quantity is less than 2 9 deg(W 3 ) 2 = 2 9 (q 2M + q M − q 2 − q) 2 . By Theorem 2.6, W 3 contains an F q n -rational absolutely irreducible component X . Note that D f has only finitely many singular points, and hence X is non-repeated and not contained in V. Arguing as in the last paragraph of the proof of Theorem 4.9, a contradiction arises. This shows m 3 = 2M , i.e. f 3 (x) = x q 2M . By means of an induction argument, we are able to extend the result of Theorem 4.10 to any Moore polynomial set of index 0, as follows. Theorem 4.11. Let f = (f 1 (x) = x, f 2 (x), . . . , f k (x)), with k > 3, be a Moore polynomial set for q and n of index 0 such that 0 < M 2 < · · · < M k . Suppose also that q > 5 and n > 13 3 M k + log q (13 · The tangent cone to U ′ f at O has equation ′ f of U f under ϕ, which has affine equation F ′ f (X 1 , . . . , X i ) = 0, where F ′ f (X 1 , X 2 , . . . , X i ) equals det       F * g (X 2 , . . . , X i , X 1 ) = det      X 2 f 2 (X 2 , X 1 ) . . . f i−1 (X 2 , X 1 ) X 3 f 2 (X 3 , X 1 ) . . . f i−1 (X 3 , X 1 ) . . . . . . . . . . . . X i f 2 (X i , X 1 ) . . . f i−1 (X i , X 1 )      = 0, where g = (f 1 (x) = x, f 2 (x), . . . , f i−1 (x)). Note that F * g (X 2 , . . . , X i , X 1 ) is homogeneous, and its dehomogenized polynomial with respect to X 1 is F g (X 2 , . . . , X i ). If g = (x, x q M , . . . , x q (i−2)M ), then by induction hypothesis U g has a non-repeated F q n -rational absolutely irreducible component not contained in V, and hence by Lemma 2.5 the same holds for For i = k, if f = (x, x q M , . . . , x q (k−2)M ), then W has a non-repeated F q n -rational absolutely irreducible component Z not contained in V. Thus, by [38,Corollary 7], Z has an affine F q n -rational point not on V, a contradiction to Theorem 4.4. U h . If g = (x, x q M , . . . , x q (i−2)M ) then f i (x) = x q (i−1)M , because i ≥ 4 4.2. Moore polynomial sets of positive index. In this section we investigate Moore polynomial sets of index t > 0, so that f 1 (x) = x q t . Proposition 4.12. Suppose that one of the following holds: • k = 3 and n > 4M 3 + 2; • k > 3, q > 5 and n > 13 3 M k + log q (13 · 2 10/3 ). If f is a Moore polynomial set for q and n of index t, then (m σ(1) , . . . , m σ(k) ) is in arithmetic progression for some σ ∈ S n . Proof. Since m i = 0 for some i, the proof is the same as in the proof of Theorem 4.6 for t = 0. Up to reordering, we can assume that the permutation σ in Proposition 4.12 satisfies σ(1) = 2, that is, f 2 (x) is separable. Proposition 4.13. Let f = (f 1 (x) = x q t , f 2 (x), f 3 (x) ) be a Moore polynomial set for q and n of index t > 0 such that f 2 (x) is separable. If max{t, M 2 } < M 3 and n > 4M 3 + 2, then f 2 (x) ∈ L n,q is scattered of index t. Proof. The proof is similar to the one of Theorem 4.9. Suppose that f 2 (x) ∈ L n,q is not scattered of index t. Then there exist λ, µ ∈ F * q n such that µ / ∈ F q , λ/µ / ∈ F q and f 2 (λ)/λ q t = f 2 (µ)/µ q t = 0. Let λ 3 = λ and define D f as above. Then D f is PGL(3, q n )-equivalent to the curve D ′ f with affine equation H ′ f (X 1 , X 2 ) = 0, where H ′ f (X 1 , X 2 ) = H f (1, X 2 + 1, X 1 ) = −λ q t f 2 (X 2 , X 1 )X q t −1 1 + f 2 (λ)X q t 2 X q M 2 −1 1 + G(X 1 , X 2 ) and G(X 1 , X 2 ) has degree at least q t +q M2 . The tangent cone to D ′ f at (0, 0) has a non-repeated F q nrational absolutely irreducible component with affine equation X 2 − µX 1 = 0, which corresponds to a non-repeated F q n -rational absolutely irreducible component of D f which is not contained in D (x,x q ,x q 2 ) . Arguing as in the proof of Theorem 4.9, the claim follows. We now use the known classification results on exceptional scattered polynomials. Corollary 4.14. Let f = (f 1 (x) = x q t , f 2 (x), f 3 (x)) be a Moore polynomial set for q and n of index t such that f 2 (x) is separable, max{t, M 2 } < M 3 , and n > 4M 3 + 2. Then: (1) t > 0 and max{t, M 2 } is not an odd prime; (2) if either t = 1, or t = 2 and q is odd, then f = (x q t , ax + x q 2t , f 3 (x)) and m 3 = 2t. Proof. Since f 2 (x) is separable, the case t = 0 cannot occur by definition. Then f 2 (x) ∈ L n,q is exceptional scattered of positive index t. (1) If max{t, M 2 } is an odd prime, then from [15,Theorem 1.4] it follows f 2 (x) = x, so that f has index 0, a contradiction. Proposition 4.13 can be extended as follows. Theorem 4.15. Let f = (f 1 (x) = x q t , f 2 (x), . . . , f k (x)), with k > 3, be a Moore polynomial set for q and n of index t > 0 such that f 2 (x) is separable. Suppose also that q > 5, max{t, M 2 } < M i for any i ≥ 3, and n > 13 3 max{M i : i ≥ 3} + log q (13 · 2 10/3 ). Then f 2 (x) ∈ L n,q is scattered of index t. Proof. It can be proved by finite induction on i ∈ {3, . . . , k} that, if h = (x q t , f 2 (x), . . . , f i (x)) and f 2 (x) ∈ L n,q is not scattered of index t, then the hypersurface U h has a non-repeated F q n -rational absolutely irreducible component not contained in V. The base i = 3 is in the proof of Proposition 4.13. For i > 3, the argument is analogous to the one in the proof of Theorem 4.11. The claim then follows again by using [38,Corollary 7]. Recalling the correspondence between Moore polynomial sets and MRD codes described in Corollary 3.5, we finally obtain Main Theorem as a consequence of Theorem 4.10, Theorem 4.11, Proposition 4.13 and Theorem 4.15. Note that, if the hypothesis of n being large enough in the aforementioned results are incorporated in the assumptions of Main Theorem, then the exceptionality of the MRD code C ⊂ L n,q can be dropped, as well as the exceptionality of the scattered property for f 2 (x) ∈ L n,q . Known examples of Moore polynomial sets This section is devoted to the description of the known examples of Moore polynomial sets corresponding to inequivalent F q n -linear MRD codes; see Table 1. The only known examples of exceptional Moore polynomial sets are the first two in Table 1. Let b : L n,q × L n,q → F q be the bilinear form given by b(f, g) = Tr q n /q n−1 i=0 a i b i , where Tr q n /q (x) = n−1 i=0 x q i , f (x) = n−1 i=0 a i x q i , g(x) = n−1 i=0 b i x q i ∈ L n,q . The Delsarte dual code of a rank metric code C ⊆ L n,q is C ⊥ = {f (x) ∈ L n,q : b(f, g) = 0, for all g(x) ∈ C}. Recall that the Delsarte dual code of an MRD code, having minimum distance greater than one, is an MRD code; see e.g. [14,17]. This yields new examples of Moore polynomial sets; see lines 4,6,8,10,12,14 in Table 1. gcd(s, n) = 1 [14,17,20] x q s , . . . , x q s(k−1) , x + δx q sk gcd(s, n) = 1, N q n /q (δ) = (−1) nk [28,35] 2t 2 x, x q s + x q s(t−1) + δ q t +1 x q s(t+1) + δ 1−q 2t−1 x q s(2t−1) q odd, N q 2t /q t (δ) = −1, gcd(s, n) = 1 [7,24,25,31,40] 2t 2t − 2 x q si : i / ∈ {0, 1, t − 1, t + 1, 2t − 1}, h1(x) = x q s − x q s(t−1) , h2(x) = δ q t +1 x q s − x q s(t+1) , h3(x) = δ 1−q 2t−1 x q s − x q s(2t−1) q odd, N q 2t (q t (δ) = −1, gcd(s, n) = 1 [7,24,25,31,40] 6 2 x, x q + δx q 4 q > 4, certain choices of δ [3,11,32] 6 4 x q , x q 2 , x q 4 , x − δ q 5 x q 3 q > 4, certain choices of δ [3,11,32] 6 2 x, x q + x q 3 + δx q 5 q odd, δ 2 + δ = 1 [13,29] 6 4 x q , x q 3 , x − x q 2 , x q 4 − δx q odd, δ 2 + δ = 1 [13,29] 7 3 x, x q s , x q 3s q odd, gcd(s, 7) = 1 [12] 7 4 x, x q 2s , x q 3s , x q 4s q odd, gcd(s, n) = 1 [12] n k f1(x), . . . , f k (x) conditions references 8 3 x, x q s , x q 3s q ≡ 1 (mod 3), gcd(s, 8) = 1 [12] 8 5 x, x q 2s , x q 3s , x q 4s , x q 5s q ≡ 1 (mod 3), gcd(s, 8) = 1 [12] 8 2 x, x q + δx q 5 q odd, δ 2 = −1 [12] 8 6 x q , x q 2 , x q 3 , x q 5 , x q 6 , x − δx q 4 q odd, δ 2 = −1 [12] 6 . Conclusions and open problems In this paper we introduce the notion of exceptional linear maximum rank distance codes of a given index, which naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework. We then classify those of index 0, and prove that those of positive index contain an exceptional scattered polynomial of the same index. We list a couple of open problems related to the obtained results. • Under the assumptions of Proposition 4.13 or Theorem 4.15, for n large enough, one may conjecture that Moore polynomial sets of positive index do not exist. Whereas, relaxing the assumption max{t, M 2 } < M i for every i ≥ 3, one should include also the second example listed in Table 1, that is the one corresponding to generalized twisted Gabidulin codes. However, a new approach seems to be needed. • A complete classification of exceptional scattered polynomials could yield to more precise results on the asymptotics of Moore polynomial sets of positive index and hence of F q n -linear MRD codes in L n,q . Definition 4. 3 . 3A Moore polynomial set f = (f 1 (x), . Theorem 4. 4 . 4The k-tuple f is a Moore polynomial set for q and n if and only if all the affine F q n -rational points of W lie on V. Let I = {i 1 = 0, i 2 , . Theorem 4 . 10 . 410Let f = (f 1 (x) = x, f 2 (x), f 3 (x)) be a Moore polynomial set for q and n of index 0 such that 0 < M 2 < M 3 , and (q,M ) / ∈ {(2, 2), (2, 4), (3, 2), (4, 2), (5, 2)}. If n > 4M 3 + 2, then f = (x, x q M , x q 2M ) with gcd(M, n) = 1.Proof. By Theorem 4.6, M 3 = 2M and max{m 2 , m 3 } = 2 min{m 2 , m 3 }. By Theorem 4.9, f 2 (x) is scattered of index 0 over F q n . Thus, by the numerical assumption on n, it follows that f 2 (x) = q M and gcd(M, n) = 1; see [8, Section 3.1] for q > 5 and [6, Section 5] for q ≤ 5. 3 ) \ {(1 : 0), (0 : 1)}. In this case, (H ′ f ) q m 3 = (ξX 2 − ηX 1 ) q m 3 up to a non-zero scalar multiple, and hence (H ′ f ) q m 3 and (H ′ f ) q m 3 +1 are not coprime. By Lemma 2.3, the intersection multiplicity of τ (Ã) and τ (B) at O is at most 2 10/3 ). Then f = (x, x q M , . . . , x q (k−1)M ) with gcd(M, n) = 1. Proof. By Theorem 4.6, M i = (i − 1)M for every i, with gcd(M, n) = 1. Also, {0, m 2 , . . . , m k } can be ordered so that they are in arithmetic progression. We prove by finite induction on i ∈ {3, . . . , k} the following fact: if h = (x, f 2 (x), . . . , f i (x)) satisfies h = (x, x q M , . . . , x q (i−1)M ), then the hypersurface U h has a non-repeated F q n -rational absolutely irreducible component not contained in V. The base i = 3 has been worked out in the proof of Theorem 4.10. For i > 3, define the map ϕ : (X 1 : . . . : X i : T ) → (T : X 2 − X 1 : X 3 : . . . X i : X 1 ), which maps (1 : 1 : 0 : . . . : 0) ∈ U f to O = (0 : . . . : 0 . . . 1), and consider the image U implies that the arithmetic progressions of the M j 's and m j 's both have ratio M . ( 2 ) 2If either t = 1, or t = 2 and q is odd, then the results in [8, Page 511] and [6, Theorem 1.4 and Corollary 1.5] imply f 2 (x) = ax + x q 2t with a = 0. The claim follows from Proposition 4.12. Table 1 : 1Known examples of Moore polynomial sets n k f1(x), . . . , f k (x) conditions references x, x q s , . . . , x q s(k−1) A few more functions that are not APN infinitely often. Y Aubry, G Mcguire, F Rodier, Finite fields: theory and applications. Providence, RI518Aubry, Y., McGuire, G., and Rodier, F. A few more functions that are not APN infinitely often. In Finite fields: theory and applications, vol. 518 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2010, pp. 23-31. 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Daniele Bartoli, Dipartimento Di Matematica E Informatica, address: [email protected], Italy EmailUniversità degli Studi di PerugiaDaniele Bartoli, Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia, Italy Email address: [email protected] Dipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli. Giovanni Zini, Ferdinando Zullo, Viale Lincoln, 5, I-81100 Caserta, ItalyEmail address: {giovanni.zini,ferdinando.zullo}@unicampania.itGiovanni Zini, Ferdinando Zullo, Dipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli", Viale Lincoln, 5, I-81100 Caserta, Italy Email address: {giovanni.zini,ferdinando.zullo}@unicampania.it
[]
[ "AGM and Jellyfish Swarms of Elliptic Curves", "AGM and Jellyfish Swarms of Elliptic Curves" ]
[ "Michael J Griffin ", "Ken Ono ", "Neelam Saikia ", "Wei-Lun Tsai " ]
[]
[]
The classical AGM produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields Fq, with q ≡ 3 (mod 4), we introduce a finite field analogue AGM Fq that spawns directed finite graphs instead of infinite sequences.
10.1080/00029890.2022.2160157
[ "https://export.arxiv.org/pdf/2110.12226v3.pdf" ]
239,768,273
2110.12226
12146341e8ba5ac9e6d1b683238e894444f3b8d2
AGM and Jellyfish Swarms of Elliptic Curves December 23, 2022 Michael J Griffin Ken Ono Neelam Saikia Wei-Lun Tsai AGM and Jellyfish Swarms of Elliptic Curves December 23, 2022arXiv:2110.12226v3 [math.NT] 22 Dec 2022 Mathematical Assoc. of America American Mathematical Monthly 121:1 3:07 a.m. jellyfishFinal.tex page 1 The classical AGM produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields Fq, with q ≡ 3 (mod 4), we introduce a finite field analogue AGM Fq that spawns directed finite graphs instead of infinite sequences. ARITHMETIC AND GEOMETRIC MEANS. Beginning with positive real numbers a 1 := a and b 1 := b, the AGM R inductively produces a sequence of pairs AGM R (a, b) := {(a 1 , b 1 ), (a 2 , b 2 ), . . . }, consisting of arithmetic and geometric means. Namely, for n ≥ 2, we let a n := a n−1 + b n−1 2 and b n := a n−1 b n−1 . For n ≥ 2, we have the elementary inequality a n ≥ b n . At a deeper level, the classical theory of the AGM R (for example, see Chapter 1 of [3]) establishes that these rapidly converging sequences have a common limit lim n→+∞ a n = lim n→+∞ b n . In 1748, Euler [3] employed AGM R ( √ 2, 1) as a remarkable device for rapidly computing digits of π. Namely, he showed that π = lim n→+∞ p n , where p n := a 2 n 1 − n i=1 2 i−2 (a 2 i − b 2 i ) . Although the first three terms p 1 = 4, p 2 = 3.18767 . . . , and p 3 = 3.14168 . . . are quite satisfying, the next two terms fact that i = √ −1 is not a real number. For such a field F q , we let φ q (·) be its quadratic residue symbol (the usual Legendre symbol · p when q = p is prime). We then define AGM Fq (a, b) for pairs a, b ∈ F × q := F q \ {0}, with a = ±b and φ q (ab) = 1. This input data gives a 1 := a and b 1 := b, and for n ≥ 2 we let a n := a n−1 + b n−1 2 and b n := a n−1 · b n−1 , (1.1) where b n is the unique square-root with φ q (a n b n ) = 1. Although a n−1 b n−1 has two square-roots, only one choice satisfies φ q (a n b n ) = 1 as φ q (−1) = −1. Therefore, we obtain a sequence of pairs Let's consider the case of F 7 . Half of the 12 pairs that appear in some AGM F 7 (a, b) form a single AGM-orbit AGM F 7 (1, 2) = {(1, 2), (5,3), (4,1), (6,5), (2,4), (3,6), . . . } (Note. The overlined pairs form a repeating orbit.). The other 6 pairs lead to this orbit after a single step. For example, we have AGM F 7 (6, 3) = {(6, 3), (1,2), (5,3), (4, 1), (6,5), (2,4), (3,6), . . . }. (1.2) The compilation J F 7 of all such sequences forms a connected directed graph. This example is typical. The compilation of the AGM Fq sequences is always a disjoint union of connected directed graphs. The nodes are admissible ordered pairs 1 (a, b), where a, b ∈ F × q , with a = ±b, and φ q (ab) = 1. Moreover, (a, b) → (c, d) is an edge if and only if c = (a + b)/2 and d = √ ab, with φ q (cd) = 1. These connected components are like jellyfish, as their 3D renderings turn out to be unit length tentacles leading to a bell head cycle. Hence, we playfully refer to the compilation of the AGM Fq sequences J Fq := J 1 ⊔ J 2 ⊔ · · · ⊔ J d(Fq) (1.3) as the jellyfish swarm for F q , where the J 1 , J 2 , . . . , J d(Fq) are the individual jellyfish which make up the swarm. Let's summarize some basic facts about AGM Fq and the jellyfish swarms J Fq . Theorem 1. If F q is a finite field with q ≡ 3 (mod 4), then the following are true. (1) The AGM Fq algorithm is well-defined. (2) The jellyfish swarm J Fq has (q − 1)(q − 3)/2 nodes. (3) Every jellyfish has a bell head with length one tentacles pointing to each node. (4) If N n (F q ) denotes the number of jellyfish with n nodes, then (q − 1) | nN n (F q ). Proof of Theorem 1. (1) If (a, b) is admissible, then we must show that the next pair (c, d) generated by AGM Fq is also admissible. It is clear that cd = 0 since ab = 0 and a = ±b. If c = ±d, then (a − b) 2 = a 2 − 2ab + b 2 = 4c 2 − 4d 2 = 0, which in turn implies the contradiction that a = b. Therefore, AGM Fq is well-defined. (2) We compute the number of admissible pairs (a, b). There are q − 1 choices for a, and (q − 3)/2 choices of b with a = ±b which additionally satisfy the quadratic residue condition φ q (ab) = 1. (3) Each AGM Fq sequence eventually enters a repeating cycle (i.e., the bell head). Suppose that (a, b) is in this orbit. Reversing AGM Fq to find its parents, say (A, B), we have A + B = 2a and AB = b 2 . Therefore, (A − B) 2 = 4a 2 − 4b 2 , and so φ q (a 2 − b 2 ) = 1 with two square-roots. If (A, B) is the parent in the cycle, then the other solution is (B, A), and is not in the cycle. Therefore, (a, b) has exactly one attached tentacle. To see that this tentacle has length 1, we use the assumption that (A, B) is in the cycle, and so has a parent of its own. Repeating the argument above, we have φ q (A 2 − B 2 ) = 1, which in turn gives φ q (B 2 − A 2 ) = −1. This means that (B, A) does not have a parent. (4) Each α ∈ F × q induces an automorphism on J Fq defined by (a, b) → (αa, αb), as α · AGM Fq (a, b) = AGM Fq (αa, αb). As there are no fixed admissible pairs provided that α = 1, we find that the orbit of a node under these automorphisms has size q − 1. As these automorphisms permute the jellyfish with fixed size, the claim follows. Theorem 1 inspires many natural questions. For example, how small (resp. large) are the jellyfish in a general swarm? This appears to be a very difficult question. For instance, there are AGM Fq -orbits that are much shorter than q, such as the length 9 AGM These examples correspond to a tiny 18 node jellyfish in J F 67 , and a gigantic 820 node jellyfish in J F 83 . As another question, what can be said about d(F q ), the number of jellyfish in J Fq ? Table 1 illustrates the oscillatory behavior of d(F q ). This erratic sequence does not appear to settle into a predictable pattern as q → +∞. Indeed, there are many astonishing examples of disproportionate consecutive values, such as d(F 479 ) = 18 and d(F 487 ) = 359. The only clear observation is that the d(F p m ) grow rapidly with m when p is fixed. For instance, we have We shall see that the theory of elliptic curves offers deep insight into these questions. d(F 3 ) = 0 → d(F 3 3 ) = 39 → d(F 3 5 ) = 1210, d(F 7 ) = 1 → d(F 7 3 ) = 1539 → d(F 7 5 ) = 876713, d(F 11 ) = 3 → d(F 11 3 ) = 8778 → d(F 11 5 ) = 25558635.y 2 = f (x) = x 3 + ax 2 + bx + c, where a, b, c ∈ F, and f has non-zero discriminant. If E(F) denotes the F-rational points of E, including the identity "point at infinity" O, then E(F) naturally forms an abelian group via the well-known "chord-tangent law." The group law can be described by asserting that three colinear points on an elliptic curve sum to the identity O. Number theorists are deeply interested in these groups of rational points. If F is a number field (i.e., a field which has finite degree over Q), then a classical theorem by Mordell and Weil asserts that E(F), the Mordell-Weil group of E/F, is finitely generated. The special case where F = Q is the subject of two frequently cited Monthly articles. The beautiful 1993 article by Silverman [17] on the representation of positive integers as sums of two rational cubes describes the intimate relationship between Ramanujan's taxi-cab numbers and positive rank elliptic curves (i.e., curves with infinitely many Q-rational points). The famous 1991 article by Mazur [12] promotes the conjectured "Modularity" of elliptic curves over Q (formerly known as the Taniyama-Weil Conjecture). This conjecture is now known to be true, largely thanks to the work of Wiles and Taylor [20,23], which was a celebrated ingredient in the proof of Fermat's Last Theorem. For the case of finite fields, it turns out that the jellyfish swarms J Fq organize elliptic curves. One can think of the nodes as spots on the jellyfish, and these spots will be mapped to curves. These swarms are coverings of networks of special Legendre elliptic curves when q ≡ 3 (mod 4) and p ≥ 7. To be precise, for λ ∈ F q \ {0, 1}, we recall the Legendre normal form elliptic curve E λ : y 2 = x(x − 1)(x − λ). (2.1) Isomorphism classes of elliptic curves are distinguished by their j-invariants, and for E λ we have j(E λ ) = 2 8 · (λ 2 − λ + 1) 3 λ 2 (λ − 1) 2 . (2.2) For an introduction to elliptic curves over finite fields, the reader may consult Chapter 4 of [22]. The jellyfish swarm J Fq organizes elliptic curves via the map Ψ Fq : J Fq → E Fq , where E Fq is the set of Legendre curves over F q , and Ψ Fq (a, b) := E λ(a,b) , (2.3) where λ(a, b) := b 2 /a 2 . For instance, (1.2) gives Ψ F 7 (AGM F 7 (6, 3)) = Ψ F 7 {(6, 3), (1, 2), (5, 3), (4, 1), (6, 5), (2, 4), (3, 6), . . . } = E 2 , E 4 , E 4 , E 4 , E 4 , E 4 , E 4 , . . . = E 2 , E 4 , . . . . As the values λ(a, b) = b 2 /a 2 cover the squares in F q \ {0, 1}, it is natural ask what special features are shared by curves of the form E λ 2 . It turns out that these curves are distinguished by the 2-Sylow subgroups of their F q -rational points. Lemma 2. Suppose that F q is a finite field with q ≡ 3 (mod 4). If λ ∈ F q \ {0, 1}, then the 2-Sylow subgroup of E λ 2 (F q ) is of the form Z/2Z × Z/2 2+b Z, where b ≥ 0. Proof of Lemma 2. Elliptic curves with four 2-torsion points (i.e., including the identity) can be written in the form E : y 2 = (x − α)(x − β)(x − γ). The non-trivial 2-torsion points correspond to the roots of the cubic. They are the points (α, 0), (β, 0) and (γ, 0). For such curves, the classical 2-descent lemma (see p. 47-49 of [10] or p. 315 of [16]) says that a non-zero point P = (x 0 , y 0 ) ∈ E(F q ) satisfies P = 2Q, where Q ∈ E(F q ), if and only if x 0 − α, x 0 − β, x 0 − γ ∈ F 2 q . Here we have α = 0, β = 1, and γ = λ 2 . Since q ≡ 3 (mod 4), then exactly one of (1, 0) and (λ 2 , 0) is in 2E λ 2 (F q ), as exactly one of ±(1 − λ 2 ) is a square. On the other hand, (0, 0) ∈ 2E λ 2 (F q ) by the 2-descent lemma because −1 is not a square. Therefore, the Z/4Z rank of E(F q ) is 1, which means that E(F q ) contains Z/4Z but not Z/4Z × Z/4Z. The J Fq organize the curves in Ψ Fq (J Fq ) as unions of explicit isogeny graphs. An isogeny between two elliptic curves is a special map Φ (called a morphism) that preserves the identity element, is given by rational functions Φ = (u(x, y), v(x, y)), and is a homomorphism on F q -points with finite kernel. The isogeny graph structure of Ψ Fq (J Fq ) is provided by the following theorem. Theorem 3. If F q is a finite field with q ≡ 3 (mod 4) and char(F q ) = p ≥ 7, then the following are true. (1) We have that Ψ Fq (J Fq ) = {E α 2 /F q : α ∈ F q \ {0, ±1}}. Moreover, each E α 2 ∈ Ψ Fq (J Fq ) has q − 1 preimages. (2) For each 1 ≤ i ≤ d(F q ), we have that Ψ Fq (J i ) is a connected graph 2 , where an edge (a n , b n ) → (a n+1 , b n+1 ) ∈ J i is the isogeny Φ n : E λ(an,bn) → E λ(a n+1 ,b n+1 ) defined by Φ n (x, y) := (a n x + b n ) 2 x(a n + b n ) 2 , − a n y(a n x − b n )(a n x + b n ) x 2 (a n + b n ) 3 . Moreover, we have that ker(Φ n ) = (0, 0) . Proof of Theorem 3. (1) For an admissible pair (a, b), we have a, b ∈ F × q , and a = ±b. Therefore, b/a ∈ F q \ {0, ±1} and Ψ Fq (J Fq ) ⊂ {E α 2 /F q : α ∈ F q \ {0, ±1}}. On the other hand, if α ∈ F q \ {0, ±1}, then one can choose (a, b) such that a = ±1 and b = ±α with φ q (ab) = 1, giving λ(a, b) = α 2 . Furthermore, each admissible pair (a, b), produces the q − 1 further admissible pairs (ka, kb), all mapping to E b 2 /a 2 . Hence, each E λ(a,b) has q − 1 preimages. (2) If (x, y) = (0, 0) ∈ E λ(an,bn) , then a brute force calculation gives (a n x + b n ) 2 x(a n + b n ) 2 (a n x + b n ) 2 x(a n + b n ) 2 − 1 (a n x + b n ) 2 x(a n + b n ) 2 − b 2 n+1 a 2 n+1 = a 2 n y 2 (a n x − b n ) 2 (a n x + b n ) 2 x 4 (a n + b n ) 6 . This proves that Φ n (x, y) ∈ E λ(a n+1 ,b n+1 ) . To verify that the map preserves the identity O = [0, 1, 0] (the point at infinity in projective space), we consider the projectivized form Φ n (x, y, z) := [ϕ 1 (x, y, z), ϕ 2 (x, y, z), ϕ 3 (x, y, z)], where ϕ 1 := (a n + b n ) × (a 4 n xy 2 + (a 4 n + a 2 n b 2 n )x 3 + 2a 3 n b n y 2 z + (2a 3 n b n + 2a n b 3 n )x 2 z − 2a n b 3 n xz 2 ), ϕ 2 := 2a 3 n b 2 n xyz − a 5 n y 3 − (a 5 n + a 3 n b 2 n )x 2 y, ϕ 3 := (a n + b n ) 3 (a 2 n y 2 z + (a 2 n + b 2 n )x 2 z − b 2 n xz 2 ). One sees that Φ n (x, y, z) preserves the point at infinity O. Furthermore, as a n + b n = 2a n+1 = 0, we find by inspection that Φ n is an isogeny with ker(Φ n ) = (0, 0) . What features are shared by the elliptic curves corresponding to the nodes of a single jellyfish? The next corollary offers the answer. (1) There is an abelian group G such that for all (a n , b n ) ∈ J i we have E λ(an,bn) (F q ) ∼ = G. Moreover, the 2-Sylow subgroup of G is Z/2Z × Z/2 2+bq (i) Z, where b q (i) ≥ 0. (2) There is a fixed "trace of Frobenius" a q (i) such that for all (a n , b n ) ∈ J i we have |E λ(an,bn) (F q )| = q + 1 − a q (i). Proof of Corollary 4. (1) For adjacent pairs (a n , b n ), (a n+1 , b n+1 ) ∈ J i , Theorem 3 (2) gives an isogeny Φ n : E λ(an,bn) → E λ(a n+1 ,b n+1 ) , with ker(Φ n ) = (0, 0) ∼ = Z/2Z. It is well-known (for example, see Exercise 5.4 on p. 153 of [16]) that isogenous elliptic curves over finite fields have the same number of rational points. In particular, the 2-Sylow subgroups of the groups of F q rational points of these two curves have the same order. Therefore, since Φ n (E λ(an,bn) (F q )) is an index 2 subgroup of E λ(a n+1 ,b n+1 ) (F q ), then as abstract groups we find that E λ(an,bn) (F q ) ∼ = E λ(a n+1 ,b n+1 ) (F q ). (2) By Theorem 3 (2), we have that Ψ Fq (J i ) is a connected isogeny graph. As mentioned above, isogenous curves over finite fields have the same number of rational points. Hence, there is a fixed integer a q (i), known as the trace of Frobenius, such that |E λ(an,bn) (F q )| = q + 1 − a q (i) for each (a n , b n ) ∈ J i . Example. By Theorem 1, J F 19 has 144 nodes. and it turns out that J F 19 = J 1 ⊔ J 2 ⊔ · · · ⊔ J 6 ⊔ J 7 ⊔ J 8 (i.e., d(F 19 ) = 8), where the jellyfish can be ordered so that J 1 , J 2 , . . . , J 6 have bell heads with cycle length 6, and J 7 and J 8 have bell heads with cycle length 18. By Theorem 3 (1), the nodes in J F 19 map to the eight Legendre curves with 18 preimages each. The 6 smaller jellyfish give the isogeny graph depicted below. This defines the 3-to-1 covering Ψ F 19 (J 1 ) = · · · = Ψ F 19 (J 6 ) = {E 6 , E 9 , E 16 , E 17 }. These Legendre curves satisfy We have a 19 (1) = · · · = a 19 (6) = 19 + 1 − |Z/2Z × Z/12Z| = −4. For J 7 and J 8 , we obtain the 9-to-1 covering E 6 (F 19 ) ∼ = E 9 (F 19 ) ∼ = E 16 (F 19 ) ∼ = E 17 (F 19 ) ∼ = Z/2Z × Z/12Z.Ψ F 19 (J 7 ) = Ψ F 19 (J 8 ) = {E 4 , E 5 , E 7 , E 11 }, with E 4 (F 19 ) ∼ = E 5 (F 19 ) ∼ = E 7 (F 19 ) ∼ = E 11 (F 19 ) ∼ = Z/2Z × Z/8Z. Therefore, we have a 19 (7) = a 19 (8) = 19 + 1 − |Z/2Z × Z/8Z| = 4. This example shows that individual jellyfish generally include many non-isomorphic curves, as the j-invariants (see (2.2)) for the smaller (resp. larger) jellyfish are j(E 9 ) = j(E 17 ) = 5 and j(E 6 ) = j(E 16 ) = 15 (resp. j(E 7 ) = j(E 11 ) = 5 and j(E 4 ) = j(E 5 ) = 15). We shall show that the number of different j-invariants, like counting types of spots on jellyfish, has taxonomic significance. Thanks to the deeper insight offered by Corollary 4, we are able to revisit the baffling numbers d(F q ), and offer a non-trivial lower bound. Theorem 5. If ε > 0, then for sufficiently large q ≡ 3 (mod 4) we have it is tempting to speculate that this lower bound is not much smaller than an optimal bound which perhaps might be of the form ≫ √ q log log(q). d(F q ) ≥ 1 2 − ε · √ q. Proof of Theorem 5. Corollary 4 guarantees that d(F q ) is at least as large as the number of distinct groups G for which E λ 2 (F q ) ∼ = G for some λ ∈ F q \ {0, 1}. For a group G, the proof of Theorem 6 establishes the existence of such a curve provided Z/2Z × Z/4Z ⊆ G and there is an E/F q for which E(F q ) ∼ = G. We can construct many such groups. If −2 √ q ≤ s ≤ 2 √ q and s ≡ q + 1 (mod 8), then let m q (s) := (q + 1 − s)/2 ≡ 0 (mod 4). A classical theorem of Rück and Voloch [15,21] guarantees that one can take G := Z/2Z × Z/m q (s)Z. For large q, this represents approximately one eighth of the integers in [−2 √ q, 2 √ q]. Therefore, if ε > 0, then for sufficiently large q we have d(F q ) ≥ 1 2 − ε √ q. (2.4) JELLYFISH SWARMS AND GAUSS' CLASS NUMBERS. The swarms J Fq offer new descriptions of the class numbers studied by Gauss, Hurwitz and Kronecker (see [5] for more on class numbers). To make this precise, recall that an integral binary quadratic form is a homogeneous degree 2 polynomial f (x, y) := ax 2 + bxy + cy 2 ∈ Z[x, y]. The Although there are infinitely many primitive binary quadratic forms with discriminant D, Gauss proved that their number of SL 2 (Z)-orbits is finite, and this number is known as Gauss' class number h(D). Gauss' class numbers lead to the more general Hurwitz-Kronecker class numbers. If N ≡ 0, 3 (mod 4), then the Hurwitz-Kronecker class number H(N ) is the class number of positive definite integral binary quadratic forms of discriminant −N, where each class C is counted with multiplicity 1/Aut(C). If −N = Df 2 , where D is a negative fundamental discriminant (i.e., the discriminant of the ring of integers of an imaginary quadratic field), then H(N ) is related to h(D) by (for example, see p. 273 of [5]) H(N ) = h(D) w(D) d|f µ(d) D d σ 1 (f /d). Here w(D) is half the number of integral units in Q( √ D), and σ s (n) denotes the sum of the s th powers of the positive divisors of n, and D · is the quadratic Dirichlet character with conductor D. Class numbers have a long and rich history. For example, class numbers play a central role in the study of quadratic forms. Indeed, if r 3 (n) denotes the number of representations of an integer n as a sum of three squares, then Gauss proved that r 3 (n) =          12H(4n) if n ≡ 1, 2 (mod 4), 24H(n) if n ≡ 3 (mod 8), r 3 (n/4) if n ≡ 0 (mod 4), 0 if n ≡ 7 (mod 8). Class numbers play even deeper roles in algebraic and analytic number theory, as they are the orders of ideal class groups of rings of integers and orders of imaginary quadratic fields. These groups themselves are the Galois groups of Hilbert class fields. For brevity, we simply say that the study of class numbers continues to drive cutting edge research today. The jellyfish swarms J Fq offer a new interpretation of these class numbers. As the nodes are jellyfish spots, it is quite gratifying to discover that class numbers represent the number of types of spots that appear in a family of jellyfish. In this analogy, the j-invariants distinguish these types of spots. Namely, for integers s, let M Fq (s) be the number of distinct j-invariants of curves in the union of jellyfish with a q (i) = s, the "Frobenius trace s family." We have the following attractive description which follows from a well-known theorem of Schoof. Theorem 6. Suppose that F q is a finite field with q ≡ 3 (mod 4) and p ≥ 7. If −2 √ q ≤ s ≤ 2 √ q is a non-zero integer with s ≡ q + 1 (mod 8), then we have H 4q − s 2 4 = M Fq (s). Example. We revisit the example of J F 19 , where J 1 , . . . , J 6 (resp. J 7 and J 8 ) are the smaller (resp. larger) jellyfish. We found earlier that the Frobenius trace -4 family is Ψ F 19 (J 1 ) ∪ · · · ∪ Ψ F 19 (J 6 ) = {E 6 , E 9 , E 16 , E 17 }. One checks (using (2.2) that j(E 9 ) = j(E 17 ) = 5 and j(E 6 ) = j(E 16 ) = 15, giving M F 19 (−4) = 2. We also found that the Frobenius trace 4 family is Ψ F 19 (J 7 ) ∪ Ψ F 19 (J 8 ) = {E 4 , E 5 , E 7 , E 11 }. As j(E 7 ) = j(E 11 ) = 5, and j(E 4 ) = j(E 5 ) = 15, we also have M F 19 (4) = 2. Therefore, since M F 19 (±4) = 2, Theorem 6 gives H 76 − (±4) 2 4 = H(15) = 2. Proof of Theorem 6. Let E/F q be an elliptic curve for which |E(F q )| ≡ 0 (mod 8) and Z/2Z × Z/2Z ⊆ E(F q ). We automatically have Z/2Z × Z/4Z ⊆ E(F q ), because these groups can always be described as a direct product of at most 2 cyclic groups. Moreover, an application of the 2-descent lemma (for example, see Proposition 3. where (x) k is the Pochhammer symbol defined by (x) k = 1 if k = 0 x(x + 1) · · · (x + k − 1) if k > 0. Gauss' theory of elliptic integrals [9, p. 182] gives I R (a, b) = π 2a · 2 F class 1 1 2 , 1 2 1 | 1 − b 2 a 2 ,(4.4) which, by letting a → (a + b)/2 and b → √ ab, also gives I R a + b 2 , √ ab = π a + b · 2 F class 1 1 2 , 1 2 1 | (a − b) 2 (a + b) 2 . (4.5) Equating these expressions, we find that AGM R leads to the identity 2 F class 1 1 2 , 1 2 1 | 1 − b 2 a 2 = 2a a + b · 2 F class 1 1 2 , 1 2 1 | (a − b) 2 (a + b) 2 , which relates 1 − b 2 /a 2 with λ(a + b, a − b) = (a − b) 2 /(a + b) 2 . This identity is a special case (i.e., α = β = 1/2 and t = (a − b)/(a + b)) of the far more general quadratic transformation formula (see [2, (3.1.11) ]) 2 F class 1 α, β 2α | 4t (1 + t) 2 (4.6) = (1 + t) 2β · 2 F class 1 β + 1 2 − α, β α + 1 2 | t 2 . Hypergeometric functions and AGM Fq . In view of the previous discussion, we seek a finite field analog of I R (a, b) := 1 2a ∞ 1 dx x(x − 1)(x − (1 − b 2 /a 2 )) . To this end, one can replace the integral over R by a sum over F q , and replace the square-root with the quadratic character φ q (·), and we can naively declare the finite field analogue to be the sum I Fq (a, b) := x∈Fq φ q (x)φ q (x − 1)φ q (x − (1 − b 2 /a 2 )). (4.7) We hope that such sums are values of hypergeometric-type functions. In his important 1984 Ph.D thesis [7], Greene defined the finite field hypergeometric functions that do the trick (see [13,14] for applications). For multiplicative characters 5 A 1 , A 2 , . . . , A n and B 1 , B 2 , . . . , B n−1 of F × q , he defined n F n−1 A 1 , A 2 , . . . , A n B 1 , . . . , B n−1 | t Fq := q q − 1 χ A 1 χ χ A 2 χ B 1 χ · · · A n χ B n−1 χ χ(t), where the sum is over the multiplicative characters of F × q , and A B is the normalized Jacobi sum A B := B(−1) q J(A, B) := B(−1) q t∈Fq A(t)B(1 − t). This definition was meant to resemble (4.3), and is based on analogies between Gauss sums and the complex Γ-function, which interpolates factorials, and the classical Gauss sum expression for Jacobi sums (when χψ is nontrivial) J(χ, Ψ) = G(χ)G(ψ) G(χψ) , which in turn emulates binomial coefficients. These functions take an attractive form when n = 2 (see p. 82 of [8]). If A, B and C are characters of F q and t ∈ F × q , then 2 F 1 A, B C | t Fq = BC(−1) q · x∈Fq B(x) · BC(1 − x) · A(1 − xt). In particular, if A = B = φ q (·) and ε q (·) is trivial, then a change of variables gives 2 F 1 (λ) Fq := 2 F 1 φ q , φ q ε q | λ Fq = − φ q (−1) q · x∈Fq φ q (x(x − 1)(x − λ)). (4.8) In analogy with Gauss' integral formulas, which give periods of elliptic curves, Greene's functions compute traces of Frobenius over F q . Indeed, if char(F q ) > 3 and λ ∈ F q \ {0, 1}, then (4.8) gives |E λ (F q )| = q + 1 + qφ q (−1) · 2 F 1 (λ) Fq . (4.9) In terms of the desired analogy, if q ≡ 3 (mod 4), then (4.7) and (4.8) gives the counterpart of (4.4) I Fq (a, b) = q · 2 F 1 (1 − b 2 /a 2 ) Fq . To complete the analogy, we require a quadratic transformation law which plays the role of (4.6). We conclude by stating this recent theorem of Evans and Greene (see Th. 2 of [6]), which precisely offers the desired analogous transformation. = A(4)φ q B(−1)G(A 2 B) · G(φ q AB) G(φ q )G(A) · B 2 (1 + t) 2 F 1 φ q AB, B φ q A | t 2 Fq . EPILOGUE. We hope that the reader agrees that the story presented here is a beautiful amalgamation of facts about elliptic curves over C and over finite fields. It is quite marvelous to find that the hypergeometric functions of Gauss (in the case of R) and of Greene (in the case of F q ) underlie different features in the theory of elliptic curves that are captured by sequences of arithmetic and geometric means. We hope that this story encourages readers to learn more about the theory of Gauss' class numbers, elliptic curves, and hypergeometry. We highly recommend D. Cox's book "Primes of the form x 2 + ny 2 " [5] and "Pi and the AGM" [3] by P. Borwein and J. Borwein. We aim to entice readers with the following tantalizing problems. Problems. (1) What can one prove about the sizes of the jellyfish in J Fq ? This question is intimately connected to the unproven Cohen-Lenstra heuristics on the expected behavior of class groups of imaginary quadratic orders. (2) Determine an "optimal" function D(q) for which d(F q ) ≥ D(q). In particular, how close to optimal is the lower bound in Theorem 2.4? Is the correct lower bound more like ≫ √ q log log(q)? (3) It would be very interesting to define variants of AGM in situations where choices of square-root are not well-defined, such as the complex field C and the finite fields F q with q ≡ 3 (mod 4). To conclude, we must confess that the AGM Fq jellyfish are merely alluring examples of creatures that inhabit the magnificent kingdom formed out of elliptic curves over finite fields. The beautiful AGM Fq sequences innocently offer glimpses of the fascinating theory of isogenies for elliptic curves over finite fields, which form networks, dubbed isogeny volcanoes. The jellyfish are examples that arise from "2-volcanoes of height 1." Isogeny volcanoes play important roles in computational number theory and cryptography. They are often employed as a means of accelerating number theoretic algorithms. They have even been used to quickly compute values of Euler's partition function [4]. This important theory has its origins in David Kohel's 1996 PhD thesis [11]. We invite interested readers to read the delightful expository article [19] by Sutherland. AGM Fq (a, b) := {(a 1 , b 1 ), (a 2 , b 2 ), . . . }. Figure 1 : 13D rendering of J F 7 F 67 (1, 17) = {(1, 17), (9, 33) . . . , (65, 15), (40, 29), . . . }, as well as those that are much longer than q, such as the length 410 AGM F 83 (1, 3) = {(1, 3), (2, 13), . . . , (37, 12), (66, 19), . . . }. Corollary 4 . 4For each 1 ≤ i ≤ d(F q ), the following are true. Figure 2 : 22D rendering of the isogeny graph of each J ∈ {J 1 , . . . , J 6 } Figure 3 : 3J F 19 swarm discriminant 3 of f is D := b 2 − 4ac. If a > 0 and D < 0, then f (x, y) is called positive definite. Furthemore, f is primitive if gcd(a, b, c) = 1. For negative discriminants D,the group SL 2 (Z) acts on Q D , the set of positive definite binary quadratic forms of discriminant D. More precisely, for any γ = ( u v r s ) , we have (f • γ) (x, y) := f (ux + vy, rx + sy). Remark. Is this lower bound close to the truth? In view of examples such asd(F 47 ) = 4 > √ 47/2 ≈ 3.4278, d(F 383 ) = 14 > √ 383/2 ≈ 9.7851, d(F 983 ) = 25 > √ 983/2 ≈ 15.6764, d(F 1907 ) = 38 > √ 1907/2 ≈ 21.8346, d(F 7703 ) = 87 > √ 7703/2 ≈ 43.8833, Mathematical Assoc. of America Theorem 7. [Th. 2 of [6]] Suppose that A, A 2 B, and φ q AB are all nontrivial characters of F × q . If t ∈ F q \ {−1}, then FqAmerican Mathematical Monthly 121:1 December 23, 2022 3:07 a.m. jellyfishFinal.tex page 14 2 F 1 A, B A 2 | 4t (1 + t) 2 There are no loops (i.e., (a, b) → (a, b)) as nodes of the form (a, a) are not allowed. © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 Mathematical Assoc. of America American Mathematical Monthly 121:1 December 23, 2022 3:07 a.m. jellyfishFinal.tex The proof of Theorem 3 shows that the swarms are graphs of 2-isogenies. We point interested readers to Sutherland's expository article[19] for more on the theory of isogeny graphs. © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 Mathematical Assoc. of America Discriminants always satisfy D ≡ 0, 1 (mod 4).January 2022] AGM AND JELLYFISH SWARMS OF ELLIPTIC CURVES These models are actually −1 quadratic twists of each other. January 2022] AGM AND JELLYFISH SWARMS OF ELLIPTIC CURVES For multiplicative characters χ, we adopt the convention that χ(0) := 0. ACKNOWLEDGMENT. The authors thank the referees, Jennifer Balakrishnan, Hasan Saad, and Drew Sutherland for comments and suggestions that improved this article. The second author thanks the Thomas Jefferson Fund and the NSF (DMS-2002265 and DMS-2055118) for their generous support, as well as the Kavli Institute grant NSF PHY-1748958. The third author is grateful for the support of a Fulbright Nehru Postdoctoral Fellowship.the isomorphism classes of elliptic curves E/F q with Z/2Z × Z/2Z ⊆ E(F q ), with the additional property that q ≡ s + 1 (mod 8), where s := q + 1 − #E(F q ).We consider the F q isomorphism classes of such curves with fixed non-zero trace of Frobenius s. If (a, b), (a ′ , b ′ ) ∈ J Fq satisfies j(E λ(a,b) ) = j(E λ(a ′ ,b ′ ) ), then either E λ(a,b) ∼ = Fq E λ(a ′ ,b ′ ) , or they are non-trivial twists of each other (see Chapter X of[16]). In the latter case, the traces of Frobenius differ in sign.Combining these facts, we have that M Fq (s) is the number of isomorphism classes of elliptic curves E/F q with Z/2Z × Z/2Z ⊆ E(F q ) and trace of Frobenius s. The rich theory of complex multiplication for elliptic curves is the bridge which connects the counts of such classes with equivalence classes of binary quadratic forms. Indeed, a well-known deep theorem of Schoof (see Section 4 of[18]) asserts that H 4q−s 2 4 equals the number of such isomorphism classes of elliptic curves. Invoking this theorem completes the proof.ANALOGIES BETWEEN HYPERGEOMETRIC FUNCTIONS.Does the classical AGM R have more in common with its finite field analogues than the inductive rules a n := a n−1 + b n−1 2 and b n := a n−1 b n−1 ?This is indeed the case. It turns out that the results offered above are a byproduct of remarkable analogies between complex hypergeometric functions and their finite field analogues. Let us explain.Hypergeometric functions and AGM R . The theory underlying AGM R (see Chapter 1 of[3]) is a story involving special integrals and their relationship with Gauss' hypergeometric functions. To make this precise, for a > b > 0, we letwhere the polynomial in the square-root in the denominator of the integrand is tantalizingly 4 close to the cubic in the Legendre curveIt is straightforward to check thatGauss discovered a beautiful formula for I R (a, b) in terms of hypergeometric functions. For α 1 , α 2 , . . . α n and β 1 , . . . , β n−1 ∈ C, these functions are defined by n F class n−1 α 1 , α 2 , . . . , α n β 1 , . . . , β n−1 | t := ∞ k=0 (α 1 ) k (α 2 ) k · · · (α n ) k (β 1 ) k · · · (β n−1 ) k t k k! , (4.3) Modularity of a certain Calabi-Yau threefold. S Ahlgren, K Ono, Montash. Math. 1293Ahlgren, S., Ono, K. (2000). Modularity of a certain Calabi-Yau threefold. Montash. Math. 129(3): 177-190. G Andrews, R Askey, R Roy, Special functions. CambridgeCambridge University PressAndrews, G., Askey, R., Roy, R. (2001). Special functions. Cambridge: Cambridge University Press. Pi and the AGM. J Borwein, P Borwein, John Wiley and SonsNew YorkBorwein, J., Borwein, P. (1998). Pi and the AGM. New York: John Wiley and Sons. Class polynomials for nonholomorphic modular functions. J H Bruinier, K Ono, A V Sutherland, J. Number Th. 161Bruinier, J. H., Ono, K., Sutherland, A. V. (2016). Class polynomials for nonholomorphic modular functions. J. Number Th. 161: 204-229. Primes of the Form x 2 + ny 2. D A Cox, John Wiley and SonsNew YorkCox, D. A. (2013). Primes of the Form x 2 + ny 2 . New York: John Wiley and Sons. A quadratic hypergeometric 2 F 1 transformation over finite fields. R Evans, J Greene, Proc. Amer. Math. Soc. 145Evans, R., Greene, J. (2017). A quadratic hypergeometric 2 F 1 transformation over finite fields. Proc. Amer. Math. Soc. 145: 1071-1076. Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields. PhD Dissertation. J Greene, Minneapolis, USAUniversity of MinnesotaGreene, J. (1984). Character sum analogues for hypergeometric and generalized hypergeometric func- tions over finite fields. PhD Dissertation. University of Minnesota, Minneapolis, USA. Hypergeometric functions over finite fields. J Greene, Trans. Amer. Math. Soc. 301Greene, J. (1987). Hypergeometric functions over finite fields. Trans. Amer. Math. Soc. 301: (1987): 77-101. Elliptic curves. D Husemöller, SpringerNew YorkHusemöller, D. (2004). Elliptic curves, New York: Springer. Introduction to Elliptic Curves and Modular Forms. N Koblitz, SpringerNew York2nd. editionKoblitz, N. (1993). Introduction to Elliptic Curves and Modular Forms, 2nd. edition, New York: Springer. Endomorphism rings of elliptic curves over finite fields. PhD Dissertation. University of California. D Kohel, Berkeley, USAKohel, D. (1996). Endomorphism rings of elliptic curves over finite fields. PhD Dissertation. Univer- sity of California, Berkeley, USA. Number theory as gadfly. B Mazur, Amer. Math. Monthly. 987Mazur, B. (1991). Number theory as gadfly. Amer. Math. Monthly. 98(7): 593-610. Values of Gaussian hypergeometric series. K Ono, Trans. Amer. Math. Soc. 3503Ono, K. (1998). Values of Gaussian hypergeometric series. Trans. Amer. Math. Soc. 350(3): 1205- 1223. The Web of Modularity. K Ono, Amer. Math. SocProvidenceOno, K. (2004). The Web of Modularity. Providence: Amer. Math. Soc. A note on elliptic curves over finite fields. H.-G Rück, Math. Comp. 49Rück, H.-G. (1987). A note on elliptic curves over finite fields. Math. Comp. 49: 301-304. The Arithmetic of Elliptic Curves. J Silverman, SpringerNew York2nd ed.Silverman, J. (2009). The Arithmetic of Elliptic Curves, 2nd ed., New York: Springer. Taxicabs and sums of two cubes. J Silverman, Amer. Math. Monthly. 1004Silverman, J. (1993). Taxicabs and sums of two cubes. Amer. Math. Monthly. 100(4): 331-340. Nonsingular plane cubic curves over finite fields. R Schoof, J. Comb. Theory Ser. A. 462Schoof, R. (1987). Nonsingular plane cubic curves over finite fields. J. Comb. Theory Ser. A. 46(2): 183-211. A V Sutherland, Isogeny volcanoes. ANTS X-Proc. Algorithmic Number Theory Symposium. Sutherland, A. V. (2013). Isogeny volcanoes. ANTS X-Proc. Algorithmic Number Theory Symposium. Berkeley: Math. Sci. Publ. pp. 507-530. Ring theoretic properties of certain Hecke algebras. R Taylor, A Wiles, Ann. of Math. 141Taylor, R., Wiles, A. (1995). Ring theoretic properties of certain Hecke algebras. Ann. of Math. 141: 553-572. A note on elliptic curves over finite fields. J F Voloch, Bull. Soc. Math. France. 116Voloch, J. F. (1988). A note on elliptic curves over finite fields. Bull. Soc. Math. France 116: 455-458. L C Washington, Elliptic Curves, Number Theory, and Cryptography. Boca RatonChapman & HallWashington, L. C. (2008). Elliptic Curves, Number Theory, and Cryptography, Boca Raton: Chapman & Hall. Modular elliptic curves and Fermat's Last Theorem. A Wiles, Ann. of Math. 141Wiles, A. (1995). Modular elliptic curves and Fermat's Last Theorem. Ann. of Math.141: 443-551. Michael J , GRIFFIN received the Ph.D. degree in Mathematics from Emory University in 2015. He is an Assistant Professor of Mathematics at Brigham University in. Provo, Utah, USA; Provo, UTBrigham Young UniversityHis research interests are in number theory. Department of Mathematics, 275 TMCB. 84602 [email protected] J. GRIFFIN received the Ph.D. degree in Mathematics from Emory University in 2015. He is an Assistant Professor of Mathematics at Brigham University in Provo, Utah, USA. His research interests are in number theory. Department of Mathematics, 275 TMCB, Brigham Young University, Provo, UT 84602 [email protected] His research interests are in number theory. KEN ONO received the Ph.D. degree in Mathematics from UCLA in 1993. He is the Thomas Jefferson Professor of Mathematics at the University of Virginia. Charlottesville, Virginia, USA; Charlottesville, VADepartment of Mathematics, University of Virginia22904 [email protected] ONO received the Ph.D. degree in Mathematics from UCLA in 1993. He is the Thomas Jefferson Pro- fessor of Mathematics at the University of Virginia, Charlottesville, Virginia, USA. His research interests are in number theory. Department of Mathematics, University of Virginia, Charlottesville, VA 22904 [email protected] NEELAM SAIKIA received the Ph.D. degree in Mathematics from the Indian Institute of Technology in. NEELAM SAIKIA received the Ph.D. degree in Mathematics from the Indian Institute of Technology in She is a Nehru-Fulbright Postdoctoral Fellow at the University of Virginia, Charlottesville. India Delhi, In, Virginia, USA; Charlottesville, VA 22904Department of Mathematics, University of VirginiaHer research interests are in number theoryDelhi, India in 2016. She is a Nehru-Fulbright Postdoctoral Fellow at the University of Virginia, Char- lottesville, Virginia, USA. Her research interests are in number theory. Department of Mathematics, University of Virginia, Charlottesville, VA 22904 Wei-Lun, TSAI received the Ph.D. degree in Mathematics from the Texas A&M University in 2020. He is a Research Postdoctoral Fellow at the University of. Virginia, Charlottesville, Virginia, USAHis researchWEI-LUN TSAI received the Ph.D. degree in Mathematics from the Texas A&M University in 2020. He is a Research Postdoctoral Fellow at the University of Virginia, Charlottesville, Virginia, USA. His research
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[ "Improved constructions of secondary structure avoidance codes for DNA sequences", "Improved constructions of secondary structure avoidance codes for DNA sequences" ]
[ "Hui Chu [email protected]@[email protected] \nSchool of Cyber Science and Technology\nKey Laboratory of Cryptologic Technology and Information Security of Ministry of Education\nShandong University\n266237QingdaoShandongChina\n", "Chen Wang \nSchool of Cyber Science and Technology\nKey Laboratory of Cryptologic Technology and Information Security of Ministry of Education\nShandong University\n266237QingdaoShandongChina\n", "Yiwei Zhang \nSchool of Cyber Science and Technology\nKey Laboratory of Cryptologic Technology and Information Security of Ministry of Education\nShandong University\n266237QingdaoShandongChina\n" ]
[ "School of Cyber Science and Technology\nKey Laboratory of Cryptologic Technology and Information Security of Ministry of Education\nShandong University\n266237QingdaoShandongChina", "School of Cyber Science and Technology\nKey Laboratory of Cryptologic Technology and Information Security of Ministry of Education\nShandong University\n266237QingdaoShandongChina", "School of Cyber Science and Technology\nKey Laboratory of Cryptologic Technology and Information Security of Ministry of Education\nShandong University\n266237QingdaoShandongChina" ]
[]
In a DNA sequence, we have the celebrated Watson-Crick complement T = A, A = T , C = G, and G = C. Given an integer m ≥ 2, a secondary structure in a DNA sequence refers to the existence of two non-overlapping reverse complement consecutive subsequences of length m, denoted as x = (x1, . . . , xm) and y = (y1, . . . , ym), such that xi = ym−i+1 for 1 ≤ i ≤ m. The property of secondary structure avoidance (SSA) forbids a sequence to contain such reverse complement subsequences, and it is a key criterion in the design of singlestranded DNA sequences for DNA computing and storage. In this paper, we improve on a recent result of Nguyen et al., by introducing explicit constructions of secondary structure avoidance codes and analyzing the capacity for any given m. In particular, our constructions have optimal rate 1.1679bits/nt and 1.5515bits/nt when m = 2 and m = 3, respectively.
10.48550/arxiv.2304.11403
[ "https://export.arxiv.org/pdf/2304.11403v1.pdf" ]
258,298,900
2304.11403
a4aee210d05e4a80ad69594441f5ab8befdf4793
Improved constructions of secondary structure avoidance codes for DNA sequences Hui Chu [email protected]@[email protected] School of Cyber Science and Technology Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education Shandong University 266237QingdaoShandongChina Chen Wang School of Cyber Science and Technology Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education Shandong University 266237QingdaoShandongChina Yiwei Zhang School of Cyber Science and Technology Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education Shandong University 266237QingdaoShandongChina Improved constructions of secondary structure avoidance codes for DNA sequences In a DNA sequence, we have the celebrated Watson-Crick complement T = A, A = T , C = G, and G = C. Given an integer m ≥ 2, a secondary structure in a DNA sequence refers to the existence of two non-overlapping reverse complement consecutive subsequences of length m, denoted as x = (x1, . . . , xm) and y = (y1, . . . , ym), such that xi = ym−i+1 for 1 ≤ i ≤ m. The property of secondary structure avoidance (SSA) forbids a sequence to contain such reverse complement subsequences, and it is a key criterion in the design of singlestranded DNA sequences for DNA computing and storage. In this paper, we improve on a recent result of Nguyen et al., by introducing explicit constructions of secondary structure avoidance codes and analyzing the capacity for any given m. In particular, our constructions have optimal rate 1.1679bits/nt and 1.5515bits/nt when m = 2 and m = 3, respectively. I. INTRODUCTION DNA sequences have become a promising medium for both computation and storage [1], [6], [7]. A DNA sequence is a quaternary sequence with the alphabet set {A, T, C, G}, which represents the four nucleotides: Adenine (A), Thymine (T ), Cytosine (C), and Guanine (G). The celebrated Watson-Crick complement indicates that T = A, A = T , C = G, and G = C. In a single-stranded DNA sequence, a secondary structure refers to the existence of two non-overlapping reverse complement consecutive subsequences, which will make the sequence fold back onto itself by complementary base pair hybridization and form a stem-loop structure (see Figure 1). The existence of stem-loop structures makes the sequence less active. Thus, in both DNA computation and DNA storage, it is desirable to use sequences avoiding secondary structure [5], [10], [13]. While there have been some dynamic programming techniques [4], [11] to efficiently check whether a given DNA sequence possesses secondary structure, and there have also been some works [9], [10] discussing the sequence design criteria for reducing the possibility of secondary structure appearances, it would be more natural to have a coding theoretic solution by directly constructing as many secondary structure avoidance (SSA) sequences as possible. Given the sequence length n and the stem length m ≥ 2, an m-SSA sequence does not contain two non-overlapping reverse complement consecutive subsequences of length m. An m-SSA code is a set of m-SSA sequences. It should be remarked that in current applications of DNA computation and storage, biochemists usually consider the secondary structure problem with stem length m ≥ 6 and loop length at least 4 [8]. In our mathematical formulation, we consider a more general setting by allowing m ≥ 2 and ignoring the parameter of the loop length (essentially it does not affect the rate of the code). The SSA sequences have been studied in [2], [3], [9], [10], [12]. In particular, the recent paper [12] applies a symbolcomposition constrained technique to construct 3-SSA codes with rate 1.3031bits/nt, which improves on the prior result 1.1609bits/nt constructed by a block concatenation method from [2]. In this paper, we further analyze the capacity and improve the constructions of SSA codes. In particular, our constructions and capacity analysis together lead to optimal rates 1.1679bits/nt and 1.5515bits/nt for the cases m = 2 and m = 3, respectively. The rest of the paper is organized as follows. In Section II we introduce the mathematical notations. In Section III we give a construction when m is odd and in particular we can explicitly calculate the rate of our construction for m ∈ {3, 5}. In Section IV we provide a method to determine the the capacity of SSA codes, which requires a brute-force search over a so-called generating set S for an m-SSA code. In Section V we further discuss the strategy to choose a good generating set S, and compute the rate of the code by a standard approach in constrained coding theory. Conclusions and discussions are given in Section VI. Let D = {A, T, C, G} be the quaternary alphabet set for DNA sequences, where the Watson-Crick complement indicates that T = A, A = T , C = G, and G = C. We use bold letters to represent the sequences and plain letters to represent the symbols in the sequences, such as x = (x 1 , x 2 , . . . , x n ). In this paper, whenever we talk about subsequences we mean consecutive subsequences. Consider two non-overlapping sub- sequences y = x[i; m] (x i , x i+1 , . . . , x i+m−1 ) and z = x[j; m] = (x j , x j+1 , . . . , x j+m−1 ) of some x, where 1 ≤ i < i + m − 1 < j ≤ n − m + 1. y and z are called reverse- complement, denoted as y = RC(z), if x i+t = x j+m−1−t for all 0 ≤ t ≤ m − 1. Definition 1. Given integers 2 < m ≤ n, a DNA sequence x ∈ D n is said to be an m-secondary structure avoidance sequence (abbreviated as an m-SSA sequence), if it does not contain any two non-overlapping reverse-complement subsequences of length m 1 . A code C ⊆ D n is called an m-SSA code if every codeword in C is an m-SSA sequence. The code rate of C is defined as 1 n log(|C|) with logarithm base 2. It measures the number of bits encoded in each DNA nucleotide. For example, C = D n has the largest rate 2bits/nt. Next we present two known constructions of 3-SSA codes from [2] and [12]. Theorem 1. [2] Set S = {AA, AC, CA, CC, T C}. Let C be a code of even length such that each codeword is formed by a block concatenation using blocks from S. Then C is a 3-SSA code of size 5 n/2 . Thus, c 3 ≥ 1 2 log 5 = 1.1609 bits/nt. Theorem 2. [12] Let C be the set of all DNA sequences such that any subsequence of length 3 contains at least one A and does not contain any T . Let f (n) be the size of such a code. It holds that f (1) = 3, f (2) = 9, f (3) = 19, and the linear recurrence relation is f (n) = f (n−1)+2f (n−2)+4f (n−3). Then C is a 3-SSA code of size ρ n , where ρ = 2.4675 is the largest real root of the corresponding characteristic equation x 3 − x 2 − 2x − 4 = 0. Thus, c 3 ≥ log(ρ) = 1.3031 bits/nt. III. A CONSTRUCTION WHEN m IS ODD In this section we propose a simple construction of m-SSA codes when m is odd. Definition 3. A DNA sequence x is called T C-m-dominant, if for every subsequence of length m, the sum of appearances of T and C is larger than m/2. Theorem 3. When m is odd, a T C-m-dominant sequence must be an m-SSA sequence. Thus a code C consisting of all T C-m-dominant sequences of length n is an m-SSA code. Proof: Consider any two non-overlapping subsequences y and z in a T C-m-dominant sequence x. In z, the sum of appearances of T and C is larger than m/2. Note that the complements of T and C are A and G, respectively. Thus, if y = RC(z), then sum of appearances of A and G should be larger than m/2 in y, which contradicts to the assumption that x is a T C-m-dominant sequence. It should be remarked that we can define AC-m-dominant, T G-m-dominant, and AG-m-dominant sequences analogously, and all of them can be shown to be m-SSA sequences. The reason we choose to use T C-m-dominant sequences as representatives is a biochemical observation that T and C are relatively cheaper and more efficient in the current DNA synthesis process [8]. Moreover, while all the four families of sequences could constitute an m-SSA code, for computing the rate it suffices to only consider one of them. Next, in order to compute the rate of our construction, we map the quaternary sequences to binary sequences by mapping T and C into 1, and mapping A and G into 0. It is routine to check this is a 2 n -to-one mapping. The set of T C-3-dominant DNA sequences are mapped into binary sequences such that every length-3 subsequence has weight at least 2, and we refer to them as good sequences. Now, let f 3 (n) be the number of good sequences, and we can derive the following linear recurrence relation. Theorem 4. Let f 3 (n) be the number of good sequences, which are binary sequences such that every length-3 subsequence has weight at least 2. For n ≥ 4 it holds that f 3 (n) = f 3 (n − 1) + f 3 (n − 3). Proof: On one hand, if a good sequence of length n starts with 1, then the 1 could be followed by any good sequence of length n − 1. This accounts for the f 3 (n − 1) part. On the other hand, if a good sequence of length n starts with 0, then it must start with 011, and then any good sequence of length n − 3 could follow. This accounts for the f 3 (n − 3) part. Therefore, the number of good sequences is approximately f 3 (n) ≈ ρ n , where ρ = 1.4656 is the largest real root of the corresponding characteristic equation x 3 − x 2 − 1 = 0. Since each good sequence corresponds to 2 n T C-m-dominant DNA sequences, we have A(n; 3) ≥ ρ n · 2 n and thus c 3 ≥ 1 + log(ρ) = 1.5514 bits/nt. Similarly, we switch to the case m = 5, and map the T C-5-dominant DNA sequences into binary sequences such that every length-5 subsequence has weight at least 3. Again by abuse of notation we refer to them as good sequences. Theorem 5. Let f 5 (n) be the number of good sequences, which are binary sequences such that every length-5 subsequence has weight at least 3. For n ≥ 11 it holds that f5(n) = f5(n−1)+f5(n−3)+2f5(n−5)−f5(n−8)−f5(n−10). Proof: Again we classify good sequences according to their prefix. • Good sequences starting with 1: f 5 (n − 1). • Good sequences starting with 00 must start with 00111, and then any good sequence of length n−5 could follow. The number of this class is then f 5 (n − 5). • Good sequences starting with 010 must start with 01011, and then any good sequence of length n − 5 could follow, except for those starting with 00. Moreover, a good sequence of length n − 5 starting with 00 must start with 00111, and is then followed by any good sequence of length n − 10. The number of this class is then f 5 (n − 5) − f 5 (n − 10). • Good sequences starting with 011, could be followed by any good sequence of length n − 3 except for those starting with 00. Moreover, a good sequence of length n − 3 starting with 00 must start with 00111, and is then followed by any good sequence of length n − 8. The number of this class is then f 5 (n − 3) − f 5 (n − 8). In sum we get the desired recurrence relation. Therefore, the number of good sequences is approximately f 5 (n) ≈ ρ n , where ρ = 1.6222 is the largest real root of the corresponding characteristic equation x 10 − x 9 − x 7 − 2x 5 + x 2 + 1 = 0. Since each good sequence corresponds to 2 n T C-5-dominant DNA sequences, we have A(n; 5) ≥ ρ n · 2 n and thus c 3 ≥ 1 + log(ρ) = 1.6980 bits/nt. Theoretically the procedure could continue. However, finding the recurrence relation even for m = 7 is already a tedious and nontrivial task. We do not bother to go on this computation. Instead, we can turn to the standard spectral radius method in constrained coding theory by computer search (see Section V). IV. UPPER BOUND ANALYSIS OF THE CAPACITY In [12], a trivial upper bound is given as c m ≤ 1 m log( 4 m 2 ), which leads to c 2 ≤ 1.5 and c 3 ≤ 1.67. In this section, we analyze the upper bound of the capacity for m-SSA codes by a standard approach in constrained coding theory, i.e., we use a state transition digraph to characterize the constrained system. We start with the following definition. Definition 5. For any S ∈ S m , Let C n (S) ⊆ D n be the set of all sequences x such that every sequence in M (x) belongs to S. Let C n (S) ⊆ D n be the set of all sequences x such that each subsequence in M (x) \ S has multiplicity at most 2m − 1 in M (x). We use C n (S m ) and C n (S m ) to denote S∈Sm C n (S) and S∈Sm C n (S), respectively. Lemma 1. Let C be the optimal m-SSA code. Then it holds that C n (S m ) ⊆ C ⊆ C n (S m ). Proof: By definition, it is routine to check every sequence in C n (S) does not contain reverse complement subsequences of length m, for any S ∈ S m . Therefore C n (S m ) ⊆ C. For any x ∈ C, let M (x) be the set of subsequences in M (x) appearing at least 2m times. We claim that M (x) is a subset of at least one S ∈ S m . Suppose otherwise, then in M (x) there exists two reverse complement sequences y and z. Note that there are at most 2m − 1 sequences have intersecting coordinates with z (calculating with multiplicity and including z itself), and y appears at least 2m times as a subsequence of x. Thus, we can find some y which does not have overlap with z, which makes x not an m-SSA sequence. Therefore, every sequence in M (x)\S appears at most 2m−1 times, so x belongs to C n (S) ⊆ C n (S m ). Next we show that for any S ∈ S m , the capacity of C n (S) and C n (S) are in fact equal. Proof: (Outline) We calculate the number of sequences in C n (S) as follows. Such a sequence contains at most a constant number of subsequences not in S, and each such subsequence appears at most a constant number of times. Therefore, out of the n coordinates we can first select the positions of the sequences not belonging to S (the number of possible ways is still a constant number only dependent on m), and then the remaining blank positions are divided into several intervals of length n 1 , n 2 , n 3 , . . . Each of the interval should be filled with sequences in C ni (S) for some length n i , and the possibility for such an interval is bounded by O(λ ni ) when n i is sufficiently large. Thus, | C n (S)| ≤ O m (1) × Π i≥1 λ n1 = O m (1) × λ n , where O m (1) represents a constant only dependent on m. Note that when m is a fixed integer and n approaches infinity, τ (m) is a constant number at most 4 m /2. Thus, the capacity of C n (S m ) is determined by the largest value among the capacities of {C n (S) : S ∈ S m }. The same holds for C n (S m ). Thus Lemma 1 and Lemma 2 together lead to the following result. To sum up, we prove that the capacity of m-SSR codes can be derived by searching for the largest capacity {C n (S) : S ∈ S m }, where C n (S) is the set of length n sequences such that every subsequence of length m belongs to S. The capacity of C n (S) can be computed by a standard constrained coding approach, to be introduced in the next section. V. CONSTRUCTIONS OF m-SSA CODES We call S a generating set for C n (S) and in this section we will discuss how to choose a good generating set S which leads to m-SSA codes with good rate. We start with the case m = 2. As has been explained in the previous section, S contains exactly one out of the following six pairs: Then the capacity of C n (S) equals log(ρ) = 1.1679 bits/nt, where ρ = 2.247 is the spectral radius of the matrix above. Since there are only 64 different choices of S for m = 2 (in fact, with some additional work one can show that there are only 8 equivalent classes), we can use a brute-force algorithm to calculate the spectral radius for every possible S. It turns out 1.1679 bits/nt is indeed optimal. For a general m, after picking the generating set S, the directed graph is formed by drawing an arc from (x 1 , . . . , x m ) to (y 1 , . . . , y m ) whenever x i+1 = y i for all 1 ≤ i ≤ m − 1, and the spectral method can be applied. Thus, what remains is to find the best S for each m, and it seems a nontrivial task. When m is odd, essentially our construction in Section III is to pick S as the set of all length-m sequences which are T C-dominant (the sum of appearances of T and C is over m/2). For m = 3, we use a brute-force search for all possible generating sets, and verify that 1.5514bits/nt is indeed optimal. For m ∈ {7, 9, 11}, our construction has rate 1.7698, 1.8131, 1.8423 bits/nt accordingly. We pose the following conjecture. When m is even, the choice of S gets complicated since we have many length-m sequences with exactly one half of the symbols to be T and C. We discuss the cases m = 4 and m = 6 in the following two subsections, respectively. A. Discussions on m = 4 In this subsection we discuss how to choose the generating set S for m = 4. Among the 4 4 = 256 sequences of length 4, 16 of them (e.g. ACGT ) are self reverse complement, and the rest are divided into 120 reverse complement pairs. A brute-force search requires checking over 2 120 possible choices of S and is thus impractical. Moreover, taking equivalence into consideration does not have much help. To choose the generating set S, we first pick those T Cdominant sequences (the sum of appearances of T and C is 4 or 3). For example, in a reverse complement pair {T CCA, T GGA} we choose T CCA and ignore T GGA. We still have to decide our choices for the sequences containing exactly a sum of two appearances of T and C. The choice is made by further analyzing the positions of T and C. Now denote T and C as 1, and denote A and G as 0. Our strategy goes on with the following steps: Here a reasonable explanation for the choice of these induced arcs within them. Following the previous strategy, the generating set S results in a rate log(3.0190) = 1.5940 bits/nt and we conjecture it to be optimal (it's impractical to perform a brute-force over all the 2 120 choices and 1.5940 is indeed the best result we have found by computer search so far). B. Discussions on m = 6 In this subsection we discuss how to choose the generating set S for m = 6. Following the same idea as in the previous subsection, we first pick all the T C-6-dominating sequences. For the sequences containing exactly a sum of three appearances of T and C, our strategy goes on with the following steps: We pause here to mention that, at this step the generating set already results in a spectral radius 3.2443 and thus a code rate log(3.2443) = 1.6979 bits/nt, which already beats the result 1.5312 bits/nt derived by [12]. We can still proceed a little bit more: • Consider the sequences of the forms 010101 and 101010. First delete the self reverse complement ones such as T ACGT A (altogether there are 16 such sequences). We still have 56 reverse complement pairs left in this class and it is still impractical to run a brute-force search. It is natural to guess that we should pick these sequences such that they have the most number of induced arcs within them. • Consider the sequences of the forms 011001 and 100110. First delete the self reverse complement ones such as T GGCCA (altogether there are 16 such sequences). We still have 56 reverse complement pairs left in this class. We haven't come up with a good strategy for the sequences in these two classes. So far our best result by brute-force search has rate log(3.3097) = 1.7267 bits/nt. VI. CONCLUSION In this paper we analyze the constructions and the capacity of m-SSR codes, which are beneficial for the applications in DNA computing and DNA storage. The rate of our construction is summarized in Table I, and in particular we have derived optimal rates when m ∈ {2, 3} with the aid of computer search. We conjecture that T C-m-dominant sequences give rise to the codes with optimal rate when m is odd, but so far there is a lack of theoretical analysis. Furthermore, we still need to analyze the strategy for choosing the generating set when m is even. Fig. 1 . 1A stem-loop secondary structure arXiv:2304.11403v1 [cs.IT] 22 Apr 2023 II. PRELIMINARY Definition 2 . 2Given an integer m ≥ 2, let A(n; m) be the largest size of an m-SSA code of length n. Define the capacity of m-SSA codes as c m = lim n→∞ 1 n log(A(n; m)). Definition 4 . 4Given m ≥ 2, let S m = {S 1 , S 2 , . . . , S τ (m) } be a family of sets, where each set S i ⊆ D m , 1 ≤ i ≤ τ (m),is a maximal set of sequences such that in S i there are no two reverse complement sequences x and y (not necessarily distinct). For example, when m = 2, each S i cannot contain the self reverse complement sequences {T A, AT, CG, GC}, and contains exactly one out of each reverse complement pairs: {T T, AA}, {T C, GA}, {T G, CA}, {AC, GT }, {AG, CT }, and {CC, GG}. Thus S 2 is a family of τ (2) = 2 6 sets. For any x ∈ D n , let M (x) = {x[i; m] : 1 ≤ i ≤ n−m+1} be the multiset of all length-m subsequences of x. Consider the two codes defined as follows. Lemma 2 . 2For any S ∈ S m , if lim n→∞ 1 n log(|C n (S)|) = λ, then it holds that lim sup n→∞ 1 n log(| C n (S)|) = λ. Theorem 6 . 6The capacities of m-SSR codes, C n (S m ), and C n (S m ) are in fact equal, and this value could be computed by searching for the largest value among {C n (S) : S ∈ S m }. {T T, AA}, {T C, GA}, {T G, CA}, {AC, GT }, {AG, CT }, and {CC, GG}.Say we choose S = {T T, T C, T G, GT, CT, CC}. C n (S) is the set of length n-sequences such that any subsequence of length 2 belongs to S. The size, or the capacity, of C n (S), can be determined by a standard constrained coding theory approach as follows. Build a directed graph with 6 vertices labelled by S, and there is a directed arc from one vertex to the other (not necessarily distinct), if the second coordinate of the former equals the first coordinate of the latter. The adjacency matrix of this directed graph is as follows (rows and columns indexed by the order T T, T C, T G, GT, CT, CC): Conjecture 1 . 1When m is odd, the generating set S with all T C-dominant length-m sequences results in m-SSA codes with the largest rate. • Between a reverse complement pair, where one is of the form 1001 and the other one is of the form 0110, we give our priority to 0110. The reason is that such vertices will have more arcs with the already chosen T C-dominant vertices (which are of the form 1111, 1110, 1101, 1011, 0111). Observe that a vertex in the form 0110 is for sure to have an incoming arc from a vertex of the form 1011, and an outgoing arc to a vertex of the form 1101, and potential connections with 0011 and 1100, if they exist. However, for a vertex in the form 1001, they only have potential connections with 0011 and 1100, but no more arcs since vertices in the form of 0100 and 0010 do not exist. Thus, we choose the vertices in the form 0110 and neglect 1001. • Between a reverse complement pair, where one is of the form 0011 and the other one is of the form 1100, it does not matter what we choose. Observe that vertices of the form 1100 do not have outgoing arcs, because 1000 and 1001 are already banned in our choice. Similarly, the vertices of the form 0011 do not have incoming edges since 0001 and 1001 are banned. While allowing these vertices in the generating set S will certainly result in more m-SSA sequences, they could be neglected when computing the capacity and the spectral radius of the adjacency matrix. • Finally there are 12 pairs left. In 6 pairs both sequences are of the form 1010 (e.g. T ACA and T GT A), and in the other 6 pairs both sequences are of the form 0101 (e.g. GCAT and AT GC). Among these 12 pairs we perform a brute-force search and find the desired choice as {CACA, T ACA, CGCA, CAT A, T ACG, CACG, ACAC, GCAC, AT AC, ACGC, GCAT, ACAT }. • Between a reverse complement pair, where one is of the form 100011 and the other one if of the form 001110, we give our priority to 001110 since it has more connections with the T C-6-dominating sequences. Similarly, we choose sequences of the forms 010110, 011010, and 011100, whereas neglect sequences of the forms 100101, 101001, and 110001, accordingly. • Sequences of the forms 000111, 111000, 001011, 110100 are self reverse complement. They can all be neglected since they have no incoming arcs or no outgoing arcs, and thus do not contribute to the rate of the code. • Between a reverse complement pair, where one is of the form 001101 and the other one if of the form 010011, we choose 001101 because sequences of the form 010011 do not have incoming arcs. Similarly, we choose sequences of the form 101100 and neglect sequences of the form 110010. TABLE I RATE OF OUR CONSTRUCTION FOR SSA CODES (BITS/NT)m 2 3 4 5 7 9 11 Rate 1.1679 1.5515 1.5940 1.6980 1.7698 1.8131 1.8423 In[12] it is defined as "of length at least m", but it is straightforward to show that the two definitions are equivalent. ACKNOWLEDGEMENTSWe deeply thank Dr. Xianger Jiang and Mr. Haoling Zhang from BGI-research, for valuable discussions on DNA chemical synthesis and introducing the biochemical backgrounds of this problem. Molecular computation of solutions to combinatorial problems. L M Adleman, Science. 2665187L. M. Adleman, "Molecular computation of solutions to combinatorial problems," Science, vol. 266, no. 5187, pp. 1021-1024, 1994. On DNA Codes With Multiple Constraints. K G Benerjee, A Banerjee, IEEE Communications Letters. 252K. G. Benerjee and A. Banerjee, "On DNA Codes With Multiple Constraints," IEEE Communications Letters, vol. 25, no. 2, pp. 365- 368, 2021. On Secondary Structure Avoiding DNA codes with Reversible and Reversible-Complement Constraints. K G Benerjee, A Banerjee, National Conference on Communications. K. G. Benerjee and A. Banerjee, "On Secondary Structure Avoiding DNA codes with Reversible and Reversible-Complement Constraints," National Conference on Communications, pp. 1-6, 2023. Predicting DNA duplex stability from the base sequence. K Breslauer, R Frank, H Blocker, L Marky, Proceedings of the National Academy of Sciences. 8311K. Breslauer, R. Frank, H. Blocker, and L. Marky, "Predicting DNA duplex stability from the base sequence," Proceedings of the National Academy of Sciences, vol. 83, no. 11, pp. 3746-3750, 1986. Efficient and Explicit Balanced Primer Codes. Y M Chee, H M Kiah, H Wei, IEEE Transactions on Information Theory. 669Y. M. Chee, H. M. Kiah, and H. Wei, "Efficient and Explicit Balanced Primer Codes," IEEE Transactions on Information Theory, vol. 66, no. 9, pp. 5344-5357, 2020. Next-generation digital information storage in DNA. G M Church, Y Gao, S Kosuri, Science. 3376102G. M. Church, Y. Gao, and S. Kosuri, "Next-generation digital infor- mation storage in DNA," Science, vol. 337, no. 6102, pp. 1628-1628, 2012. Towards practical, high-capacity, lowmaintenance information storage in synthesized DNA. N Goldman, P Bertone, S Chen, C Dessimoz, E M Leproust, B Sipos, E Birney, Nature. 4947435N. Goldman, P. Bertone, S. Chen, C. Dessimoz, E. M. LeProust, B. Sipos, and E. Birney, "Towards practical, high-capacity, low- maintenance information storage in synthesized DNA," Nature, vol. 494, no. 7435, pp. 77-80, 2013. . X Jiang, H Zhang, personal communicationX. Jiang and H. Zhang, personal communication. DNA codes that avoid secondary structures. O Milenkovic, N Kashyap, Proceedings. International Symposium on Information Theory. International Symposium on Information TheoryO. Milenkovic and N. Kashyap, "DNA codes that avoid secondary struc- tures," Proceedings. International Symposium on Information Theory, pp. 288-292, 2005. On the design of codes for DNA computing. O Milenkovic, N Kashyap, Coding and CryptographyO. Milenkovic and N. Kashyap, "On the design of codes for DNA computing," Coding and Cryptography, pp. 100-119, 2006. Fast algorithms for predicting the secondary structure of single stranded RNA. R Nussinov, A B Jacobson, Proceedings of the National Academy of Sciences. 7711R. Nussinov and A. B. Jacobson, "Fast algorithms for predicting the secondary structure of single stranded RNA," Proceedings of the National Academy of Sciences, vol. 77, no. 11, pp. 6309-6313, 1980. On the design of codes for DNA computing: Secondary structure avoidance codes. T T Nguyen, K Cai, H M Kiah, D T Dao, K A S Immink, arXiv:2302.13714arXiv preprintT. T. Nguyen, K. Cai, H. M. Kiah, D. T. Dao, and K. A. S. Immink. "On the design of codes for DNA computing: Secondary structure avoidance codes," arXiv preprint arXiv:2302.13714, 2023. Mutually Uncorrelated Primers for DNA-Based Data Storage. S M H T Yazdi, H M Kiah, R Gabrys, O Milenkovic, IEEE Transactions on Information Theory. 649S. M. H. T. Yazdi, H. M. Kiah, R. Gabrys, and O. Milenkovic, "Mutually Uncorrelated Primers for DNA-Based Data Storage," IEEE Transactions on Information Theory, vol. 64, no. 9, pp. 6283-6296, 2018.
[]
[ "Confronting Instanton Perturbation Theory with QCD Lattice Results", "Confronting Instanton Perturbation Theory with QCD Lattice Results" ]
[ "A Ringwald \nDeutsches Elektronen-Synchrotron DESY\nHamburgGermany\n", "F Schrempp \nDeutsches Elektronen-Synchrotron DESY\nHamburgGermany\n" ]
[ "Deutsches Elektronen-Synchrotron DESY\nHamburgGermany", "Deutsches Elektronen-Synchrotron DESY\nHamburgGermany" ]
[]
We exploit a recent lattice investigation (UKQCD) on the topological structure of the (quenched) QCD vacuum, in order to gain information on crucial building blocks of instanton perturbation theory. A central motivation is to further constrain our previous predictions of instanton-induced hard scattering processes. First, we address the generic problem of extracting quantitative information from cooled lattice data. We find a new scaling variable, interpreted as a "cooling radius", which allows to combine lattice data for a whole range of lattice spacings and cooling sweeps. This variable strongly helps to extract information on the uncooled distributions of interest. After performing the continuum extrapolation of the instanton size and instanton-anti-instanton distance distributions, we find striking agreement with the theoretical predictions from instanton-perturbation theory, for instanton sizes < ∼ 0.5 fm and distances > ∼ 0.5 fm, respectively. These results imply first direct support for the validity of the known valley interaction between instantons and anti-instantons.
10.1016/s0370-2693(99)00682-6
[ "https://export.arxiv.org/pdf/hep-lat/9903039v1.pdf" ]
1,113,094
hep-lat/9903039
4a4466bde2e181499c18f720ce0c9ee1923678a5
Confronting Instanton Perturbation Theory with QCD Lattice Results 31 Mar 1999 A Ringwald Deutsches Elektronen-Synchrotron DESY HamburgGermany F Schrempp Deutsches Elektronen-Synchrotron DESY HamburgGermany Confronting Instanton Perturbation Theory with QCD Lattice Results 31 Mar 1999arXiv:hep-lat/9903039v1 We exploit a recent lattice investigation (UKQCD) on the topological structure of the (quenched) QCD vacuum, in order to gain information on crucial building blocks of instanton perturbation theory. A central motivation is to further constrain our previous predictions of instanton-induced hard scattering processes. First, we address the generic problem of extracting quantitative information from cooled lattice data. We find a new scaling variable, interpreted as a "cooling radius", which allows to combine lattice data for a whole range of lattice spacings and cooling sweeps. This variable strongly helps to extract information on the uncooled distributions of interest. After performing the continuum extrapolation of the instanton size and instanton-anti-instanton distance distributions, we find striking agreement with the theoretical predictions from instanton-perturbation theory, for instanton sizes < ∼ 0.5 fm and distances > ∼ 0.5 fm, respectively. These results imply first direct support for the validity of the known valley interaction between instantons and anti-instantons. 1. Non-abelian gauge theories like QCD are known to exhibit a rich vacuum structure. The latter includes topologically non-trivial fluctuations of the gauge fields, carrying an integer topological charge Q. The simplest building blocks of topological structure are instantons (Q = +1) and anti-instantons (Q = −1) which are well-known explicit solutions of the euclidean field equations in four dimensions [1]. Instantons are widely believed to play an important rôle in various long-distance aspects [2] of QCD: First of all, they may provide a solution of the famous U A (1) problem [3] (m η ′ ≫ m η ), with the corresponding pseudoscalar mass splitting related to the topological susceptibility in the pure gauge theory by the well-known Witten-Veneziano formula [4]. Moreover, a number of authors have attributed a strong connection of instantons with chiral symmetry breaking [5,2] as well as the hadron and glueball spectrum. However, there are also very important short-distance implications [6][7][8][9][10] of QCD instantons: They are known to induce certain processes which violate chirality in accord with the general axialanomaly relation [3] and which are forbidden in conventional perturbation theory. Of particular interest in this context is the deep-inelastic scattering (DIS) regime. Here, hard instanton-induced processes may both be calculated [8][9][10] within instanton-perturbation theory and possibly be detected experimentally [7,11,12,13]. Since an experimental discovery of instanton-induced events would clearly be of basic significance, our ongoing theoretical and phenomenological study of the discovery potential of instanton-induced DIS events is well motivated. On the other hand, there has been much recent activity in the lattice community to "measure" topological fluctuations in lattice simulations [14] of QCD. Being independent of perturbation theory, such simulations provide "snapshots" of the QCD vacuum including all possible nonperturbative features like instantons. They may be exploited to provide crucial support for important prerequisites of calculations in DIS, like the validity of instanton-perturbation theory and the dilute instanton-gas approximation for small instantons of size ρ ≤ O(0.3) fm. Along these lines, we were able [9] to translate the lattice constraints into a "fiducial" kinematical region for our predictions of the instanton-induced DIS cross-section based on instanton-perturbation theory. The purpose of the present letter is to present a more thorough confrontation of predictions from instanton-perturbation theory with recent lattice simulations. Specifically, we concentrate on recent high-statistics results from the UKQCD collaboration [15,16], try to carefully discuss the effects of cooling and perform an extrapolation to the continuum before drawing our conclusions. 2. Let us start by considering various important quantities relevant for instanton (I)-induced scattering processes, that may both be calculated in I-perturbation theory and measured in lattice simulations. In I-perturbation theory one expands the path integral for the generating functional of the Green's functions about the known, classical instanton solution A µ = A (I) µ + . . . instead of the trivial vacuum A (0) µ = 0 (pQCD) and obtains a corresponding set of modified Feynman rules. Like in conventional pQCD, the strong gauge coupling α has to be small. The I-induced contribution to hard scattering processes is typically encoded in the total cross-section [9,10] associated with an I-induced partonic subprocess q ′ p ⇒ X, σ (I) (q ′ p ⇒ X) = d 4 R ∞ 0 dρ ∞ 0 dρ D(ρ) D(ρ) dU e − 4π α Ω U, R 2 ρρ , ρ ρ {. . .} (1) ×F ( −q ′ 2 ρ) F ( −q ′ 2 ρ)F ( −p 2 ρ) F ( −p 2 ρ) e i (p+q ′ )·R . σ (I) involves integrations over all I [I]-"collective coordinates", including the I [I]-sizes ρ [ρ], the II-distance 1 4-vector R µ and the II relative colour orientation U. With each initial-state parton, there is an associated "form factor" [8,9], F (x) = x K 1 (x)          ∼ π/(2 x) exp(−x) for x → +∞; = 1 for x = 0; oscillating for x imaginary. (2) An important quantity entering Eq. (1) as generic weight is the (reduced) I-size distribution in the dilute gas approximation, D(ρ). It is known in the framework of I-perturbation theory for small α(µ r ) ln(ρ µ r ), where µ r is the renormalization scale. After its pioneering evaluation at 1-loop [3] for N c = 2 and its generalization [17] to arbitrary N c , it is meanwhile available [18] in 2-loop renormalization-group (RG) invariant form, i.e. D −1 dD/d ln(µ r ) = O(α 2 ), D(ρ) = d MS ρ 5 2π α MS (µ r ) 2 Nc exp − 2π α MS (µ r ) (ρ µ r ) β 0 +(β 1 −4 Ncβ 0 ) α MS (µr ) 4π ,(3) d MS = 2 e 5/6 π 2 (N c − 1)!(N c − 2)! e −1.511374 Nc+0.291746 n f (Ref. [19]); (4) β 0 = 11 3 N c − 2 3 n f ; β 1 = 34 3 N 2 c − 13 3 N c − 1 N c n f ; γ 0 = 3 N 2 c − 1 N c .(5) For the case n f = 0, relevant for our comparison with lattice results below, the reduced size distribution D(ρ), Eq. (3), equals the proper size distribution, i.e. the number of instantons per unit volume per unit size, dn I /d 4 x dρ, (in the dilute gas approximation). On the other hand, for n f > 0, the proper size distribution reads dn I d 4 x dρ ≃ D m (ρ) = D(ρ) n f i=1 (ρ m i (µ r )) (ρ µ r ) n f γ 0 α MS (µr ) 4π ,(6) for small ρ m i (µ r ), where m i (µ r ) are the running quark masses. However, even for n f > 0, only the reduced size distribution D(ρ) appears in cross-sections like Eq. (1), since for small m i , the mass-dependent factor in Eq. (6) is cancelled by corresponding terms from the external quarks. The powerlaw behaviour of the (reduced) I-size distribution, D(ρ) ∼ ρ β 0 −5+O(α) ,(7) generically causes the dominant contributions to the I-size integrals to originate from the infrared (IR) regime (large ρ) and thus often spoils the applicability of I-perturbation theory. In deep-inelastic scattering, however, one parton q ′ say, carries a spacelike virtuality, such that the contribution of large instantons to the integrals (1) is exponentially suppressed (c. f. Eq. (2)), F ( −q ′ 2 ρ) F ( −q ′ 2 ρ) ∝ e −Q ′ (ρ+ρ) ; for Q ′ = −q ′ 2 > 0,(8) and the predictivity of I-perturbation theory is retained for sufficiently large Q ′ . A second important ingredient into Eq. (1) is the function Ω(U, R 2 /(ρρ), ρ/ρ), appearing in the exponent with a large numerical coefficient 4π/α. It incorporates the effects of final-state gluons. Within strict I-perturbation theory, it is given in form of a perturbative expansion [9,20] for large R 2 , while in the so-called II-valley approximation 1 [21] Ω is associated with an analytically known closed expression 2 [23] for the interaction between I andĪ, Ω = α/(4π)S[A II µ ] − 1, Ω U(v, z, . . .), R 2 ρρ , ρ ρ = 1 2 − 2 z 1 + v 2 Σ 1 + 1 2 − 2 z + 4 z 2 1 + v 2 2 + 2 v 2 Σ 2 (9) with Σ 1 = −2 1−λ 4 +4 λ 2 ln(λ) (λ 2 −1) 3 ; Σ 2 = 1−λ 2 +(1+λ 2 ) ln(λ) (λ 2 −1) 3 ; λ = 1 2 ξ + 1 2 √ ξ 2 − 4; ξ = R 2 ρρ + ρ ρ + ρ ρ .(10) The II-valley interaction (9) only depends on two of the four angle variables necessary in general to parametrize the relative color orientation U (see e.g. Ref. [24]). In terms of these variabels (v = cos α, z = cos(δ) 2 ) and an additional azimuthal angle variable φ, the integration over the SU(3) group measure then takes the form [10] dU = 4 π 1 −1 | v | 1 − v 4 dv π 0 dφ 1 2 1−2 v cos(φ)+v 2 1+v 2 0 dz 1 − 2 v cos(φ) + v 2 = 1.(11) With these prerequisites in mind, let us next consider the group-averaged distribution of II pairs, dn II /d 4 x d 4 R dρ dρ. This quantity is of basic interest for I-induced scattering processes (1), crucially depends on the II-interaction Ω and may be measured on the lattice. For the pure gauge theory 3 (n f = 0) and in the dilute-gas approximation, it takes the form [9,10] dn II d 4 x d 4 R dρ dρ ≃ D II (R, ρ, ρ) = D(ρ) D(ρ) dU exp − 4π α MS (s II / √ ρρ) Ω U, R 2 ρρ , ρ ρ ,(12) with the scale factor s II = O(1) parametrizing the residual scheme dependence. In the deep-inelastic regime, relevant for instanton searches at HERA, the collective coordinate integrals involved in the hard cross-section (1) are known to be dominated by a unique saddle point [9]. Therefore, the Bjorken variables (Q ′ , x ′ ) of the I-induced subprocess in e ± P scattering are effectively related in a one-to-one correspondence to the collective coordinates as (c. f. also Eqs. (1), (8)), ρ ∼ ρ ∼ 1/Q ′ and R 2 ∼ 1/(p + q ′ ) 2 ⇒ ρρ/R 2 ∼ (p + q ′ ) 2 /Q ′2 = 1/x ′ − 1.(13) Restrictions on the collective coordinates obtained from confronting predictions of I-perturbation theory with lattice results may thus be directly translated into a fiducial kinematical region for the predicted rate of I-induced processes at HERA [9]. This fact illustrates the great importance of lattice simulations for I-induced scattering processes and largely motivates the present analysis. 3. Before turning to our concrete analysis, we first have to address various topical problems associated with lattice data on instantons. The recent quenched QCD results (n f = 0) from the UKQCD collaboration [15,16], which we shall use here, involve a lattice spacing a = 0.05 − 0.1 fm and (roughly constant) volumes V = l 3 space · l time of 32 3 64 a 4 at β(a) ≡ 6/g 2 0 (a) = 6.4, 24 3 48 a 4 at β = 6.2 and 16 3 48 a 4 at β = 6.0. In principle, such a lattice allows to study the properties of an ensemble of topological gauge field fluctuations (I's and I's) with sizes a < ρ < V 1/4 . However, in the raw data, ultraviolet fluctuations of wavelength O(a) dominate and completely mask the topological effects which are expected to be of much larger size. In order to make the latter visible, a certain "cooling" procedure has to be applied first. Cooling is a technique for removing the high-frequency nontopological excitations of the gauge-field, while affecting the topological fluctuations of much longer wavelength ρ ≫ a comparatively little. After cooling, I's and I's can clearly be identified as bumps in the topological charge density and the Lagrange density. By applying sophisticated pattern-recognition algorithms, various distributions characterizing the ensemble of I's and I's may be extracted. However, during cooling instantons are also removed; either if they are very small (lattice artifacts) or through IĪ annihilation. In Ref. [15], so called under-relaxed (or slow) cooling is used to reduce the latter problem. Generically, the cooling procedure introduces substantial uncertainties in the extraction of physical information on the topological structure of the QCD vacuum. No objective criterium is yet available on the quantitative amount of "vacuum damage" introduced by it (see Refs. [14,15]). Since one is interested in the physics of the uncooled vacuum, it appears suggestive (at first) to try and extrapolate [14] the measured distributions to the limit of a vanishing number of cooling sweeps, n cool → 0. However, to do so would be to ignore the fact that the pattern-recognition procedures become increasingly unreliable near that limit, since the extracted instanton density appears to strongly increase for decreasing n cool . Below a certain minimal number of cools, (n cool ) min , even the total topological charge Q = n I − n I starts to deviate from the plateau value it has for a whole range of larger n cool . In view of these objections, we follow the alternative strategy of Ref. [15] to extract information about the physical vacuum. The method incorporates certain scaling studies that are important for the continuum limit a → 0 and consists in the following steps: First, at β = 6.0, say, one picks the smallest number of cooling sweeps for which the pattern-recognition starts to become reliable and the cooling history of the total topological charge enters a plateau. For straight cooling, one finds (n cool ) min ≃ 10 corresponding to ≃ 23 under-relaxed cools [16]. The next step is to search for further, so-called equivalent pairs (β, n cool ), for which shape and normalization of specific distributions (expressed in physical units 4 ) essentially remain the same (see also Ref. [25]). The result of Ref. [15] were the pairs (β, n cool ) = (6.0, 23), (6.2, 46), (6.4, 80). With these equivalent, scaling data sets, the continuum limit is then performed with hopefully little "vacuum damage" due to cooling. Yet, the crucial question remains at this point, whether the choice (β, n cool ) = (6.0, 23) as a "starting pair" for the equivalent data is somehow distinguished, i. e. physically meaningful. In this context, let us present an intriguing line of arguments: "Equivalent" data should first of all be characterized by a similar amount of cooling. For n cool cooling sweeps, an effective "cooling radius" r cool in physical units 4 may be defined, via a (qualitative) random walk argument, as r cool 2 r 0 = const ( √ n cool ) 1+δ a 2 r 0 ; δ small.(14) The three equivalent pairs (β, n cool ) above, indeed, turn out to have a very similar value of r cool (see Table 1). (β, n cool ) (6.0, 23) (6.2, 46) (6.4, 80) √ n cool Amazingly, the usefulness of the variable (14) seems to be much greater. In Fig. 1, we illustrate that the average instanton size ρ in physical units from Ref. [15], for all available values of (β, n cool ), fits onto a single smooth curve as function of r cool in Eq. (14) with δ small. In contrast, one obtains three widely spaced curves, if ρ /2r 0 is displayed as a function of n cool alone for fixed β = 6.0, 6.2 and 6.4, respectively. The same scaling with r cool holds practically for all quantities extracted in Ref. [15], characterizing the instanton ensemble on the lattice. Perhaps the most striking feature of Fig. 1 is the clear indication of a plateau corresponding to ρ ≃ 0.5 fm. Let us try to provide some interpretation for it. Suppose, the localized fluctuations on the lattice are a mixture of two distinct ensembles, separated by a certain gap in their respective average sizes. For example, one expects fluctuations that are lattice artifacts typically located and for all available values of n cool as a function of the effective cooling radius (14). There is a clear indication of a plateau corresponding to ρ ≃ 0.5 fm, which is interpreted in the text. The quantity r 0 denotes the Sommer scale 4 with 2 r 0 ≃ 1 fm. around smaller sizes O(few · a), while the physical instanton sizes are peaking around ρ ≃ r 0 ≃ 0.5 fm. By gradually increasing the cooling radius r cool , we first filter out the smaller artifacts, leading clearly to a corresponding increase in the effective ρ . When the radius of our filter further increases, one then expects a plateau in ρ to indicate that the small artifacts have been mostly wiped out, while the much larger physical instantons are not yet strongly affected. Eventually, the instantons will also start to be erased, leading again to an increase in ρ . In conclusion, we think that the plateau region is the correct regime for extracting physical information about instantons. The equivalent data sets of Ref. [15] are precisely located around the plateau. Our scaling variable r cool allows to combine lattice data for a whole range of lattice spacings a and cools n cool , which strongly helps to increase the separation power between the a-dependent artifacts and the instantons of a certain physical size. In future analyses, it may be worthwhile to determine the unknown constant in the definition (14) of r cool , e. g. by means of calibration via cooling an isolated instanton until it disappears. 4. We are now ready to discuss the continuum limit of the UKQCD data for the size distribution of I's and I's, dn I+I /d 4 x dρ, and to compare it with the dilute-gas prediction D I+I (ρ) = 2 D(ρ) of I-perturbation theory, Eq. (3), for n f = 0. We note, first of all, that the (small) residual a dependence of the equivalent data for the size distribution in physical units may be parametrized as (2 r 0 ) 5 dn I+I d 4 x dρ = function ρ ρ(a) . (15) The continuum limit of Eq. (15) can thus be performed quite reliably, by simply rescaling the arguments ρ → ρ(0) / ρ(a) · ρ. Here, ρ(0) /2r 0 = 0.518(5) denotes the continuum limit of the weakly varying average ρ values, ρ(a) . As in Ref. [15], ρ(a) /2r 0 was extrapolated linearly in (a/r 0 ) 2 . The resulting continuum size distribution obtained by the above rescaling from the three equivalent data sets is displayed in Fig. 2. It scales nicely. (3) is apparent. The 3-loop form of α MS with Λ MS n f =0 from ALPHA [28] was used. (a) Log-log plot to exhibit the expected power law ∼ ρ 6 and the agreement in magnitude for small ρ over a wide range of µ r . The dashed error band results from varying Λ MS n f =0 and µ r within its error and given range, respectively; (b) For µ r = 1.2/ρ, the agreement extends up to the peak. We are now ready to perform a quantitative comparison with I-perturbation theory. The corresponding prediction (3) is a power law for small ρ, i. e. approximately D ∼ ρ 6 for n f = 0. Due to its 2-loop RG-invariance the normalization of D I+I (ρ) is practically independent of the renormalization scale µ r over a wide range. It is strongly dependent on r 0 Λ MS , for which we take the most recent, accurate lattice result by the ALPHA collaboration [28], 2 r 0 Λ MS n f =0 = (238 ± 19) MeV fm. In Fig. 2 (a) we display both this parameter-free prediction (3) of I-perturbation theory and the continuum limit of the UKQCD data in a log-log plot, to clearly exhibit the expected power law in ρ. The agreement in shape 5 and normalization for ρ/2r 0 < ∼ 0.3 is striking, indeed, notably in view of the often criticized cooling procedure and the strong sensitivity to Λ MS . One may argue that a choice ρµ r = O(1) is theoretically favoured, since it tends to suppress ln(ρµ r ) corrections. Indeed, as apparent from Fig. 2 (b), the agreement extends in this case up to the peak around ρ/2r 0 ≃ 0.5. At this point, an interesting way to proceed is to define a non-perturbative "instanton-scheme" for α by the requirement that the theoretical expression (3) for the size distribution D I+I (ρ) precisely reproduces the corresponding lattice data, ρ 5 dn I+I d 4 x dρ ≡ 2 d MS   2π α I ( 1 ρ )   2 Nc exp   − 2π α I ( 1 ρ )   s β 0 + α I ( 1 ρ ) 4π (β 1 −4 Ncβ 0 ) I .(16) By comparing in the perturbative regime with the MS form (3) for µ r = s/ρ, we arrive at a standard scheme conversion formula, α MS (s/ρ) = α I (1/ρ) − β 0 2π ln(s/s I ) α I (1/ρ) 2 + O(α I (1/ρ) 3 ),(17) with s I = Λ MS Λ I and α MS (s I /ρ) ≃ α I (1/ρ).(18) Next, we note that as a function of α I , D I+I (ρ) has a peak structure like the lattice data, with maximum at α I |max = π N c 2 1 + 1 + 1 Nc ln(s I )(2β 0 − β 1 2Nc ) .(19) The scale factor s I of the I-scheme is non-perturbatively and uniquely determined by the peak value of the lattice data for the size distribution. Using the very good Gaussian fit to all data in Fig. 2 (b), (2 r 0 ) 5 dn I+I d 4 x dρ |fit = (aρ) 6 exp (−(cρ) 2 ) (1 + (bρ) p ) ;          a we find s I = 1.18 and α I |max = 0.98. In the perturbative regime, the I-scheme is thus quite close to the MS-scheme (c. f. Eq. (18)). We are now able to extract for each data point of the I + I size distribution a corresponding value of α I , as illustrated in Fig. 3. Actually, besides the asymptotically free solution [α I ] to Eq. (16), there is a second one [α (2) I ] (dotted line in Fig. 3), with a monopole-like relation to α I , α I (µ) ∝ (µ/Λ) β0/2Nc in the UV-limit. This observation may be of interest in the context of Ref. [29]. I · α I = O(1),(2) It is remarkable that in the non-perturbative long-distance regime, ρ/2r 0 > ∼ 0.5, α I (1/ρ) from the I + I size distribution tends to approach α qq (1/ρ) defined via the static qq potential, V qq , 4 3 α qq (1/r) r 2 ≡ dV qq (r) dr ≃ σ + π 12 1 r 2 r→∞ ⇒ σ,(23) with string tension σ. This is illustrated in Fig. 3 (b). In summary, α I as non-perturbatively defined through Eq. (16), is close to the MS-scheme in the perturbative short-distance region and to the qq-scheme in the non-perturbative long-distance regime. Figs. 2 (b) and 3 (b) illustrate that the transition between perturbative and confinementsensitive physics seems to be rather sharp and located where the instanton density peaks, around ρ ≃ 0.5 fm. The above results also suggest the following working hypothesis [30], which might considerably enlarge the region of applicability of predictions like (1) and (12) for observables associated with sufficiently dilute instantons. Whenever one encounters the I-size distribution D(ρ) within the perturbative calculus, one may try to replace the indefinitely growing perturbative expression D(ρ) ∼ ρ β 0 −5 for the I-size distribution by the functional form actually extracted from the lattice (see e.g. Eq. (20)). Obviously, such an attempt eliminitates the problematic IR divergencies of I-size integrals. Due to the strong peaking of the measured size distribution (20), only I's of size ρ ≃ 0.5 fm effectively contribute to the observables of interest 6 and the dilute gas approximation may well continue to hold qualitatively up to the peak around 0.5 fm. 5. As an immediate first application of this strategy, let us turn to the group-averaged distribution of II-pairs, D II (R, ρ, ρ), in the dilute gas approximation, Eq. (12). While the conservative prediction of instanton perturbation theory is restricted to ρ, ρ < ∼ 0.3 − 0.5 fm, UKQCD lattice results [15,16] are available up to now only for the II-distance distribution, integrated over all values of ρ, ρ, dn II d 4 x d 4 R = dρ dρ dn II d 4 x d 4 R dρ dρ .(24) This is the point to exploit the above working hypothesis: We now replace in the theoretical expression for the II-distance distribution (24) In this region, we make use of the non-perturbatively extracted coupling in the I-scheme (Fig. 3), with α MS s II √ ρρ ⇒ α I s II s I 1 √ ρρ ; s I = 1.18,(25) as a smooth continuation of the perturbative MS coupling (c. f. Eq. (18)). In the quantitative evaluation of Eqs. (24) and (12), we perform two integrals of the group integration and the two integrals over the I-and I-sizes numerically. Before being able to compare this theoretical prediction of the II-distance distribution with lattice results, we have to perform the continuum limit of the respective UKQCD data [16]. This we do in complete analogy to the method described in Sect. 4., Eq. (15), for the I-size distribution. Unfortunately, the equivalent data for β = 6.4 are missing in this case [16]. The equivalent data for the remaining two β values, (β, n cool ) = (6.0, 23) and (6.2, 46), again are well compatible with a scaling law, like Eq. (15), dn II d 4 x d 4 R dn II d 4 x d 4 R | R large = function R r(a) .(26) By extrapolating the scaling factor λ(a) ≡ r(a)/r | β=6.2 linearly in (a/r 0 ) 2 , we find from requiring best overall scaling (26) of the two equivalent data sets, λ II (a) = 1.32−69.75(a/2r 0 ) 2 . The resulting "continuum data" obtained from the simple rescaling of the arguments, R → r(0)/r(a) R = λ(0)/λ(a) R, are displayed in Fig. 4 and exhibit good scaling. The solid line in Fig. 4 denotes the theoretical prediction discussed above with the scale factor s II = 0.82 being O(1) as expected. Figure 4: Continuum limit of "equivalent" UKQCD data [16] for the II-distance distribution, dn II /d 4 x d 4 R, normalized to its value at R/2r 0 = 1.5, at (β, n cools ) = (6.0, 23) [✷], (6.2, 46) [•]. As discussed in the text, the solid line denotes the theoretical prediction from instanton-perturbation theory based on the valley form of the II-interaction. We note a very good agreement with the lattice data down to II-distances R/ ρ ≃ 1, corresponding to ξ > ∼ 3 (c. f. Eq. (10)). These results imply first direct support for the validity of the valley interaction (9) between instantons and anti-instantons. Figure 1 : 1The average instanton size from Ref. [15] in physical units for β = 6.0 [✷], 6.2 [•], 6.4 [△] Figure 2 : 2Continuum limit of "equivalent" UKQCD data[15] for the (I + I)-size distribution at (β, n cools ) = (6.0, 23) [✷], (6.2, 46) [•],(6.4, 80) [△]. The striking agreement with 2 D(ρ) of Iperturbation theory from Figure 3 : 3(a) Running coupling α I in the "instanton" scheme, extracted via Eq. (16) from the I + I size distribution. The same UKQCD data as inFig. 2have been used. The dotted line denotes a possible second solution vanishing in the IR limit. The excellent agreement with the corresponding 3-loop perturbative α MS (s I /ρ) ≃ α I (1/ρ) down to 1/ρ ≃ 0.4 GeV is apparent (solid line). (b) At large distances, α I from (a) appears to be quite close to α qq defined via the static qq potential, with (4/3r 2 ) α qq (1/r) approaching the string tension σ for large r and[27] 2 r 0 √ σ = (472.4 ± 4.3) MeV fm.vanishing in the IR-limit and increasing like α along with Eq. (12) the perturbative form for D(ρ) by the Gaussian fit (20) to the UKQCD lattice data. We are left with only one free parameter, the scale factor s II = O(1) in the running coupling α MS (s II / √ ρρ). On account of the peaking of the size distributions in Eq. (12), the running coupling at a momentum scale around O(0.3) GeV enters significantly. Table 1 : 1The "cooling radius" for the equivalent data sets is similar. Both an instanton and an anti-instanton enter here, since cross-sections result from taking the modulus squared of an amplitude in the single I-background. Alternatively, one may view the cross-section (1) as a discontinuity of the q ′ p forward elastic scattering amplitude in the II-background[9]. For the most attractive relative II-orientation the form of the II-valley action was first given in Ref.[22].3 For n f > 0, there is an additional fermionic interaction factor | d 4 x κ † 0 I (x) [i D (II) ] κ 0 I (x − R) | 2 n f under the group integral in Eq.(12), with κ 0 being the fermionic zero modes. For consistency and minimization of uncertainties, one should use only a single dimensionful quantity to relate lattice units and physical units. Throughout our analysis, all dimensions are therefore expressed by the so-called Sommer scale[26,27] r 0 , with 2 r 0 ≃ 1 fm, which we prefer over the string tension[15]. The agreement in shape up to ρ ≃ 0.3 fm has been noted by us already previously[9]. Here, we have extended the analysis to the continuum limit and, in particular, verified the agreement in normalization. This philosophy is partly also inherent in the I-liquid model[2]. AcknowledgementsWe thank Hans Joos for constant encouragement and many inspiring discussions on instantons. To Martin Lüscher we owe constructive criticism and helpful discussions on the results of the ALPHA collaboration. Moreover, we also acknowledge discussions with Fritz Gutbrod, John Negele and Gerrit Schierholz, notably on cooling. Last not least, we wish to thank Mike Teper for many helpful communications and suggestions, as well as for providing us with partly unpublished data from his work[15]. . A Belavin, A Polyakov, A Schwarz, Yu Tyupkin, Phys. Lett. B. 5985A. Belavin, A. Polyakov, A. Schwarz and Yu. Tyupkin, Phys. Lett. B 59 (1975) 85. . T Schäfer, E V Shuryak, Rev. Mod. Phys. 70323T. Schäfer and E.V. Shuryak, Rev. Mod. Phys. 70 (1998) 323. . G Hooft, Phys. Rev. Lett. 378G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; . Phys. Rev. D. 143432Phys. Rev. D 14 (1976) 3432; . Phys. Rev. D. 182199Erratum)Phys. Rev. D 18 (1978) 2199 (Erratum). . E Witten, Nucl. Phys. B. 156269E. 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[ "Artificial intelligence-driven digital twin of a modern house demonstrated in virtual reality", "Artificial intelligence-driven digital twin of a modern house demonstrated in virtual reality" ]
[ "Elias Mohammed Elfarri \nDepartment of Engineering Cybernetics\nNorwegian University of Science and Technology\n\n", "Adil Rasheed \nDepartment of Engineering Cybernetics\nNorwegian University of Science and Technology\n\n", "Omer San \nSchool of Mechanical and Aerospace Engineering\nOklahoma State University\n\n" ]
[ "Department of Engineering Cybernetics\nNorwegian University of Science and Technology\n", "Department of Engineering Cybernetics\nNorwegian University of Science and Technology\n", "School of Mechanical and Aerospace Engineering\nOklahoma State University\n" ]
[]
A digital twin is a powerful tool that can help monitor and optimize physical assets in real-time. Simply put, it is a virtual representation of a physical asset, enabled through data and simulators, that can be used for a variety of purposes such as prediction, monitoring, and decision-making. However, the concept of digital twin can be vague and difficult to understand, which is why a new concept called "capability level" has been introduced. This concept categorizes digital twins based on their capability and defines a scale from zero to five, with each level indicating an increasing level of functionality. These levels are standalone, descriptive, diagnostic, predictive, prescriptive, and autonomous. By understanding the capability level of a digital twin, we can better understand its potential and limitations. To demonstrate the concepts, we use a modern house as an example. The house is equipped with a range of sensors that collect data about its internal state, which can then be used to create digital twins of different capability levels. These digital twins can be visualized in virtual reality, allowing users to interact with and manipulate the virtual environment. The current work not only presents a blueprint for developing digital twins but also suggests future research directions to enhance this technology. Digital twins have the potential to transform the way we monitor and optimize physical assets, and by understanding their capabilities, we can unlock their full potential.
10.1109/access.2023.3265191
[ "https://export.arxiv.org/pdf/2212.07102v2.pdf" ]
254,636,603
2212.07102
372ae35a76695bf5c9ea255a633db4d634338a57
Artificial intelligence-driven digital twin of a modern house demonstrated in virtual reality Elias Mohammed Elfarri Department of Engineering Cybernetics Norwegian University of Science and Technology Adil Rasheed Department of Engineering Cybernetics Norwegian University of Science and Technology Omer San School of Mechanical and Aerospace Engineering Oklahoma State University Artificial intelligence-driven digital twin of a modern house demonstrated in virtual reality Digital twinsVirtual realityArtificial intelligencePhysics-based modelingData-driven modeling A digital twin is a powerful tool that can help monitor and optimize physical assets in real-time. Simply put, it is a virtual representation of a physical asset, enabled through data and simulators, that can be used for a variety of purposes such as prediction, monitoring, and decision-making. However, the concept of digital twin can be vague and difficult to understand, which is why a new concept called "capability level" has been introduced. This concept categorizes digital twins based on their capability and defines a scale from zero to five, with each level indicating an increasing level of functionality. These levels are standalone, descriptive, diagnostic, predictive, prescriptive, and autonomous. By understanding the capability level of a digital twin, we can better understand its potential and limitations. To demonstrate the concepts, we use a modern house as an example. The house is equipped with a range of sensors that collect data about its internal state, which can then be used to create digital twins of different capability levels. These digital twins can be visualized in virtual reality, allowing users to interact with and manipulate the virtual environment. The current work not only presents a blueprint for developing digital twins but also suggests future research directions to enhance this technology. Digital twins have the potential to transform the way we monitor and optimize physical assets, and by understanding their capabilities, we can unlock their full potential. Introduction A digital twin (DT) is a virtual replica of a physical asset, enabled through data and simulations, that can be used for realtime monitoring, optimization, and decision-making [43]. The motivation to study DTs is rooted in the potential cost savings and efficiency gains they offer. In Fig. 1, we can see the concept of a DT. The physical asset is located in the top right side of the figure, equipped with various sensors that provide real-time big data. However, this data has limited spatio-temporal resolution and does not tell about the future state of the asset. To complement the measurement data, models are used to create a digital representation of the asset. If the DT can provide the same information as the physical asset, it can be utilized for informed decision-making and optimal control. The green arrows in the figure show real-time data exchange and analysis. To perform risk assessment, what-if analysis, uncertainty quantification, and process optimization, the DT can be run in an offline setting for scenario analysis. It is then called digital siblings. The grey box and arrows represent the digital sibling. Additionally, the DT predictions can be archived during the lifetime of the asset and can be used for designing a next generation of assets, in which the concept is referred to as digital threads. The authors in [45] present a DT capability level scale adapted from a DNV GL report [19] that divides a DT into six distinct levels. These levels are 0-Standalone, 1-Descriptive, 2-Diagnostic, 3-Predictive, 4-Prescriptive and 5-Autonomous (Fig. 2). Standalone DT can exist even before the asset is built and can consist of solid models. When the asset is in place and is equipped with sensors, data can be streamed in real-time to create a descriptive DT, giving more insight into the state of the asset. When analytics tools are applied to the incoming data stream to diagnose anomalies, the DT advances to a diagnostic level. At the first three levels, the DT can provide information/insight only about the past and present. However, a predictive DT can describe the future state of the asset. Using the predictive DT, one can do scenario analysis to provide recommendations to push the asset to the desired state. This is then referred to as prescriptive level. Lastly, the asset updates the DT at the autonomous level, and the DT controls the asset autonomously. We will refer to this setting as the capability level framework for DTs from now onwards. Within the field of DTs there is no consensus as to what qualifies as a DT application [46]. Organizations and sectors operate with different definitions of DTs that are simply too vague and generic to provide any indication of the current capabilities of the DT [5]. Due to the ambiguity of the definitions of DTs, researchers and practitioners may dismiss them as mere marketing hype. As a result, once the excitement and the inevitable backlash have passed, interest in and use of this promising technology may not reach its full potential [58]. Furthermore within the built environment, a BIM of a building relates somehow to the DT of the asset, but many still struggle to see the liaison between the two [13]. More recently some encouraging advancement has started to happen in the field for e.g. [55] improves Digital Siblings Figure 1: Schematic of a digital twin, and digit sibling concept. The top-right most box with a house represents any asset equipped with sensors acquiring big data. The data is processed using models to improve the spatio-temporal resolution for instilling physical realism in the digital twin. Information from the digital twin can be used for informed decision making and public engagement. Additionally it can be used for optimally controlling the asset. The green arrows signify real-time data / information transfer. The same architecture can also be used for conducting offline hypothetical scenario analysis, in which case can be called digital siblings. X X X X Matterport X - - - - Openspace X - - - - Revit X - - - - Invicara X X - - - TwinView X X - X - the construction efficiency to ensure the infrastructure needs of urban development using the BIM model. In [31] the focus is on the modelling methodology used by the energy domain to support the development of a DT for a multi-functional building element. The authors in [56] argue that a comprehensive perception of physical systems is the preconditions for DTs implementation while in [62] it is demonstrated that DT technologies can enable efficient and responsive planning and control of facility management activities by providing real-time status of the building assets. Researchers in [1] studied the structural system integrity using finite element method in historical masonry buildings using the concept of the DT. It can be easily realized that the capability level scale concept has the potential to make the DT related communication more standardized and can help compare different existing DT solutions available in the market. A brief overview of the capability of existing so-lutions in the market is presented in Table 1. It shows a clear lack of high capability level DTs in the industry. The connection between the DT capability level and the service that each company provides depends on what the focus of the provider is. For instance, Matterport [33] and Openspace are virtual tour services that focus on providing 360 degrees photogrammetry for building management [34,40], and therefore only qualify as a standalone DT service. Note that there are many other similar standalone services in the market [16,39]. Revit is a BIM modeling software and is therefore only limited to constructing 3D models of a building [4]. Invicara and TwinView have an ambition of allowing customers to combine BIM models with IoT sensor data integration qualifying both companies for descriptive DT, with the addition of Twinview providing predictive DT capabilities [53,27]. None of the above-mentioned applications qualify as diagnostic or prescriptive DTs. To this end, the cur-2 Prescriptive Can provide prescription or recommendations based on what if / risk analysis and uncertainty quantification. Autonomous Can replace the user by closing the control loop to make decisions and execute control actions on the system autonomously. rent work • introduces the concept of DT and its capability level in the context of built environment. • presents the basic ingredients to get started with building a DT. • combines the power of AI, advanced sensor technologies, and virtual reality (VR) to develop the DT which is used as a way to communicate the concept and its values. • proposes future research directions to enhance the capability of DTs. It is important to stress that the objective of this paper is not to present a detailed analysis of the diverse class of data used in this project. Instead, we are using data, and their analysis to demonstrate the potential value of DTtechnology. To the best of our knowledge, this has never been attempted before at the fidelity addressed in the current article. This paper is structured as follows. Section 2 presents the relevant theory that has been used to develop the DT. We then outline the methodology of the work in Section 3, namely how the data was generated, how the models were trained, and how they were evaluated. In Section 4 we present the results and discuss them. Finally, we summarise our findings and outline future work in Section 5. Theory This section gives a brief description of the concepts, algorithms, and tools utilized to develop and demonstrate the DT concept using the capability level framework. Since one of the goals of this work is to provide a blueprint for developing DT of any asset from scratch, for the sake of completion, we have also included some theories (eg. collaborative filtering and big data cybernetics) which we have not yet used in the current work. 3D Model Representation As explained earlier, a standalone DT is a virtual description of an asset, disconnected from its physical counterpart. In the current context the house might not even exist at the inception of a standalone DT. However, the standalone DT can give the stakeholders a feel of the building and its environment, enabling them to make informed decisions. In creating the standalone DT we involved three basic steps, 3D modeling, rendering/texturing, and creating a virtual representation of the building. These are explained in the following section. 3D Modeling The term 3D modeling is the process of using a software tool to construct a 3D representation of an object (house in the current context). The 3D model also called the Computer Aided Design (CAD) model consisting of point clouds, edges, and surfaces giving the illusion of physical objects. The 3D models can be utilized for engineering analysis using Computational Fluid Dynamics (CFD) and Finite Element Methods (FEM). 3D Rendering 3D rendering is a computer graphics process in which surfaces of a 3D model are overlayed with textures such that the 3D object achieves a photorealistic appearance [60]. 3D rendering can be broken down into three steps; visual texturing, lighting, and detailing. Visual textures refer to the visual perception of a spatial surface with a variety of details such as color, orientation, intensity, size, shape, and density [18,22]. In addition, lighting, reflection, and shadows can all be generated in 3D rendering software, and the light-absorbing characteristics of the materials are integrated into the texture. Detailing is the last step and requires a designer to carefully sculpt wear and tear, imperfections, dents, and other details into the surfaces, giving the model a more lifelike impression. For texturing to be interpreted across different platforms, it is usually represented in the form of a uv-map. uv-mapping is projecting a 2D image onto a 3D surface, where the "uv" part refers to the axes of the image projection. Virtualization using 3D Game Engine A 3D game engine is a software development environment that effectively allows for the rapid development of interactive 3D experiences and games. A game engine is known to support a programming environment, 2D and/or 3D rendering, accurate physics engines, and many more well-optimized features that would take much time for a single developer to create on their own. Thanks to commercially available game engines, it is easy for an individual to only focus on the specifics of their own game, simulation, or experience [25]. Creating a standalone DT of an asset with 3D modeling and 3D rendering allows only for a static 3D model while using a 3D game engine through scripting language allows for state management of the 3D asset. This enables real-time evolution of the DT with respect to the asset it represents. Virtual Reality The concept of VR is not recent and can be defined as a model of the real world that is maintained in real-time, sounds and feels real with the possibility to directly and realistically manipulate the environment. Today there exist many affordable VR solutions generally consisting of a headset and complimentary controllers, that either utilize their own internal hardware or external processing power to render the virtual environment. Such a representation of a virtual environment through the usage of VR hardware compliments very well the visualization of a DT asset, which allows for a more realistic representation and feel of the asset. Time Series Prediction and Forecasting Model A time series can be defined as sequenced data consisting of real-valued continuous numerical observations that are a function of time [30]. The data collected in the current work comes from sensors sampled at regularly spaced intervals; thus, the data can be viewed as continuous-valued but discrete in time. Time series predictions and time series forecasting, while being slight variations of the same thing, can often be confused to mean the same thing [15]. In the context of machine learning (ML) and this work, a time series predictions model will refer to a regression model capable of predicting unknown or unseen values based on present information. On the other hand, a time series forecasting model is a regression model capable of making future predictions based on learned trends and seasonality, amongst other things. AutoRegressive Integrated Moving Average ARIMA, a time series forecasting model, is a versatile model that can capture both the linear and non-linear patterns in the data, as well as handle different types of seasonality and trend. The ARIMA model consists of three components: Autoregression (AR), Integration (I), and Moving Average (MA). AR models the relationship between an observation and a number of lagged observations. The idea behind this is that the current value of a time series is a function of its past values. AR models can be used to capture linear patterns in the data. Moving Average (MA) models the relationship between the errors of the time series. The idea behind this is that the errors in a time series are correlated with the errors of the previous time points. MA models can be used to capture non-linear patterns in the data. Integration (I) is a technique used to transform a non-stationary time series into a stationary one. A stationary time series has a constant mean and variance over time, which makes it easier to model. Integration is achieved by taking the difference between consecutive observations. ARIMA models are typically denoted as ARIMA(p, d, q), where p, d, and q are integers that represent the order of the AR, I, and MA components, respectively. Prophet Prophet [17] is a popular time series forecasting model designed to handle seasonality, holiday effects, and other time series features that are commonly encountered in real-world data. It is based on a decomposable time series model that can capture trend, seasonality, and holiday effects using piecewise linear models. Prophet is also robust to missing data and can handle outliers and changes in trend. Additionally, it offers a wide range of customizable options and hyperparameters to fine-tune the model performance. Overall, Prophet has gained popularity due to its ease of use, flexibility, and ability to provide accurate and reliable forecasts for a variety of time series applications. Long Short Term Memory Networks LSTM [23] is a type of RNN architecture that is specifically designed to handle time series data. LSTM networks are capable of learning and remembering long-term dependencies in time series data, making them well-suited for a wide range of applications such as speech recognition, natural language processing, and temperature prediction. Unlike traditional RNNs, which have a simple structure and can suffer from the vanishing gradient problem, LSTMs use a memory cell and a set of gates to selectively store, retrieve, and forget information. The memory cell can retain information over long time periods, while the gates control the flow of information into and out of the cell. This allows LSTMs to capture complex patterns and dependencies in time series data, making them a powerful tool for time series forecasting and prediction. Gradient Boosting Machines GBM is a powerful ML algorithm used for both regression and classification tasks. It works by iteratively building an ensemble of weak decision trees, where each new tree is trained to correct the errors made by the previous trees in the ensemble. GBM is a form of boosting, which means it improves the accuracy of the model by focusing on the misclassified samples in each iteration. The algorithm works by minimizing a loss function, such as mean squared error or log loss, using gradient descent. GBM has several advantages over other ML algorithms, such as the ability to handle missing data and outlier detection, and it is less prone to overfitting. GBM has become a popular choice for solving complex ML problems. GBM were one of the first sequential ensemble methods of its kind created by Friedman and have evolved to many of the state-of-the-art treebased algorithms such as XGBoost, CatBoost and LightGBM [9,42,29]. Stacking Stacking uses output predictions of base models as input to a second-level model, usually called the meta-learner. However, one cannot simply train the base models on the full training data, generate predictions on the full test set and then output these for the second-level training. This would not lead to the benefits that stacking provides. Instead, K-folds of the dataset is created. Then each model is fitted on K − 1 of the training set and predicts only on 1 K of the data set, this is done for K 4 iterations until all data appears on the test set. All K predictions of a single model are concatenated into the size of the original test set vector, this is done for each of the models. Finally, all these vectors are fed together as features to the metaregressor, which produces the final predictions [21,2]. Empirical evidence shows that model stacking makes the model more robust to changes in the data set, allowing for better generalization [11]. This is because the stacking deduces the bias in a model on a particular data set, and then corrects said bias in the meta-learner [57]. Weight Averaging A simple but powerful way to create a strong predictor is by using parallel ensembling. One way to think of parallel ensemble learning is weighting the predictions of multiple different models. Another way of parallel ensemble predictions is to have multiple models of the same type e.g. LSTM, but each LSTM model has different hyperparameters or is fed with different features, and then their predictions are weighted to get the final prediction. One can combine and experiment with endless types of parallel ensembling, as each dataset might work very well with a specific kind. In Equation 1 assuming predictions from N different models, then the final predictionŷ f is the weighted average of all the individual models represented byŷ i . y f = N i=1ŷ i w i N i=1 w i (1) where the weights w i given by w i = 1 ( 1 N N k=1 (y val k −ŷ val k ) 2 ) p(2) are calculated using the predictions on the validation set (represented with the subscript val ) Note that the tunable hyperparameter p can be chosen to be any value greater than zero. Sun Position Prediction Model The approximate algorithm that is used in the current implementation of the predictive DT is inspired by [20] which is taken from Montenbruck's book on algorithms about astronomical phenomena. The precision of these calculations is in the range of 01.03.1900 till 28.02.2100 as stated by the author [14]. The resulting sun position algorithm in Unity is accurate enough that it can be used for external lighting simulations on the house. In order to calculate the azimuth and altitude of the sun accurately, several parameters are required to convert the input date into the correct format. These parameters include JD, JC, t UT , t S R , t S RUT , and t LS R . Additionally, the ecliptic coordinates of the sun, λ and β, are calculated using the values of L 0 and M 0 . The ecliptic coordinate system is used to represent the apparent positions of any solar system object. These coordinates, together with Ω, are then used to calculate the equatorial coordinates, α and δ. This enables us to determine the position of the sun relative to the earth, with the earth located at the origin of the equatorial coordinate system. Finally, the coordinates are converted into azimuth and altitude angles, φ and θ, which provide the output sun coordinates relative to a stationary point on the earth's surface. In our case, the stationary point is the longitude and latitude positions of the house. First, in order to calculate the number of days relative to the reference date and time of January 1, 2000, 12h Universal Time (J2000), we need to calculate the Julian Date. The Julian Date is represented by JD and can be calculated using the following equations: JD = 367Y − 7 4 (Y + 9 12 M) + 275 9 M + D − 730531.5 This relates to the Julian Century as JC = JD 36525 Furthermore using Julian Century, Sidereal time (given in hours) can be defined, which is the meridian of Greenwich at midnight (00h) for a given date as well as the conversion from sidereal time of the Greenwich meridian for Universal Time can be calculated as follows. t S R = 6.6974 + 2400.0513JC t S RUT = t S R + 366.2422 365.2422 t UT Where 365.2422 is the length of a tropical year given in days. Finally, the local sidereal time for a geographical longitude L can be found as such. t LS R = t S RUT + L Local sidereal time will come in handy later when calculating the altitude and azimuth of the sun. Firstly number of days d from J2000 for the input date is given as. JD d = JD + t UT 24 Which is needed to calculate the relative centuries from the reference time. JC d = JD d 36525 Using the Julian century of the input date the sun's mean longitude and its mean anomaly can be calculated. Using the Equation of center of the sun the ecliptic longitude can be found. Note that the ecliptic latitude is approximately zero (β ≈ 0). λ = L 0 + C 5 There are many intermediate calculations that need to be done in order to get the correct position of the sun relative to where a person's relative geographical longitude and latitude. Consequently in order to find the azimuth and altitude it is important to find the Sun's equatorial coordinates, right ascension α and declination δ which are both relying on the obliquity of the ecliptic Ω. Ω = 23.439 − 0.013JC d α = arctan(tan(λ) cos(Ω)) δ = arctan(sin(α) sin(Ω)) Now it is possible to proceed to find the horizontal coordinates for the sun for a given input of geographical longitude L and latitude B. First the hour angle of the object is given as. HA = t LS R − α Resulting in the final Equations for the altitude of the sun θ and the azimuth φ respectively, where the algorithmic version of the azimuth φ is calculated using Arctan2 as suggested by [61]. These angles are required is utilized to correctly render the objects in the DT. Recommender Systems Recommender systems are widely used for information filtering, providing users with valuable insights from relevant data sources. These insights can be inferred from the data or concatenated from a collection of data features. User-Based Collaborative Filtering Collaborative filtering is an information filtering technique that predicts a user's interests or behavior by collecting data from many users. The underlying assumption is that if person A behaves similarly to person B in a specific context, then person A might behave similarly to person B in another context than a randomly chosen person from the population. Companies such as Amazon, Netflix, YouTube, and other services extensively use recommender systems to learn about user preferences and provide personalized recommendations based on their behavior and similarities with other users [47]. However, the main challenge with this approach is that it requires a lot of data about user behavior, not necessarily from the user in question, but from the entire user base from which the data is collected. Moreover, the cold start problem arises when a new user registers and has not provided any interaction yet, making it impossible to provide personalized recommendations [8]. At the beginning, the algorithm may not be very accurate but becomes more precise as more data is collected while the user is active. Control Systems The aim of cybernetics in autonomous DTs is to guide the house towards an optimal set point. To achieve this, the system's output is continuously monitored and compared against a reference. The difference between the two, referred to as the error signal, is fed back to the controller, which generates a system input to bring the output set-point closer to the reference. As more sensors and communication technologies become available, larger volumes of real-time data, i.e., big data, are being generated. However, the quantity of interest may not be directly measurable, and it becomes a challenge to extract and understand the relevant information to be used for control purposes. Big Data Cybernetics is a new field of research that aims to address this challenge in a real-time control context. The first step involves interpreting the big data using well-understood physics-based models. The difference between the observation and the physics-based model is called the interpretable residual. In the second step, interpretable datadriven modeling approaches are used to model and analyze this residual. The remaining uninterpretable residual is then modeled using more complex black box models like Deep Neural Networks, which generally represent noise that can be discarded. This approach is known as Hybrid Analysis and Modeling (HAM), which continuously loops with the availability of new data resulting in ever-improving models. HAM is a promising new approach that aims to combine existing physicsbased models with interpretable and non-interpretable big datadriven modeling techniques. Fig. 3 illustrates the looped steps of the HAM process and the overall Big Data Cybernetics philosophy. HAM models can be utilized not only in the context of the interpretability of big data, but also in the Controller block. They can be incorporated alongside model-free control algorithms such as reinforcement learning (RL), or with model predictive control (MPC) algorithms, to improve the overall control performance of the system. Method and Set-up This section gives the details of setting up DTs at different capability levels. Set-up for Standalone DT The asset in this case consists of a house, its surroundings terrain, and objects inside the house. Creating a standalone DT involved creating CAD models as a starting point. Here we give a detail of the steps involved in building the standalone DT. 3D CAD Model of the House The house's 3D CAD model was created using Autodesk Revit software based on the 2D floor plans, as shown in Fig. 4. The three floors were stacked on top of each other, and details such as doors and windows were accurately placed. To distinguish different components of the house, each surface was tagged and textured appropriately. This tagging ensured that components could be identified accurately at a later stage if necessary. The scene was rendered using Autodesk 3DS Max and converted into a format compatible with the Unity Game Engine. Highquality textures were obtained from Architextures [3], a library designed for architectural models, and imported into 3DS Max. The textures were then applied to the uv-maps of walls, floors, and other surfaces to achieve the final look of the building. Unity Game Engine for Interaction with the 3D Objects The Unity Game Engine offers a C# .NET programming environment, 3D rendering capabilities, accurate physics engines, and optimized features that help developers create games more efficiently. Importing CAD models into Unity allows for real-time interaction with objects and instills physical realism through precise physics engines. If the exact physics engine is not available, it can be implemented within the Unity framework using programming languages such as C# or Python. While Revit and 3DS Max are integrated by default, setting up the connection between 3DS Max and Unity requires additional steps. Unity can be linked with 3DS Max through a middleware called FBX Exporter, which was developed collaboratively by Unity and Autodesk to facilitate the workflow between their programs [52]. To use the FBX Exporter, it must be added as a plugin to the Unity project after installing all necessary software. Once the 3DS Max model is exported into Unity, some final setup is required. This includes selecting the 3D model in Unity, going under Materials" in the Unity inspector, and checking off Use External Materials" under Location" to import the textures from 3DS Max. Finally, clicking on Generate Colliders" under the model enables Unity to treat the house as a physical object and interact with other physical objects in compliance with Newtonian mechanics. This work uses Unity Version 2020.3.16f1 with the Universal Render Pipeline version 10.5.1 (URP). Unity offers three different graphics rendering pipelines: the standard pipeline is a general-purpose tool with limited features and options; the High Definition Render Pipeline (HDRP) focuses on creating high-end graphics that are supported only by cutting-edge graphics cards and machinery; and URP is customizable and provides optimized graphics across a broad range of platforms [12]. URP is preferred for this project because the DT is intended to run on low-end android VR platforms, such as the Oculus Quest 2. Additionally, the setup requires the following add-on packages from the Unity Asset Store: Each of which supports some particular module of the setup, for example, the Oculus XR Plugin makes it easier to integrate the application in Unity with the Oculus Quest 2 with minimal setup. While NuGet is used to manage .NET-specific libraries. • Unity also allows the possibility of bringing different 3D models together to live in the same environment. This could be a car, or furniture, as seen in Fig. 5. Note that when importing models in Unity, it is crucial to be aware of the relative scale/unit associated with the model. Other 3D objects To create the paintings on the walls, we used 3DS Max to design the image frames and add images of the actual paintings. For other furniture pieces like the carpet, TV table, and sofa, we downloaded models from open-source websites such as Polantis [41], which provides free CAD and BIM 3D objects that resemble furniture from brands like IKEA. To ensure optimal performance, it's important to use 3D models with a low polygon count. High-polygon models can create a performance bottleneck, but it's possible to use 3DS Max to reduce the polygon count of downloaded models without compromising their appearance. It's worth noting that Unity 3D objects work on a meter scale by default, while external 3D models may be generated using other units of measurement like centimeters or inches. Therefore, it's important to be aware of the units of measurement used in external models and scale them appropriately before importing them into Unity. Terrain Creating terrain in Unity based on a height map of Trondheim is not a straightforward task. However, the Norwegian government provides accurate height maps of Norwegian terrain with up to 1m × 1m resolution. One can apply for this data at Kartverket website, by selecting the specific region one wants to extract from the map of Norway and sending a request [28]. The data retrieved from Kartverket needs to be preprocessed from its original file format GeoTIFF to RAW, which is the input format that Unity supports for height maps. Here is an enumerated instruction list to convert heightmap data into a format usable in Unity: 1. Apply for heightmap data from Kartverket. 2. Import all the GeoTIFF files to the software QGis. 3. Merge the GeoTIFF files into one common one using QGis merge option (Raster → Miscellaneous → Merge) use output data type as UInt16 and click run. 4. Export file by converting it to a BIL file, which is a variant of the RAW file type. This is done by navigating to (Raster → Convert → Translate). 5. Click on Advanced Parameters, and add the following commands "-ot UInt16 -outsize 2049 2049" into "Additonal command-line parameters [optional]", note outsize needs to be scaled to be in bit format +1 i.e. 1025,2049,4097 etc. 6. Click on "Converted → Save to File" and change "save as type" to BIL files (*.bil). 7. Click run, go to the folder where you saved the BIL file, create a copy and change filetype to ".raw". 8. Finally upload the extracted RAW file into a Unity Terrain object using the height map property of the object. The 3D model of the house is placed in the location based on its real location in terms of longitude and latitude, where the top left corner of the terrain height map functions as the origin in a Unity grid, such that the house can be precisely positioned. Orientation of the House The Unity environment also simulates a day-night-cycle, including a highly accurate algorithm for the sun. Here orientation and altitude of the house are essential to synchronize the house's position relative to the sun simulation. For the orientation, the house needed to be rotated 203 • from the cardinal north direction of a compass. This was to align the front of the house correctly with the sun's movement. Since the setup of the standalone DT also includes a correct height map terrain of the outside environment in Trondheim, the house was elevated to the correct altitude, which is 211m. Set-up for Descriptive DT At this stage, the physical house has been constructed, and a standalone DT has been developed. The next step is to refine the DT to match the physical house and equip it with a diverse class of sensors. A data stream is then established to convert the standalone DT into a descriptive DT. More details regarding the placement of the sensors can be found in Fig. 4, and are explained in detail in the following sections. Netatmo Weather Station The Netatmo Weather Station is located in the living room, as shown in Fig. 4, and records data once every five minutes [36]. The station measures temperature, humidity, CO2 concentration, noise levels, and pressure. While the weather station can be customized and equipped with many more sensors, it is only equipped to monitor the aforementioned data. All Netatmo endpoints are located at "https://api.netatmo.com". Before accessing the data, it is important to first make a POST request for an access token by adding the "/oauth2/token" endpoint to the Netatmo API link. The request must include the grant type, client id, client secret, username, password, and scope [35]. Once the request is successful, Netatmo grants access to the current state of the sensor via an access token. Philips Hue The Philips Hue sensors are located throughout the house, as depicted by the yellow dots in Fig. 4. A total of 16 Philips Hue lights were used in the project. Information about the brightness, state of the light (on/off), colors, and any other relevant information about the lights can be retrieved in real-time using the Philips Hue API. The API that hosts the endpoints is "https://api.meethue.com". To refresh an access token, one must provide a valid refresh token as part of the request and perform a POST request to the "/v2/oauth2/token" endpoint. Assuming a valid access token, one can access the light data by making a GET request to the "/bridge/<USER ID>/lights" endpoint, where USER ID is an id granted by Philips Hue when an account is registered. Like with Netatmo, the access token must be provided as part of the GET request to obtain the data. Disruptive Technologies Sensors A total of twenty-eight sensors for disruptive technologies are positioned on all floors, as shown in Fig. 4. Among them are 16 temperature sensors, with one placed outside the house to track external temperatures. There are also five proximity sensors, with one used to detect the opening and closing of the fireplace doors, while the remaining four monitor the openclose status of other doors. Four water sensors detect any water leakage in different locations, while three humidity sensors also record the temperature. Moreover, all sensors can function as touch sensors, meaning that if one is pressed, it could trigger an event. This provides significant freedom for experimentation, such as triggering an email or an action inside the disruptive technologies platform. The authentication process for the sensors is similar to that of Netatmo but with an additional security layer. To obtain an access token, the email and key id provided at the time of account registration must be encrypted using the HS256 hash algorithm. The encrypted token must then be sent using a POST request to the "https://identity.disruptivetechnologies.com/oauth2/token" endpoint, along with an assertion and grant type [50]. With a valid access token in a GET request, various data can be obtained using the endpoint "https://api.disruptive-technologies.com/v2", with parameter variations provided in the documentation [51]. Note that data is updated at a frequency of every fifteen or five minutes or when a significant change occurs. Weather Conditions The house's local weather data, including wind speed and direction, is obtained by making requests to the "api.met.no" weather forecast API, which is operated by the Norwegian Meteorological Institute. While there are several endpoints available, we use the "Nowcast 2.0" endpoint for our specific use case. To make a successful request to this endpoint, a User-Agent identity is required to identify the purpose of the request [26]. In order to visualize the wind direction and speed, a Unity vignette is created, which appears around the house. This visual effect is designed to show a wavy white line moving across the sky, indicating the wind's direction and velocity. The vignette is a useful tool for conveying the wind's speed and direction at any given moment, making it easier to understand the weather conditions outside. Set-up for Diagnostic DT Once the data stream is established in the descriptive DT, data analysis should give additional insight into the asset and this should be presented to the end user in an easy-to-understand format. The set-up presented here for the diagnostic DT enables that. Figure 6: How to use the Oculus controllers in VR. If the play area is big enough, one is also able to physically move around or rotate without using the thumbsticks. Pointing at an interactable object with the right hand controller and clicking on the "A" button triggers events. Virtual Reality User Interface The VR setup uses an Oculus Quest 2, where a UI "tablet" is implemented on the user's left hand. This way, the user can move the menu in and out of sight. The main focus of the UI is to give the user a sense of empowerment, curiosity, and, most importantly, feedback, which are all ideas derived from the Octalysis Framework for Gamification [10]. The empowerment and feedback come directly from the real-time control that a user feels when they can, for example, slide the time of the day and see the weather change to reflect reality. Furthermore, this responsive system triggers an exploratory curiosity in the user, making them want to seek out the remaining contents of the UI system. Recall that the diagnostic DT is mainly about monitoring and troubleshooting, meaning that sparking the user's interest in seeking diagnostic information is as important as presenting the information itself. In Fig. 7 one can see the enumeration of images featuring different navigation panels of the UI system. Fig. 7a is the main menu which is at the center of all the other monitoring and troubleshooting systems for the diagnostic DT. Fig. 7b features the sensor inspector, providing critical information related to condition monitoring. By pointing at spherical probes around the house and scanning them with the "A" button of the right-hand controller, the sensor inspector displays relevant information relating to that probe. Real-time data from the specific sensor implemented for the descriptive DT is then revealed in the sensor inspector panel. Note that the probe positions are supposed to reflect the real three-dimensional positions of the sensors from Fig. 4. Fig. 7c shows the UI for the sun control panel. The sliders can be used to simulate the lighting conditions depending on the time of the year. Fig. 7d demonstrates how data from multiple sensors can be combined to create a heat map that gives a better feeling of the indoor environment. Temperature Heat Map The temperature heat map is a visualization aspect of the diagnostic DT that conveniently presents valuable information. The heat map does not represent the temperature on the room's floor, but the temperature distribution of the temperature probes around the room. The heat map in Fig. 25 assumes that the heat distributes itself in a plane, and therefore is using a euclidean distance equation (Equation 3) for path-finding to render the temperature gradient radially outwards onto a Unity shader with the path limited to a reach based on α 2 . A shader is an object that communicates how to correctly color pixels onto a material in Unity [54]. The implemented algorithm takes the current warmest and coldest temperature sensor measurements in a room and uses that interval to weight the radial output of the temperature gradient. The color mapping from temperature to color is the same as the one in Fig. 21, i.e. if the minimum temperature is 16.5 • C and maximum 25.7 • C then that would be the blue and red colors respectively, and the other temperatures would fall within the color space in between. d (p, q) = α 2 n i=1 (q i − p i ) 2(3) Fog Particle Effects Another possible way to monitor the real-time data coming into Unity from the various data sources is by using the particle system in Unity. In the current set-up, the red fog has its opacity adjusted based on if the real-time CO2 concentration is within a certain ppm interval, while the temperature is being mapped to color to represent the heat. The CO2 and temperature representations in Fig. 20 and 21 are implemented to visualize the Netatmo Weather Station placed in the living room, seen in Fig. 4. The temperature reading is first transformed into an HSV value to be mapped into a color, and then from HSV into an RGB that is finally displayed as fog in Unity. It is important to note that the temperature intervals can be redefined, but for this example, the temperature interval is set to be between zero and 40 degrees Celsius. Set-up for Predictive DT This section is about the setup and methodology used to achieve a predictive DT. At this level, the DT can predict the system's future states or performance and support prognostic capabilities. To demonstrate the DT's predictive capability, we consider two cases, one related to the prediction of the inner state of the house in terms of the indoor temperature and another related to the external state in terms of the available solar potential. The reason for choosing the first case is that knowing the evolution of inside temperature can help develop cost and energy-effective control strategies. The second case is of great relevance for a country like Norway, where complex terrain can significantly affect solar exposure. This section also demonstrates two entirely different approaches to modeling that are relevant to DT-like technologies. One is pure data-driven modeling (DDM), while the other is physics-based modeling (PBM). DDM is effective when the physics governing a process is not entirely known, is computationally expensive to solve in real-time, or the values of physical parameters are not known accurately. PBM is more effective when the physics is known, or there is a need for generalization in unseen situations for which no data exists. We will use a DDM to predict the indoor second-floor temperature based on past experiences because the state-of-the-art building simulation model can not precisely describe the dynamics of the buildings, and the exact composition of the built material is unknown. On the other hand, we will use a PBM for sun position prediction because the equations governing the sun's movement are well-known and can be used to deterministically simulate any situation. Data Collection and Preprocessing First, the raw time series data (temperature, humidity, proximity) from Disruptive Technologies sensors are aligned and sampled at five-minute intervals to generate a multivariate timeseries corresponding to 90 days, out of which 70 days were used for training, 10 for validation, and 10 for testing. The training of the ML models is done on a computer with the following specifications: Intel(R) Xeon(R) Gold 6140M CPU @ 2.30GHz for a CPU and NVIDIA Quadro RTX 5000 for GPU. The GPU is mainly used to speed up the training of the LSTM utilizing NVIDIA's CUDA library [37,38]. Forecasting and Prediction Pipelines As seen in Fig. 8, a weight-averaging ensemble is used as the final model. Gradient boosting machines and the stacked model worked very well with a differencing transform of the target, while the LSTM model performed better with standard data normalization. For the Prophet model, providing extra features increased performance. Also based on the RMSE validation score, hyperparameters of every single model are optimized using the Optuna Python library. The final forecasts on the validation and test set are achieved by using the weighted average based on the validation RMSE of each model. All the forecast models except for LSTM are setup as incremental forecasting models, mainly because none of them support an instant multi-output target. In the prediction model, the data is preprocessed in a similar fashion to the forecasting model. In practice, the prediction pipeline in Fig. 9 can be used in two ways. First, it can be used to predict the potential state of an out-ofcommission sensor, assuming that the fireplace sensor in the room is still operational with data X. Or it can be used to fill in the eight forecasted temperature sensors using the forecasted outputŷ of Fig. 8, where in the context of Fig. 9 the forecasted fireplace data is denoted as inputX. Therefore assuming a temperature sensor deployed in the second floor dies after a few years, instead of replacing it, these sensors can be accurately predicted based on the fireplace sensor. Otherwise, the prediction model can be seen as an extension of the forecasting model. Hyperparameters of the LSTM The neural network is simply constructed of a LSTM layer, followed by one-dimensional global max pooling into a dense output layer. The following hyperparameters are used, hyperparameters not explicitly mentioned here can be assumed to be the default of the Keras/Tensorflow library: Set-up for Prescriptive Digital Twin This section is about the setup and methodology used to achieve the prescriptive DT. The prescriptive DT is the fourth capability of a DT pipeline. In this level, the DT is able to make recommendations based on what-if? scenarios, and through that support uncertainty quantification. What-if scenarios in our case can be potential weather forecasts or historical weather conditions in the past that may have led to certain behaviors. Behaviors can certainly be used to motivate future recommendations, given if such a situation would again occur sometime in the future. The recommendations can be used to give experts a decision support system, but for our case, such a system might be used for uncertainty quantification of the unpredictable parts of the future forecasts, that have been modeled and highlighted in the predictive DT. Collaborative Filtering There are only eight recorded fireplace events for 102 days. This is a very small dataset to work from, making it hard to conclude anything in particular about the user's fireplace lighting behavior. Furthermore, no data has been collected from other homeowners and there are therefore no other users, from whom we can obtain our recommendations to the specific user. Hence there is insufficient data, as the sparsity of data comes from that most of the days the fireplace is simply not used. The motivation for demonstrating this particular method comes from its potential to become valuable, as more data is collected from this house and other locations. The collaborative filtering for fireplace lighting events is set up with U homeowners (users) as the rows and V days (items) as columns. Note each day has specific information about the outdoor temperature of that certain day. In the collaborative filtering matrix from Fig. 11, the fireplace timesteps for every single day are extracted and used as a row for a specific user. The timestep for when the fireplace was approximately turned on is inferred by finding the biggest difference between two points on a specific day. The timestep for a lit fireplace for a specific day can be seen in the first row, as a red vertical line in Fig. 11. The user-based collaborative (UBCF) pipeline as implemented is visualized in Fig. 10. Given that an unknown in-put outdoor temperature forecast or scenario is given to the pipeline, first, the RMSE between the input and each column-specific outdoor temperature is calculated to find the most similar scenario to our input. For simplicity all users are assumed to have experienced the same outdoor temperature in Trondheim, therefore for users outside of Trondheim, RMSE specific to that user needs to be calculated. The remaining pipeline would still work the same way. Resulting in the RMSE weight vector w RMS E seen in Equation 4. ∑ = � ∑ = , � ∈ ℝ × � = 1 ( 2 1 ∑ =1 ( − � ) 2 ) The recommendation pipeline seen in Fig. 10 is supposed to be for the first user. In parallel with finding w RMS E for the user, the Pearson correlation of all the fireplace lighting event actions of said user is calculated against the other users in the population, assuming these exist. Then the highest correlated users based on some threshold are extracted as a weight vector w pearson seen in Equation 5. Where χ pq is the set of days both users p and q that have fireplace behavioral data. Knowing the most similar scenarios to the input scenario and the most correlated users to the user we wish to make recommendations for, now we extract a matrix denoted T. This matrix is the region where both the green and red rectangles cross as seen in Fig. 10. Each row of said matrix is weighted by the RMSE score of the specific user in a weighted average. This gives the predicted fireplace lighting behavior of each individual user, given that specific scenario. That particular prediction vector t predicted is then weighted by the most similar users to the particular user we are recommending, to produce the final recommendation time to light the fireplace t n+1 . w RMS E =          1 ( √ 1 N N k=1 (T item1 k −T input ) 2 ) . . . 1 ( √ 1 N N k=1 (T itemV k −T input ) 2          (4) w pearson =            j∈χ 1i (t 1 j −t 1 )(t i j −t i ) √ j∈χ 1i (t 1 j −t 1 ) 2 √ j∈χ 1i (t i j −t i ) 2 . . . j∈χ 1U (t 1 j −t 1 )(t U j −t U ) √ j∈χ 1U (t 1 j −t 1 ) 2 √ j∈χ 1U (t U j −t U ) 2            (5) Set-up for Autonomous Digital Twin In the current work we have not demonstrated the autonomous capability of the DT. Results and Discussions In this section, we aim to showcase the potential and value of various types of DTs. To illustrate each capability level, we begin with a scenario that presents specific challenges, and then use the DT to demonstrate how these challenges can be addressed effectively. Standalone DT Scenario: A potential homebuyer is interested in purchasing a yet-to-be-constructed house. The real estate company provides the buyer with a tour of the construction site (Fig. 12) and several 2D sketches (Fig. 4) depicting the different levels of the future house. Unfortunately, the provided documents do not offer any insight into how the neighborhood will look once the construction is complete. Despite the lack of information, the buyer commits to the purchase. Later on, the buyer is invited to customize the house, but once again, lacks insight into how their choices will look and feel in the real world. In this scenario, a virtual tour using a standalone DT could provide the buyer with the necessary information to make a more informed decision, and improve communication between the seller and buyer. Solution: The image provided by the real estate agent in Fig. 12 only provides a limited idea of the site before construction. However, with a standalone DT, it is possible to create a more immersive experience of the interior and exterior environment. This would allow the buyer to make informed decisions about customizing the interior, estimating solar potential, and assessing the recreational activities in the area. By using a DT, the buyer can gain a better understanding of the property and make more informed decisions. It is apparent that Figs. 13 and 14 provide more comprehensive information in contrast to Fig. 12. Figs. 13a and 13b offer a better understanding of the surroundings after construction, facilitating the selection of building materials that complement the natural environment. The available space in the driveway, as shown in Fig. 13c, can assist in planning which car sizes can be accommodated, or whether the driveway design should be altered to allow for larger cars. In addition, the balcony lighting simulation (illustrated in Fig. 13d) can provide an estimation of the sunlight availability for any day and time of the year. Furthermore, visualization of the interior environment (depicted in Fig. 14) can aid in optimal placement of lighting fixtures, selection of wall colors, flooring, furniture, and other relevant objects. Descriptive DT Scenario: Suppose a standalone DT of the house was available, allowing the buyer to make an informed decision and customize the house to their liking. With digitalization in mind, advanced sensors and controllers were installed, providing realtime information about various aspects of the house, including indoor air quality, water leakage, door status, occupancy, security breaches, and external weather conditions. The homeowner wants to monitor the house remotely while away and desires more insight into the indoor environment while present inside the house. Solution: We now show how a descriptive DT can provide additional information regarding the house using real-time data from the installed sensors. In Figs. 15 and 16, the real-time states of the house are displayed, based on the data collected by various sensors and lights. The presented data is raw and unprocessed, intended for human interpretation to draw relevant conclusions. Fig. 17 provides an example of rendering the prevailing weather conditions on specific days. Additionally, Fig. 18 shows real-time data from sensors measuring CO2, humidity, temperature, and paintings, which are otherwise not visually representable in the descriptive DT. The homeowner regularly updates the data from the sensors, and the DT database stores this information. This enables the DT to reflect the up-to-date state of the objects in the house, such as the name of the selected painting. Although monitoring the name of a painting may seem trivial, the same Figure 11: Collaborative filtering matrix. Note that other than the house that we have collected data from in row one, the remaining houses and their data is artificial for the sake of demonstrating user based collaborative filtering. concept can be applied to other objects in the house, such as monitoring resource inventory levels or the amount of wood remaining in the fireplace. Advanced object detection and classification algorithms are necessary to monitor such details. Diagnostic DT Scenario: Imagine two scenarios, one where the homeowner is physically present inside the house and the other where the homeowner is remotely located and does not have physical access to the house. With the multitude of sensors installed in the house, the diagnostic DT provides not only real-time updates on the current state of the house, as with the descriptive DT, Figure 12: A visit to the site before the deal of the house was finalized. but also critical alerts in case of any changes. When the homeowner is physically present, the diagnostic DT can be utilized to gain insight into the internal environment using virtual reality to visualize air quality, noise, or temperature maps, which are not visible to the naked eye. On the other hand, when the homeowner is remotely located, the diagnostic DT can provide analytics on the current situation in the house, such as temperature increases, room occupancy, and other relevant information by fusing data from various sensor sources. Remote Location When the homeowner is remotely located, they can diagnose whether a room is occupied or not by analyzing temperature and door sensors related to a specific room. Fig. 19 shows a body heat of the occupant, and by using simple diagnostic tools, these peaks corresponding to occupancy can be automatically detected and communicated. Thus, one can diagnose that the room was occupied between 11:00-14:00. Inside The House When the homeowner is present inside the house, they can visualize diagnostic information such as temperature and CO2 density, as shown in Figs. 21 and 20, respectively. This information can be used to gain insight into those aspects of the indoor environment which are otherwise invisible to the naked eye, such as CO2 and CO concentration. Note that at the descriptive level, these data could only be presented as numbers, while the diagnostic DT allows for visualization and interpretation of the data to gain a deeper understanding of the indoor environment. Predictive DT Scenario: The homeowner now has access to a significant amount of high-quality sensor data from both past and present, but is interested in knowing the future state of the house to plan for more efficient and cost-effective utilization of energy. They may also want to know about the availability of natural sunlight due to potential future developments, such as the construction of a high-rise building. Temperature prediction and forecasting To fill in missing temperature values for eight sensors when only the fireplace sensor is available, the predictive model pipeline can be used, as shown in Fig. 22. This ensures that the temperature in the room can still be tracked even if some sensors go out of commission in the future. Fig. 23 shows how different timeseries forecasting models predict the future temperature profile at the fireplace (2Fireplace) during a typical day when the fireplace is routinely lit. The black line represents the profile used to train the model for the next 24 hours. The weight averaging ensemble model was found to be the best for making accurate forecasts. Once the future temperature profile is available, it can be used to predict the temperature profiles at other sensor locations. Fig. 24 presents the weight averaging ensemble predictions at all the other locations, showing good agreement with what was later observed. Finally, these point measurements are converted into contour plots and projected onto the floor surface for visualization purposes in Fig. 25. Fig. 26 shows the predictive sun model for each month, for informing decision-making around sun hours for the entire year. Sun Position Prediction Model Performance Furthermore, Fig. 27 shows the same algorithm used in VR. The significance of knowing about how many sun-hours, a house gets in the span of a day is a key factor for any potential homeowner. Therefore having a PBM that accurately displays that future prediction both as a bar plot of the entire year as well as viewing a specific day visually within the VR setup, allows for better decision-making around the biggest investment that the majority of people go through. While the bar plot does show you the sun hours, the value of the visual demonstration is that it fills in the gaps by showing exactly which part of the house will be exposed to the sun. For instance, if the terrace or balcony does not get a lot of sun exposure, some buyers would become disinterested in the property. Prescriptive DT Scenario: Now that the homeowner has a predictive DT, he can foresee possible future scenarios based on past data or physics simulations. However, he is not only interested in the forecasts but specific recommendations in the house based on his past behavior. Such as what time of day is it recommended for the user to turn on the fireplace, given the temperature profile the next day. Perhaps there is also a cluster of neighbors with a DT setup, and the recommendations can be supported by all of them if they share a similar behavioral pattern. In Fig. 28 we are trying to predict day 23 of the 102 registered days in the dataset, using the seven other existing scenarios where the fireplace is turned on. The RMSE threshold is set to 1.5, and the days with RMSE under 1.5 are then days one, two, and 25, which resemble the outside weather of day 23 the most out of the existing samples. There isn't any other user data, such that the final recommendation is only based on the user's weighted average of other similar days, and not the Pearson correlation. Therefore the recommendation uses only the RMSE part of the UBCF pipeline in Fig. 10. We can see the recommendation for turning on the fireplace based on previous observations is very close. Autonomous DT Scenario: Now imagine that the homeowner has an accurate and computationally efficient model for predicting the future state of the house and its surroundings under the influence of input changes that the homeowner can affect. These models in combination with advanced control algorithms can be used to get humans completely out of the loop. The asset can continuously update the DT while the latter can control the asset to push it towards a preset optimal operating condition. Despite the ability to develop control algorithms and the availability of remotely controllable equipment like the balance ventilation system and heat pump, no effort was made to demonstrate this capability. There are several reasons for that, which we briefly mention here. Due to a lack of time, all the algorithms and modeling tools presented could not be rigorously tested. Furthermore, using black-box neural networkbased methods for predictions complicates the matter. Unless the model's working is humanly interpretable, it is not wise to use them in a controlled setting. Doing so could risk the safety Figure 23: 2Fireplace temperature forecast. Given 24 hours of past fireplace sensor data from Disruptive Technologies temperature sensor "2Fireplace" (black graph), these models are constructed to forecast 24 hours into the future of said sensor. The best forecasting model is the weight averaging ensemble (blue graph), weighing the contribution of each model seen in Fig. 8. Note that the random walk is not part of the forecasting model, but merely a way to demonstrate that the models can learn a pattern better than a baseline coin flip forecast. The ground truth is the red graph where the models are compared to that. Fig. 23, we can feed it to the weighted average prediction model from Fig. 9, similarly to the one used to output Fig. 22 and generate forecasts for the remaining second floor temperatures. Conclusion and Future Work In this work, we exploited the power of artificial intelligence, advanced sensor technologies, and virtual reality to develop a fully functional and high capability level digital twin (DT) of a modern house. The work involved creating a realistic 3D model of the house that is not only good for visualization but also for conducting engineering simulations. This corresponded to a standalone DT. The physical house was then equipped with a diverse class of sensors, and the corresponding digital representation of the house was updated accordingly. A real-time data acquisition pipeline was established to update the state of the DT with any changes in the state of the physical house, resulting in a descriptive DT. Analytics tools were applied to the incoming data to detect critical changes, resulting in a diagnostic DT. These three levels of DT were not capable of giving any information about the future state as they all relied on the incoming data. At the next level, i.e., the predictive DT, an ensemble of pure data-driven timeseries forecasting models was built and trained to predict the future state of the house accurately. In addition, a physics-based modeling approach to predict the sun's movement and its obstruction by the local terrain was also implemented to predict the solar potential for any time in the future. At the prescriptive DT level, it was argued how data/insight from similar houses in the neighborhood could be utilized using collaborative filtering. It also demonstrated how a prescriptive DT could learn about the user's own behavior to support future recommendations. Finally, the autonomous DT wove all the subsystems together by closing the control loop. The main contributions of the work can be enumerated as follows: • We demonstrated the concept and value of DT [43] and its capability levels [45]. Although the asset chosen for the demonstration was a modern house, the workflow proposed is generic in nature. • We have shown the concept and value of DT [43], as well as its capability levels [45]. While we used a modern house as an example, the proposed workflow is applicable to various contexts. • We have highlighted the significance of a diverse set of data and two distinct modeling approaches (physics-based and data-driven) to enhance the physical realism of DTs. • We have demonstrated how computer graphics, specifically virtual reality technology developed with game engines, can significantly enhance the capability levels of DTs. • Through an interactive graphical interface in virtual reality, we have showcased the potential of DTs not only for remote monitoring but also for remote interaction with assets. • Our proposed workflow and future research directions outlined in the following section can serve as a guide for developing DTs from scratch. While we have highlighted the strengths of our work, we have also identified areas where improvements can be made. One advantage of the DT framework we developed is its modular nature, which allows for individual capabilities to be extended and improved without compromising the functionality of others. We list these areas for improvement below: • Standalone DT: At this capability level, we manually created a 3D model of the house and its furnishings, which can be a bottleneck for scaling DT technology to encompass multiple houses in a neighborhood. However, this issue can be addressed with image-based photogrammetry, as demonstrated in recent works [59,24]. Additionally, solid models are typically represented by textured tessellated polygon surfaces, whose number can be reduced without an observable degradation in quality to enable the DT to run on low-end, affordable computing devices. • Descriptive DT: Geometric change detection [48] can be implemented in a descriptive DT to keep track of geometric changes within the house using cost-effective solutions like RGB cameras, limited communication bandwidth, and storage. Furthermore, real-time satellite data can be used to more accurately describe the external environment, such as cloud cover. • Diagnostic DT: Principal component analysis or autoencoder can be used for detecting deviations from the norm in heterogeneous multivariate data to detect anomalies. Other sensor data, like temperature, humidity, noise, and air quality, are only measured in a few discrete locations. A simple interpolation scheme was implemented to create heatmaps, which can be improved by sensitizing inverse distance weighting with door states and wall corner locations [49]. Additionally, optimal sensor placement strategies [32] can be evaluated for more efficient use of sensors and reconstruction. • Predictive DT: In the current project, we used either a purely physics-based model or a data-driven model to predict the external and internal state of the house, but both approaches have inherent weaknesses, as discussed in [45]. Recent works [7,6] have shown how a hybrid modeling approach can address these weaknesses and make accurate and more certain predictions, making it ideal for modeling partially understood physics and addressing the issues of input parameter uncertainties. For instance, [44] has already shown the applicability of accurately modeling heat transfer in an aluminum extraction process, which is similar to the building energy modeling considered in our work. • Prescriptive DT: At this level, we used the DT to provide recommendations based on learning from the behavior of the same house. However, we faced challenges due to the lack of available data for training ML algorithms, as the house was newly constructed. Collaborative filtering or self-organizing maps can be useful to learn from the performance of older houses for which data exists. • Autonomous DT: We could not practically demonstrate the full potential of a fully autonomous DT due to concerns of voiding equipment guarantees, such as the balance ventilation system and heat pumps. While the smart lights could be controlled remotely, no data was recorded to develop a control strategy for lighting. Therefore, recording data regarding the lighting preferences of occupants and developing a controller to satisfy those preferences would be interesting. Alternatively, research on the psychological effects of lighting on occupants can be integrated into the au-tonomous DT. Additionally, making the models on which decisions are made humanly interpretable is a challenge worth addressing before realizing a fully autonomous DT. The concept of DT is rapidly advancing and this work, along with the future research directions proposed, represents only a small piece of the larger puzzle. However, this work has produced an extensible DT framework that can be valuable for educational purposes and for testing new techniques that can help make DT indistinguishable from its physical counterpart. Acknowledgments The project owes a great deal to the generous support of Disruptive Technologies, who provided advanced sensors for temperature, proximity, humidity, and water detection. These sensors played a critical role in demonstrating the potential of nextgeneration digital twins for the built environment. We are also grateful for the prize awarded by Tekna, which was invested in the development and demonstration of the project's VR component using the Oculus Quest 2 virtual reality headset. Figure 2 : 2Description of capability levels of a digital twin. L 0 = 0280.466 + 36000.770JC d M 0 = 357.529 + 35999.050JC d The sun's equation of center C is given as C = (1.915 − 0.005JC d ) sin(M 0 ) + 0.020 sin(2M 0 ) θ = arcsin(sin(B) sin(δ) + cos(B) cos(HA)) φ = arctan − sin(HA) tan(δ) cos(B) − sin(B) cos(HA) •Figure 4 : 4Toolchain Win Linux x64 1.0.0 • Unity UI 1.0.0 • XR Legacy Input Helpers 2.2D floor plans of the target house with all the sensors and devices setup for the project. All sensors are identified by their label and color as there are temperature, proximity, humidity, water detection and light sensors. All sensors are integrated into the Unity application environment where the BIM model of the house lives. ( a ) aVirtual living room. (b) Real living room. Figure 5 : 5Comparison of furnished living room. Figure 7 : 7Navigating UI panels in Oculus Quest 2 VR. • Dense layer neurons: 64 Figure 8 : 8High level view of the forecasting pipeline. Figure 9 : 9High level view of the prediction pipeline. Figure 13 : 13Demonstration of the external environment using a standalone DT. (a) Entrance. (b) Bathroom. (c) Living room. (d) Living room. Figure 14 : 14Demonstration of the internal environment using a standalone DT.(a) Virtual office door. (b) Real office door. Figure 15 : 15Real-time states of 2OfficeDoor, 2Stair, HueColorLamp2 located in the second floor, observed 21.02.2022. Note that the color of the spheres follows the same standards set inFig. 4and their positions represent the 3D position of the sensor in the physical asset. Figure 16 : 16Demonstration of various door and light sensor states in the descriptive DT. Observations recorded throughout the day on 21.05.2022. clear temperature rise in the office when the door was opened. Since the sensor was placed under the table, it can sense the (a) Rain observed 24.03.2022. (b) Snow observed 03.04.2022. (c) Fog observed 25.04.2022. (d) Cloudy observed 03.05.2022. (e) Clear sky observed 21.04.2022. (f) Wind effect. Figure 17 : 17Demonstration of weather conditions taking place in the descriptive DT. Real-time temperature/humidity data. (d) Another pastel painting. Figure 18 :Figure 19 : 1819Information from the remaining sensors and other objects, obtained using a virtual reality interface. Diagnostic information from fusing office temperature and office door proximity sensor. Scenario one recorded 21.01.2022. Figure 20 :Figure 21 : 2021Visualizing CO2 concentration from Netatmo Weather Station as fog in the Unity Game Engine based on 4 predefined intervals. Recorded 27.11.2021. Visualizing indoor temperature from Netatmo Weather station as fog by converting temperature in Celsius to an RGB color representation. 15.6 • C was observed 30.11.2021 at 07:30 AM, 20 • C at the same date 10:30 AM and 35 • C was not an observation but a simulated scenario to display how that would look like in the event of such an occurrence. Figure 22 : 22Temperature predictions at different sensor locations when the temperature profile at the fireplace is known. This approach can be used to fill in missing data due to sensor malfunction or should need to decommission some of the sensors arises in the future. TheFig.shows the performance of different regression models. The best model is the weight averaging ensemble fromFig. 9, which weighs the contribution of each model based on their validation set RMSE. Figure 24 : 24Using the output of the fireplace forecaster in Figure 25 :Figure 26 : 2526Using the forecasting model to forecast the next 24 hours in the future (29.03.2022), visualized in the temperature heat map made for the diagnostic DT. The visualization shows how the model believes that the temperature on the second floor will develop the next day.-DQ )HE 0DU $SU 0D\ -XQ -XO $XJ 6HS 2FW 1RY 'Predictive sun position model used to give a range of sunlight for the last day of each month in the year 2022 from the location of the house. Figure 27 :Figure 28 : 2728Sunlight simulation in Unity using the sun position algorithm. This observation was made at 07.03.2022. Note that the view is from the balcony of the virtual house placed in the correct altitude, rotation and geographic location on a terrain generated based on a Trondheim height map from Kartverket. Recommendation to turn on the fireplace. Performance for day 23 of the dataset. 0 corresponds to midnight and the timestep corresponds to 5 minutes resulting in 288 data points, which corresponds to the 24th hour of the inhabitants or at the least make the manufacturer's guarantee on the equipment null and void. Table 1 : 1Comparing current work with industry DT services.Comparing Digital Twin Solutions Capability Standalone Descriptive Diagnostic Predictive Prescriptive Current work X Figure 10: User based collaborative filtering pipeline.Input Outdoor Forecast or What If? Scenario 0.48 RMSE 2.13 RMSE 5.75 RMSE 1.43 RMSE 2.45 RMSE 1.00 Pearson Correlation 0.84 Pearson Correlation … … -0.74 Pearson Correlation } Find K Nearest Neighbour Similar Recorded Outdoor Temperatures To Input Outdoor Temperature less than an RMSE Find K Nearest Neighbour Similar Users To a Specific User Based on the Pearson Correlation (The First User In This Example) } Weight Averaging kNN predicted timesteps weighted by Pearson correlation score Weight Averaging kNN columns of historical fireplace events for similar user weighted by RMSE score Predicted timestep for each user Final Recommendation For HomeOwner 1 + … … … … … Homeowner (user) Days (items) 08.01.2022 09.01.2022 10.01.2022 11.01.2022 19.04.2022 … Fireplace Not Used Fireplace Not Used … … … … Fireplace Not Used Fireplace Not Used 16:20:00 (196) 06:40:00 (80) 07:15:00 (87) 16:30:00 (198) 15:35:00 (187) 07:55:00 (95) Fireplace Not Used Fireplace Not Used 20:50:00 (250) Fireplace Not Used Fireplace Not Used ARI MA AutoRegressive Integrated Moving Average D/M/Y Input Day/Month/Year RMS E Root Mean Squared Error t LS R Local Sidereal Time t S RUT Sidereal Time Greenwich for Universal time t S R FEM Finite Element Method GBM Gradient Boosting Machine HAM Hybrid Analysis and Modeling HDRP High Definition Render Pipeline HSV Hue Saturation ValueNomenclature α Right Ascension β Sun's Ecliptic Latitude δ Declination λ Sun's Ecliptic Longitude Ω The Earth's Obliquity of the Ecliptic φ Azimuth from North θ Altitude B Input Latitude C Sun's Center HA Hour Angle of Object JC Julian Century JD Julian Date L Input Longitude L 0 Sun's Mean Longitude M 0 Sun's Mean Anomaly Sidereal Time Greenwich t UT Universal Time ADAM Adaptive Moment Estimation AI Artificial Intelligence API Application Program Interface AR AutoRegressive BIM Building Information Model CAD Computer Aided Design CFD Computational Fluid Dynamics DDM Data-Driven Modeling DT Digital Twin I Integration IoT Internet of Things LSTM Long Short Time Memory MA Moving Average ML Machine Learning MPC Model Predictive Control MSE Mean Squared Error PBM Physics-Based Modeling RGB Red Green Blue RL Reinforcement Learning RNN Recurrent Neural Network UBCF User-Based Collaborative Filtering UI User Interface URP Universal Render Pipeline VR Virtual Reality Development of the simulation model for Digital Twin applications in historical masonry buildings: The integration between numerical and experimental reality. 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[ "PACKING TOPOLOGICAL ENTROPY FOR AMENABLE GROUP ACTIONS", "PACKING TOPOLOGICAL ENTROPY FOR AMENABLE GROUP ACTIONS" ]
[ "Dou Dou ", "Dongmei Zheng ", "Xiaomin Zhou " ]
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Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper, we will give a systematically study to the packing topological entropy for a continuous G-action dynamical system (X, G), where X is a compact metric space and G is a countable infinite discrete amenable group.We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. And then we obtain an entropy inequality concerning amenable packing entropy. Finally we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure µ coincides with the metric entropy if either µ is ergodic or the system satisfies a kind of specification property.2020 Mathematics Subject Classification. Primary: 37B40, 28D20, 37A15.
10.1017/etds.2021.126
[ "https://arxiv.org/pdf/2010.14719v2.pdf" ]
225,094,309
2010.14719
af42ceb302e93f9e96378ec87ea27ad57685dce6
PACKING TOPOLOGICAL ENTROPY FOR AMENABLE GROUP ACTIONS Dou Dou Dongmei Zheng Xiaomin Zhou PACKING TOPOLOGICAL ENTROPY FOR AMENABLE GROUP ACTIONS Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper, we will give a systematically study to the packing topological entropy for a continuous G-action dynamical system (X, G), where X is a compact metric space and G is a countable infinite discrete amenable group.We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. And then we obtain an entropy inequality concerning amenable packing entropy. Finally we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure µ coincides with the metric entropy if either µ is ergodic or the system satisfies a kind of specification property.2020 Mathematics Subject Classification. Primary: 37B40, 28D20, 37A15. Introduction In 1973, in his profound and lasting paper [3], Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension, which is now known as Bowen topological entropy or dimensional entropy. For dynamical systems over compact Hausdorff spaces, Bowen showed that Bowen topological entropy on the whole space coincides with the Adler-Konheim-McAndrew topological entropy defined through open covers. Bowen topological entropy can be viewed as a dynamical analogy of Hausdorff dimension and it exhibits very deep connections with dimension theory in dynamical system and multifractal analysis ever since its appearance (see, for example, [19]). It is natural to consider the analogous concepts in dynamical systems for other forms of dimensions. For pointwise dimension (of a measure), its dynamical correspondence is the Brin-Katok's local entropy ( [4]). For packing dimension, its dynamical correspondence is the packing topological entropy, which was introduced by Feng and Huang [8]. Applying the methods in geometric measure theory, they also provided variational principles for Bowen topological entropy and packing topological entropy. Other works on packing topological entropy can be found in [27], where the packing topological entropy for certain noncompact subsets was considered. In this paper, we will focus on packing topological entropy in the framework of countable discrete amenable group actions. Amenable packing entropy and local entropies. Let (X, G) be a G−action topological dynamical system, where X is a compact metric space with metric d and G is a topological group acting continuously on X. Throughout this paper, we always assume that G is a countable infinite discrete amenable group unless it is specially mentioned. Recall that a countable discrete group G is amenable if there is a sequence of non-empty finite subsets {F n } of G which are asymptotically invariant, i.e. lim n→+∞ |F n gF n | |F n | = 0, for all g ∈ G. Such sequences are called Følner sequences. One may refer to [14,18] for more details on amenable groups and their actions. A Følner sequence {F n } in G is said to be tempered if there exists a constant C > 0 which is independent of n such that | k<n F −1 k F n | ≤ C|F n |, for any n. Let ε > 0 and x in X, we denote by B F (x, ε) = {y ∈ X : d F (x, y) < ε} and B F (x, ε) = {y ∈ X : d F (x, y) ≤ ε}, which are respectively the open and closed (F -)Bowen balls with center x and radius ε. When we want to clarify the underlying metric d, we also denote the above balls by B F (x, ε, d) and B F (x, ε, d). For a Z-action (or N-action) topological dynamical system (X, T ) (T is the homeomorphism (or the continuous onto map) on X), let F = {0, 1, · · · , n − 1} := [0, n − 1]. The (F -)Bowen balls will be written as B n (x, ε, T ) or B n (x, ε, d). Definition 1.1. Let {F n } be a sequence of finite subsets of G with |F n | → ∞ (need not to be Følner). For Z ⊆ X, s ≥ 0, N ∈ N and ε > 0, define P (Z, N, ε, s, {F n }) = sup i exp − s|F n i | , where the supremum is taken over all finite or countable pairwise disjoint families {B Fn i (x i , ε)} such that x i ∈ Z, n i ≥ N for all i. The quantity P (Z, N, ε, s, {F n }) does not increase as N increases, hence the following limit exists: Define P(Z, ε, s, {F n }) = inf{ +∞ i=1 P (Z i , ε, s, {F n }) : +∞ i=1 Z i ⊃ Z}. It is easy to see that if Z ⊆ +∞ i=1 Z i , then P(Z, ε, s, {F n }) ≤ +∞ i=1 P(Z i , ε, s, {F n }). There exists a critical value of the parameter s, which we will denote by h P top (Z, ε, {F n }), where P(Z, ε, s, {F n }) jumps from +∞ to 0, i.e., P(Z, ε, s, {F n }) = 0, s > h P top (Z, ε, {F n }), +∞, s < h P top (Z, ε, {F n }). It is not hard to see that h P top (Z, ε, {F n }) increases when ε decreases. We call h P top (Z, {F n }) := lim which are called the upper local entropy and the lower local entropy of µ over Z (w.r.t {F n }), respectively. For Z-action or N-action case, if the sequence {F n } is chosen to be F n = [0, n − 1] (hence {F n } is naturally a Følner sequence), the local entropies of a topological dynamical system (X, T ) will be denoted by h loc µ (Z, T ) and h loc µ (Z, T ), respectively. By the Brin-Katok's entropy formula for amenable group actions (see [26]), if µ is in addition G-invariant and {F n } is a tempered Følner sequence which satisfies the growth condition (1.2) lim n→+∞ |F n | log n = +∞, then the values of the upper and lower local entropies over the whole space X coincide with the measure-theoretic entropy of the system (X, G). We will prove the following variational principle between amenable packing entropy and upper local entropy. Theorem 1.3. Let (X, G) be a G−action topological dynamical system and G a countable infinite discrete amenable group. Let {F n } be a sequence of finite subsets in G satisfying the growth condition (1.2). Then for any non-empty Borel subset Z of X, h P top (Z, {F n }) = sup{h loc µ (Z, {F n }) : µ ∈ M (X), µ(Z) = 1}. 1.2. Amenable packing entropy inequalities via factor maps. Let (X, G) and (Y, G) be two G-action topological dynamical systems. A continuous map π : (X, G) → (Y, G) is called a homomorphism or a factor map from (X, G) to (Y, G) if it is onto and π • g = g • π, for all g ∈ G. We also say that (X, G) is an extension of (Y, G) or (Y, G) is a factor of (X, G). For a subset Z of X, we denote by h U C top (Z, {F n }) the upper capacity topological entropy of Z (defined in Section 2). We will show in section 2 that the packing entropy can be estimated via the upper capacity topological entropy with parameters (Proposition 2.7). This is a dynamical version of the fact that the packing dimension can be defined via the upper Minkowski dimension. Applying Proposition 2.7, we can prove the following packing entropy inequalities for factor maps. Theorem 1.4. Let G be a countable infinite discrete amenable group and π : (X, G) → (Y, G) be a factor map between two G-action topological dynamical systems. Let {F n } be any tempered Følner sequence in G satisfying the growth condition (1.2). Then for any Borel subset E of X, (1.3) h P top (π(E), {F n }) ≤ h P top (E, {F n }) ≤ h P top (π(E), {F n }) + sup y∈Y h U C top (π −1 (y), {F n }). We remark here that for Z-actions, the inequalities was proved in [24]. But for amenable group actions, except employing Bowen's idea in [2] and the quasi-tiling techniques developed by Ornstein and Weiss [18], we need a crucial covering lemma for amenable groups built by Lindenstrauss while proving pointwise theorems for amenable groups in [15]. Amenable packing entropy for certain subsets. Let M (X, G) and E(X, G) be the collection of G-invariant and ergodic G-invariant Borel probability measures on X respectively. Since G is amenable, M (X, G) and E(X, G) are both non-empty. For µ ∈ M (X, G), let h µ (X, G) denote the measuretheoretic entropy of (X, G) w.r.t. µ. For µ ∈ M (X, G) and a Følner sequence {F n } in G, let G µ,{Fn} be the set of generic points for µ (w.r.t. {F n }), which is defined by G µ,{Fn} = {x ∈ X : lim n→+∞ 1 |F n | g∈Fn f (gx) = X f dµ, for any f ∈ C(X)}. For simplicity, we write G µ,{Fn} by G µ when there is no ambiguity on {F n }. But we should note that for different Følner sequence {F n }, the corresponding G µ may not coincide. When µ is ergodic and the Følner sequence {F n } is tempered, G µ has full measure for µ (see Remark 5.1). But when µ is not ergodic, the set G µ could be empty. For the case G = Z, Bowen [3] proved that the Bowen topological entropy of G µ equals the measure-theoretic entropy of µ if µ is ergodic. Pfister and Sullivan [20] extended Bowen's result to the system with the so called g-almost product property for invariant Borel probability measure µ. And the results for amenable group action version were proved by Zheng and Chen [26] and Zhang [23] respectively. The g-almost product property (see [20]) is an extension of the specification property for Z-systems and was generalized to amenable systems in [23] (called almost specification property there). It was shown in [23] that weak specification implies almost specification for amenable systems. We leave the detailed definitions of almost specification and weak specification in section 5. We will prove for packing entropy the following theorem. Theorem 1.5. Let (X, G) be a G−action topological dynamical system with G a countable infinite discrete amenable group and let µ ∈ M (X, G) and {F n } be a Følner sequence in G satisfying the growth condition (1.2). If either µ is ergodic and {F n } is tempered or (X, G) satisfies the almost specification property, then h P top (G µ , {F n }) = h µ (X, G). (1.4) In geometric measure theory, a set is said to be regular (or "dimension-regular") if it has equal Hausdorff and packing dimensions ( [22]). As a counterpart in dynamical systems, we have the following definition. Definition 1.6. A subset is said to be regular in the sense of dimensional entropy (or regular, for simplicity) if it has equal Bowen entropy and packing entropy. An affirmative task in the study of packing entropy is to compute the dimensional entropies (including packing entropy and its dual, Bowen entropy) for various subsets and to make clear which subsets are regular. From Theorem 1.5 and results in [26,23] for Bowen entropy, under the conditions in Theorem 1.5, the set G µ is regular since both of its dimensional entropies equal the measure-theoretic entropy for µ. Organization of the paper. In section 2, we will give some properties of amenable packing entropy including connections with Bowen entropy and upper capacity entropy. In section 3, we will devote to the proof of Theorem 1.3, the variational principle between amenable packing entropy and upper local entropy. This extends Feng and Huang's result in [8] from Z-actions to amenable group actions. In section 4, we are going to prove the packing entropy inequalities for factor maps (Theorem 1.4). Finally in section 5, we will give the proof of Theorem 1.5 and provide some examples to discuss the regularity for certain subsets. These examples include subsets of symbolic dynamical system and fibers of the (T, T −1 ) transformation. Some detailed computations via these examples are included in Appendix A. Properties of amenable packing entropy Due to the definition of packing entropy in section 1, it is not hard to prove that the packing entropy has the following properties. Proposition 2.1. Let {F n } be a sequence of finite subsets in G with |F n | → ∞, Z, Z and Z i (i = 1, 2, · · · ) be subsets of X. (1) If Z ⊆ Z , then h P top (Z, {F n }) ≤ h P top (Z , {F n }). (2) If Z ⊆ +∞ i=1 Z i , then for any ε > 0, h P top (Z, ε, {F n }) ≤ sup i≥1 h P top (Z i , ε, {F n }). Hence h P top (Z, {F n }) ≤ sup i≥1 h P top (Z i , {F n }). (3) If {F n k } is a subsequence of {F n }, then h P top (Z, {F n k }) ≤ h P top (Z, {F n }). In the following we recall the definition of amenable Bowen topological entropy which was introduced in [25]. Let {F n } be a sequence of finite subsets in G with |F n | → ∞. For Z ⊆ X, s ≥ 0, N ∈ N and ε > 0, define M(Z, N, ε, s, {F n }) = inf i exp(−s|F n i |), where the infimum is taken over all finite or countable families {B Fn i (x i , ε)} such that x i ∈ X, n i ≥ N and M(Z, s, {F n }) = 0, s > h B top (Z, {F n }), +∞, s < h B top (Z, {F n }). Next we will compare packing topological entropy with Bowen topological entropy. Let {Z i } +∞ i=1 be any covering of Z. For each i, for any n ∈ N and ε > 0, let {B Fn (x i,j , ε)} N i j=1 be a disjoint subfamily of {B Fn (x, ε)} x∈Z i with maximal cardinality N i . Then N i i=1 B Fn (x i,j , 3ε) ⊇ Z i . So M(Z i , n, 3ε, s, {F n }) ≤ N i e −|Fn|s ≤ P (Z i , n, ε, s, {F n }), and hence M(Z i , 3ε, s, {F n }) ≤ P (Z i , ε, s, {F n }). Thus M(Z, 3ε, s, {F n }) ≤ +∞ i=1 M (Z i , 3ε, s, {F n }) ≤ +∞ i=1 P (Z i , ε, s, {F n }), which deduces that M(Z, 3ε, s, {F n }) ≤ P(Z, ε, s, {F n }). Since s < h B top (Z, {F n }), we have M(Z, s, {F n }) = +∞. So 1 ≤ M(Z, 3ε, s, {F n }) ≤ P(Z, ε, s, {F n })h µ (X, G) ≤ h P top (Z, {F n k }) ≤ h P top (Z, {F n }). Let ε > 0, Z ⊆ X and F ∈ F (G). A subset E ⊆ Z is said to be an (F, ε)-separated set of Z, if for any two distinct points x, y ∈ E, d F (x, y) > ε. Let s F (Z, ε) denote the largest cardinality of (F, ε)-separated sets for Z. A subset E ⊆ X is said to be an (F, ε)-spanning set of Z, if for any x ∈ Z, there exists y ∈ E with d F (x, y) ≤ ε. Let r F (Z, ε) (sometimes we use r F (Z, ε, d) to indicate the accompanied metric d) denote the smallest cardinality of (F, ε)-spanning sets for Z. For a sequence of finite subsets {F n } in G with |F n | → ∞, the upper capacity topological entropy of Z is defined as h U C top (Z, {F n }) = lim ε→0 lim sup n→+∞ 1 |F n | log s Fn (Z, ε) = lim ε→0 lim sup n→+∞ 1 |F n | log r Fn (Z, ε). The second equality comes from the following simple fact: r Fn (Z, 2ε) ≤ s Fn (Z, 2ε) ≤ r Fn (Z, ε). (2.1) For convention we denote by h U C top (Z, ε, {F n }) = lim sup n→+∞ 1 |F n | log s Fn (Z, ε), and then h U C top (Z, {F n }) = lim ε→0 h U C top (Z, ε, {F n }) . We note here that for the case Z = X and {F n } is a Følner sequence, the quantity h U C top (X, {F n }) coincides with h top (X, G), the topological entropy of (X, G). Proposition 2.4. Let {F n } be a sequence of finite subsets in G satisfying the growth condition (1.2), then for any subset Z of X and any ε > 0, h P top (Z, ε, {F n }) ≤ h U C top (Z, ε, {F n }). Hence h P top (Z, {F n }) ≤ h U C top (Z, {F n }). Proof. Let ε > 0 be fixed. The proposition is obvious for the case h P top (Z, ε, {F n }) = 0. Since {F n } satisfies the growth condition lim n→+∞ |Fn| log n = +∞, k≥1 e |F k |(t−s) converges. Let M = k≥1 e |F k |(t−s) . There must be some k ≥ N such that m k > 1 M e |F k |t , otherwise the above sum is at most k≥1 1 M e |F k |t e −|F k |s = 1. So s F k (Z, ε) ≥ m k > 1 M e |F k |t and hence h U C top (Z, ε, {F n }) = lim sup k→+∞ 1 |F k | log s F k (Z, ε) ≥ t, which deduces that h P top (Z, ε, {F n }) ≤ h U C top (Z, ε, {F n }). As a corollary, we have Corollary 2.5. If {F n } is a Følner sequence that satisfies the growth condition (1.2), then h P top (X, {F n }) = h top (X, G). Proof. By Corollary 2.3, for any µ ∈ M (X, G), h µ (X, G) ≤ h P top (X, {F n }). Applying the variational principle for amenable topological entropy (cf. [17,21] ), we have h P top (X, {F n }) ≥ h top (X, G). By Proposition 2.4, we have h P top (X, {F n }) ≤ h top (X, G). Remark 2.6. (1) By Theorem 1.1 of [7] (see also [25]), if {F n } is a tempered Følner sequence and satisfies the growth condition (1.2), then h P top (X, {F n }) = h top (X, G) = h B top (X, {F n }). (2) If we replace X by a G-invariant compact subset, the above equality also holds and hence any G-invariant compact subset is regular (when the Følner sequence {F n } is tempered and satisfies the growth condition (1.2)). At the end of this section, we will give further relations between packing entropy and upper capacity topological entropy. Proposition 2.7. Let ε > 0, Z be a subset of X and let {F n } be a sequence of finite subsets in G satisfying the growth condition (1.2). (1) h P top (Z, ε, {F n }) ≤ inf{sup i≥1 h U C top (Z i , ε, {F n }) : Z = ∪ ∞ i=1 Z i }. Hence h P top (Z, {F n }) ≤ inf{sup i≥1 h U C top (Z i , {F n }) : Z = ∪ ∞ i=1 Z i }. (2) For any δ > 0, there exists a cover ∪ ∞ i=1 Z i = Z (which depends on both ε and δ) such that h P top (Z, ε, {F n }) + δ ≥ sup i≥1 h U C top (Z i , 3ε, {F n }). Proof. For any Z = ∪ ∞ i=1 Z i , by Propositions 2.1 and 2.4, h P top (Z, ε, {F n }) ≤ sup i≥1 h P top (Z i , ε, {F n }) ≤ sup i≥1 h U C top (Z i , ε, {F n }). Hence h P top (Z, ε, {F n }) ≤ inf{sup i≥1 h U C top (Z i , ε, {F n }) : Z = ∪ ∞ i=1 Z i }. For the opposite direction, we may assume that h P top (Z, ε, {F n }) < ∞. Let δ > 0 be fixed and set s = h P top (Z, ε, {F n }) + δ. By the definition of the amenable packing entropy, we have that P(Z, ε, s, {F n }) = 0. Then there exists a cover ∪ ∞ i=1 Z i ⊇ Z such that i≥1 P (Z i , ε, s, {F n }) < 1. For each Z i , when N is large enough, we have P (Z i , N, ε, s, {F n }) < 1. Let E be any (F N , 3ε)-separated subset of Z i . Noting that the closed Bowen balls B F N (x i , ε) (x i ∈ E ⊂ Z i ) are pairwise disjoint, we have |E|e −s|F N | = x i ∈E e −s|F N | ≤ P (Z i , N, ε, s, {F n }) < 1. Hence s F N (Z i , 3ε) < e s|F N | , which leads to h U C top (Z i , 3ε, {F n }) ≤ s. Thus we have h P top (Z, ε, {F n }) + δ ≥ sup i≥1 h U C top (Z i , 3ε, {F n }). Remark 2.8. Proposition 2.4 is motivated from the equivalent definition of packing dimension through Minkowski dimension in geometric measure theory, which was due to Tricot [22] (see also [16] for reference). It is unclear for us whether it holds that h P top (Z, {F n }) = inf{sup i≥1 h U C top (Z i , {F n }) : Z = ∪ ∞ i=1 Z i }. A variational principle for amenable packing entropy In this section we will give the proof of our Theorem 1.3, the variational principle for amenable packing topological entropy. We assume in this section {F n } is a sequence of finite subsets in G and satisfies the growth condition (1.2). Lower bound. Proposition 3.1. Let Z ⊆ X be a Borel set, then h P top (Z, {F n }) ≥ sup{h loc µ (Z, {F n }) : µ ∈ M (X), µ(Z) = 1}. For the proof, we need the following classical 5r-Lemma in geometric measure theory (cf. [16], Theorem 2.1). Lemma 3.2 (5r-Lemma). Let (X, d) be a compact metric space and B = {B(x i , r i )} i∈IB = {B(x i , r i )} i∈I of pairwise disjoint balls in B such that B∈B B ⊆ i∈I B(x i , 5r i ). We also need the following lemma, which comes from the definition of the packing entropy directly. Lemma 3.3. Let E ⊂ X and s > 0. Then for any 0 < ε 1 < ε 2 , P (E, ε 2 , s, {F n }) ≤ P (E, ε 1 , s, {F n }). Proof of Proposition 3.1. Let µ ∈ M (X) with µ(Z) = 1 and assume h loc µ (Z, {F n }) > 0. Let 0 < s < h loc µ (Z, {F n }). Then there exist ε, δ > 0 and a Borel set A ⊂ Z with µ(A) > 0 such that for every x ∈ A, it holds that h µ (x, ε, {F n }) > s + δ, where h µ (x, ε, {F n }) := lim sup n→+∞ − 1 |Fn| log µ(B Fn (x, ε)). Let E ⊂ A be any Borel set with µ(E) > 0. Define E n = {x ∈ E : µ(B Fn (x, ε)) < e −|Fn|(s+δ) }, n ∈ N. Then +∞ n=N E n = E for any N ∈ N, and hence µ( +∞ n=N E n ) = µ(E). Fix N ∈ N, there exists n ≥ N such that µ(E n ) ≥ 1 n(n + 1) µ(E). Fix such n and consider the family of Bowen balls {B Fn (x, ε 5 ) : x ∈ E n }. By the 5r-Lemma 3.2 (we use metric d Fn instead of d), there exists a finite pairwise disjoint family {B Fn (x i , ε 5 )} with x i ∈ E n such that ∪ i B Fn (x i , ε) ⊃ ∪ x∈En B Fn (x, ε 5 ) ⊃ E n . Thus P (E, N, ε 5 , s, {F n }) ≥ P (E n , N, ε 5 , s, {F n }) ≥ i e −|Fn|s = e |Fn|δ i e −|Fn|(s+δ) ≥ e |Fn|δ i µ(B Fn (x i , ε)) ≥ e |Fn|δ µ(E n ) ≥ e |Fn|δ n(n + 1) µ(E). By the growth condition (1.2) of the sequence {F n }, we have e |Fn|δ n(n + 1) → +∞, as n → +∞. Letting N → +∞, we obtain that P (E, ε 5 , s, {F n }) = +∞. Note that this equality holds for every Borel set E ⊂ A with µ(E) > 0. Let {A i } ∞ i=1 be any covering of A. Then by Lemma 3.3, i P (A i , ε 10 , s, {F n }) ≥ i P (A i , ε 5 , s, {F n }). Since there must exist some A i such that A i ∩ A (which is a Borel set now) contains a Borel subset E ⊂ A i ∩ A with µ(E) > 0, we have i P (A i , ε 10 , s, {F n }) ≥ P (E, ε 5 , s, {F n }) = +∞. Thus P(Z, ε 10 , s, {F n }) ≥ P(A, ε 10 , s, {F n }) = +∞, which deduces that h P top (Z, {F n }) ≥ h P top (Z, ε 10 , {F n }) ≥ s. Since s is chosen arbitrarily in (0, h µ (Z, {F n })), we finally show that h P top (Z, {F n }) ≥ h loc µ (Z, {F n }) . This finishes the proof of Proposition 3.1. Upper bound. The following Proposition is the upper bound part of the variational principle. In fact it is valid for any analytic set Z. Recall that a set in a metric space is said to be analytic if it is a continuous image of the set N of infinite sequences of natural numbers. In a Polish space, the collection of analytic subsets contains Borel sets and is closed under countable unions and intersections (c.f. Chapter 11 of [11]). Proposition 3.4. Let Z ⊆ X be an analytic set with h P top (Z, {F n }) > 0. For any 0 < s < h P top (Z, {F n }), there exists a compact set K ⊆ Z and µ ∈ M (K) such that h loc µ (K, {F n }) ≥ s. The following lemma is needed. Lemma 3.5. Let Z ⊆ X and s, ε > 0. Assume that P(Z, ε, s, {F n }) = +∞. Then for any given finite interval (a, b) ⊂ R with a ≥ 0 and any N ∈ N, there exists a finite disjoint collection {B Fn i (x i , ε)} such that x i ∈ Z, n i ≥ N and i exp(−|F n i |s) ∈ (a, b). Proof. See Lemma 4.1 of [8]. Proof of Proposition 3.4. Since Z is analytic, there exists a continuous surjective map φ : N → Z. Let Γ n 1 ,n 2 ,...,np = {(m 1 , m 2 , . . .) ∈ N : m 1 ≤ n 1 , m 2 ≤ n 2 , . . . , m p ≤ n p } and let Z n 1 ,...,np = φ(Γ n 1 ,...,np ). For 0 < s < h P top (Z, {F n }), take ε > 0 small enough to make 0 < s < h P top (Z, ε, {F n }) and take t ∈ (s, h P top (Z, ε, {F n })). Followed by Feng-Huang's steps (which are inspired by the work of Joyce and Preiss [12] on packing measures), we will construct inductively the following data. (D-1) A sequence of finite sets (K i ) +∞ i=1 with K i ⊂ Z. (D-2) A sequence of finite measures (µ i ) +∞ i=1 with each µ i supported on K i . (D-3) A sequence of integers (n i ) +∞ i=1 and a sequence of positive numbers (γ i ) +∞ i=1 . (D-4) A sequence of integer-valued functions (m i : K i → N) +∞ i=1 . Moreover, the sequences (K i ), (µ i ), (n i ), (γ i ) and (m i (·)) will be constructed to satisfy the following conditions. (C-1) For each i, the family V i := {B(x, γ i )} x∈K i is disjoint. Each element in V i+1 is a subset of B(x, γ i /2) for some x ∈ K i . (C-2) For each i, K i ⊂ Z n 1 ,...,n i and µ i = y∈K i e −|F m i (y) |s δ y with 1 < µ 1 (K 1 ) < 2. (C-3) For each x ∈ K i and z ∈ B(x, γ i ), B F m i (x) (z, ε) ∩ y∈K i \{x} B(y, γ i ) = ∅ (3.1) and µ i (B(x, γ i )) = e −|F m i (x) |s ≤ y∈E i+1 (x) e −|F m i+1 (y) |s < (1 + 2 −(i+1) )µ i (B(x, γ i )), (3.2) where E i+1 (x) = B(x, γ i ) ∩ K i+1 . We will give the construction later. Suppose the sequences (K i ), (µ i ), (n i ), (γ i ) and (m i (·)) have been constructed. By (3.2), for V i ∈ V i , µ i (V i ) ≤ µ i+1 (V i ) = V ∈V i+1 ,V ⊂V i µ i+1 (V ) ≤ (1 + 2 −(i+1) )µ i (V i ). Using the above inequalities repeatedly, we have for any j > i and any V i ∈ V i , µ i (V i ) ≤ µ j (V i ) ≤ j n=i+1 (1 + 2 −n )µ i (V i ) ≤ Cµ i (V i ), (3.3) where C := +∞ n=1 (1 + 2 −n ) < +∞. Letμ be a limit point of (µ i ) in the weak * topology. Let K = +∞ n=1 i≥n K i . Thenμ is supported on K. Furthermore, K ⊂ +∞ p=1 Z n 1 ,...,np . By the continuity of φ, applying Cantor's diagonal argument, we can show that +∞ p=1 Z n 1 ,...,np = +∞ p=1 Z n 1 ,...,np . Hence K is a compact subset of Z. By (3.3), for any x ∈ K i , e −|F m i (x) |s = µ i (B(x, γ i )) ≤μ(B(x, γ i )) ≤ Cµ i (B(x, γ i )) = Ce −|F m i (x) |s . In particular, 1 ≤ x∈K 1 µ 1 (B(x, γ 1 )) ≤μ(K) ≤ x∈K 1 Cµ 1 (B(x, γ 1 )) ≤ 2C. Note that K ⊂ x∈K i B(x, γ i /2). By (3.1), the first part of (C-3), for each x ∈ K i and z ∈ B(x, γ i ),μ (B F m i (x) (z, ε)) ≤μ(B(x, γ i /2)) ≤ Ce −|F m i (x) |s . For each z ∈ K and i ∈ N, z ∈ B(x, γ i /2) for some x ∈ K i . Hencẽ µ(B F m i (x) (z, ε)) ≤ Ce −|F m i (x) |s . Define µ =μ/μ(K). Then µ ∈ M (K). For each z ∈ K, there exists a sequence k i ↑ +∞ such that µ(B F k i (z, ε)) ≤ Ce −|F k i |s /μ(K). Hence h loc µ (K, {F n }) ≥ s. Now the only thing left is to give the inductive construction of the data (K i ), (µ i ), (n i ), (γ i ) and (m i (·)). The inductive steps are as follows. Step 1. Construct the data K 1 , µ 1 , n 1 , γ 1 and m 1 (·). Since t ∈ (s, h P top (Z, ε, {F n })), we have that P(Z, ε, t, {F n }) = +∞. Let H = ∪{U ⊂ X : U is open, P(Z ∩ U, ε, t, {F n }) = 0}. Then by the separability of X, H is a countable union of the open sets U 's. Hence Define µ 1 = x∈K 1 e −|F m 1 (x) |s δ x , where δ x denotes the Dirac measure at x. Take γ 1 > 0 sufficiently small such that for any function z : K 1 → X with max x∈K 1 d(x, z(x)) ≤ γ 1 , we have for each x ∈ K 1 , P(Z ∩ H, ε, t, {F n }) = 0. Let Z = Z \ H = Z ∩ (X \ H). If P(Z ∩ U,B(z(x), γ 1 ) ∪ B F m 1 (x) (z(x), ε) ∩ ∪ y∈K 1 \{x} B(z(y), γ 1 ) ∪ B F m 1 (y) (z(y), ε) = ∅. (3.4) Since K 1 ⊂ Z , we have P(Z∩B(x, γ 1 /4), ε, t, {F n }) ≥ P(Z ∩B(x, γ 1 /4), ε, t, {F n }) > 0 for each x ∈ K 1 . Therefore we can pick a sufficiently large n 1 ∈ N so that Z n 1 ⊃ K 1 and P(Z n 1 ∩ B(x, γ 1 /4), ε, t, {F n }) > 0 for each x ∈ K 1 . Step 2. Construct the data K 2 , µ 2 , n 2 , γ 2 and m 2 (·). By (3.4), the family of balls {B(x, γ 1 )} x∈K 1 are pairwise disjoint. For each x ∈ K 1 , since P(Z n 1 ∩ B(x, γ 1 /4), ε, t, {F n }) > 0, similar as Step 1, we can construct a finite set E 2 (x) ⊂ Z n 1 ∩B(x, γ 1 /4) and an integer-valued function m 2 : E 2 (x) → N∩[max{m 1 (y) : y ∈ K 1 }, +∞) such that To see this, we fix x ∈ K 1 . Denote V = Z n 1 ∩ B(x, γ 1 /4). Let Set V = V \ H x . Then as in Step 1, we can show that P(V , ε, t, {F n }) = P(V, ε, t, {F n }) > 0. Moreover, for any open set U ⊂ X, either V ∩ U = ∅ or P(V ∩ U, ε, t, {F n }) > 0. Since s < t, it holds that P(V , ε, s, {F n }) = +∞. By Lemma 3.5, we can find a finite set E 2 (x) ⊂ V and a map m 2 : E 2 (x) → N ∩ [max{m 1 (y) : y ∈ K 1 }, +∞) such that (2-b) holds. Notice that if an open set U satisfies U ∩ E 2 (x) = ∅, then U ∩ V = ∅. So P(Z n 1 ∩ U, ε, t, {F n }) ≥ P(V ∩ U, ε, t, {F n }) > 0. Hence (2-a) holds. Since the family {B(x, γ 1 )} x∈K 1 is disjoint, so is the family {E 2 (x)} x∈K 1 . Define K 2 = ∪ x∈K 1 E 2 (x) and µ 2 = y∈K 2 e −|F m 2 (y) |s δ y . By (3.4) and (2-b), the elements in {B F m 2 (y) (y, ε)} y∈K 2 are pairwise disjoint. Hence we can take 0 < γ 2 < γ 1 /4 such that for any function z : K 2 → X satisfying that max x∈K 2 d(x, z(x)) < γ 2 , we have for x ∈ K 2 , B(z(x), γ 2 ) ∪ B F m 2 (x) (z(x), ε) ∩ ∪ y∈K 2 \{x} B(z(y), γ 2 ) ∪ B F m 2 (y) (z(y), ε) = ∅. (3.5) Choose a sufficiently large n 2 ∈ N so that Z n 1 ,n 2 ⊃ K 2 and P(Z n 1 ,n 2 ∩ B(x, γ 2 /4), ε, t, {F n }) > 0 for each x ∈ K 2 . Step 3. Next we suppose that the data K i , µ i , n i , γ i and m i (·) (i = 1, . . . , p) have been constructed. We will construct the data K p+1 , µ p+1 , n p+1 , γ p+1 and m p+1 (·). Assume that we have constructed K i , µ i , n i , γ i and m i (·) for i = 1, . . . , p. And assume that for any function z : K p → X with d(x, z(x)) < γ p for all x ∈ K p , we have B(z(x), γ p ) ∪ B F mp(x) (z(x), ε) ∩ ∪ y∈Kp\{x} B(z(y), γ p ) ∪ B F mp(y) (z(y), ε) = ∅ (3.6) for each x ∈ K p ; and Z n 1 ,...,np ⊃ K p and P(Z n 1 ,...,np ∩ B(x, γ p /4), ε, t, {F n }) > 0 for each x ∈ K p . Note that the family of balls {B(x, γ p )} x∈Kp are pairwise disjoint. For each x ∈ K p , since P(Z n 1 ,...,np ∩B(x, γ p /4), ε, t, {F n }) > 0, similar as Step 2, we can construct a finite set E p+1 (x) ⊂ Z n 1 ,...,np ∩ B(x, γ p /4) and an integer-valued function m p+1 : E p+1 (x) → N ∩ [max{m p (y) : y ∈ K p }, +∞) such that (3-b) the elements in the family {B F m p+1 (y) (y, ε)} y∈E p+1 (x) are pairwise disjoint and µ p ({x}) < y∈E p+1 (x) e −|F m p+1 (y) |s < (1 + 2 −(p+1) )µ p ({x}). By (3.6) and (3-b), the family {B F m p+1 (y) (y, ε)} y∈K p+1 is disjoint. Hence we can take 0 < γ p+1 < γ p /4 such that for any function z : K p+1 → X with max x∈K p+1 d(x, z(x)) < γ p+1 , we have for each x ∈ K p+1 , B(z(x), γ p+1 )∪B F m p+1 (x) (z(x), ε) ∪ y∈K p+1 \{x} B(z(y), γ p+1 ) ∪ B F m p+1 (y) (z(y), ε) = ∅. (3.7) Choose a sufficiently large n p+1 ∈ N so that Z n 1 ,...,n p+1 ⊃ K p+1 and P(Z n 1 ,...,n p+1 ∩ B(x, γ p+1 /4), ε, t, {F n }) > 0 for each x ∈ K p+1 . Then we finish the required construction and complete the proof of the Proposition. Packing entropy inequalities for factor maps In this section, we will prove Theorem 1.4. The proof is a combination of Bowen's method in Theorem 17 of [2] and quasi-tiling techniques for amenable groups. Preliminaries for amenable groups. Let G be a countable infinite discrete amenable group. Let A and K be two nonempty finite subsets of G. Recall that B(A, K), the K-boundary of A, is defined by B(A, K) = {g ∈ G : Kg ∩ A = ∅ and Kg ∩ (G \ A) = ∅}. For δ > 0, the set A is said to be (K, δ)-invariant if |B(A, K)| |A| < δ. We say a sequence of non-empty finite subsets {F n } of G becomes more and more invariant if for any δ > 0 and any non-empty finite subset K of G, F n is (K, δ)invariant for sufficiently large n. An equivalent condition for the sequence {F n } to be a Følner sequence is that {F n } becomes more and more invariant (see [18]). LetF be a collection of finite subsets of G. It is said to be δ-disjoint if for every A ∈F there exists an A ⊂ A such that |A | ≥ (1 − δ)|A| and such that A ∩ B = ∅ for every A = B ∈F. The following is a covering lemma for amenable groups by Lindenstrauss. Requirement 1. For every i,F i, * = {F i,j } N i j=1 satisfies | k <k F −1 i,k F i,k | ≤ C|F i,k |, for k = 2, . . . , N i . Requirement 2. Denote F i, * = ∪F i, * . The finite set sequences F i, * satisfy that for every 1 < i ≤ M and every 1 ≤ k ≤ N i , | i <i DF −1 i , * F i,k | ≤ (1 + δ)|F i,k |. Assume that A i,j is another array of finite subsets of G with F i,j A i,j ⊂ F for some finite subset F of G. Let A i, * = ∪ j A i,j and α = min 1≤i≤M |DA i, * | |F | . Then the collection of subsets of F , F = {F i,j a : 1 ≤ i ≤ M, 1 ≤ j ≤ N i and a ∈ A i,j } has a subcollection F that is 10δ 1/4 -disjoint such that | ∪ F| ≥ (α − δ 1/4 )|F |. Proof of Theorem 1.4. Let G, {F n } and π : (X, G) → (Y, G) be as in Theorem 1.4 and E ⊂ X be a Borel set. Let d and ρ be the compatible metrics on X and Y , respectively. For any ε > 0, there exists δ > 0 such that for any x 1 , x 2 ∈ X with d(x 1 , x 2 ) ≤ δ, one has ρ(π(x 1 ), π(x 2 )) ≤ ε. Now let {y i } k i=1 ⊂ π(E) be any (F n , ε)-separated set of π(E) and choose for each i a point x i ∈ π −1 (y i ) ∩ E, then {x i } k i=1 forms an (F n , δ)-separated set of E. Hence s Fn (π(E), ε) ≤ s Fn (E, δ) and h U C top (π(E), ε, {F n }) ≤ h U C top (E, δ, {F n }).∪ ∞ i=1 E i = E such that h P top (E, δ/3, {F n }) + η ≥ sup i≥1 h U C top (E i , δ, {F n }). Then we have h P top (π(E), ε, {F n }) ≤ sup i≥1 h P top (π(E i ), ε, {F n }) (by Proposition 2.1 (2)) ≤ sup i≥1 h U C top (π(E i ), ε, {F n }) (by Proposition 2.4) ≤ sup i≥1 h U C top (E i , δ, {F n }) (noting that (4.1) also holds for each E i ) ≤ h P top (E, δ/3, {F n }) + η, which implies that h P top (π(E), {F n }) ≤ h P top (E, {F n }) . To get the upper bound, we need to prove the following inequality for amenable upper capacity topological entropy, which extends a result by Bowen ([2], Theorem 17) to amenable group actions. Theorem 4.2. Let G be a countable infinite discrete amenable group and π : (X, G) → (Y, G) be a factor map between two G-action topological dynamical systems. Let {F n } be any tempered Følner sequence in G satisfying the growth condition (1.2). Then for any subset E of X, (4.2) h U C top (E, {F n }) ≤ h U C top (π(E), {F n }) + sup y∈Y h U C top (π −1 (y), {F n }). Proof. If sup y∈Y h U C top (π −1 (y), {F n }) = ∞ then there is nothing to prove. So we assume that a := sup y∈Y h U C top (π −1 (y), {F n }) < ∞. To verify (4.2), we need some preparations along the following three steps. Step 1. Construct F i,j , the array of subsets of G. Fix τ > 0. For any ε > 0, let 0 < δ < min{ε, 1/100} be small enough. Let C > 0 be the constant in the tempered condition (1.1) for the Følner sequence {F n } and let D = {e G } ⊂ G, where e G is the identity element of G. Let M > 0 be large enough as in Lemma 4.1 corresponding to δ, C and D. For each y ∈ Y , choose m(y) ∈ N such that for any n ≥ m(y), 1 |F n | log r Fn (π −1 (y), ε, d) ≤ h U C top (π −1 (y), {F n }) + τ ≤ a + τ. (4.3) Here recall that r Fn (π −1 (y), ε, d) denotes the smallest cardinality of (F n , ε)-spanning sets for π −1 (y). Let E y be an (F m(y) , ε)-spanning set of π −1 (y) with the smallest cardinality |E y | = r F m(y) (π −1 (y), ε, d). Denote U y = {p ∈ X : there exists q ∈ E y such that d F m(y) (p, q) < 2ε}, which is an open neighborhood of π −1 (y). Since γ>0 π −1 (B(y, γ, ρ)) = π −1 (y), we have (X \ U y ) γ>0 π −1 (B(y, γ, ρ)) = ∅. Hence by the finite intersection property of compact sets, there is a W y := B(y, γ y , ρ) for some γ y > 0 such that U y ⊃ π −1 (W y ). Since Y is compact, there exist y 1,1 , · · · , y 1,r 1 such that W y 1,1 , · · · , W y 1,r 1 cover Y . List the Følner sets in the collection {F m(y 1,k ) : 1 ≤ k ≤ r 1 } by F n 1,1 , F n 1,2 , · · · , F n 1,N 1 , where n 1,1 < n 1,2 < · · · < n 1,N 1 . Note that N 1 = #{F m(y 1,k ) : 1 ≤ k ≤ r 1 } ≤ r 1 . For each y ∈ Y , choose m(y) > n 1,N 1 such that (4.3) holds for any n ≥ m(y). Repeat the above process, we can obtain y 2,1 , · · · , y 2,r 2 ∈ Y such that W y 2,1 , · · · , W y 2,r 2 cover Y . We then list the Følner sets in the collection {F m(y 2,k ) : 1 ≤ k ≤ r 2 } by F n 2,1 , F n 2,2 , · · · , F n 2,N 2 , where n 2,1 < n 2,2 < · · · < n 2,N 2 . Note that N 2 = #{F m(y 2,k ) : 1 ≤ k ≤ r 2 } ≤ r 2 . Repeating the above steps inductively, we can obtain for each 1 ≤ i ≤ M , (1) a collection of points y i,1 , · · · , y i,r i ∈ Y such that W y i,1 , · · · , W y i,r i cover Y ; (2) a collection of Følner sets {F n i,1 , F n 2,2 , · · · , F n i, N i }(= {F m(y i,k ) : 1 ≤ k ≤ r i }) with n i,1 < n i,2 < · · · < n i,N i and N i ≤ r i . From the above construction, for each 1 ≤ i ≤ M − 1, n i,N i < n i+1,1 . Moreover, n i+1,1 can be chosen sufficiently large compared with n i,N i such that for every 1 < i ≤ M and every 1 ≤ k ≤ N i , | i <i F −1 n i , * F n i ,k | ≤ (1 + δ)|F n i ,k |. (4.4) For simplification, we denote F i,j = F n i,j for each 1 ≤ i ≤ M and 1 ≤ j ≤ N i , which is the array of G we required. Step 2. Produce quasi-tilings from F i,j . Let η 1 be a common Lebesgue number of the family of open covers {W y i,1 , · · · , W y i,r i } with respect to the metric ρ. Denote η = η 1 2 . Let N be large enough such that for every n > N , F n is (F i, * ∪ {e G }, δ)-invariant for all 1 ≤ i ≤ M . For each y ∈ Y and n > N , let A i,j ={a ∈ F n : F i,j a ⊂ F n and there exists 1 ≤ k ≤ r i such that F m(y i,k ) = F i,j and B(ay, η, ρ) ⊆ W y i,k }. We note here that A i,j depends on y. For any g ∈ F n \B(F n , F i, * ∪{e G }), we have F i, * g ⊂ F n . Since η 1 (= 2η) is a Lebesgue number of {W y i,1 , · · · , W y i,r i }, B(gy, η, ρ) is contained in some W y i,k and then g ∈ A i, * . Hence A i, * ⊇ F n \ B(F n , F i, * ∪ {e G }) for each 1 ≤ i ≤ M , and α = min 1≤i≤M |DA i, * | |F n | = min 1≤i≤M |A i, * | |F n | > 1 − δ.= {F i,j a : 1 ≤ i ≤ M, 1 ≤ j ≤ N i and a ∈ A i,j }, we can find by Lemma 4.1 a subcollection F which is 10δ 1/4 -disjoint and | ∪ F| ≥ (α − δ 1/4 )|F n | ≥ (1 − δ − δ 1/4 )|F n |. (4.5) In fact, this subcollection F is a 10δ 1/4 quasi-tiling of F n subordinate to y ∈ Y (see for example [14] for the detail definition of quasi-tiling). There may exist overlaps between elements in F. Since F is 10δ 1/4 -disjoint, there exits T ⊂ T for each T ∈ F such that |T |/|T | ≥ 1 − 10δ 1/4 and the collection {T : T ∈ F} is disjoint. Denote this new collection by F . By (4.5), | ∪ F| ≤ T ∈F |T | ≤ 1 1 − 10δ 1/4 T ∈F |T | ≤ 1 1 − 10δ 1/4 |F n | (4.6) and | ∪ F | = T ∈F |T | ≥ (1 − 10δ 1/4 )(1 − δ − δ 1/4 )|F n |. (4.7) Step 3. Cover π −1 B Fn (y, η, ρ) through F n -Bowen balls in (X, G). Claim. When δ is sufficiently small, for any y ∈ Y and n > N , there exists l(y) > 0 and v 1 (y), v 2 (y), · · · , v l(y) (y) ∈ X such that l(y) i=1 B Fn (v i (y), 4ε, d) ⊇ π −1 (B Fn (y, η, ρ)) and l(y) ≤ exp (a + 2τ )|F n | . Proof of the Claim. For each T = F i,j a ∈ F, since a ∈ A i,j , by the construction of A i,j , there exists some point in {y i,1 , y i,2 , · · · , y i,r i }, denoted by y T , such that B(ay, η, ρ) ⊆ W y T and F m(y T ) = F i,j . In the following we will recover the F n -orbits of (X, G) from the T -orbits. Let E ⊂ X be any finite ε-spanning set under the metric d. For any sequence of points {z T } T ∈F with each z T ∈ E y T and any sequence of points {z g } g∈Fn\∪F with each z g ∈ E, let V (y; {z T }, {z g }) := {u ∈ X : d T (u, a −1 z T ) < 2ε for all T = F i,j a ∈ F, d(gu, z g ) < 2ε for all g ∈ F n \ ∪F }. It is not hard to verify that (B Fn (y, η, ρ)), i.e. the family {V (y; {z T }, {z g }) : z T ∈ E y T , T ∈ F, z g ∈ E, g ∈ F n \ ∪F } forms an open cover of π −1 (B Fn (y, η, ρ)). We also note that some of V (y; {z T }, {z g })'s may be empty. {z T },{zg} V (y; {z T }, {z g }) ⊇ π −1 We pick any point v({z T }, {z g }) in each non-empty V (y; {z T }, {z g }), then B Fn (v({z T }, {z g }), 4ε, d) ⊇ V (y; {z T }, {z g }). Enumerate these v({z T }, {z g })'s by y 1 , y 2 , · · · , y l(y) . We then obtain y, η, ρ)). Now the only thing left is to estimate l(y). Clearly, l(y) i=1 B Fn (y i , 4ε, d) ⊇ π −1 (B Fn (l(y) ≤ T ∈F |E y T | · g∈Fn\∪F |E| = T ∈F r F m(y T ) (π −1 (y), ε, d) · |E| |Fn|−|∪F | ≤ exp T ∈F |F m(y T ) |(a + τ ) + (|F n | − | ∪ F |) log |E| ≤ exp 1 1 − 10δ 1/4 (a + τ ) + (1 − (1 − 10δ 1/4 )(1 − δ − δ 1/4 )) log |E| |F n | (by (4.6) and (4.7)) ≤ exp (a + 2τ )|F n | (when δ is sufficiently small). Now we proceed to prove (4.2). For any subset E of X, let H be an (F n , η)-spanning set of π(E) with minimal cardinality r Fn (π(E), η, ρ). Then by the above claim, the set R = {v i (y) : 1 ≤ i ≤ l(y), y ∈ H} forms an (F n , 4ε)-spanning set of E, since y∈H l(y) i=1 B Fn (v i (y), 4ε, d) ⊇ y∈H π −1 (B Fn (y, η, ρ)) ⊇ π −1 π(E) ⊇ E. Hence r Fn (E, 4ε, d) ≤ r Fn (π(E), η, ρ) · exp (a + 2τ )|F n | . (4.9) This deduces that h U C top (E, {F n }) ≤ h U C top (π(E), {F n }) + a + 2τ . Letting τ tend to 0, (4.2) is proved. Proof of the upper bound of h P top (E, {F n }). What we need in fact is the inequality (4.9) in the proof of Theorem 4.2. From the fact (2.1), we firstly covert inequality (4.9) into This implies that h U C top (E, 8ε, {F n }) ≤ h U C top (π(E), η, {F n }) + a + 2τ. By Proposition 2.7 (2) again, for any δ > 0, there exists a cover ∪ ∞ i=1 V i = π(E) such that h P top (π(E), η/3, {F n }) + δ ≥ sup i≥1 h U C top (V i , η, {F n }). Using the similar argument in the proof of the lower bound, Remark 5.1. If µ ∈ E(X, G) and {F n } is a tempered Følner sequence then µ(G µ ) = 1. To show this, let {f i } ∞ i=1 be a countable dense subset of C(X) and denote by h P top (E, 8ε, {F n }) ≤ sup i≥1 h P top (π −1 (V i ), 8ε, {F n }) ≤ sup i≥1 h U C top (π −1 (V i ), 8ε, {F n }) ≤ sup i≥1 h U C top (V i , η, {F n }) + a + 2τ ≤ h P top (π(E), η/3, {F n }) + δ + a + 2τ. Hence h P top (E, {F n }) ≤ h P top (π(E), {F n }) + a + 2τ. Since τ > 0 is arbitrary, we finally obtain h P top (E, {F n }) ≤ h P top (π(E), {F n }) + sup y∈Y h U C top (π −1 (y), {F n }).X i = {x ∈ X : lim n→+∞ 1 |F n | g∈Fn f i (gx) = X f i dµ}. By the pointwise ergodic theorem, µ(X i ) = 1. Hence G µ = ∩ ∞ i=1 X i has full measure. The system (X, G) is said to have almost specification property if there exists a nondecreasing function g : (0, 1) → (0, 1) with lim r→0 g(r) = 0 (a mistake-density function) and a map m : (0, 1) → F (G)×(0, 1) such that for any k ∈ N, any ε 1 , ε 2 , · · · , ε k ∈ (0, 1), and any x 1 , x 2 , · · · , x k ∈ X, if F i is m(ε i )-invariant, i = 1, 2, · · · , k and {F i } k i=1 are pairwise disjoint, then 1≤i≤m B(g; F i , x i , ε i ) = ∅, where B(g; F, x, ε) := y ∈ X : |{h ∈ F : d(hx, hy) > ε}| ≤ g(ε)|F | , the Bowen ball allowing a mistake with density g(ε). [23] that the weak specification implies the almost specification. Recall that in [6] the system (X, G) (the group G need not to be amenable) has weak specification if for any ε > 0 there exists a nonempty finite subset F of G with the following property: for any finite collection F 1 , · · · , F m of finite subsets G with Remark 5.2. It was shown in F F i ∩ F j = ∅ for 1 ≤ i, j ≤ m, i = j, and for any collection of points x 1 , · · · , x m ∈ X, there exists a point y ∈ X such that d(gx i , gy) ≤ ε for all g ∈ F i , 1 ≤ i ≤ m, i.e. 1≤i≤m B F i (x i , ε) = ∅. In this section, we will prove Theorem 1.5, i.e. if µ ∈ M (X, G), the Følner sequence {F n } satisfies the growth condition (1.2) and either µ is ergodic and {F n } is tempered or (X, G) has almost specification, then h P top (G µ , {F n }) = h µ (X, G). (5.1) The idea of the proof comes from Pfister and Sullivan [20] (see also [26,23] for amenable group actions). Upper bound for h P top (G µ , {F n }) . In the following we are going to prove h P top (G µ , {F n }) ≤ h µ (X, G) assuming that the Følner sequence {F n } satisfies the growth condition (1.2). For µ ∈ M (X, G), let {K m } m∈N be a decreasing sequence of closed convex neighborhoods of µ in M (X) such that m∈N K m = {µ}. Let A n,m = {x ∈ X : 1 |F n | g∈Fn δ x • g −1 ∈ K m }, for m, n ∈ N, and R N,m = {x ∈ X : for any n > N, 1 |F n | g∈Fn δ x • g −1 ∈ K m }, for m, N ∈ N. Then for any m, N ≥ 1, R N,m = n>N A n,m and G µ ⊆ k>N R k,m . For ε > 0 and Z ⊆ X, recall that s Fn (Z, ε) denotes the maximal cardinality of any (F n , ε)-separated subset of Z. Hence for any η > 0, there exists 0 < ε 1 such that for any 0 < ε < ε 1 , there exists M = M (ε) ∈ N such that, lim sup n→+∞ 1 |F n | log s Fn (A n,m , ε) < h µ (X, G) + η, whenever m ≥ M . Especially, lim sup n→+∞ 1 |F n | log s Fn (A n,M , ε) < h µ (X, G) + η. (5.3) Joint (5.2) and (5.3) together, for any 0 < ε < ε 1 , we have that for any N ∈ N, lim sup n→+∞ 1 |F n | log s Fn (R N,M , ε) < h µ (X, G) + η. Since for any N ∈ N, G µ ⊆ N ≥N R N,M , by Propositions 2.1 and 2.4, h P top (G µ , ε, {F n }) ≤ sup N ≥N h P top (R N,M , ε, {F n }) ≤ sup N ≥N lim sup n→+∞ 1 |F n | log s Fn (R N,M , ε), which follows that h P top (G µ , ε, {F n }) ≤ h µ (X, G) + η. Letting ε → 0 and then η → 0, we obtain that h P top (G µ , {F n }) ≤ h µ (X, G). 5.1.2. Lower bound for h P top (G µ , {F n }). For the case µ is ergodic and {F n } is tempered, since µ(G µ ) = 1, Corollary 2.3 gives the lower bound. For the case µ ∈ M (X, G) and the system (X, G) has almost specification property, the proof of the lower bound becomes rather complicated because of the quasitiling techniques for amenable groups. But it was shown in [23] that h B top (G µ , {F n }) = h µ (X, G). Hence by Proposition 2.2, we obtain h P top (G µ , {F n }) ≥ h µ (X, G). G-symbolic dynamical system. Let A be a finite set with cardinality |A| ≥ 2 and A G = {(x g ) g∈G : x g ∈ A} be the G-symbolic space over A. Consider the left action of G on A G : g (x g ) g∈G = (x gg ) g∈G , for all g ∈ G and (x g ) g∈G ∈ A G . (A G , G) forms a G-symbolic dynamical system or a G-acting full shift (over A). For any non-empty closed G-invariant subset X of A G , the subsystem (X, G) is called a subshift. For x = (x g ) g∈G and a finite subset F ⊂ G, denote by x| F = (x g ) g∈F ∈ A F the restriction of x to F and denote by [x| F ] = {ω ∈ A G : ω g = x g for all g ∈ F } (which is called a cylinder). Fix any tempered Følner sequence {F n } of G with F 0 = {e G } F 1 F 2 . . . and n F n = G. Note that {F n } satisfies the growth condition (1.2) automatically. We can then define a metric d on A G associated to {F n } by the following: d(x, y) = 1, if x and y are not equal on F 0 ; e −|Fn| , n = max{k : x| F k = y| F k }. (5.4) To discuss the regularity for subsets of (A G , G), we need to consider the relation between Bowen entropy (packing entropy) and the corresponding Hausdorff dimension (packing dimension). Before that, we recall the definitions of Hausdorff dimension and packing dimension (cf. [16]). Definition 5.3. Let (X, d) be a compact metric space. Let 0 ≤ s < ∞. For E ⊂ X and ε > 0, put P s ε (E) = sup i (diamB i ) s where the supremum is taken over all disjoint families of closed balls {B i } such that diamB i ≤ ε and the centres of the B i 's are in E. Then set P s (E) = lim ε→0 P s ε (E) (since P s ε (E) is non-decreasing on ε) and define P s (E) = inf{ ∞ i=1 P s (E i ) : E = ∞ i=1 E i }. The packing dimension of E is then defined by dim P (E) = inf{s : P s (E) = 0} = sup{s : P s (E) = ∞}. Let H s (E) = lim ε→0 H s ε (E), where H s ε (E) = inf{ ∞ i=1 (diamE i ) s : E ⊂ ∞ i=1 E i , diamE i ≤ ε}. The Hausdorff dimension of E is then defined by dim H (E) = inf{s : H s (E) = 0} = sup{s : H s (E) = ∞}. Comparing with the definitions of dimensional entropies, we have the following proposition. Proposition 5.4. Let the Følner sequence {F n } satisfy the following two conditions: (1) F m F n ⊂ F m+n for each m, n ∈ N; (2) lim Proof. See Appendix A.1. Remark 5.5. (1) Due to [9], the sequence of finite subsets {F n } of G satisfying condition (1) in Proposition 5.4 is called a regular system. If G is a finitely generated group and let F n be the collection of elements in G with word length (w.r.t. a finite symmetric generating subset) no more than n, then {F n } satisfies condition (1). (2) There are examples of amenable groups which admit Følner sequences satisfying conditions in Proposition 5.4. An abelian example of the group G is Z d , with F n = [−n, n] d . An non-abelian example of the group G is the dihedral group, with F n which is chosen to be the collection of elements with word length no more than n (see Example 6.4.11 of [5] By Remark 2.6, any non-empty closed G-invariant subset X of A G is regular in the sense of dimensional entropy. Hence we have Random walk in random scenery (RWRS) is a class of stationary random processes which are well-studied both in probability theory and ergodic theory. RWRS provides measure-theoretic models with amazingly rich behavior (see [10,1]). Among the class of RWRS, {T, T −1 } transformation, although seems simple, is possibly the best known in the history of ergodic theory since it is a natural example of a K-automorphism that is not Bernoulli ( [13]). In spite of its measure-theoretic aspect, we will consider the topological model of the {T, T −1 } transformation and investigate subsets of the topological system. (T (x)) i = x i+1 , for any x = (x i ) i∈Z ∈ A Z . The {T, T −1 } transformation, denoted by S, on A Z × A Z is defined by S(x, y) = T (x), T (y) , if y 0 = 1; T −1 (x), T (y) , if y 0 = −1. Then S n (x, y) = T ω(y,n) (x), T n y for n ∈ Z, where ω(y, n) :=      n−1 j=0 y j , if n > 0; 0, if n = 0; − −1 j=n y j , if n < 0. Clearly for the system (A Z × A Z , S), the acting group G here is the integer group Z. The Følner sequence {F n } is chosen naturally to be F n = {0, 1, · · · , n − 1} : = [0, n − 1]. Define a metric ρ on A Z × A Z by ρ (x, y), (x , y ) = max{d(x, x ), d(y, y )}, where d is the metric on A Z defined by d(x, y) = 2 −n , where n = min{|i| : x i = y i }. We note that the metric d here is different from (5.4). Let π : A Z × A Z → A Z be the projection to the second coordinate. Then it induces a factor map between (A Z × A Z , S) and (A Z , T ). For any y ∈ A Z , let E y := A Z × {y} denote the fiber of y under the factor map π. We denote by Proof. See Appendix A.3. In this appendix we will give the detailed proofs of Propositions 5. (x i , 1)} with x i ∈ E, n i ≥ N and i B Fn i (x i , 1) ⊃ E such that i e −s|Fn i | < δ. Notice that Assume ε ∈ (e −|F k | , e −|F k−1 | ] for some k ∈ N and let η > 0 be fixed. By condition (2), there exists N > k such that |F n | |F n−k | < 1 + η, whenever n > N . B Fn i (x i , 1) = {y ∈ A G : ρ(gx i , gy) < 1, ∀g ∈ F n i } = {y ∈ A G : (gx i ) e G = (gy) e G , ∀g ∈ F n i } = [x i | Fn i ] (A.1) and diam[x i | Fn i ] = e −|Fn i | ≤ e −|F N | . By (A.2), there exists N > N such that M(E, N, ε, s, {F n }) > 1. Let {E i } ∞ i=1 be any countable family that covers E and diamE i < e −|F N +k | for each i. By the definition of the metric d, diamE i = e −|Fn i | for some n i > N + k. Choose any point x i ∈ E i , then E i ⊂ [x i | Fn i ]. Noticing that B F n i −k (x i , ε) = {y ∈ A G : ρ(gx i , gy) < ε, for all g ∈ F n i −k } = {y ∈ A G : ρ(gx i , gy) ≤ e |F k | , for all g ∈ F n i −k } = {y ∈ A G : (gx i )| F k = (gy)| F k , for all g ∈ F n i −k } = {y ∈ A G : x i | F k F n i −k = y| F k F n i −k } = [x i | F k F n i −k ] ⊇ [x i | Fn i ] ( since F k F n i −k ⊆ F n i ), (A.3) the family {B F n i −k (x i , ε)} ∞ i=1 also covers E (with each n i − k > N ). So ∞ i=1 e −s|F n i −k | ≥ M(E, N, ε, s, {F n }) > 1. And hence ∞ i=1 e − s 1+η |Fn i | = ∞ i=1 e − s 1+η |Fn i | |F n i −k | |F n i −k | > ∞ i=1 e −s|F n i −k | > 1,} ∞ i=1 of E such that ∞ i=1 P (E i , ε, s, {F n }) < δ. For each i, we can find N i ∈ N sufficiently large such that P (E i , N i , ε, s, {F n }) < P (E i , ε, s, {F n }) + δ 2 i . Let {B i,j } ∞ j=1 be a family of disjoint closed balls in A G (with centers x i,j ∈ E i and diamB i,j ≤ e −|F N i | ) such that P s e −|F N i | (E i ) ≤ ∞ j=1 (diamB i,j ) s + δ 2 i . From the definition of the metric d, diamB i,j = e −|Fn i,j | for some n i,j ≥ N i . Noticing that B i,j = B(x i,j , e −|Fn i,j | ) = [x i,j | Fn i,j ] ⊇ B Fn i,j (x i,j , ε) (here we have assumed ε < 1), {B Fn i,j (x i,j , ε)} ∞ j=1 is also a pairwise disjoint family. Hence P s (E) ≤ ∞ i=1 P s (E i ) ≤ ∞ i=1 P s e −|F N i | (E i ) ≤ ∞ i=1 ∞ j=1 e −s|Fn i,j | + δ 2 i ≤ ∞ i=1 P (E i , N i , ε, s, {F n }) + δ < ∞ i=1 P (E i , ε, s, {F n }) + δ 2 i + δ < 3δ. Thus P s (E) = 0 and dim P E ≤ s. Since s > h P top (E, {F n }) is arbitrary, we obtain dim P E ≤ h P top (E, {F n }). Next we will show h P top (E, {F n }) ≤ dim P E. Assume h P top (E, {F n }) > 0, otherwise there is nothing to prove. Let 0 < s < h P top (E, {F n }). Then there exists 0 < ε < 1 such that h P top (E, ε, {F n }) > s. Assume ε ∈ (e −|F k | , e −|F k−1 | ] for some k ∈ N and let η > 0 be fixed. Similar to Part 1, by condition (2), there exists N > k such that |F n | |F n−k | < 1 + η, whenever n > N. Let {E i } ∞ i=1 be any countable family that covers E, then ∞ i=1 P (E i , ε, s, {F n }) ≥ P(E, ε, s, {F n }) = ∞. For each i, let N i > N be sufficiently large such that P s 1+η e −|F N i | (E i ) < P s 1+η (E i ) + 1 2 i . Let {B Fn i,j (x i,j , ε)} ∞ j=1 be a disjoint family with x i,j ∈ E i and n i,j ≥ N i for each j such that X α,β ⊂ {0, 1} G is defined by X α,β = {(x g ) g∈G : x g = 0 if g / ∈ H}. P (E i , N i , ε, s, {F n }) < ∞ j=1 e −s|Fn i,j | + 1 2 i . Thus we have ∞ i=1 P s 1+η (E i ) > ∞ i=1 P s 1+η e −|F N i +k | (E i ) − 1 ≥ ∞ i=1 ∞ j=1 e − s 1+η |F n i,j +k | − 1 = ∞ i=1 ∞ j=1 e − s 1+η |F n i,j +k | |Fn i,j | |Fn i,j | − 1 ≥ ∞ i=1 ∞ j=1 e −s|Fn i,j | − 1 > ∞ i=1 P (E i , N i , ε, s, {F n }) − 1 2 i − 1 ≥ ∞ i=1 P (E i , Let µ ∈ M (X α,β ) such that µ([x| Fn ]) = 1 2 |H∩Fn| for every x ∈ X α,β and n ∈ N. Noticing that B Fn (x, 1) = [x| Fn ] (see (A.1)), we have h loc µ (X α,β , {F n }) = To prove h B top (X α,β , {F n }) ≤ α log 2, by Proposition 5.4, we only need to prove that dim H (X α,β ) ≤ α log 2. Let E n = {(x g ) g∈G : x g = 0 if g / ∈ H ∩ F n }, (A.4) which is a subset of X α,β with cardinality #E n = 2 |H∩Fn| . Note that x∈En [x| Fn ] ⊇ X α,β and diam[x| Fn ] = e −|Fn| for each x ∈ E n . Let δ > 0 and N ∈ N be fixed. When n > N , Thus dim H (X α,β ) ≤ (α + δ) log 2, which deduces that dim H (X α,β ) ≤ α log 2. Finally we will prove that h P top (X α,β , {F n }) ≤ β log 2. Let ε > 0 such that ε ∈ (e −|F k | , e −|F k−1 | ] for some k ∈ N. Noticing that [x| F n+k ] ⊆ B Fn (x, ε) (see (A.3), here condition (1) of Proposition 5.4 is used), one can check that the set E n+k , defined by (A.4), is an (F n , ε)-spanning set of X α,β . Hence r Fn (X α,β , ε) ≤ 2 |H∩F n+k | . Then lim sup n→+∞ 1 |F n | log r Fn (X α,β , ε) ≤ lim sup n→+∞ |H ∩ F n+k | |F n | log 2 = β log 2, where for the last equality we use condition (2) Let δ > 0 be fixed. For any ε > 0, there exists k ∈ N such that for any x, z ∈ A Z , whenever x i = z i for every |i| ≤ k, it holds that d(x, y) < ε. Hence for any interval Note that there exist infinitely many n ∈ N such that M (y, n − 1) − m(y, n − 1) + 2k + 1 n < lim inf This finishes the proof of (2) of Proposition 5.10. (G) denote the collection of finite subsets of G. For F ∈ F (G), let d F be the metric defined by d F (x, y) = max g∈F d(gx, gy), for x, y ∈ X. P (Z, ε, s, {F n }) = lim N →+∞ P (Z, N, ε, s, {F n }). (Z, ε, {F n }) the amenable packing topological entropy (amenable packing entropy or packing entropy, for short) of Z (w.r.t. the Følner sequence {F n }). Let M (X) denote the collection of Borel probability measures on X. Definition 1.2. Let {F n } be a sequence of finite subsets of G with |F n | → ∞. For µ ∈ M (X) and Z ∈ B(X) (the Borel σ-algebra on X), denote by h loc µ (Z, {F n }) |F n | log µ(B Fn (x, ε))dµ, B Fn i (x i , ε) ⊇ Z. The quantity M(Z, N, ε, s, {F n }) does not decrease as N increases and ε decreases, hence the following limits exist: M(Z, ε, s, {F n }) = lim N →+∞ M(Z, N, ε, s, {F n }) and M(Z, s, {F n }) = lim ε→0 M(Z, ε, s, {F n }). The Bowen topological entropy h B top (Z, {F n }) is then defined as the critical value of the parameter s, where M(Z, s, {F n }) jumps from +∞ to 0, i.e., Assume h P top (Z, ε, {F n }) > 0 and let 0 < t < s < h P top (Z, ε, {F n }). Then P (Z, ε, s, {F n }) ≥ P(Z, ε, s, {F n }) = +∞. Thus for any N , there exists a countable pairwise disjoint family {B Fn i (x i , ε)} with x i ∈ Z and n i ≥ N for all i such that 1 < i e −|Fn i |s . For each k, let m k be the number of i with n i = k. Then we have i e −|Fn i |s = k≥N m k e −|F k |s . be a family of closed (or open) balls in X. Then there exists a finite or countable subfamily ε, t, {F n }) = 0 for some open set U , thenP(Z ∩ U, ε, t, {F n }) ≤ P(Z ∩ U, ε, t, {F n }) + P(Z ∩ H, ε, t, {F n }) = 0. So U ⊂ H and then Z ∩ U = ∅. Hence for any open set U ⊂ X, either Z ∩ U = ∅ or P(Z ∩ U, ε, t, {F n }) > 0. Because P(Z, ε, t, {F n }) ≤ P(Z , ε, t, {F n }) + P(Z ∩ H, ε, t, {F n }) = P(Z , ε, t, {F n }), we have P(Z , ε, t, {F n }) = P(Z, ε, t, {F n }) = +∞. Then P(Z , ε, s, {F n }) = +∞.By Lemma 3.5, we can find a finite set K 1 ⊂ Z , an integer valued function m 1 (x) on K 1 such that the collection {B F m 1 (x) (x, ε)} x∈K 1 is disjoint and x∈K 1 e −|F m 1 (x) |s ∈ (1, 2). ( 2 2-a) for each open set U with U ∩ E 2 (x) = ∅, P(Z n 1 ∩ U, ε, t, {F n }) > 0; (2-b) the elements in the family {B F m 2 (y) (y, ε)} y∈E 2 (x) are pairwise disjoint and µ 1 ({x}) < y∈E 2 (x) e −|F m 2 (y) |s < (1 + 2 −2 )µ 1 ({x}). H x := ∪{U ⊂ X : U is open and P(V ∩ U, ε, t, {F n }) = 0}. (3-a) for each open set U with U ∩ E p+1 (x) = ∅, P(Z n 1 ,...,np ∩ U, ε, t, {F n }) > 0; Lemma 4. 1 ( 1Lindenstrauss's covering lemma, Corollary 2.7 of[15]). For any δ ∈ (0, 1/100), C > 0 and finite D ⊂ G, let M ∈ N be sufficiently large (depending only on δ, C and D). Let F i,j be an array of finite subsets of G where i = 1, . . . , M and j = 1, . . . , N i , with the following two requirements: 2.7 (2), for any η > 0, there exists a cover 5 . 5Amenable packing entropy for certain subsets 5.1. The set of generic points. Recall that for µ ∈ M (X, G) and a Følner sequence {F n } in G, the set of generic points for µ (w.r.t. {F n }) is defined by G µ = {x ∈ X , for any f ∈ C(X)}. |F n | log s Fn (R N,m , ε) ≤ lim sup n→+∞ 1 |F n |log s Fn (A n,m , ε), for any m, N ≥ 1. n | log s Fn (A n,m , ε) ≤ h µ (X, G). Fn = 1 . 1Then for anyE ⊂ A G , h B top (E, {F n }) = dim H E and h P top (E, {F n }) = dim P E,where the dimensions dim H and dim P are both under the metric d defined in (5.4). Corollary 5 . 7 . 57Let the Følner sequence {F n } satisfy the conditions in Proposition 5.4. Then any non-empty closed G-invariant subset X of A G is both regular in the sense of dimensional entropy and dimension-regular (under the Følner sequence {F n } and metric d). Let 0 ≤ α < β ≤ 1 and A = {0, 1}. Let H ⊂ G such that lim inf n→+∞ |H ∩ F n | |F n | = α and lim sup n→+∞ |H ∩ F n | |F n | = β, i.e. H is a subset of G with lower density α and upper density β w.r.t. {F n }. Now we define X α,β ⊂ {0, 1} G by X α,β = {(x g ) g∈G : x g = 0 if g / ∈ H}. Assume in addition that {F n } satisfies the conditions in Proposition 5.4, then we have Proposition 5.8. h B top (X α,β , {F n }) = α log 2 and h P top (X α,β , {F n }) = β log 2. Hence X α,β is not regular. Proof. See Appendix A.2. 5.3. Fibers of {T, T −1 } transformation. Topological {T, T −1 } transformation). Let A = {1, −1} and as convention we denote the shift map on A Z by T which is defined by h P top (E y , S) = h P top (E y , {F n }) and h B top (E y , S) = h B top (E y , {F n }), the packing and Bowen entropies of E y for the Z-system (A Z × A Z , S) respectively. For n > 0, denote by M (y, n) = max 0≤i≤n ω(y, i) and m(y, n) = min 0≤i≤n ω(y, i). Proposition 5.10. (1) h P top (E y , S) = lim sup n→+∞ M (y,n)−m(y,n) n h top (A Z , T ); (2) h B top (E y , S) = lim inf n→+∞ M (y,n)−m(y,n) n h top (A Z , T ).Here h top (A Z , T )(= log 2) is the topological entropy of the symbolic dynamical system (A Z , T ). Hence E y is regular if and only if lim With similar discussion as (A.3), B Fn i,j (x i,j , ε) ⊇ [x i,j | F n i,j +k ] and hence {[x i,j | F n i,j +k ]} ∞ j=1 is a disjoint family of closed balls with diam[x i,j | F n i,j +k ] = e −|F n i,j +k | ≤ e −|F N i +k | . Therefore P s 1+η e −|F N i +k | (E i ) ≥ ∞ j=1 e − s1+η |F n i,j +k | . (α+δ) log 2|Fn| = e log 2|Fn|( |H∩Fn| |Fn| −α−δ) .Since lim inf n→+∞ |H∩Fn| |Fn| = α, we have H (α+δ) log 2 e −|F N | (X α,β ) = 0 and H (α+δ) log 2 (X α,β of Proposition 5.4. Thush P top (X α,β , {F n }) ≤ h U C top (X α,β , {F n }) ≤ β log 2.A.3. Proof of Proposition 5.10. Proof. (1). For any ε > 0 and x ∈ A Z , in the system (A Z × A Z , S), we have B n (x, y), ε, ρ ∩ E y = (x , y) : ρ S i (x , y), S i (x, y) < ε, for 0 ≤ i ≤ n − 1 = (x , y) : d T ω(y,i) (x ), T ω(y,i) x < ε, for 0 ≤ i ≤ n − 1 = B [m(y,n−1),M (y,n−1)] x, ε, d × {y}. (A.5) B [m,n] (x, ε, d) ⊇ [x| [−k+m,k+n] ], where [x| [−k+m,k+n]] := {z ∈ A Z : z i = x i , for every − k + m ≤ i ≤ k + n} is the cylinder in A Z . Let E n = {x ∈ A Z : x i = 1 if i / ∈ [−k + m(y, n − 1), k + M (y, n − 1)]}.Consider the family {B n (x, y), ε, ρ ∩ E y } x∈En . Apparently, it covers E y , since B n (x, y), ε, ρ ∩ E y = B [m(y,n−1),M (y,n−1)] x, ε, d × {y} (by (A.5)). Hence for any N ∈ N, we have for any n ≥ N , M(E y , N, ε, s, {F n }) ≤ (#E n )e + δ) log 2, we can deduce that M(E y , N, ε, s, {F n }) = 0. Letting N → +∞, ε → 0 and then δ → 0, from the definition of the Bowen entropy, we can conclude that h B top (E y , S) ≤ lim inf n→∞ M (y, n) − m(y, n) n h top (A Z , T ). whenever ε is sufficiently small. This implies thath P top (Z, ε, {F n }) ≥ s. Letting ε → 0, we have h P top (Z, {F n }) ≥ s. Hence h B top (Z, {F n }) ≤ h P top (Z, {F n }). Corollary 2.3. Let µ ∈ M (X, G), Z ⊆ X with µ(Z) = 1 and {F n } be a Følner sequence in G, then h µ (X, G) ≤ h P top (Z, {F n }). Proof. Let {F n k } be a subsequence of {F n } which is tempered and satisfies the growth condition (1.2). By Theorem 1.2 of [26], h µ (X, G) ≤ h B top (Z, {F n k }). Together with Proposition 2.2 and (3) of Proposition 2.1, we have Now we are able to apply Lemma 4.1: the temperedness assumption for {F n } makes Requirement 1 hold and (4.4) makes Requirement 2 hold. From the collection of subsets of F n ,F ).By Proposition 5.4, we have Proposition 5.6. Let the Følner sequence {F n } satisfy the conditions in Proposition 5.4. Then any subset E ⊂ A G is regular in the sense of dimensional entropy if and only if E is dimension-regular (under the Følner sequence {F n } and metric d). 4, 5.8 and 5.10. A.1. Proof of Proposition 5.4.Let E be a subset of the compact metric space (A G , d) as defined in Section 5.2. Recall that in Proposition 5.4 the Følner sequence {F n } satisfies the following two conditions:(1) F m F n ⊆ F m+n for each m, n ∈ N;(2) limFn = 1.We divide the proof of Proposition 5.4 into two parts.Part 1. Proof of h B top (E, {F n }) = dim H E. Proof. Let s > h B top (E, {F n }). From the definition of Bowen entropy, we have M(E, ε, s, {F n }) = 0, for any ε > 0. Hence for any N > 0, M(E, N, 1, s, {F n }) = 0. For any δ > 0, there exists a countable familiy {B Fn in→∞ F n+1 Since the family of cylinders{[x i | Fn i ]} covers E, we have H s e −|F N | (E) ≤ i e −s|Fn i | < δ. Therefore H s e −|F N | (E) = 0 and then H s (E) = 0. This means that dim H E ≤ s. Since s > h B top (E, {F n }) is arbitrary, h B top (E, {F n }) ≥ dim H E. Now we will show h B top (E, {F n }) ≤ dim H E. If h B top (E, {F n }) = 0,then there is nothing to prove. Assume h B top (E, {F n }) > 0 and let 0 < s < h B top (E, {F n }), then there exists 0 < ε < 1 such that M(E, ε, s, {F n }) > 1. (A.2) which implies that H s 1+η (E) ≥ H s 1+η e −|F N +k | (E) ≥ 1. Thus dim H E ≥ s 1+η . Letting η → 0, we have dim H E ≥ s. Since 0 < s < h B top (E, {F n }) is chosen arbitrarily, we obtain that h B top (E, {F n }) ≤ dim H E. Part 2. h P top (E, {F n }) = dim P E.Proof. We show h P top (E, {F n }) ≥ dim P E firstly. Let s > h P top (E, {F n }). Then for any 0 < ε < 1 it holds that h P top (E, ε, {F n }) < s and hence P(E, ε, s, {F n }) = 0. So for any δ > 0, there exists a countable covering {E i ε, s, {F n }) − 2 = ∞, which implies that P 1+η (E) = ∞ and then dim P E ≥ s 1+η . Letting η → 0, we have dim P E ≥ s. Since 0 < s < h P top (E, {F n }) is chosen arbitrarily, we obtain that h P top (E, {F n }) ≤ dim P E.A.2. Proof of Proposition 5.8. Recall that for Proposition 5.8 we let 0 ≤ α < β ≤ 1, A = {0, 1} and let H ⊂ G such that lim inf n→+∞ |H ∩ F n | |F n | = α and lim sup n→+∞ |H ∩ F n | |F n | = β.s |H∩Fn| = α log 2, and similarly, loc µ (X α,β , {F n }) ≥ β log 2. Applying Theorem 3.1 of [25] (the variational principle for amenable Bowen entropy) and Theorem 1.3 (the variational principle for amenable packing entropy) respectively, we have h B top (X α,β , {F n }) ≥ α log 2 and h P top (X α,β , {F n }) ≥ β log 2.X α,β lim ε→0 lim inf n→+∞ − 1 |F n | log µ(B Fn (x, ε))dµ ≥ X α,β lim inf n→+∞ − 1 |F n | log µ([x| Fn ])dµ = lim inf n→+∞ − 1 |F n | log 1 2 h s Fn (E, 8ε, d) ≤ s Fn (π(E), η, ρ) · exp (a + 2τ )|F n | . where for the last inequality we use the simple facts that M (y, n − 1) ≤ M (y, n) ≤ M (y, n − 1) + 1 and m(y, n − 1) − 1 ≤ m(y, n) ≤ m(y, n − 1). Hence by Theorem 1.3, we haveLet E × {y} be any (n, ε)-separated set for E y . Note that from (A.5) E must be an ([m(y, n − 1), M (y, n − 1)], ε)-separated set for A Z . We have Relative complexity of random walks in random sceneries. J Aaronson, Ann. Probab. 406J. Aaronson, Relative complexity of random walks in random sceneries, Ann. Probab., 40 (2012), no. 6, 2460-2482. Entropy for group endomorphisms and homogeneous spaces. R Bowen, Trans. Amer. Math. Soc. 153R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. Topological entropy for noncompact sets. R Bowen, Trans. Amer. Math. Soc. 184R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125- 136. On local entropy. M Brin, A Katok, Lecture Notes in Mathematics. 1007SpringerM. Brin and A. Katok, On local entropy, Lecture Notes in Mathematics, vol.1007, Springer, Berlin, 1983, 30-38. T Ceccherini-Silberstein, M Coornaert, Cellular automata and groups. BerlinSpringer-VerlagT. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Springer-Verlag, Berlin, 2010. Homoclinic groups, IE groups, and expansive algebraic actions. N P Chung, H Li, Invent. Math. 1993N.P. Chung and H. Li, Homoclinic groups, IE groups, and expansive algebraic actions, Invent. Math., 199 (2015), no. 3, 805-858. A note on dimensional entropy for amenable group actions. D Dou, R Zhang, Topol. Methods Nonlinear Anal. 512D. Dou and R. Zhang, A note on dimensional entropy for amenable group actions, Topol. Methods Nonlinear Anal., 51 (2018), no. 2, 599-608. Variational principles for topological entropies of subsets. D J Feng, W Huang, J. Funct. Anal. 2638D.J. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal. 263 (2012), no. 8, 2228-2254. Topological entropy of group and semigroup actions. K.-H Hofmann, L N Stoyanov, Adv. Math. 115K.-H. Hofmann and L.N. Stoyanov, Topological entropy of group and semigroup actions, Adv. Math. 115 (1995), 54-98. Random walk in random scenery: a survey of some recent results. F Hollander, J E Steif, IMS Lecture Notes Monogr. Ser. 48Inst. Math. Statist.F. den Hollander and J. E. Steif, Random walk in random scenery: a survey of some recent results, Dynamics & stochastics, 53-65, IMS Lecture Notes Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, 2006. T Jech, Springer Monographs in Mathematics. SpringerSet theoryT. Jech, Set theory, in: Springer Monographs in Mathematics, Springer, 2003. On the existence of subsets of finite positive packing measure, Mathematika. H Joyce, D Preiss, 42H. Joyce, D. Preiss, On the existence of subsets of finite positive packing measure, Mathematika, 42 (1995), 15-24. . S A Kalikow, T , T − , Ann. of Math. 115transformation is not loosely BernoulliS. A. Kalikow, T, T −1 transformation is not loosely Bernoulli, Ann. of Math., 115 (1982), 393-409. D Kerr, H Li, Ergodic theory: independence and dichotomies. SpringerD. Kerr and H. Li, Ergodic theory: independence and dichotomies, in: Springer Monographs in Mathematics, Springer, 2016. Pointwise theorems for amenable groups. E Lindenstrauss, Invent. Math. 146E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146 (2001), 259-295. P Mattila, Geometry of Sets and Measures in Euclidean Spaces. Cambridge University PressP. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. The variational principle. J M Ollagnier, D Pinchon, Studia Math. 722J.M. Ollagnier and D. Pinchon, The variational principle, Studia Math., 72 (1982), no. 2, 151-159. Entropy and isomorphism theorems for actions of amenable groups. D S Ornstein, B Weiss, J. Anal. Math. 48D.S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141. Dimension Theory in Dynamical Systems. . B Ya, Pesin, Contemporary Views and Applications. Chicago, ILUniversity of Chicago PressYa. B. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, University of Chicago Press, Chicago, IL, 1997. On the topological entropy of saturated sets. C.-E Pfister, W G Sullivan, Ergod. Th. Dynam. Sys. 273C.-E. Pfister and W.G. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Sys., 27 (2007), no. 3, 929-956. Variational characterization of topological pressure of the amenable groups of transformations. A M Stepin, A T Tagi-Zade, Dokl. Akad. Nauk SSSR. 2543in RussianA.M. Stepin and A.T. Tagi-Zade, Variational characterization of topological pressure of the amenable groups of transformations, Dokl. Akad. Nauk SSSR, 254 (1980), no. 3, 545-549 (in Russian). Two definitions of fractional dimension. C Tricot, Math. Proc. Cambridge Philos. Soc. 911C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc., 91 (1982), no. 1, 57-74. Topological pressure of generic points for amenable group actions. R Zhang, J. Dynam. Differential Equations. 304R. Zhang, Topological pressure of generic points for amenable group actions, J. Dynam. Differ- ential Equations, 30 (2018), no. 4, 1583-1606. A formula of packing pressure of a factor map. C Zhao, E Chen, X Hong, X Zhou, Entropy. 1910C. Zhao, E. Chen, X. Hong and X. Zhou, A formula of packing pressure of a factor map, Entropy, 19 (2017), no. 10, Paper No. 526, 9 pp. Bowen entropy for actions of amenable groups. D Zheng, E Chen, Israel Journal of Mathematics. 212D. Zheng and E. Chen, Bowen entropy for actions of amenable groups, Israel Journal of Mathe- matics, 212 (2016), 895-911. Topological entropy of sets of generic points for actions of amenable groups. D Zheng, E Chen, Sci. China Math. 615D. Zheng and E. Chen, Topological entropy of sets of generic points for actions of amenable groups, Sci. China Math., 61 (2018), no. 5, 869-880. Packing entropy and divergence points. X Zhou, E Chen, W Cheng, Dynamical Systems: An International Journal. 273X. Zhou, E. Chen and W. Cheng, Packing entropy and divergence points, Dynamical Systems: An International Journal, 27 (2012), no. 3, 387-402. . D Dou, Nanjing, Jiangsu, 210093Department of Mathematics, Nanjing UniversityP.R. China Email address: [email protected]. Dou: Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, P.R. China Email address: [email protected]
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[ "Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity", "Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity", "Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity", "Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity", "Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity", "Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity" ]
[ "Anthony L Shoup \nMilagro Collaboration\n\n\nDepartment of Physics and Astronomy\nUniversity of California\n92697IrvineCA\n", "Anthony L Shoup \nMilagro Collaboration\n\n\nDepartment of Physics and Astronomy\nUniversity of California\n92697IrvineCA\n", "Anthony L Shoup \nMilagro Collaboration\n\n\nDepartment of Physics and Astronomy\nUniversity of California\n92697IrvineCA\n" ]
[ "Milagro Collaboration\n", "Department of Physics and Astronomy\nUniversity of California\n92697IrvineCA", "Milagro Collaboration\n", "Department of Physics and Astronomy\nUniversity of California\n92697IrvineCA", "Milagro Collaboration\n", "Department of Physics and Astronomy\nUniversity of California\n92697IrvineCA" ]
[]
Previous works have shown that water Cherenkov detectors have superior sensitivity to those of scintillation counters as applied to detecting extensive air showers (EAS). This is in large part due to their much higher sensitivity to EAS photons which are more than five times more numerous than EAS electrons. Large area water Cherenkov detectors can be constructed relatively cheaply and operated reliably. A sparse detector array has been designed which uses these types of detectors to substantially increase the area over which the Milagro Gamma Ray Observatory collects EAS information. Improvements to the Milagro detector's performance characteristics and sensitivity derived from this array and preliminary results from a prototype array currently installed near the Milagro detector will be presented.
null
[ "https://export.arxiv.org/pdf/astro-ph/9907214v1.pdf" ]
117,782,517
astro-ph/9907214
2fc587fbe15badd9c1716cf796b96d4e04a56129
Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity Jul 1999 Anthony L Shoup Milagro Collaboration Department of Physics and Astronomy University of California 92697IrvineCA Use of Instrumented Water Tanks for the Improvement of Air Shower Detector Sensitivity Jul 1999arXiv:astro-ph/9907214v1 15 Previous works have shown that water Cherenkov detectors have superior sensitivity to those of scintillation counters as applied to detecting extensive air showers (EAS). This is in large part due to their much higher sensitivity to EAS photons which are more than five times more numerous than EAS electrons. Large area water Cherenkov detectors can be constructed relatively cheaply and operated reliably. A sparse detector array has been designed which uses these types of detectors to substantially increase the area over which the Milagro Gamma Ray Observatory collects EAS information. Improvements to the Milagro detector's performance characteristics and sensitivity derived from this array and preliminary results from a prototype array currently installed near the Milagro detector will be presented. Introduction The field of Very High Energy (VHE) gamma-ray astronomy has exploded in recent years, mainly pushed by the development of more sensitive telescopes. The emphasis has been to lower energy thresholds, improve angular and energy resolutions and most importantly hadronic cosmic ray background rejection. Considerable efforts have also been made to develop telescopes which detect VHE extensive air showers (EAS) which have secondaries that survive to ground level, such as Milagro and the Tibet Array. If reasonable sensitivity at VHE energies can be achieved with these detectors, they will offer powerful capabilities, such as full overhead sky coverage both day and night regardless of weather and skylight conditions. This would allow much higher temporal coverage of sources that are already known to be highly variable, such as Active Galactic Nuclei. The Milagro detector is progressing toward reaching the necessary VHE sensitivity. It is a large (60m x 80m x 8m) water pond instrumented with 723 8" photomultiplier tubes (pmts) in two layers. These pmts detect the Cherenkov light produced by EAS secondaries passing through the optically clear water. Its high altitude (2650m) and sensitivity to both photonic and leptonic EAS components give it an energy threshold such that for zenith traversing sources the peak primary energy will be 1 TeV. After calibrations it will have good angular resolution and hadronic cosmic ray rejection (see McCullough 1999 for more details). To improve the sensitivity of the current Milagro detector, 172 instrumented, large area (5m 2 ), water Cherenkov detectors (tanks) will be deployed around the pond to effectively extend its active area. As discussed below, this will improve both the energy and angular resolution of Milagro and increase its hadronic cosmic ray rejection, thus improving its overall VHE sensitivity. It can also be used to increase Milagro's efficiency for detecting EAS below 1.0 TeV which have core positions significantly away from the Milagro pond. Water Tank Detector & Array The criteria for selecting a detector design that will improve the performance of EAS experiments are: low cost and low maintence (a large ground area needs to be covered, typically at a remote high altitude site), high sensitivity to EAS secondary particles, and good timing and particle density resolution. Previous works (Yodh 1996) showed that water Cherenkov detectors have superior sensitivity to those of scintillation counters for detecting EAS secondaries. Thus the tank design proposed here satisfies these design criteria, although the particle density resolution is somewhat poor. On average the pmt signal is about 100 photoelectrons for a through-going vertical muon. Figure 1 displays a crossectional view of a tank showing the position of the top-mounted, downwardlooking 8" pmt and the Tyvek-lined bottom, sides, and floating top. This position of the pmt gives a fairly uniform response across the full tank, although it does degrade the timing resolution somewhat compared to a bottom mounted, upward looking position. Due to its active material, water, the tank is sensitive to both the photonic and leptonic components of EAS as opposed to plastic scintillator based detectors which are mainly sensitive to the leptonic component. The Tyvek lining provides a diffusivly reflective inner surface with > 90% reflectivity at the important wavelengths determined from convoluting the Cherenkov photon spectrum and pmt quantum efficiency (wavelengths around 350 nm). Monte Carlo generated data was used to determine the performance characteristics of these tanks, and the improvement of the sensitivity of the Milagro detector generated by using these tanks. Corsika was used for generating simulated EAS and the Geant package was used to simulate the tank and Milagro detector responses (see (Westerhoff 1998) for more details). Monte Carlo Estimates of Milagro Performance Improvements The information acquired with the tanks discussed above can be used in two separate ways. First, it may improve the angular and energy reconstruction resolutions of EAS which trigger the Milagro pond detector by making additional independent shower front timing measurements and by improving the EAS core position resolution for EAS whose cores done not strike the pond. A simple multiplicity trigger condition of 50 pond pmts being hit by an EAS was used as a pond-trigger in simulations. Second, the information can be used to increase the effective area of the Milagro detector by using it in a combined pond-tank trigger. Improvements in Pond Triggered Events From simulation, on average about 24 tank pmts are hit per event where a hit is the detection of 1 or more photoelectrons. The occupancy (fraction of the time a given pmt or tank is hit) for pond pmts is about 30% and for tanks is about 10%. The tanks have fewer low pulse height hits than the pond pmts (below 30 pes) but about the same number of large pulseheight hits (above 30 pes). As seen in Figure 2, our simulations predict that using the tank array in reconstructing EAS core positions can improve the position resolution tremendously for EAS whose core positions are off the pond. This improvement is crucial for EAS energy determination and is also important in EAS angle determination because the pmt hit times must be corrected for EAS shower front curvature about the core position. Current ongoing studies of Monte Carlo generated EAS show that a good core position resolution should improve the hadronic cosmic ray rejection capabilities of Milagro as well. Figure 2: Plot of median core position error versus core distance from center of Milagro pond using tanks (circles) and not using tanks (crosses). Improvements in the angular reconstruction resolution of EAS is displayed in Figure 3. The improvement is maximal for low multiplicity (number of pmts in Milagro which detect light) events which are typically low primary energy EAS. It is also maximal for EAS whose cores land far from the pond. Improvements in Trigger Sensitivity Including tank acquired information within the Milagro trigger condition can increase the efficiency for seeing low energy events. This is clearly seen in Figure 4 which displays a plot of the effective area of the Milagro detector for three types of triggers. The pond-only trigger is a requirement that at least 50 pmts be hit by the EAS. The tank+pond trigger is that either the pond trigger be satisfied or that at least 5 tanks be hit by the EAS. The tank-only trigger is that at least 5 tanks be hit and that less than 50 pond pmts be hit by the EAS. The tank-only trigger is included to explicitly show the contribution of the tanks to the effective area. Those events obtained by using the tanks in a trigger condition have an average pond pmt multiplicity of 20 and an angular resolution of about 2.5 o . This resolution is significantly worse than the resolution of pondtriggered events (< 1.0 o ) but is adequate for doing coincident searches with most BATSE-detected Gamma Ray Bursts and for photon counting analyses where the event angles are not used. Results from a Prototype Array A prototype tank array has been installed near the Milagro pond to study the response of the water tanks to typical EAS that trigger Milagro. The array consists of 11 tanks built with commercially available polyethylene storage tanks. The installed pmts are of the same type as those in the Milagro pond (Hamamatsu R5912). The tanks are at various distances from the pond which will enable us to study their response as a function of EAS core distance and particle density. The tank hit multiplicity with at pond trigger requirement of approximately 120 hit pond pmts is 2.5. Results from these prototypes will also be presented. Figure 1 : 1Schematic of example water Cherenkov tank. Key features are top mounted, downward looking pmt and Tyvek lined inner surfaces (units are feet). The Milagro inspired tank array has 172 tanks placed on a square grid with a spacing of 15 m, giving a full array area of 200m x 200m centered on the Milagro pond. Figure 3 : 3Plots of median angle error versus number of pmt hits in top layer of Milagro pond and versus EAS core distance from center of Milagro pond. Circles are values when tanks are used and crosses are value when tanks are not used. Figure 4 : 4Plot of effective area versus energy. Dashed is pond-only trigger, dotted is tank-only trigger, and solid is pond-tank trigger. SummaryFrom the above simulation results one can see the predicted large improvement to both the angular and core position resolutions of the Milagro detector using information acquired by a spare array of instrumented Cherenkov water tanks. This improvement is mainly for EAS whose cores do not fall directly on the Milagro pond. Since the sensitivity of an VHE detector is proportional to its angular precision, this improvement will have a large positive effect on Milagro's sensitivity. The greatly improved core position resolution will increase Milagro's sensitivity to various source spectral characteristics.This research was supported in part by the National Science Foundation, the U. S. Department of Energy Office of High Energy Physics, the U. S. Department of Energy Office of Nuclear Physics, Los Alamos National Laboratory, the University of California, the Institute of Geophysics and Planetary Physics, The Research Corporation, and CalSpace. HE 6.1.02, these ICRC procedings. J F Mccullough, Westerhoff, 19th Texas Symposium on Relativistic Astrophysics procedings. 75McCullough, J.F. 1999, HE 6.1.02, these ICRC procedings. Westerhoff et al. 1998, 19th Texas Symposium on Relativistic Astrophysics procedings. Yodh, G.B. 1996, Space Science Reviews, 75: 199-212.
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[ "Improving Recommendation Fairness via Data Augmentation", "Improving Recommendation Fairness via Data Augmentation" ]
[ "Lei Chen ", "Le Wu ", "Kun Zhang [email protected] ", "Richang Hong ", "Defu Lian [email protected] ", "Zhiqiang Zhang ", "Jun Zhou [email protected] ", "Meng Wang ", "Lei Chen ", "Le Wu ", "Kun Zhang ", "Richang Hong ", "Defu Lian ", "Zhiqiang Zhang ", "Jun Zhou ", "Meng Wang ", "\nHefei University of Technology\nHefeiChina\n", "\nHefei University of Technology\nHefeiChina\n", "\nHefei University of Technology\nHefeiChina\n", "\nHefei University of Technology\nHefeiChina\n", "\nUniversity of Science and Technology of China\nHefei, Hangzhou, HangzhouChina, China, China\n", "\nInstitute of Artificial Intelligence\nComprehensive National Science Center\nHefei University of Technology\nHefei, Hefei, HefeiChina, China\n" ]
[ "Hefei University of Technology\nHefeiChina", "Hefei University of Technology\nHefeiChina", "Hefei University of Technology\nHefeiChina", "Hefei University of Technology\nHefeiChina", "University of Science and Technology of China\nHefei, Hangzhou, HangzhouChina, China, China", "Institute of Artificial Intelligence\nComprehensive National Science Center\nHefei University of Technology\nHefei, Hefei, HefeiChina, China" ]
[ "Proceedings of the ACM Web Conference 2023 (WWW '23)" ]
Collaborative filtering based recommendation learns users' preferences from all users' historical behavior data, and has been popular to facilitate decision making. Recently, the fairness issue of recommendation has become more and more essential. A recommender system is considered unfair when it does not perform equally well for different user groups according to users' sensitive attributes (e.g., gender, race). Plenty of methods have been proposed to alleviate unfairness by optimizing a predefined fairness goal or changing the distribution of unbalanced training data. However, they either suffered from the specific fairness optimization metrics or relied on redesigning the current recommendation architecture. In this paper, we study how to improve recommendation fairness from the data augmentation perspective. The recommendation model amplifies the inherent unfairness of imbalanced training data. We augment imbalanced training data towards balanced data distribution to improve fairness. Given each real original user-item interaction record, we propose the following hypotheses for augmenting the training data: each user in one group has a similar item preference (click or non-click) as the item preference of any user in the remaining group. With these hypotheses, we generate "fake" interaction behaviors to complement the original training data. After that, we design a bi-level optimization target, with the inner optimization generates better fake data to augment training data with our hypotheses, and the outer one updates the recommendation model * Le Wu is the corresponding author. parameters based on the augmented training data. The proposed framework is generally applicable to any embedding-based recommendation, and does not need to pre-define a fairness metric. Extensive experiments on two real-world datasets clearly demonstrate the superiority of our proposed framework. We publish the source code at https://github.com/newlei/FDA.CCS CONCEPTS• Information systems → Collaborative filtering; • Human-centered computing → User models.
10.1145/3543507.3583341
[ "https://export.arxiv.org/pdf/2302.06333v2.pdf" ]
256,827,410
2302.06333
4c030a94a735d1beb0dc09abd4bfea86d84faca1
Improving Recommendation Fairness via Data Augmentation May 1-5, 2023. May 1-5, 2023 Lei Chen Le Wu Kun Zhang [email protected] Richang Hong Defu Lian [email protected] Zhiqiang Zhang Jun Zhou [email protected] Meng Wang Lei Chen Le Wu Kun Zhang Richang Hong Defu Lian Zhiqiang Zhang Jun Zhou Meng Wang Hefei University of Technology HefeiChina Hefei University of Technology HefeiChina Hefei University of Technology HefeiChina Hefei University of Technology HefeiChina University of Science and Technology of China Hefei, Hangzhou, HangzhouChina, China, China Institute of Artificial Intelligence Comprehensive National Science Center Hefei University of Technology Hefei, Hefei, HefeiChina, China Improving Recommendation Fairness via Data Augmentation Proceedings of the ACM Web Conference 2023 (WWW '23) the ACM Web Conference 2023 (WWW '23)Austin, TX, USA 2023; Austin, TX, USAMay 1-5, 2023. May 1-5, 202310.1145/3543507.3583341ACM ISBN 978-1-4503-9416-1/23/04. . . $15.00 ACM Reference Format:. ACM, New York, NY, USA, 9 pages.user modelingfairnessdata augmentationfair recommendation Collaborative filtering based recommendation learns users' preferences from all users' historical behavior data, and has been popular to facilitate decision making. Recently, the fairness issue of recommendation has become more and more essential. A recommender system is considered unfair when it does not perform equally well for different user groups according to users' sensitive attributes (e.g., gender, race). Plenty of methods have been proposed to alleviate unfairness by optimizing a predefined fairness goal or changing the distribution of unbalanced training data. However, they either suffered from the specific fairness optimization metrics or relied on redesigning the current recommendation architecture. In this paper, we study how to improve recommendation fairness from the data augmentation perspective. The recommendation model amplifies the inherent unfairness of imbalanced training data. We augment imbalanced training data towards balanced data distribution to improve fairness. Given each real original user-item interaction record, we propose the following hypotheses for augmenting the training data: each user in one group has a similar item preference (click or non-click) as the item preference of any user in the remaining group. With these hypotheses, we generate "fake" interaction behaviors to complement the original training data. After that, we design a bi-level optimization target, with the inner optimization generates better fake data to augment training data with our hypotheses, and the outer one updates the recommendation model * Le Wu is the corresponding author. parameters based on the augmented training data. The proposed framework is generally applicable to any embedding-based recommendation, and does not need to pre-define a fairness metric. Extensive experiments on two real-world datasets clearly demonstrate the superiority of our proposed framework. We publish the source code at https://github.com/newlei/FDA.CCS CONCEPTS• Information systems → Collaborative filtering; • Human-centered computing → User models. INTRODUCTION Recommender systems automatically help users to find items they may like, and have been widely deployed in our daily life for decision-making [14,37,40]. Given the original user-item historical behavior data, most recommendation models design sophisticated techniques to learn user and item embeddings, and try to accurately predict users' unknown preferences to items [27,40,45]. Recently, researchers argued that simply optimizing recommendation accuracy lead to unfairness issues. Researchers have found that current recommender systems show apparent demographic performance bias of different demographic groups [16,17]. Career recommender systems tend to favour male candidates compared to females even though they are equally qualified [39]. Besides, recommendation accuracy performance shows significant differences between advantaged user group and disadvantaged user group [41,43]. Since the unfairness phenomenon has been ubiquitous in recommender systems and user-centric AI applications, how to define fairness measures and improve fairness to benefit all users is a trending research topic [24,46]. Among all fairness metrics, group based fairness is widely accepted and adopted, which argues that the prediction performance does not discriminate particular user groups based on users' sensitive attributes (e.g., gender, race) [15,20]. Given the basic ideas of group fairness, researchers proposed various group fairness measures and debiasing models to improve fairness, such as fairness regularization based models [3,53], sensitive attribute disentangle models [58], and adversarial techniques to remove sensitive attributes [52]. E.g., researchers proposed different recommendation fairness regularization terms and added these terms in the loss function for collaborative filtering [53]. In summary, these works alleviate unfairness in the modeling process with the fixed training data and achieve better fairness results. Table 1: An illustration of unfairness of recommendation accuracy performance on MovieLens dataset. We divide users into two subgroups based on the gender attribute. For each group, we calculate the distribution of clicked items of the corresponding group based on the training data, as well as the hit items on Top-K recommendation from two widely used recommendation models. Then, we measure the distribution differences of the two groups with JS divergence. We consider two typically recommendation models (BPR [49] and GCCF [40]). We consider the corrected hit from Top-K ranking list, denoted as "Top-20-Hit" and "Top-50-Hit". The recommendation accuracy performance has larger JS divergence compared to the training data, showing recommendation models exacerbate unfairness from training data. BPR [49] JS divergence GCCF [ In fact, researchers agree that unfairness in machine learning mainly comes from two processes. Firstly, the collected historical data shows imbalanced distribution among different user groups or reflects the real-world discrimination [9,30]. After that, algorithms that learn the typical patterns amplify imbalance or bias inherited from training data and hurt the minority group [48,52]. To show whether current recommendation algorithms amplify unfairness, let us take the MovieLens dataset as an example (detailed data description is shown in Section 5). We divide users into two subgroups based on the sensitive attribute gender. Given the training data, for each subgroup, we calculate the distribution over all items based on clicked items of users in this group. Then, we measure the distribution differences of the two sub user groups with Jensen-Shannon (JS) divergence [44]. After employing recommendation algorithms for Top-K recommendation, we measure the distribution difference of hit items of the two user groups. As can be seen from Table 1, the training data shows the preference distributions of the two groups are different. Both of the two recommendation models (i.e., BPR [49] and GCCF [40]) show larger group divergences of recommendation accuracy results compared to the divergence value of the training data. As a result, these two groups receive different benefits from the recommendation algorithms. Since all recommendation models rely on the training data for model optimization, comparing with the huge works on model-level fairness research, how to consider and improve recommendation fairness from the data perspective is equally important. When considering fairness from data perspective, some previous works proposed data resampling or data reweighting techniques to change data distribution [30,48]. As a particular group of users (e.g., females) are underrepresented in the training data, BN-SLIM is designed to resample a balanced neighborhood set of males and females for neighborhood-based recommendation [9]. Besides, given a specific fairness measure in recommendation, researchers made attempts to add virtual users via a unified optimization framework [48]. These previous works show the possibility of improving fairness from the data perspective. However, they either suffered from applying to specific recommendation models or relied on a specific predefined fairness metric, limiting the generality to transferring to current RS architecture. In this paper, we study the problem of designing a model-agnostic framework to improve recommendation fairness from data augmentation perspective. Given the original training data of user-item implicit feedback, we argue that the augmented training data are better balanced among different user groups, such that RS algorithms could better learn preferences of different user groups and avoid neglecting preferences of the minority groups. As a result, we argue the augmented data should satisfy the following hypotheses: for any user's two kinds of preference (a click record or a non-click record) in one group, there is another user in the remaining group (users with opposite sensitive attribute value) that has a similar item preference. With these hypotheses, we propose a general framework of Fairness-aware Data Augmentation (FDA ) to generate "fake" data that complement the original training data. We design a bi-level optimization function with the inner and outer loop to optimize FDA . The proposed FDA is model-agnostic and can be easily integrated to any embedding-based recommendation. Besides, FDA does not rely on any predefined specific fairness metrics. Finally, extensive experiments on two real-world datasets clearly demonstrate the superiority of our proposed framework. RELATED WORK Fairness Discovery and Measures. As machine learning technologies have become a vital part of our daily lives with a high social impact, fairness issues are concerned and raised [33,46]. Fairness refers to not discriminating against individuals or user groups based on sensitive user attributes [19,42]. One increasing requirement is how to define and measure fairness. Current fairness metrics can be categorized into individual fairness and group fairness. Individual fairness refers to producing similar predictions for similar individuals, and group fairness argues not discriminating a particular user group based on the sensitive attribute [5,7,52,54]. Among all group fairness metrics, Demographic Parity ( ) and Equality of Opportunity ( ) are widely accepted [12,25]. DP requires that each particular demographic group has the same proportion of receiving a particular outcome [19,56]. According to specific tasks, researchers have proposed specific demographic parity measures [10,35,56]. A notable disadvantage of demographic parity lies in ignoring the natural differences across groups. To this end, EO is proposed for an equal proportion of receiving a particular outcome conditioned on the real outcome [2,25,47]. In other words, a model is fair if the predicted results and sensitive attributes are independently conditioned on the real outcome [25]. Fairness aware Models. Building on the mathematical fairness metrics, some researchers design task-oriented model structures to meet fairness requirements. Current model based fairness approaches can be classified into regularization based approaches [1,18,21,29], causal based approaches [36,38,50], adversarial learning based approaches [4,52,57], and so on. These methods ensure that the outcome of models can meet fairness requirements, and the modification of model structures heavily relies on specific fairness definitions. E.g., regularization-based approaches add one fairness constraint to achieve a specific fairness at a time. Researchers need to design various constraints for achieving different fairness requirements [3,18,55]. Different from alleviating unfairness in the modeling process, some researchers attempted to solve fairness problems from the data perspective. Early works tried to directly remove the sensitive attribute from the training data [11,34]. Some researchers argue that unfairness comes from the imbalanced training data of the protected and unprotected groups, and employ bagging and balance groups in each bag to build stratified training samples [30]. Besides, perturbation approaches change the training data distribution with some prior assumptions of sensitive attributes, input features and labels. After that, a perturbed distribution of disadvantaged groups is used to mitigate performance disparities [22,32,51]. Most of these data modification approaches are designed for classification tasks with abundant data samples, in which each data sample is independent. In CF based recommender systems, users and items are correlated with sparse interaction data. Therefore, current approaches of modifying data distribution in the classification task could not be easily adapted for the recommendation task with limited observed user behavior data. Recently, the authors [48] propose two metrics that capture the polarization and unfairness in recommendation. The framework needs to take one of the proposed metrics as the optimization direction, in order to generate corresponding antidote new user profiles for the selected metric. By proposing the concept of a balanced neighborhood, the authors [9] borrow the idea of data sampling and design corresponding regularization to control neighbor distribution of each user. The regularization is applied to the sparse linear method (SLIM) to improve the outcome fairness [9]. The above two models explore the possibility of improving recommendation fairness by changing the training data distribution. However, they either need to define/introduce specific fairness metrics or are only suitable for a particular kind of recommendation model with well-designed heuristics. Therefore, the problem of how to design a general fairness framework from the data perspective that is suitable for different recommendation backbones and multiple fairness metrics is still under explored. PRELIMINARY In a recommender system, there are two sets of entities: a user set (| | = ) and an item set (| | = ), we denote the useritem interaction matrix as = [ ] × . We consider the common implicit feedback scenario. If user has clicked item , then = 1 indicates user likes the item , otherwise = 0. For each user , her clicked item set is denoted as = { : = 1}. As most modern recommender systems are built on embedding based architecture, we focus on embedding based recommendation models. Generally speaking, there are two key components for recommendation. First, a recommendation model Rec employs an encoder Enc to project users and items into the embedding space, formulated as E = ( , ) = [e 1 , .., e , ..., e , ..., e + ], where e is user 's embedding, e is item 's embedding. Then, the predicted preferenceˆcan be calculated withˆ= e e . Learning high-quality user and item embeddings has become the key of modern recommender systems. There are two typical classes of embedding approaches: the classical matrix factorization models [45,49] and neural graph-based models [27,40]. Given a binary sensitive attribute ∈ {0, 1}, denotes user 's attribute value. We divide the user set into two subsets: 0 and 1 . If = 0, then ∈ 0 , otherwise ∈ 1 . Please note that as we do not focus on any specific recommendation models, we assume the embedding based recommendation models are available, such as matrix factorization models [45] or neural graph-based models [40]. Our goal is to improve recommendation fairness with relatively high accuracy from the data augmentation perspective. Notations Description U, V userset | | = , itemset | | = ∈ {0, 1} a binary sensitive attribute 0 , 1 user group ∈ 0 if = 0 ∈ 1 if = 1 0 , 1 users, 0 ∈ 0 , 1 ∈ 1 0 , 1 , 0 , 1 real items 0 ,¯1,¯0,¯1 fake items , real positive data, real negative datā ,¯fake positive data, fake negative data THE PROPOSED FRAMEWORK In this section, we first introduce two hypotheses in our proposed FDA framework. Then, we show how to optimize two hypotheses given a recommendation model. Hypotheses for Generating Fake Data We argue that the augmented data should be balanced between two user groups, such that RS could learn latent preferences of different groups without neglecting the minority group. Since users have two kinds of behaviors, i.e., click (positive behavior) and non-click (negative behavior), we corresponding propose two hypotheses to augment data towards balanced distribution. In other words, for each behavior data ( 0 , , ) of user 0 ( 0 ∈ 0 ), item , and the implicit feedback value , we hope there is a user in the remaining group 1 that shows the same preference value to a similar item. Under this assumption, we can improve data balance of different user groups by generating fake behavior data that complements the training data. Specifically, the first hypothesis focuses on the positive behavior among two groups ( 0 and 1 ). ! ∈ ! " ∈ " ! > # ! ! # ! $ ! # ! % ! positive negative " > # " " # " $ " # " % " positive negative . . ! ≅ ! , . . " ≅ " , ! ∈ ! " ∈ " ! , > # " " ̅ # " $ ! & # " % " positive negative " , > # ! ! # ! % ! positive negative ̅ # ! $ " & Hypothesis 1: generating fake positive data to balance positive data distribution of two groups Hypothesis 2: generating fake negative data to balance negative data distribution of two groups ! ∈ ! " ∈ " ! > # ! ! # ! $ ! # ! % ! positive negative " > # " " # " $ " # " % " positive negative . . ! ≅ ! , . . " ≅ " , ! ∈ ! " ∈ " " > # " ! , # " $ " ̅ # " % ! positive negative ! > # ! " , ̅ # ! % " positive negative # ! $ ! real data generated fake data < ! , ! , ! > < " , " , " > < " , " , " > < ! , ! , ! > < " , ! , , " > < ! , " , , ! > < " , " , ! , > < ! , ! , " , > Figure 1: The overall architecture of the proposed FDA framework. By not changing the recommendation model, our key idea is to improve data balance between different user groups via data augmentation. For each real data ( 0 , , ) of user 0 ( 0 ∈ 0 ), item , and the implicit feedback value of this user-item pair from the training data, we hope there is a user in the remaining group 1 that shows the same preference value to a similar item¯. As the detailed implicit feedback value can be positive or unknown, we turn the above idea into two hypotheses. In the upper part of the figure, according to Hypothesis 1, we generate two kinds of fake positive examples (¯0¯1 and¯1¯0 ). Correspondingly, based on the generated fake data, we can obtain fake records (< 0 ,¯1, 0 >, < 1 ,¯0, 1 >). As shown in the lower part of the figure, according to Hypothesis 2, we generate two kinds of fake negative data (¯0¯1 and¯1¯0 ), and obtain fake records (< 0 , 0 ,¯1 >, < 1 , 1 ,¯0 >). Hypothesis 1. Assume that there is user 0 ∈ 0 , and item 0 is one of user 0 's clicked items 0 ∈ 0 . There should exist user 1 ∈ 1 that also clicks a similar item¯0 ≃ 0 . Similarly, if user 1 ∈ 1 has clicked item 1 ∈ 1 , user 0 ∈ 0 should also click a similar item¯1 ≃ 1 . This hypothesis can be formulated as follows: ∀ 0 ∈ 0 , 1 ∈ 1 , 0 0 = 1, ℎ¯1¯0 = 1; ℎ 0 ≃¯0. (1) ∀ 1 ∈ 1 , 0 ∈ 0 , 1 1 = 1, ℎ¯0¯1 = 1; ℎ 1 ≃¯1. (2) In the above equations,¯1¯0 and¯0¯1 are fake interactions data that does not appear in the training data. The second hypothesis focuses on the negative behavior among two groups ( 0 and 1 ): Hypothesis 2. If user 0 ∈ 0 does not click item 0 ∈ − 0 , there should also exist user 1 ∈ 1 that does not click similar item¯0 ≃ 0 . Correspondingly, if user 1 ∈ 1 does not click item 1 ∈ \ 1 , user 0 ∈ 0 should not click similar item¯1 ≃ 1 . This hypothesis can be formulated as follows: ∀ 0 ∈ 0 , 1 ∈ 1 , 0 0 = 0, ℎ¯1¯0 = 0; ℎ 0 ≃¯0. (3) ∀ 1 ∈ 1 , 0 ∈ 0 , 1 1 = 0, ℎ¯0¯1 = 0; ℎ 1 ≃¯1. (4) In the above equations,¯1¯0 and¯0¯1 are fake interactions. By employing Hypothesis 1& 2, we can adjust the training data to make sure that the augmented training data are balanced for different groups. Optimization For Fake Data After generating fake data, we focus on augmenting the original training data for implicit feedback based recommendation. In implicit feedback based recommendation, pairwise learning has been widely used [27,40,49]. For each user , if item is clicked by user , and item is not clicked by user , then this clicked item is more relevant than a non-clicked item , which can be formulated as > . As a result, the clicked item should be assigned higher prediction value compared to the predicted preference of any non-clicked item : ∀ ∈ , ∈ − :ˆ>ˆ (5) . ., e e > e e . Based on users' two sensitive attribute values and two kinds of implicit feedback behaviors, we can obtain four types of interactions (i.e., positive interactions for users from two groups: 0 0 and 1 1 , negative interactions for users from two groups: 0 0 and 1 1 .) to support Hypothesis 1 and Hypothesis 2. The original training data contains two types of interactions as: = {< 0 , 0 , 0 >, < 1 , 1 , 1 >},(7) where user 0 belongs to user group 0 . Item 0 is a positive feedback of user 0 , and item 0 is a negative feedback of user 0 . Similarly, for user 1 ∈ 1 , item 1 is a positive feedback and item 1 is a negative feedback. Next, we introduce how to construct the remaining two types of fake interactions and optimize them. Optimization of Hypothesis 1. Hypothesis 1 focuses on clicked items and encourages the positive behavior distribution of two groups are balanced. For each , we can generate the corresponding fake data 1 according to Hypothesis 1 as: = {< 0 , 0 , 0 >, < 1 , 1 , 1 >},(8)1 = {< 1 ,¯0, 1 >, < 0 ,¯1, 0 >},(9) where 0 ≃¯0 and 1 ≃¯1. 1 denotes fake positive interaction data. Note that, given any two real positive behavior data ( 0 0 and¯1 1 ), 1 also contains two corresponding fake positive behavior data (¯1¯0 and¯0¯1 ). Therefore, we have the following expressions on 1 : 0 > 1 1¯1 > 0 0 .(10) Similar to Eq.(5), we can formulate Eq.(10) with the following optimization goal:ˆ1¯0 >ˆ1 1ˆ0¯1 >ˆ0 0 ,(11) which can also be calculated as follows: e 1ē 0 > e 1 e 1 e 0ē 1 > e 0 e 0 .(12) Optimization of Hypothesis 2. Hypothesis 2 focuses on nonclicked items and encourages the non-click behavior of two user groups to be balanced. For each triplet from , we can generate the corresponding fake behavior data 2 according to Hypothesis 2: = {< 0 , 0 , 0 >, < 1 , 1 , 1 >},(13)2 = {< 1 , 1 ,¯0 >, < 0 , 0 ,¯1 >},(14) where 0 ≃¯0 and 1 ≃¯1. Therefore, we have the following goal on 2 : 1 > 1¯0 0 > 0¯1 ,(15) Similarly, we can turn the above goal into optimization functions as:ˆ1 1 >ˆ1¯0ˆ0 0 >ˆ0¯1,(16) which can be calculated as follows: e 1 e 1 > e 1ē 0 e 0 e 0 > e 0ē 1 . We construct corresponding fake data for each hypothesis, then we integrate all fake data from 1 and 2 : = { 1 , 2 } = {< 1 ,¯0, 1 >, < 0 ,¯1, 0 >, < 1 , 1 ,¯0 >, < 0 , 0 ,¯1 >}.(18) With the implicit feedback, Bayesian Personalized Ranking (BPR) is widely used for learning the pairwise based optimization function [45,49]. We also adopt BPR loss to optimize the fake data generation process: min L = − ∑︁ < , , > ∈ ( (ˆ−ˆ)) (19) = − ∑︁ < 1 ,¯0, 1 > ∈ 1 ( (ˆ1¯0 −ˆ1 1 )) − ∑︁ < 0 ,¯1, 0 > ∈ 1 ( (ˆ0¯1 −ˆ0 0 )) − ∑︁ < 1 , 1 ,¯0> ∈ 2 ( (ˆ1 1 −ˆ1¯0 )) − ∑︁ < 0 , 0 ,¯1> ∈ 2 ( (ˆ0 0 −ˆ0¯1 )). The key challenge in Eq. (19) lies in estimating the fake data . A direct approach is to define a similarity function among items, and then find similar items within a predefined threshold. However, with sparse user-item click behavior data, directly computing item similarities from user-item interaction matrix is not only time-consuming, but also not accurate. We propose to find similar items (¯and¯) from continuous embedding space. An intuitive idea is to utilize the well-trained embeddings from the recommendation model. Note that, recommender systems transform user/item ID to continuous embedding space: E = ( , ). We therefore define and find similar items (¯and¯) based on continuous embedding space E. In order to satisfy the similarity requirement, first of all, the similar items need to lie within the original embedding distribution. Otherwise, it will seriously affect the recommendation accuracy. Inspired by adversarial examples and poisoning attacks [6], we employ a non-random perturbation to generate similar items in continuous embedding E. For each item , we add small random noise to the item original embedding e , then we can construct the corresponding similar item embeddingē . The similar item can be formulated as follows: e = e + , ℎ ∥ ∥ ≤(20) The noise is bounded in a small range , and the operator ≤ enforces the constraint for each dimension of . Since is a small "unseen" noise, it is natural thatē is similar to e andē also lies within the original embedding distribution. This method can meet the requirements of Hypothesis 1 and Hypothesis 2. By combining the similar item requirement in Eq. (20) and the optimization of the fake data in Eq. (19), the loss function on fake data can be changed into the embedding form as: where 0 , 1 , 0 , 1 respectively denote the small noises adding to corresponding item embeddings, and Θ denotes the combination of all small noises in the fake data. With fixed embeddings E from any recommender model, we optimize the Θ to generate fake data. The Overall Bi-Level Optimization After generating fake training data, we intend to integrate these fake data with original training data as augmented data. We could not use all the fake data for data augmentation as too many fake data records would dramatically modify the original data distribution, leading to decrease accuracy. To better trade-off, we develop a random mask operation to inject fake data. The mask operation can be formulated as follows: = 1, 0, else and ∑︁ =1 ≤ ,(22) where denotes the maximum number of selected fake data in each update. Obviously, should be less than the number of items . Since only part of the fake data is selected for training, the impact on recommendation accuracy can be controlled. After fake data records are selected by the mask operation, we combine the selected fake data and original training data to construct the augmented data . Next, we retrain the recommendation model on the augmented data. This training process of a recommender model is the same as the recommender model except that the data input is augmented. In short, the overall training process of FDA involves two iterative steps: generating fake data based on the previous recommendation embeddings and training recommendation models given updated fake data. The two iterative steps can be combined by solving the following bi-level optimization problem: min E min Θ L = ∑︁ − ( ([ 0 * ¯1¯0 + (1 − 0 ) * ˆ1 1 ] −ˆ1 1 )) − ( ([ 1 * ¯0¯1 + (1 − 1 ) * ˆ0 0 ] −ˆ0 0 )) − ( (ˆ1 1 − [ 0 * ¯1¯0 + (1 − 0 ) * ˆ1 1 ])) − ( (ˆ0 0 − [ 1 * ¯0¯1 + (1 − 1 ) * ˆ0 0 ])),(23) where = ∪ is the augmented data. From this formulation, we can observe that our proposed FDA involves two levels of optimization. Discussion Model Analysis. Given the above bi-level optimization process of FDA , we now analyze why FDA can achieve a better balance between recommendation accuracy and fairness without changing the recommendation architecture (i.e., the outer loop that updates recommendation parameters). In the inner minimization loop, our key idea is to encourage for each triple behavior of the user, there is a user in the remaining group that shows the same preference value to a similar item. As users have two kinds of preferences (i.e., click and non-click), the key idea turns to two hypotheses for generating fake data. Therefore, the augmented data that contains both the training data and the fake data are more balanced compared to the original training data. For the original training data, the recommendation results are unfairer as recommendation results would amplify unbalance inherited from the input data. With more balanced data, the recommender models can output better fairness metrics even though the recommendation part does not model fairness. Besides, as we constrain the fake data generated by adding a small random noise, and the fake data is controlled by a limited ratio with random mask operation, the recommendation accuracy can also be guaranteed. Compared to other data balance or data augmentation based fairness enhanced recommendation models, FDA shows the advantage of generally applicable to embedding based recommendation backbones and does not need to predefine a specific group fairness metric. Extension to Multiple Sensitive Values. In FDA , similar as many fairness aware approaches, we start with the binary sensitive attribute values [9,25,53]. When dealing with a sensitive attribute with K ( > 2) values, a naive extension to multiple sensitive attribute values is to encourage that: for each user's one kind of behavior to an item in one group, we encourage that there are other users in each remaining group that show the same behavior to a similar item. Therefore, for each hypothesis, we can generate − 1 fake data, and use the bi-level optimization with both original and the selected fake data with mask operation. EXPERIMENTS 5.1 Experimental Setup Datasets. We conduct experiments on two publicly available datasets: MovieLens [26] and LastFM [13,52]. For MovieLens 1 , we adopt the same data splitting strategy as previous works for fair recommendation [8,52]. We treat the items that a user's rating is larger than 3 as positive feedback. Moreover, we randomly select 80% of records for training and the remaining 20% records for the test. LastFM is a large music recommendation dataset 2 . We treat the items that a user plays as the positive feedback. To ensure the quality of the dataset, we use the 10-core setting to ensure that users (items) have at least 10 interaction records. We split the historical records into training, validation, and test parts with the ratio of 7:1:2. Besides the user-item interaction records, these two datasets also have the user profile data, including gender (two classes) for users. Similar as previous works, we treat gender as the sensitive attribute and divide users into two subgroups. The statistics of these two datasets are shown in Table 3. Evaluation Metrics. Since we focus on the trade-off between fairness and recommendation accuracy, we need to evaluate two aspects and report the trade-off results. On the one hand, we employ two widely used ranking metrics for recommendation accuracy: HR [23] and NDCG [31] to evaluate the Top-K recommendation. Larger values of HR and NDCG mean better recommendation accuracy performance. On the other hand, we adopt two group fairness metrics: Demographic Parity (DP) [56] and Equality of Opportunity (EO) [25] to evaluate fairness. DP and EO evaluate group fairness from different aspects. For both fairness metrics, the smaller values mean better fairness results. The following equation calculates DP measure: = 1/ ∑︁ ∈ | ∈ 0 1 ∈ − ∈ 1 1 ∈ | ∈ 0 1 ∈ + ∈ 1 1 ∈ , where 0 denotes user group with sensitive attribute = 0, and 1 denotes user group with sensitive attribute = 1. is Top-K ranked items for user . Please note that as DP forcefully requires similarly predicted results across different groups, it naturally neglects the natural preference differences of user clicked patterns [53]. EO is proposed to solve the limitation of DP [12,24]. Specifically, it requires similar Table 4: Recommendation accuracy and fairness performance on MovieLens with varying Top-K values. We compare all fairness-aware models, the best results are presented in bold font and the second best results are presented in underline. The performance improvement of our model against the best baseline is significant under the paired-t test. predicted results across different groups conditioned on user real preferences. We calculate EO as follows: = 1/ ∑︁ ∈ | ∈ 0 1 ∈ & ∈ − ∈ 1 1 ∈ & ∈ | ∈ 0 1 ∈ & ∈ + ∈ 1 1 ∈ & ∈ , where is items that user clicks on the test data. Because not all predicted results have corresponding ground true labels, we calculate EO metric only on the testing data. A smaller EO means there is less unfairness, as the two groups receive similar recommendation accuracy. Baselines. The baseline models can be divided into two categories: recommendation based models BPR [49] and GCCF [40], and the fairness-oriented models BN-SLIM [9], data augmentation model (FairUser) [48] and the fairness regularization based model [53]. FairUser adds virtual users to achieve fairness. As its optimization process is very time-consuming and needs to compute the full user-item matrix at each iteration, we only test FairUser on MovieLens dataset. Similar as the fairness regularization based model [53], we add different group fairness metrics (DP and EO) as regularization terms into recommendation based models. E.g., BPR_EO denotes treating EO as the regularization term for the base recommendation model of BPR. Our proposed framework FDA can be applied on different recommendation backbones. We select BPR [49], NCF [28] and GCCF [40] as recommendation backbones show state-of-the-art performance. We use FDA_BPR, FDA_NCF and FDA_GCCF to denote the variants of our proposed framework with different recommendation backbones. Parameter Setting. Our implementations are based on Pytorch-GPU 1.6.0. The embedding size is set as 64 for FDA. All the parameters are differentiable in the objective function, and we use the Adam optimizer to optimize the model. In Eq.(23), the initial learning rate of Adam is 0.001 for the outer minimization and the inner minimization. Table 4 and 5 report the overall results on two datasets. We have several observations from these two tables. FDA achieves the best performance from the two aspects: 1) improving EO and DP concurrently; 2) the trade-off between recommender accuracy and fairness. Overall Performance on Two Datasets First, when comparing each fairness metric, our proposed FDA outperforms other models on both DP and EO group fairness metrics. FairUser and BN-SLIM show worse performance than BPR (not considering fairness) on EO metric. BPR_EO and GCCF_EO can improve EO and DP concurrently. But, BPR_EO and GCCF_EO is worse than FDA. DP and EO reflect the group fairness from two aspects. FDA achieves good results of multiple group metrics on all datasets. Thus, FDA can well alleviate the unfairness problem and achieve better fairness performance. Second, when comparing the trade-off between recommender accuracy and fairness, all fairness-aware baselines perform worse on recommendation accuracy performance than FDA. In other words, all fairness-aware baselines cause a larger decrease in recommender accuracy. Therefore, while achieving better fairness performance, FDA has the least damage to accuracy on different datasets. We conclude that FDA can reach the best balance between accuracy and fairness. Third, no matter the base backbone model is BPR, NCF or GCCF, FDA can improve fairness and show better recommendation performance. FDA_BPR shows worse performance than FDA_GCCF. This is due to the fact that the base model (i.e., BPR) in FDA_BPR does not Model Analyses Effects of fake data numbers. The setting of plays an important role to control the maximum number of fake data. We conduct experiments on different , as shown in Figure 2. Specifically, we show different ratios of maximum fake data and the number of all items ( / ). Since the fake data is similar to the real data, the fake data does not seriously affect the recommendation accuracy. When the ratio increases from 0.1 to 0.6, the recommended performance of FDA_BPR and FDA_GCCF does not decrease nearly. Among all ratios, we can find that FDA achieves a better balance on the fairness and recommender accuracy when the ratio equals 0.3 for FDA_BPR and 0.4 for FDA_GCCF on LastFM. The trend is similar on MovieLens. Due to the page length limit, the results on MovieLens are not reported. Measuring the distribution differences of recommendation results. As shown in Table 1, we employ the Jensen-Shannon (JS) divergence to measure differences of two user groups (i.e., 0 and 1 ) on the larger LastFM. We pay attention to Top-20 and Top-50 ranked items, denoted as "Top-20" and "Top-50". We also consider the corrected Top-20 and Top-50 ranked items, denoted as "Top-20-Hit" and "Top-50-Hit" to measure the recommendation accuracy differences between the two groups. As shown in Figure 3, FDA can also improve fairness performance compared to its original recommendation backbone. Also, we observe that JS divergence of "Top-20" and "Top-50" is smaller than that of "Top-20-Hit" and "Top-50-Hit". This is reasonable as "Top-K" divergence measures ranking differences between the two groups and does not take recommendation accuracy into consideration. In contrast, "Top-K-Hit" measures the recommendation accuracy differences between the two groups. Nevertheless, FDA can improve these two metrics. Ablation study. In this part, we investigate the effectiveness of each hypothesis: Hypothesis 1 and Hypothesis 2 of our proposed FDA framework. The results are illustrated in Table 6. From this table, we can obtain the following observations. First, each single hypothesis (Hypothesis 1 or Hypothesis 2) can help the model achieve comparable performance, indicating the usefulness of our proposed hypothesis. Second, compared with the performance of models with a single hypothesis, models with both of them (the entire FDA framework) have better performance, demonstrating the necessity of both hypotheses. As these two hypotheses consider different kinds of users' click or non-click behavior, combining them together reaches the best performance. CONCLUSION In this paper, we studied the recommendation fairness issue from data augmentation perspective. Given the original training data, we proposed a FDA framework to generate fake user behavior data, in order to improve recommendation fairness. Specifically, given the overall idea of balanced data, we proposed two hypotheses to guide the generation of fake data. After that, we designed a bi-level optimization target, in which the inner optimization generates better fake data and the outer optimization finds recommendation parameters given the augmented data that comprises both the original training data and the generated fake data. Please note that, FDA can be applied to any embedding based recommendation backbones, and does not rely on any specific fairness metrics. Extensive experiments on two real-world datasets clearly showed FDA is effective to balance recommendation accuracy and fairness under different recommendation backbones. ( (e 0 (e 1 + 1 ) − e 0 e 0 )) − ∑︁ ( (e 1 (e 0 + 0 ) − e 1 e 1 )) − ∑︁ ( (e 0 e 0 − e 0 (e 1 + 1 ))) − ∑︁ ( (e 1 e 1 − e 1 (e 0 + 0 ))), Figure 2 : 2Performance under different ratios on LastFM. Figure 3 : 3The distribution differences of recommendation results with two recommendation backbones on LastFM. Table 2 : 2Mathematical Notations Table 3 : 3Statistics of the two datasets.Datasets Users Items Traning Records Density MovieLens 6,040 3,952 513,112 2.150% LastFM 139,371 60,081 4,017,311 0.048% Table 5 : 5Recommendation accuracy and fairness performance on LastFM with varying Top-K values. The performance improvement of our model against the best baseline is significant under the paired-t test.Model K=10 K=20 K=30 K=40 K=50 HR↑ NDCG↑ DP↓ EO↓ HR↑ NDCG↑ DP↓ EO↓ HR↑ NDCG↑ DP↓ EO↓ HR↑ NDCG↑ DP↓ EO↓ HR↑ NDCG↑ DP↓ EO↓ BPR 0.1323 0. Table 6 : 6Ablation study of the two modules of FDA: Hypothesis 1 and Hypothesis 2 .perform as well as the base graph embedding model GCCF. On the whole, for different recommendation backbone models and different experimental settings, FDA can effectively balance accuracy and fairness. This demonstrates the flexibility and effectiveness of FDA. Fourth, we observe the improvements of fairness on MovieLens are not so obvious as results on LastFM. In other words, eliminating unfairness on LastFM is easier than that on MovieLens. We guess a possible reason is that the sparsity and number of users are different on these two datasets.Model Hypothesis 1 Hypothesis 2 MovieLens LastFM HR@20 NDCG@20 DP@20 EO@20 HR@20 NDCG@20 DP@20 EO@20 FDA_BPR ✓ ✕ 0.2706 0.2462 0.6099 0.6564 0.1971 0.1613 0.6017 0.6202 FDA_BPR ✕ ✓ 0.2721 0.2360 0.5841 0.6682 0.1951 0.1552 0.5585 0.6016 FDA_BPR ✓ ✓ 0.2716 0.2321 0.5731 0.6562 0.1965 0.1573 0.5392 0.5656 FDA_GCCF ✓ ✕ 0.2650 0.2397 0.6078 0.6610 0.1850 0.1511 0.5905 0.6021 FDA_GCCF ✕ ✓ 0.2876 0.2463 0.5824 0.6696 0.1906 0.1499 0.5647 0.5991 FDA_GCCF ✓ ✓ 0.2857 0.2487 0.5786 0.6593 0.1976 0.1580 0.5450 0.5693 https://grouplens.org/datasets/movielens/ 2 http://ocelma.net/MusicRecommendationDataset/lastfm-360K.html ACKNOWLEDGEMENTSThis work is supported in part by grants from the National Key Re Learning optimal and fair decision trees for non-discriminative decision-making. 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[ "https://github.com/newlei/FDA.CCS" ]
[ "JWST constraints on the UV luminosity density at cosmic dawn: implications for 21-cm cosmology", "JWST constraints on the UV luminosity density at cosmic dawn: implications for 21-cm cosmology" ]
[ "Sultan Hassan \nCenter for Cosmology and Particle Physics\nDepartment of Physics\nNew York University\n726, 10003Broadway, New YorkNYUSA\n\nCenter for Computational Astrophysics, Flatiron Institute\n162 5th Ave10010New YorkNYUSA\n\nDepartment of Physics & Astronomy\nUniversity of the Western Cape\n7535Cape TownSouth Africa\n", "Christopher C Lovell \nInstitute of Cosmology and Gravitation\nUniversity of Portsmouth\nBurnaby RoadPO1 3FXPortsmouthUK\n\nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n", "Piero Madau \nDepartment of Astronomy & Astrophysics\nUniversity of California\nSanta Cruz95064CAUSA\n", "Marc ", "Rachel S Somerville \nCenter for Computational Astrophysics, Flatiron Institute\n162 5th Ave10010New YorkNYUSA\n", "Blakesley Burkhart \nCenter for Computational Astrophysics, Flatiron Institute\n162 5th Ave10010New YorkNYUSA\n\nDepartment of Physics and Astronomy\nRutgers University\n136 Frelinghuysen Rd08854PiscatawayNJUSA\n", "Keri L Dixon \nInstitute for Computational Science\nUnited Arab Emirates 12 Center for Astro, Particle and Planetary Physics (CAP 3 )\nNew York University\nAbu Dhabi, New York University Abu Dhabi 13PO Box 129188Abu Dhabi\n\nUniversity of Zurich\nCH-8057ZurichSwitzerland\n", "Robert Feldmann ", "Tjitske K Starkenburg \nDepartment of Physics & Astronomy and CIERA\nNorthwestern University\n1800 Sherman Ave60201EvanstonILUSA\n", "John F Wu \nSpace Telescope Science Institute\n3700 San Martin Dr21218BaltimoreMD\n\nDepartment of Physics & Astronomy\nJohns Hopkins University\n3400 N Charles St21218BaltimoreMD\n", "Christian Kragh Jespersen \nDepartment of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA\n", "Joseph D Gelfand \nInstitute for Computational Science\nUnited Arab Emirates 12 Center for Astro, Particle and Planetary Physics (CAP 3 )\nNew York University\nAbu Dhabi, New York University Abu Dhabi 13PO Box 129188Abu Dhabi\n\nUniversity of Zurich\nCH-8057ZurichSwitzerland\n", "Ankita Bera \nDepartment of Physics\nPresidency University\n86/1 College Street700073KolkataIndia\n", "\nHuertas-Company\n7, 8\n", "\nInstituto de Astrofísica de Canarias\nLa LagunaTenerifeSpain\n", "\nUniversidad de la Laguna\nLa LagunaTenerifeSpain\n", "\nLERMA -Observatoire de Paris, PSL\nUniversité Paris-Cité\nParisFrance\n" ]
[ "Center for Cosmology and Particle Physics\nDepartment of Physics\nNew York University\n726, 10003Broadway, New YorkNYUSA", "Center for Computational Astrophysics, Flatiron Institute\n162 5th Ave10010New YorkNYUSA", "Department of Physics & Astronomy\nUniversity of the Western Cape\n7535Cape TownSouth Africa", "Institute of Cosmology and Gravitation\nUniversity of Portsmouth\nBurnaby RoadPO1 3FXPortsmouthUK", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Department of Astronomy & Astrophysics\nUniversity of California\nSanta Cruz95064CAUSA", "Center for Computational Astrophysics, Flatiron Institute\n162 5th Ave10010New YorkNYUSA", "Center for Computational Astrophysics, Flatiron Institute\n162 5th Ave10010New YorkNYUSA", "Department of Physics and Astronomy\nRutgers University\n136 Frelinghuysen Rd08854PiscatawayNJUSA", "Institute for Computational Science\nUnited Arab Emirates 12 Center for Astro, Particle and Planetary Physics (CAP 3 )\nNew York University\nAbu Dhabi, New York University Abu Dhabi 13PO Box 129188Abu Dhabi", "University of Zurich\nCH-8057ZurichSwitzerland", "Department of Physics & Astronomy and CIERA\nNorthwestern University\n1800 Sherman Ave60201EvanstonILUSA", "Space Telescope Science Institute\n3700 San Martin Dr21218BaltimoreMD", "Department of Physics & Astronomy\nJohns Hopkins University\n3400 N Charles St21218BaltimoreMD", "Department of Astrophysical Sciences\nPrinceton University\n08544PrincetonNJUSA", "Institute for Computational Science\nUnited Arab Emirates 12 Center for Astro, Particle and Planetary Physics (CAP 3 )\nNew York University\nAbu Dhabi, New York University Abu Dhabi 13PO Box 129188Abu Dhabi", "University of Zurich\nCH-8057ZurichSwitzerland", "Department of Physics\nPresidency University\n86/1 College Street700073KolkataIndia", "Huertas-Company\n7, 8", "Instituto de Astrofísica de Canarias\nLa LagunaTenerifeSpain", "Universidad de la Laguna\nLa LagunaTenerifeSpain", "LERMA -Observatoire de Paris, PSL\nUniversité Paris-Cité\nParisFrance" ]
[]
An unprecedented array of new observational capabilities are starting to yield key constraints on models of the epoch of first light in the Universe. In this Letter we discuss the implications of the UV radiation background at cosmic dawn inferred by recent JWST observations for radio experiments aimed at detecting the redshifted 21-cm hyperfine transition of diffuse neutral hydrogen. Under the basic assumption that the 21-cm signal is activated by the Lyα photon field produced by metal-poor stellar systems, we show that a detection at the low frequencies of the EDGES experiment may be expected from a simple extrapolation of the declining UV luminosity density estimated at z 14 by JWST early galaxy data. Our findings raise the intriguing possibility that a high star formation efficiency at early times may trigger the onset of intense Lyα emission at redshift z 18 and produce a cosmic 21-cm absorption signal 200 Myr after the Big Bang.
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[ "https://export.arxiv.org/pdf/2305.02703v1.pdf" ]
258,479,943
2305.02703
30acfefffd3908f513697edb8868cd5d16def519
JWST constraints on the UV luminosity density at cosmic dawn: implications for 21-cm cosmology May 5, 2023 Sultan Hassan Center for Cosmology and Particle Physics Department of Physics New York University 726, 10003Broadway, New YorkNYUSA Center for Computational Astrophysics, Flatiron Institute 162 5th Ave10010New YorkNYUSA Department of Physics & Astronomy University of the Western Cape 7535Cape TownSouth Africa Christopher C Lovell Institute of Cosmology and Gravitation University of Portsmouth Burnaby RoadPO1 3FXPortsmouthUK Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Piero Madau Department of Astronomy & Astrophysics University of California Santa Cruz95064CAUSA Marc Rachel S Somerville Center for Computational Astrophysics, Flatiron Institute 162 5th Ave10010New YorkNYUSA Blakesley Burkhart Center for Computational Astrophysics, Flatiron Institute 162 5th Ave10010New YorkNYUSA Department of Physics and Astronomy Rutgers University 136 Frelinghuysen Rd08854PiscatawayNJUSA Keri L Dixon Institute for Computational Science United Arab Emirates 12 Center for Astro, Particle and Planetary Physics (CAP 3 ) New York University Abu Dhabi, New York University Abu Dhabi 13PO Box 129188Abu Dhabi University of Zurich CH-8057ZurichSwitzerland Robert Feldmann Tjitske K Starkenburg Department of Physics & Astronomy and CIERA Northwestern University 1800 Sherman Ave60201EvanstonILUSA John F Wu Space Telescope Science Institute 3700 San Martin Dr21218BaltimoreMD Department of Physics & Astronomy Johns Hopkins University 3400 N Charles St21218BaltimoreMD Christian Kragh Jespersen Department of Astrophysical Sciences Princeton University 08544PrincetonNJUSA Joseph D Gelfand Institute for Computational Science United Arab Emirates 12 Center for Astro, Particle and Planetary Physics (CAP 3 ) New York University Abu Dhabi, New York University Abu Dhabi 13PO Box 129188Abu Dhabi University of Zurich CH-8057ZurichSwitzerland Ankita Bera Department of Physics Presidency University 86/1 College Street700073KolkataIndia Huertas-Company 7, 8 Instituto de Astrofísica de Canarias La LagunaTenerifeSpain Universidad de la Laguna La LagunaTenerifeSpain LERMA -Observatoire de Paris, PSL Université Paris-Cité ParisFrance JWST constraints on the UV luminosity density at cosmic dawn: implications for 21-cm cosmology May 5, 2023Draft version Typeset using L A T E X twocolumn style in AASTeX631 An unprecedented array of new observational capabilities are starting to yield key constraints on models of the epoch of first light in the Universe. In this Letter we discuss the implications of the UV radiation background at cosmic dawn inferred by recent JWST observations for radio experiments aimed at detecting the redshifted 21-cm hyperfine transition of diffuse neutral hydrogen. Under the basic assumption that the 21-cm signal is activated by the Lyα photon field produced by metal-poor stellar systems, we show that a detection at the low frequencies of the EDGES experiment may be expected from a simple extrapolation of the declining UV luminosity density estimated at z 14 by JWST early galaxy data. Our findings raise the intriguing possibility that a high star formation efficiency at early times may trigger the onset of intense Lyα emission at redshift z 18 and produce a cosmic 21-cm absorption signal 200 Myr after the Big Bang. INTRODUCTION A number of observational facilities are currently or will soon probe the epoch of cosmic dawn (z > 10), when the first stars and galaxies are expected to have formed. Results from these facilities are expected to place important constraints on the first astrophysical sources of radiation, including their number density, ionising emissivity, as well as the physics of their formation. * E-mail: [email protected] NHFP Hubble Fellow. The James Webb Space Telescope (JWST) is the current flagship space-based infrared observatory, and was specifically designed to probe the epoch of first light as a one of the main scientific goals (Robertson 2022). One of the first tantalising results from early-release JWST data has been the discovery of very high redshift candidate galaxies in NIRCam imaging (e.g. Adams et al. 2022;Atek et al. 2023;Bradley et al. 2022;Castellano et al. 2022a,b;Donnan et al. 2022Finkelstein et al. 2022Finkelstein et al. , 2023Morishita & Stiavelli 2023;Naidu et al. 2022a;Pérez-González et al. 2023;Yan et al. 2023). Not only are these galaxies at much higher redshifts than any galaxy discovered previ-ously by the Hubble Space Telescope (HST), but they are also surprisingly bright. Almost all galaxy formation models struggle to reproduce the number densities of these bright early systems . Additionally, after performing SED fitting on the measured fluxes, many authors obtain high stellar masses (Donnan et al. 2022;Labbe et al. 2022) that may be in tension with the astrophysics of early galaxy formation (Boylan-Kolchin 2022; Lovell et al. 2023), although some recent works suggest no tension (McCaffrey et al. 2023;Prada et al. 2023;Yung et al. 2023). Given these challenges, it is important to seek out independent measurements of the source population at very early epochs. The cosmic microwave background (CMB) spectrum is predicted to show an absorption feature at frequencies below 150 MHz imprinted when the Universe was flooded with Lyα photons emitted from the very first stars and before it was reheated and reionized (Madau et al. 1997;Tozzi et al. 2000). The depth and timing (frequency) of the global 21-cm signal carry a wealth of information about the nature of the first sources and the thermal state of the intergalactic medium (IGM), and can constrain the physics of the very early Universe. The Experiment to Detect the Global EoR Signature (EDGES) team has reported a controversial detection (Bowman et al. 2018) of a flattened absorption profile in the sky-averaged radio spectrum, centered at 78 MHz and with an anomalous amplitude of 0.5 K, placing the birth of the first astrophysical sources at z ∼ 18. Explaining such a deep absorption trough requires new exotic physics during cosmic dawn, such as the interaction between dark-matter and baryons (see e.g., Barkana 2018;Muñoz et al. 2018;Slatyer & Wu 2018), or an excess radio background (e.g., Fraser et al. 2018;Pospelov et al. 2018;Feng & Holder 2018;Ewall-Wice et al. 2018;Fialkov & Barkana 2019;Ewall-Wice et al. 2020). It has also been argued (see e.g., Hills et al. 2018;Bradley et al. 2019;Singh & Subrahmanyan 2019;Sims & Pober 2020;Singh et al. 2022) that the EDGES signal may not be of astrophysical origin (e.g. due to pure systematics). While both JWST results based on early NIRCam observations in this extreme redshift regime and the EDGES detection are highly uncertain, it is interesting to discuss the implications of a bright UV radiation background at cosmic dawn for 21-cm cosmology. In this Letter, we attempt to answer the following question: Can the young galaxies detected by JWST at the highest redshifts provide enough Lyα radiation to mix the hyperfine levels of neutral hydrogen and produce a global 21-cm signal at the redshfts, z ∼ 16 − 18, as probed by the EDGES experiment? We stress that, as in Madau (2018), our analysis focuses on the timing of the EDGES signal and on the constraints imposed by the required Wouthuysen-Field coupling strength on the UV radiation background at first light, and does not attempt to explain the claimed absorption trough altogether. All values were obtained by integrating the observed UV luminosity function down to the cutoff. In the figure we also plot the UV luminosity density required to produce an EDGES-like 21-cm signal at 16 < z < 19 in the 'minimal coupling' regime (Madau 2018). This constraint is imposed on the background Lyα flux by the Wouthuysen-Field mechanism that mixes the hyperfine levels of neutral hydrogen and is key to the detectability of 21-cm absorption from the epoch of first light (Madau et al. 1997). It yields a UV luminosity density in the range ρ UV = 10 24.52 − 10 25 (18/(1 + z)) 1/2 erg s −1 Mpc −3 Hz −1 for a proportionality constant (g = 0.06) that relates the number density of Lyα photons to the ionizing emissivity (see Equation 10 and the associated text in Madau 2018 for more details). The grey dashed solid line shows Madau 2018's fitting function (log 10 (ρ UV /ergs −1 Mpc −3 Hz −1 ) = (26.30± 0.12)+(−0.130±0.018)(z−6)) using a magnitude cut-off M UV = −13, and the light grey shaded area quantifies the fitting uncertainty (1-σ level). Regardless of the magnitude cutoff, we find a general consistency between the early and more recent measurements of the UV luminosity density by HST and JWST. At the limit M UV = −18 (left panel in Figure 1), HST and JWST data indicate a rapid decline in ρ UV to- wards early epochs consistent with the expected ρ UV evolution from constant star formation efficiency models ), but inconsistent with the constraints imposed by a possible EDGES-like 21-cm signal centered at redshift z ∼ 17. Data from Bouwens et al. (2022b, magenta stars) shows an increase in ρ UV at z ∼ 12, which may be particularly attributable to cosmic variance arising from the relatively small fields probed at this redshift (Moster et al. 2011;Steinhardt et al. 2021;Thomas et al. 2023;Harikane et al. 2023). By extrapolating to fainter magnitudes and integrating down to M UV = −13, we find that all measurements suggest a milder evolution in ρ UV . This suggests that the high redshift constraints by JWST in the redshift range of z ∼ 8 − 12 and an EDGES-like signal (red box) at z ∼ 17 may all be consistent with an extrapolation of the declining galaxy UV luminosity density measured at z ∼ 4−10 by HST (grey shaded area). It is useful at this stage to understand what form of the galaxy UV luminosity function (UVLF) would be required to explain a 21-cm feature at z ∼ 16. Using the EDGES constraints (log 10 ρ UV /[erg s −1 Mpc −3 Hz −1 ] = 24.6±0.24) following Madau (2018), and fixed Schechter function parameters M UV = −19.0 and α = −2.35 (following fits provided by Harikane et al. 2023), we derive a φ parameter of log 10 φ = −4.671 +0.240 −0.246 . This fit, shown as the solid red line in Figure 2, lies below the UVLF at z ∼ 12 calculated by Harikane et al. (2023) from spectroscopically confirmed candidates (dashed blue line), as well as the upper limit at z ∼ 16 presented by Harikane et al. (2023, blue arrow). The fit has a higher normalization, however, when compared with an extrapolation to z ∼ 16 of the Schechter fitting parameters provided in Harikane et al. (2023) 3 . Future deep JWST surveys are expected to better probe the z > 12 Universe ) and may provide a definitive test of these predictions. CONCLUSION The epoch of first light provides a unique window to the earliest astrophysical sources of radiation and their impact on the IGM. In this work, we have focused on the possibility that a high star formation efficiency at early times -as implied by early JWST results -may Harikane et al. (2023) Schechter function fitting parameters obtained from the spectroscopically confirmed candidates in the redshift range z ∼ 9 − 12 at z ∼ 12 (dashed line) and extrapolated to z ∼ 16 (dashed line). The upper limit (blue arrow) is obtained from photometric estimates at z ∼ 16 by Harikane et al. (2023). The 21-cm signal constraint predicts a much higher number of galaxies than the extrapolation of Harikane et al. (2023)'s results from z ∼ 9 − 12 by approximately an order of magnitude at the faint end. Future JWST surveys will provide more measurements at z > 12 that can be used to test these predictions. trigger the onset of intense Lyα emission at redshift z = 16 − 18 and produce a cosmic 21-cm absorption signal 200 Myr after the Big Bang. We have shown that a radio signal at the frequencies probed by the EDGES experiment may be consistent with an extrapolation of the evolving galaxy UV luminosity density measured at 4 z 12 by deep HST and JWST observations. By extrapolating the currently measured JWST luminosity functions down to M UV = −13, we find that all the observational data suggests a steady and mild evolution of ρ UV , and imply there may have been enough Lyα photons produced to trigger a global 21-cm signal at z ∼ 17. Using a semi-analytical model of reionization, Bera et al. (2022) have recently shown that reproducing this mild evolution of ρ UV requires much higher contribution from faint galaxies, since massive galaxies at high-z are rare. We note that, using a fixed star formation efficiency linked to the halo mass function predicted by ΛCDM would lead to a significant drop in the UV luminosity density beyond z > 12, a decrease that is not observed (Sun & Furlanetto 2016;Harikane et al. 2018;Harikane et al. 2023;Mason et al. 2018Mason et al. , 2023. The high UV luminosity density inferred at early times may require a revision of the standard astrophysics of early galaxy formation Mason et al. 2023;Dekel et al. 2023), including the impact of dust (Ferrara et al. 2022;Nath et al. 2023), a top heavy initial mass function or a high AGN fraction (Inayoshi et al. 2022;Yung et al. 2023), exotic sources such as primordial black holes or population III stars (Liu & Bromm 2022;Wang et al. 2022;Hütsi et al. 2023;Yuan et al. 2023), or modifications to the cosmological model (Haslbauer et al. 2022;Menci et al. 2022;Maio & Viel 2023;Biagetti et al. 2023;Dayal & Giri 2023;Melia 2023). During the preparation of this letter, Meiksin (2023) has independently found that the new JWST data could provide enough Lyα photons to decouple the spin temperature from that of the CMB by redshift 14. In our case, we have focused on redshifts (z ∼ 16−18) reported by the EDGES experiment. However, the redshift of the 21cm global signal, in princple, is set by the existence of sufficient Lyα photons at these early epochs, which will potentially be constrained by future JWST surveys. Figure 1 1shows estimates of the UV luminosity density, ρ UV , from HST (Oesch et al. 2018, faint cyan traingles; Ishigaki et al. 2018, faint orange hexagons; Bouwens et al. 2015, faint blue pentagons), JWST/NIRCam 1 (Donnan et al. 2023, lime circles; Harikane et al. 2022, open green diamonds; Bouwens et al. 2022b, magenta stars) and JWST/NIRSpec (Harikane et al. 2023, filled green diamonds), and quoted in the legend for different magnitude faint-end cutoffs of M UV = −18 (left panel) and M UV = −13 (right panel) 2 . Figure 1 . 1The galaxy UV luminosity density, ρUV, from HST(Oesch et al. 2018, faint cyan traingles, Ishigaki et al. 2018, faint orange hexagons, Bouwens et al. 2015, faint blue pentagons) and JWST, lime circles, Harikane et al. 2022, open green diamonds, Bouwens et al. 2022b, magenta stars), using the measured luminosity functions from z = 4 to z = 14 for different magnitude cutoff MUV = −18 (left panel), and MUV = −13 (right panel). The recently confirmed spectroscopic measurements reported byHarikane et al. (2023) are shown as filled green diamonds. The red box depicts the UV luminosity density needed to produce an EDGES-like 21-cm signal at 16 < z < 19 in the 'minimal coupling' regime(Madau 2018). If JWST luminosity function measurements could be extrapolated down to MUV = −13, the ensuing luminosity density would match the gradual redshift evolution predicted by Madau (2018, grey line), providing enough Lyα background photons to mix the hyperfine levels of neutral hydrogen 200 Myr after the Big Bang. Figure 2 . 2Predictions of the UV luminosity function at z ∼ 16 obtained by fitting the Schechter function φ parameter (log 10 φ = −4.671 +0.240 −0.246 ) to the 21cm Wouthuysen-Field coupling constraint at redshift 16 (log 10 ρUV/erg s −1 Mpc −3 Hz −1 = 24.6 ± 0.24) at fixed M UV = −21.15 and α = −2.35 (solid red line). The blue curves show the We omit results at z > 15 based on a candidate galaxy that was recently spectroscopically confirmed to be at lower redshift(Naidu et al. 2022b;Zavala et al. 2023;Arrabal Haro et al. 2023).2 We extrapolate only down to M UV = −13 since most of the reconstructed UVLFs from measurements and theory do not show a cutoff at magnitudes brighter than this limit (e.g. seeBouwens et al. 2022a). These fits were obtained from z ∼ 9 − 12 spectroscopically confirmed candidates (see their Equations 7, 8 and 9). ACKNOWLEDGEMENTSThe authors acknowledge helpful discussions with Adam Lidz, Martin Rey, Ingyin Zaw, Andrea Maccio, Anthony Pullen and Patrick Breysse. SH acknowledges support for Program number HST-HF2-51507 provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, incorporated, under NASA contract NAS5-26555. CCL acknowledges support from a Dennis Sciama fellowship funded by the University of Portsmouth for the Institute of Cosmology and Gravitation. PM acknowledges support from NASA TCAN grant 80NSSC21K0271 and the hospitality of New York University Abu Dhabi during the completion of this work. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. Part of this work was completed during the KITP GALEVO23 workshop for data-driven astronomy. . N J Adams, C J Conselice, L Ferreira, 10.48550/arXiv.2207.11217arXiv:2207.11217Adams, N. J., Conselice, C. J., Ferreira, L., et al. 2022, arXiv:2207.11217, doi: 10.48550/arXiv.2207.11217 Spectroscopic verification of very luminous galaxy candidates in the early universe. Arrabal Haro, P Dickinson, M Finkelstein, S L , 10.48550/arXiv.2303.15431Tech. rep. Arrabal Haro, P., Dickinson, M., Finkelstein, S. 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[ "OPTIMAL BOUNDARY CONTROL FOR A CAHN-HILLIARD-NAVIER-STOKES EQUATIONS", "OPTIMAL BOUNDARY CONTROL FOR A CAHN-HILLIARD-NAVIER-STOKES EQUATIONS" ]
[ "Manika Bag ", "Tania Biswas ", "Sheetal And ", "Dharmatti " ]
[]
[]
In In this work, we study an optimal boundary control control problem for a Cahn-Hilliard-Navier-Stokes(CHNS) system in a two-dimensional bounded domain. The CHNS system consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn-Hilliard equation for the relative concentration of the fluids. An optimal control problem is formulated as the minimization of a cost functional subject to the controlled CHNS system where the control acts on the boundary of the Navier-Stokes equations. We first prove that there exists an optimal boundary control. Then we establish that the control-to-state operator is Fréchet differentiable and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint system.
null
[ "https://export.arxiv.org/pdf/2305.02501v1.pdf" ]
258,480,045
2305.02501
4c767261bd2d57e46a809a7c7beacdb1a4a6fe63
OPTIMAL BOUNDARY CONTROL FOR A CAHN-HILLIARD-NAVIER-STOKES EQUATIONS 4 May 2023 Manika Bag Tania Biswas Sheetal And Dharmatti OPTIMAL BOUNDARY CONTROL FOR A CAHN-HILLIARD-NAVIER-STOKES EQUATIONS 4 May 2023Cahn-Hilliard-Navier-Stokes systemBoundary controlOptimal controlLagrange multipliersLagrangian principle Mathematics Subject Classification (2020): 49J2049J5076B99 In In this work, we study an optimal boundary control control problem for a Cahn-Hilliard-Navier-Stokes(CHNS) system in a two-dimensional bounded domain. The CHNS system consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn-Hilliard equation for the relative concentration of the fluids. An optimal control problem is formulated as the minimization of a cost functional subject to the controlled CHNS system where the control acts on the boundary of the Navier-Stokes equations. We first prove that there exists an optimal boundary control. Then we establish that the control-to-state operator is Fréchet differentiable and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint system. problems related to the CHNS system. Mathematically, boundary control problems are harder to deal with, specifically obtaining the optimality conditions, as higher regularity of the solution is often required. Also, the mathematical analysis becomes more challenging since the CHNS system is a highly nonlinear coupled system. To the best of our knowledge, most of the analytical work related to optimal control problems for the CHNS system is devoted to the case of distributed control. This work is the first contribution to the analytic study of the boundary control problem for the CHNS system. The system under consideration in this work is the following controlled CHNS system where control acts on the boundary of the Navier-Stokes equation as a time-dependent Dirichlet boundary condition: (1.1)                              ϕ t + u · ∇ϕ = ∆µ, in Ω × (0, T ), µ = −∆ϕ + F ′ (ϕ), Here u(x, t) is the average velocity of the fluid, and ϕ(x, t) is the relative concentration of the fluid. Also, Ω is a bounded domain in R 2 , with a sufficiently smooth boundary ∂Ω. The density is taken as matched density, i.e., constant density, which is equal to 1. Moreover, µ denotes the chemical potential, π denotes the pressure, ν denotes the viscosity, F is a double-well potential, and h is the control acting on the boundary of the domain. Furthermore, µ is the first variation of the Helmholtz free energy functional E(ϕ) := Ω 1 2 |∇ϕ| 2 + F(ϕ(x)) dx,(1.2) where F is a double-well potential of the regular type. A typical example of regular F is F(s) = (s 2 − 1) 2 , s ∈ R. (1.3) We now discuss some of the works available in the literature for the solvability of system (1.1) when h = 0. In [8], the authors established the existence and uniqueness of weak solutions and strong solutions in 2 and 3 dimensions in the case of regular potential (for strong solution global in 2D and local in time in 3D). The authors have proved the existence and uniqueness results for the case of singular potential in [1,22]. In [20], the authors have studied the asymptotic behavior where they have proved the existence of global and exponential attaractor. For the system (1.1), when the time-dependent Dirichlet boundary condition for the Navier-Stokes equation is considered, in [5], we have proved the existence, uniqueness of weak solution and existence of strong solution. Moreover, we refer [ [2,3,4,19] and references therein] for more generalized models by considering nonconstant viscosity, general density, thermodynamically consistent model, compressible fluids, moving contact lines, etc. In this work, our aim is to study the optimal boundary control related to the system (1.1). Let us now briefly mention some related literature on optimal control problems. Optimal control problems for Cahn-Hilliard equations have been studied by several mathematicians in [ [10,39,11,28,40], and the references therein] for both the cases of distributed and boundary control. The authors have considered a distributed optimal control problem for Navier-Stokes equations in [23,24,14,9], to name a few. Whereas in [25,16,17,18], the authors have studied boundary optimal control problem for the Navier-Stokes equation. In [16], the 2D Navier-Stokes equation on the unbounded domain is considered with control acts on the boundary. In the seminal paper by the same authors [18], the optimal boundary control for the Navier-Stokes system in 3D has been studied. Turning to optimal control problems for the CHNS system, there are a few analytical works available in the literature that we would like to mention. The robust control problem for the local CHNS system is investigated in [35], and the optimal control with state constraints is considered in [34]. For the nonlocal CHNS system, we mention [6,7,13,12] for regular potential and singular potential, respectively. In [6], the authors have investigated an enstrophy minimization problem and also a data assimilation type of problem where control acts as initial data. In all the other works mentioned above, a distributed optimal control problem has been studied in terms of minimizing a standard tracking type cost functional. Regarding the numerical studies, optimal control problems of semi-discrete CHNS system for various cases like distributed and boundary control, with nonsmooth Landau-Ginzburg energies and with non-matched fluid densities are studied in [27,29,30]. These works considered the local Cahn-Hilliard-Navier-Stokes equations for their numerical studies. In this work, we consider the control as Dirichlet boundary data for u in the system (1.1). Now with the required well-posedness result of (1.1) in hand [see [5] for details], the road is paved for studying optimal control problems associated with the system (1.1). We restrict ourselves to dimension 2 as the regularity of the solution required to study optimal control problems is available in dimension 2 only. The control problem under investigation in this paper reads as follows: Let us consider the quadratic cost functional J (u, ϕ, h) = 1 2 T 0 u − u Q 2 + 1 2 T 0 ϕ − ϕ Q 2 + 1 2 u(T ) − u Ω 2 L 2 (Ω) + 1 2 ϕ(T ) − ϕ Ω 2 L 2 (Ω) + 1 2 T 0 h 2 L 2 (∂Ω) . (1.4) So the optimal control problem is formulated as follows: (OCP) min h∈U ad J (u, ϕ, h), subject to (1.1). Here u Q , ϕ Q , u Ω , ϕ Ω are the target functions such that u Q ∈ L 2 (0, T ; L 2 div (Ω)), ϕ Q ∈ L 2 (Ω × (0, T )), u Ω ∈ L 2 div (Ω), ϕ Ω ∈ L 2 (Ω) . So the motivation of the problem (OCP) is that we want to find the best control h from the set of admissible controls such that the corresponding optimal solution of (1.1) is as close as possible to the target state. The last term in (1.4) is the effort by the control that we have to pay in order to reach the final state. Throughout this paper, we assume T is given finite final time and set Q = Ω × (0, T ), Σ = ∂Ω × (0, T ). To overview the work of this paper: in the state problem (1.1), h acts as boundary control, which is hypothesized to belong to a suitable closed convex bounded subspace U ad of a suitable Banach space U [which will be specified later, see (3.1), (3.4)]. In the (1.4), (u, ϕ) is the strong solution of state problem (1.1) corresponding to time-dependent Dirichlet boundary data u| ∂Ω = h. The main results of this paper are summarized as follows: (1) We establish the existence of an optimal boundary control for the problem (OCP The plan of the paper is as follows: in the next section, we give some preliminaries about function spaces and operators used in the sequel. We also recall the well-posedness results required in this work and give a brief sketch of a proof that the strong solution depends continuously on the boundary data. This result is crucial in proving the existence of optimal control which is tackled in section 3. Section 4 is devoted to studying the linearized system, which comes naturally when one wants to establish the differentiability of control to state operator. Finally, in section 5, we characterize the optimal control with the help of adjoint system and establish the well-posedness of the adjoint system. Preliminary Functional Setup. Let Ω be a bounded subset of R 2 with sufficiently smooth boundary ∂Ω. We introduce the functional spaces that will be useful in the paper. G div := u ∈ L 2 (Ω; R 2 ) : div u = 0, u · n ∂Ω = 0 , Above spaces are defined for s > 0. With usual convention, the dual space of H s (Ω) is denoted by H −s (Ω). Let us denote · and (·, ·) the norm and the scalar product, respectively, on L 2 div and G div . The duality between any Hilbert space X and its dual X ′ will be denoted by ·, · . We know that V div is endowed with the scalar product V div := u ∈ H 1 0 (Ω; R 2 ) : div u = 0 , L 2 div := u ∈ L 2 (Ω; R 2 ) : div u = 0 , H s div := u ∈ H s (Ω; R 2 ) : div u = 0 , L 2 (Ω) := L 2 (Ω; R),(u, v) V div = (∇u, ∇v) = 2(Du, Dv), for all u, v ∈ V div . The norm on V div is given by u 2 V div := Ω |∇u(x)| 2 dx = ∇u 2 . Since Ω is bounded, the embedding of V div ⊂ G div ≡ G ′ div ⊂ V ′ div is compact. Linear and Nonlinear Operators. Let us define the Stokes operator A : D(A) ∩ G div → G div by A = −P∆, D(A) = H 2 (Ω) ∩ V div , where P : L 2 (Ω) → G div is the Helmholtz-Hodge orthogonal projection. Note also that we have Au, v = (u, v) V div = (∇u, ∇v), for all u ∈ D(A), v ∈ V div . Let u, v be two vector-valued functions. Then ∇(u · v) = (∇u) · v + (∇v) · u = v T (∇u) + u T (∇v). Assumption 2.1. We take the following assumptions on F : (1) There exist C 1 > 0, C 2 > 0 such that F ′′ (s) ≤ C 1 |s| p−1 + C 2 , for all s ∈ R, 1 ≤ p < ∞ and a.e., x ∈ Ω. (2) F ∈ C 2 (R) and there exists C 3 > 0 such that F ′′ (s) ≥ −C 3 , for all s ∈ R, a.e., x ∈ Ω. (3) There exist C ′ 3 > 0, C 4 ≥ 0 and r ∈ (1, 2] such that |F ′ (s)| r ≤ C ′ 3 |F(s)| + C 4 , for all s ∈ R. (4) F ∈ C 3 (R) and there exists C 5 > 0, |F ′′′ (s)| ≤ C 5 (1 + |s| q ) for all s ∈ R where q < +∞. (5) F (ϕ 0 ) ∈ L 1 (Ω). Well-Posedness of Stokes Equations u e ∈ H 1 (0, T ; L 2 div (Ω)) ∩ L ∞ (0, T ; H 1 div (Ω)) ∩ L 2 (0, T ; H 2 div (Ω)), such that T 0 u e (t) 2 H 2 div dt ≤ c T 0 h(t) 2 H 3 2 (∂Ω) dt, (2.3) T 0 ∂ t u e (t) 2 L 2 div dt ≤ c T 0 ∂ t h(t) H − 1 2 (∂Ω) dt. (2.4) Moreover if h satisfies h ∈ L 2 (0, T ; H 5 2 (∂Ω)) ∩ L ∞ (0, T ; H 3 2 (∂Ω)), ∂ t h ∈ L 2 (0, T ; H 1 2 (∂Ω)) (2.5) then Stokes' equations (2.1) admits a strong solution u e ∈ H 1 (0, T ; H 1 div ) ∩ L 2 (0, T ; H 3 div ), (2.6) such that T 0 u e (t) 2 H 3 div dt ≤ c T 0 h(t) 2 H 5 2 (∂Ω) dt, T 0 ∂ t u e (t) 2 H 1 div dt ≤ c T 0 ∂ t h(t) H 1 2 (∂Ω) dt. Proof. The proof of the above theorem can be found in [32,33]. Well-posedness of state problem. In this section, we will state the results on well-posedness and existence of strong solution of the system (1.1). Definition 2.3. Let u 0 ∈ L 2 div (Ω), ϕ 0 ∈ H 1 (Ω) and T > 0 be given. Let F satisfy the Assumption 2.1, and h satisfies (2.2). Let us also assume u e solves (2.1). A pair (u, ϕ) is said to be a weak solution of the system (1.1) if u = u + u e and • (u, ϕ) satisfies                  u ∈ L ∞ (0, T ; G div (Ω)) ∩ L 2 (0, T ; V div (Ω)) u t ∈ L 2 (0, T ; V ′ div (Ω)) ϕ ∈ L ∞ (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) ϕ t ∈ L 2 (0, T ; (H 1 (Ω)) ′ ) µ ∈ L 2 (0, T ; H 1 (Ω)). (2.7) • for every v ∈ V div , every ψ ∈ H 1 and for a.e t ∈ (0, T ) we have ∂ t u(t), v − ν(∆u(t), v) + ((u(t) + u e (t)) · ∇(u(t) + u e (t)), v) = (µ∇ϕ, v) − ∂ t u e , v , (2.8) ∂ t ϕ(t), ψ + (u(t) + u e (t)) · ∇ϕ(t), ψ = (∆µ(t), ψ) (2.9) Theorem 2.4. Let u 0 ∈ L 2 div (Ω) and ϕ 0 ∈ H 1 (Ω). Let F satisfy the Assumption 2.1 and h satisfies (2.2). Moreover, let us assume the compatibility condition u 0 ∂Ω = h t=0 . (2.10) Then, for a given T > 0, there exists a unique pair (u, ϕ) which satisfies (2.8) and (2.9). Theorem 2.5. Let the assumptions of the theorem (2.4) holds. Then there exists a unique weak solution (u, ϕ) to the system (1.1) , where u = u + u e , u as defined in theorem (2.4). Moreover, (u, ϕ) satisfy u ∈ L ∞ (0, T ; L 2 div ) ∩ L 2 (0, T ; H 1 div ), ϕ ∈ L ∞ (0, T ; H 1 ) ∩ L 2 (0, T ; H 2 ), u t ∈ L 2 (0, T ; H 1 div ′ ), ϕ t ∈ L 2 (0, T ; (H 1 ) ′ ), µ ∈ L 2 (0, T ; H 1 ). Proposition 2.6. Let (u 1 , ϕ 1 ) and (u 2 , ϕ 2 ) be two pair of weak solution of the system (1.1) with boundaries h 1 , h 2 and initial datum (u 10 , ϕ 10 ), (u 20 , ϕ 20 ) respectively. Define u := u 1 − u 2 and ϕ := ϕ 1 − ϕ 2 , where u i = u i − u ei , u ei is the solution of "lifting operator" with h i on the boundary, i = 1, 2. Then there exists a constant C > 0 such that u L ∞ (0,T,G div )∩L 2 (0,T,V div ) + ϕ L ∞ (0,T,H 1 )∩L 2 (0,T,H 2 ) + ∇µ L 2 (0,T ;L 2 ) ≤ C h L ∞ (0,T,H 1 2 (∂Ω))∩L 2 (0,T,H 3 2 (∂Ω)) + ∂ t h L 2 (0,T,H − 1 2 (∂Ω)) , (2.11) where µ = µ 1 − µ 2 , h = h 1 − h 2 and C depends on u i0 , ϕ i0 , i = 1, 2 . Proof. Note that in [5] section 5, we have proved u L ∞ (0,T,G div )∩L 2 (0,T,V div ) + ϕ L ∞ (0,T,H 1 )∩L 2 (0,T,H 2 ) ≤ C h L ∞ (0,T,H 1 2 (∂Ω))∩L 2 (0,T,H 3 2 (∂Ω)) + ∂ t h L 2 (0,T,H − 1 2 (∂Ω)) . But similar calculations and observations will lead to (2.11). Theorem 2.7 ([5] , Theorem 6.1). Let F satisfies assumption (2.1) and h satisfies (2.5). For a given pair (u 0 , ϕ 0 ) ∈ H 1 div × H 2 , there exists a unique pair (u, ϕ) which is the weak solution of the system (1.1) and also satisfies u ∈ L ∞ (0, T ; H 1 div ) ∩ L 2 (0, T ; H 2 div ) ϕ ∈ L ∞ (0, T ; H 2 ) ∩ L 2 (0, T ; H 3 ∩ H 4 ). Remark 2.8. In [5], authors did not estimate the time derivatives u t , ϕ t . One can easily show that u t ∈ L 2 (0, T ; L 2 div (Ω)), ϕ t ∈ L 2 (0, T ; H 1 (Ω)). In the next section, we will discuss the continuous dependence of a strong solution. We will give a brief sketch of proof as it will follow similarly from [5], section 5, with a slight modification in estimates. Continuous dependence of strong solution. Let (u 1 , ϕ 1 ) and (u 2 , ϕ 2 ) be two weak solutions of the system with non-homogeneous boundaries h 1 and h 2 and initial conditions u i0 , ϕ i0 , for i = 1, 2 respectively. Then denote the differences u = u 1 − u 2 , ϕ = ϕ 1 − ϕ 2 where u i = u i − u ei , u ei is the solution of (2.1) corresponding to boundary h i , for i = 1, 2. Note that, u e := u e1 − u e2 satisfies the equation (2.1) with the boundary h = h 1 − h 2 . Then (u, ϕ) satisfies:                                  ϕ t + u · ∇ϕ 1 + u 2 · ∇ϕ + u e · ∇ϕ 1 + u e2 · ∇ϕ = ∆μ, in Ω × (0, T ), µ = −∆ϕ + F ′ (ϕ 2 ) − F ′ (ϕ 1 ), in Ω × (0, T ), u t − ν∆u + (u · ∇)u e1 + (u e1 · ∇)u + (u · ∇)u 1 + (u 2 · ∇)u e + (u e · ∇)u 2 + (u e · ∇)u e1 + (u e2 · ∇)u e + ∇π =μ∇ϕ 1 + µ 2 ∇ϕ − ∂ t u e , in Ω × (0, T ), div u = 0, in Ω × (0, T ), ∂ϕ ∂n = 0, ∂μ ∂n = 0, on Σ, u = 0, on Σ, u(x, 0) = u 0 , ϕ(x, 0) = ϕ 0 , in Ω. (2.12) We have the following continuous dependence result for the strong solution of (1.1): Proposition 2.9. Let h i ∈ U and (ϕ i , u i ) be the strong solution of the system (1.1) with corresponding boundary data h i with initial data (ϕ i0 , u i0 ) for i = 1, 2. Let h = h 1 − h 2 . Then there exist a constant C > 0 such that: u L ∞ (0,T ;H 1 div (Ω))∩L 2 (0,T ;H 2 (Ω)) + ϕ L ∞ (0,T ;H 2 (Ω))∩L 2 (0,T ;H 2 (Ω)) ≤ C h U ,(2. 13) where the norm linear space U defined in (3.1) with the norm, · U , given as (3.2). Proof. We multiply the equation (2.12) 3 by Au and get the following 1 2 d dt ∇u 2 + ν Au 2 = ( (u · ∇)u e1 , Au ) + ( (u e1 · ∇)u, Au ) + ( (u · ∇)u 1 , Au ) + ( (u 2 · ∇)u e , Au ) + ( (u e · ∇)u 2 , Au ) + ( (u e · ∇)u e1 , Au ) + ( (u e2 · ∇)u e , Au ) + ( ∇π, Au ) + (μ∇ϕ 1 , Au ) + ( µ 2 ∇ϕ, Au ) + ( ∂ t u e , Au ) (2.14) Similarly multiplying by ∆ 2 ϕ to the equation (2.12) 1 we get 1 2 d dt ∆ϕ 2 + ∆ 2 ϕ 2 = − (u · ∇ϕ 1 , ∆ 2 ϕ) − (u 2 · ∇ϕ, ∆ 2 ϕ) − (u e · ∇ϕ 1 , ∆ 2 ϕ) − (u e2 · ∇ϕ, ∆ 2 ϕ) + (∆μ, ∆ 2 ϕ) (2.15) Now estimating each term of the right-hand side of (2.14) using Sobolev inequality, Hölder inequality, Young's inequality and keeping in mind the regularity of (ϕ i , u i ) and u ei we obtain 1 2 d dt ∇u 2 + ν Au 2 ≤ ν 2 Au 2 + C 2 ∇u 2 2 + h 1 2 H 3 2 + h 2 2 H 3 2 + ϕ 1 2 H 3 + 5 ν µ 2 2 H 2 h 2 U + C h 1 2 H 3 2 + u 1 2 H 2 div ∇u 2 . (2.16) Similarly, estimating the right-hand side of (2.15), we get 1 2 d dt ∆ϕ 2 + ∆ 2 ϕ 2 ≤ 1 2 ∆ 2 ϕ 2 + C 3 u 2 2 H 2 + ∇ϕ 1 2 + 3 h 2 2 H 3 2 h 2 U + C ∆ϕ 2 . (2.17) Adding the inequalities (2.16), (2.17) then integrating the resulting inequality between 0 to t and applying Gronwall's lemma we get sup t∈[0,T ) ∇u(t) 2 + ∆ϕ(t) 2 + t 0 ( Au(t) 2 + ∆ 2 ϕ(t) 2 )dt ≤ C h 2 U (2.18) for all t ∈ [0, T ), where C is a positive constant depends on (ϕ i , u i ) V , h i U , ϕ i0 H 2 , u i0 H 1 div , i = 1, 2, Which immidiately gives (2.13). Optimal Control We introduce the space U := {h(x, t) : h ∈ L ∞ (0, T ; H 3 2 (∂Ω)) ∩ L 2 (0, T ; H 5 2 (∂Ω)), ∂ t h ∈ L 2 (0, T ; H 1 2 (∂Ω)}, (3.1) with the norm given by h U = h L ∞ (0,T ;H 3 2 (∂Ω)) + h L 2 (0,T ;H 5 2 (∂Ω)) + ∂ t h L 2 (0,T ;H 1 2 (∂Ω)) . (3.2) Remark 3.1. If ∂Ω is sufficiently smooth boundary of Ω then from [36] we have the following embedding L ∞ (0, T ; H 3 2 (∂Ω)) ∩ L 2 (0, T ; H 5 2 (∂Ω)) ֒→ C([0, T ]; H 1 2 +δ ) ֒→ C([0, T ] × ∂Ω),(3.3) for any 0 < δ ≤ 1. Therefore the compatibility condition h(x, 0) = u 0 | ∂Ω is well-defined and also satisfied for every solution, u of (1.1). For a sufficiently large positive constant L, we define the set of admissible boundary control space as follows: U ad := {h ∈ U | h L ∞ (0,T ;H 3 2 (∂Ω))∩L 2 (0,T ;H 5 2 (∂Ω)) ≤ L, ∂ t h L 2 (0,T ;H 1 2 (∂Ω) ≤ L}. (3.4) Let us define W := C([0, T ]; L 2 div (Ω)) ∩ L 2 (0, T ; H 1 div (Ω)) × C([0, T ], H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) , (3.5) V := C([0, T ]; H 1 div (Ω)) ∩ L 2 (0, T ; H 2 div (Ω)) ∩ H 1 (0, T ; L 2 div (Ω)) × C([0, T ], H 2 (Ω)) ∩ L 2 (0, T ; H 4 (Ω)) ∩ H 1 (0, T ; H 1 (Ω)) ,(3.6) which denote the space of global weak and strong solution of the system (1.1) respectively. Now let us define control to state operator S : U → V by h → (u, ϕ), where (u, ϕ) is the strong solution of (1.1) corresponding to boundary control h and with initial data (u 0 , ϕ 0 ) ∈ H 1 div × H 2 . Therefore we can now define a reformulated cost functional J : U → [0, ∞) byJ (h) := J (u, ϕ, h) = J (S(h), h). (3.7) In the following theorem, we prove that the optimal control problem (OCP) has a solution that gives the existence of an optimal control. Theorem 3.2. (Existence of an optimal control) Let the initial data (u 0 , ϕ 0 ) ∈ H 1 div × H 2 , and F satisfy the Assumption 2.1. Let the target functionals u Q ∈ L 2 (0, T ; L 2 div (Ω)), ϕ Q ∈ L 2 (Ω × (0, T )), u Ω ∈ L 2 div (Ω), ϕ Ω ∈ L 2 (Ω). There exists a control h * ∈ U ad such that J (u * , ϕ * , h * ) = min h∈U ad J (u, ϕ, h), where (u * , ϕ * ) is the unique solution of the state problem (1.1) corresponding to boundary control h * . Proof. Let j = inf h∈U ad J (u, ϕ, h). As J (u, ϕ, h) ≥ 0 for all h ∈ U ad so j ≥ 0. Then there exist a sequence h n ∈ U ad such that lim n→∞ J (u n , ϕ n , h n ) = j where S(h n ) = (u n , ϕ n ). Since {h n } is bounded, there exist a subsequence still denoted by {h n } such that h n → h * weakly in L 2 (0, T ; H 5 2 (∂Ω)), (3.8) h n → h * weak* in L ∞ (0, T ; H 3 2 (∂Ω)), (3.9) ∂ t h n → ∂ t h * weakly in L 2 (0, T ; H 1 2 (∂Ω),(3.10) for some h * ∈ U. Now we show that h * ∈ U ad . To do that we only need to show h * U ≤ L. We know that if h n → h * weakly in . , then h * ≤ lim inf n→∞ h n . Therefore from ( ) ≤ L. Again using the fact that any norm in a Banach space is weak* -lower semicontinuous, we have h * L ∞ (0,T ;H 3 2 (∂Ω)) ≤ L. Now from continuous dependence of strong solution, Proposition 2.9, we conclude that there exists a subsequence (u n , ϕ n ) such that u n → u * weak * in L ∞ (0, T ; H 1 div ), u n → u * weakly in L 2 (0, T ; H 2 div ), u n → u * strongly in L 2 (0, T ; L 2 div ), ϕ n → ϕ * weak * in L ∞ (0, T ; H 2 ), ϕ n → ϕ * weakly in L 2 (0, T ; H 4 ), ϕ n → ϕ * strongly in L 2 (0, T ; H 1 ). Using the above convergence result on (u n , ϕ n ), we can pass to the limit in the weak formulation of the system (1.1) and the limit (u * , ϕ * ) satisfy the weak formulation of (1.1) with initial condition (u 0 , ϕ 0 ) ∈ H 1 div × H 2 , and boundary condition u * = h * . Since (u * , ϕ * ) ∈ V and h * ∈ U, therefore S(h * ) = (u * , ϕ * ). Since J is weakly lower semi-continuous in V × U, we have J (u * , ϕ * , h * ) ≤ lim inf n→∞ J (u n , ϕ n , h n ). We have from the above convergence lim inf n→∞ J (u n , ϕ n , h n ) = j Therefore, j ≤ J (u * , ϕ * , h * ) ≤ lim inf n→∞ J (u n , ϕ n , h n ) = j Hence we conclude that, J (u * , ϕ * , h * ) = min h∈U ad J (u, ϕ, h), which yields that (u * , ϕ * , h * ) is a solution of optimal control problem. The h * obtained in the Theorem 3.2 is called an optimal control and the corresponding solution (u * , ϕ * , h * ) is called the optimal solution. Linearize system In this section, we will derive the linearized system and establish the existence uniqueness results. Then we will show that the Frechet derivative of the control to state operator satisfies the linearized system. Now, let for a fixed controlĥ ∈ U, S(ĥ) = (û,φ) ∈ V be the strong solution of the system (1.1) corresponding to boundary dataĥ. Let a control η ∈ U −{ĥ} be given. To prove Fréchet differentiability of control to state operator, we consider the following system, which we obtain by linearising the state problem (1.1) around (û,φ). In this process we take u =û + w, ϕ =φ + ψ, π =π + π. Substituting this, we obtain the following linearized system as                                  w t − ν∆w + (û · ∇)w + (w · ∇)û + ∇π = −∆φ∇ψ − ∆ψ∇φ + F ′ (φ)∇ψ + F ′′ (φ)ψ∇φ, in Q div w = 0, in Q, ψ t + w · ∇φ +û · ∇ψ = ∆µ ψ , in Q, µ ψ = −∆ψ + F ′′ (φ)ψ, in Q, w = η,w ∈ L ∞ (0, T ; L 2 div (Ω)) ∩ L 2 (0, T ; H 1 div (Ω)) ∩ H 1 (0, T ; V ′ div (Ω)), ψ ∈ L ∞ (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; (H 1 ) ′ (Ω)). Proof. We prove the theorem using the Faedo-Galerkin approximation scheme. Let us consider the families of functions (u k ) and (γ k ), eigenfunctions of the Stokes operator and the Naumann operator −∆ + I respectively. We consider n-dimensional subspaces U n := u 1 , · · · , u n and Ψ n = ψ 1 , · · · , ψ n , spanned by n-eigenfunctions, and orthogonal projection on these spaces, P n := P Un and P n = P Ψn . We look for the functions w n (t, x) = w n (t, x) + w e (t, x) = i=n i=0 a i (t)u i (x) + w e (t, x), ψ n (t, x) = i=n i=0 b i (t)γ i (x), which solves the following approximated problem a.e. in [0, T ] and for i = 1, ..., n ∂ t w n , u i + (∇w n · ∇u i ) + ((û · ∇)(w n + w e ), u i ) + (((w n + w e ) · ∇)û, u i ) = −(∆φ∇ψ n , u i ) − (∆ψ n ∇φ, u i ) + (ψ n F ′′ (φ)∇φ, u i ) − ∂ t w e , u i for all u i ∈ U n , in Ω, (4.2) ∂ t ψ n , γ i + ((w n + w e ) · ∇φ, γ i ) + (û · ∇ψ n ), γ i ) = (∆µ n , γ i ), for all γ i ∈ Ψ n , in Ω, where w e is a solution of (2.1) with w e (t) = η(t) on ∂Ω and a.e t ∈ [0, T ]. This is a Cauchy problem of the system of 2n ordinary differential equations with 2n unknowns a i , b i , which can be solved by the Cauchy-Lipschitz theorem. So we get a unique solution (w n , ψ n ) of approximated system (4.2)-(4.3). Now we prove some apriori estimates for approximated solution (w n , ψ n ) independent of n. Let us take u i = w n in (4.2) and γ i = ψ n − ∆ψ n in (4.3), and multiply the equation (4.1) 4 with −∆µ n + ∆ψ n , and adding these equations we get 1 2 d dt ( w n 2 + ∇ψ n 2 + ψ n 2 ) + ν ∇w n 2 + ∆ψ n 2 + ∇µ n 2 ≤ Ω [(û · ∇(w n + w e )) w n ]dx + Ω [(w n + w e ) · ∇û) w n ]dx − Ω ∆φ(∇ψ n · w n )dx − Ω ∆ψ n (∇φ · w n )dx + ω ψ n F ′′ (φ)(∇φ · w n )dx − Ω ∂ t w e · w n dx + Ω ((w n + w e ) · ∇φ) ∆ψ n dx + Ω (û · ∇ψ n ) ∆ψ n dx − Ω F ′′ (φ)ψ n ∆µ n dx + Ω F ′′ (φ)ψ n ∆ψ n dx − Ω ((w n + w e ) · ∇φ) ψ n . (4.6) Now we estimate each term of the right-hand side of the above inequality individually. We will use the following abbreviation in the rest of the proof: C is a generic constant that may depend on the norm of (û,φ), η but not on n. Using Hölder inequality, Young's inequality, and Agmon's inequality repeatedly give the following series of estimates: Ω (û · ∇w e ) w n dx ≤ û L 4 ∇w n w e L 4 ≤ C û 2 H 1 div η 2 H 3 2 + ν 8 ∇w n 2 (4.7) Ω (w n · ∇û) w n dx ≤ w n ∇û L 4 w n L 4 ≤ C û 2 H 2 div w n 2 + ν 8 ∇w n 2 (4.8) Ω (w e · ∇û) w n dx ≤ w e L 4 ∇w n û L 4 ≤ C w e 2 H 1 div û 2 H 1 div + ν 8 ∇w n 2 ≤ C η 2 H 3 2 û 2 H 1 div + ν 8 ∇w n 2 (4.9) Ω ∆φ(∇ψ n · w n )dx ≤ ∆φ L ∞ ∇ψ n w n ≤ 1 2 ∆φ 2 L ∞ ∇ψ n 2 + 1 2 w n (4.10) Ω ∆ψ n (∇φ · w n ) ≤ ∆ψ n ∇φ L ∞ w n ≤ 3 4 ∇φ 2 L ∞ w n 2 + 1 12 ∆ψ n 2 (4.11) ω ψ n F ′′ (φ)(∇φ · w n )dx = Ω F ′ (φ)∇ψ n · w n dx ≤ F ′ (φ) L ∞ ∇ψ n w n ≤ C ∇ψ n 2 + 1 2 w n 2 (4.12) Ω ∂ t w e · w n dx ≤ ∂ t w e w n ≤ 1 2 ∂ t h 2 H 1 2 + 1 2 w n 2 (4.13) Ω (w n · ∇φ)∆ψ n dx ≤ 1 12 ∆ψ n 2 + 3 4 ∇φ 2 L ∞ w n 2 (4.14) Ω (w e · ∇φ)∆ψ n dx ≤ 1 12 ∆ψ n 2 + 3 4 ∇φ 2 w e 2 ≤ 1 12 ∆ψ n 2 + 3 4 ∇φ 2 η 2 H 3 2 (∂Ω) (4.15) Ω (û · ∇ψ n )∆ψ n dx ≤ C û 1 2 û 1 2 H 2 ∇ψ n ∆ψ n ≤ 1 12 ∆ψ n 2 + 3 4C û 2 H 2 ∇ψ n 2 (4.16) Ω (w n · ∇φ)ψ n dx ≤ ν 8 ∇w n 2 + C φ 2 L 4 ∇ψ n 2 (4.17) Ω (w e · ∇φ)ψ n dx ≤ w e L ∞ ∇φ ψ n ≤ C w e 1 2 w e 1 2 H 2 ∇φ ψ n ≤ C η 2 H 3 2 (∂Ω) + ∇φ 2 ψ n 2 (4.18) Ω F ′′ (φ)ψ n ∆µ n dx = Ω F ′′′ (φ)ψ n (∇φ · ∇µ n )dx + Ω F ′′ (φ)∇ψ n · ∇µ n dx ≤ 1 2 ∇µ n 2 + C 4 F ′′′ (φ)∇φ L ∞ ψ n 2 + C 4 F ′′ (φ) 2 L ∞ ∇ψ n 2 (4.19) Ω F ′′ (φ)ψ n ∆ψ n dx ≤ 1 12 ∆ψ n 2 + 3 4C F ′′ (φ) 2 L ∞ ψ n 2 (4.20) Adding all the estimates of the right-hand side of (4.6), we obtain 1 2 d dt ( w n 2 + ψ n 2 H 1 ) + ν 2 ∇w n 2 + 2 3 ∆ψ n 2 + 1 2 ∇µ n 2 ≤C(2 û 2 H 1 div + 3 4 ∇φ 2 + 3 2 ) η 2 U + C û 2 H 2 + ∆φ 2 L ∞ + ∇φ 2 L ∞ + φ 2 L 4 + ∇ϕ 2 + F ′′′ (φ)∇φ L ∞ + F ′′ (φ) 2 L ∞ ( w n 2 + ψ n 2 H 1 ) (4.21) Observe that, with the help of Theorem 2.5 the map t → G(û(t),φ(t)) belongs to L 1 (0, T ) where G(û(t),φ(t)) := û(t) 2 H 2 + ∆φ(t) 2 L ∞ + ∇φ(t) 2 L ∞ + φ(t) 2 L 4 + ∇φ(t) 2 + F ′′′ (φ(t))∇φ(t) L ∞ + F ′′ (φ(t)) 2 L ∞ Thus owing to Gronwall's lemma yields w n L ∞ (0,T,G div )∩L 2 (0,T,V div ) ≤ C η U , ψ n L ∞ (0,T,H 1 )∩L 2 (0,T,H 2 ) ≤ C η U . By comparison in (4.2), (4.3) we can indeed get uniform bound on ∂ t w n and ∂ t ψ n as ∂ t w n L 2 (0,T ;V ′ div ) ≤ C η U ∂ t ψ n L 2 (0,T ;(H 1 (Ω)) ′ ) ≤ C η U . Then we can get a pair (w, ψ) such that w ∈ L ∞ (0, T ; G div ) ∩ L 2 (0, T ; V div ) ∩ H 1 (0, T ; V ′ div ), ψ ∈ L ∞ (0, T ; H 1 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; (H 1 (Ω)) ′ ). which is the weak limit of (w n , ψ n ). Then we can pass to limit as n → ∞ in (4.3) -(4.3) and also able to verify the pair (w, ψ) with w = w + w e satisfy the weak formulation of (4.1). Since the system (4.1) is a linear system, uniqueness of weak solutions follows easily. Differentiability of Control to State Operator. Consider the control to state operator S : U → V with given initial data (u 0 , ϕ 0 ) ∈ H 1 div × H 2 . As V ⊆ W, we can consider S from U to a weaker space W. Definition 4.2. We define S : U → W is Fréchet differentiable in U if for anyĥ ∈ U, there exist a linear operator S ′ (ĥ) : U → W such that lim η U →0 S(ĥ + η) − S(ĥ) − S ′ (ĥ)(η) W η U = 0 (4.22) for any arbitrary small perturbation η ∈ U \ {ĥ}. In the next theorem, we prove the Fréchet differentiability of the control to state operator S. Theorem 4.3. Let F ∈ C 4 (R) and satisfies the assumption (2.1). Also, assume (u 0 , ϕ 0 ) be a given initial data. Then the control to state operator S : U → W is Fréchet differentiable. Moreover, for anyĥ ∈ U, its Fréchet derivative S ′ (ĥ) is given by S ′ (ĥ)(η) = (w, ψ), ∀η ∈ U \ {ĥ}, where (w, ψ) is the unique weak solution of the linearized system (4.1) with control η, which is linearized around a strong solution (û,φ) of the system (1.1) with controlĥ. Proof. Letĥ be a given fixed control and (û,φ) = S(ĥ) be the strong solution of (1.1) with controlĥ. Let (u, ϕ) be the strong solution of the system (1. 1) with controlĥ + η. Let z = u −û, ξ = ϕ −φ. Then (z, ξ) satisfies,                              z t − ν∆z + z · ∇z +û · ∇z + z · ∇û + ∇π z = µ ξ ∇φ + µ ξ ∇ξ + µφ∇ξ, in Q, ξ t + z · ∇ξ +û · ∇ξ + z · ∇φ = ∆µ ξ , in Q, div z = 0, in Q, where π z = π u − πû. Now from Proposition 2.6, continuous dependence of weak solutions, we have µ ξ = −∆ξ + F ′ (ϕ) − F ′ (φ), in Q, z| ∂Ω = η, on Σ, ∂ξ ∂n ∂Ω = 0 = ∂µ ξ ∂n ∂Ω on Σ, z(0) = 0, ξ(0) = 0, in Ω.z 2 L ∞ (0,T ;L 2 )∩L 2 (0,T ;H 1 div ) + ξ 2 L ∞ (0,T ;H 1 )∩L 2 (0,T ;H 2 ) + ∇µ ξ 2 ≤ M η 2 U . (4.24) Let us define y = z − w, ρ = ξ − ψ where (w, ψ) is the solution of linearised system (4.1) corresponding to non zero boundary for velocity as η. Then, (y, ρ) satisfies,                              y t − ν∆y + z · ∇z +û · ∇y + y · ∇û + ∇π y = µ ρ ∇φ + µ ξ ∇ξ + µφ∇ρ, in Q, ρ t + y · ∇φ +û · ∇ρ + z · ∇ξ = ∆µ ρ , in Q, We have from Theorem 4.1 µ ρ = −∆ρ + F ′ (ϕ) − F ′ (φ) − F ′′ (φ)ψ, in Q, div y = 0, in Q,(w, ψ) W ≤ C η U . We aim to show (y, ρ) W η U → 0 as η U → 0, which will directly imply (4.22). For that, we multiply the equation (4.25) 1 by y to yield 1 2 d dt y 2 + ν ∇y 2 = −(z · ∇z, y) − (y · ∇û, y) + (µ ρ ∇φ, y) + (µ ξ ∇ξ, y) + (µφ∇ρ, y) := 6 i=1 I i (4.26) Again we multiply the equation (4.25) 2 and (4.25) 3 by ρ − ∆ρ and −∆µ ρ + ∆ρ respectively and adding them together we get 1 2 d dt ρ 2 H 1 + ∇µ ρ 2 + ∆ρ 2 = −(y · ∇φ, ρ) − (z · ∇ξ, ρ) + (y · ∇φ, ∆ρ) + (û · ∇ρ, ∆ρ) + (z · ∇ξ, ∆ρ) + (F ′ (ϕ) − F ′ (φ) − F ′′ (φ)ψ, ∆ρ) − (F ′ (ϕ) − F ′ (φ) − F ′′ (φ)ψ, ∆µ ρ ) := 7 i=1 J i (4.27) We estimate the right-hand side of (4.26) by using Hölder inequality, Agmon's inequality, Young's inequality, and Sobolev inequality and derive the following, |I 1 | ≤ C z L 4 ∇z y L 4 ≤ C z 4 H 1 + ν 8 ∇y 2 |I 2 | ≤ y L 4 ∇û y L 4 ≤ C ∇û 2 y 2 + ν 8 ∇y 2 |I 3 | ≤ ∇µ ρ φ L ∞ y ≤ 1 4 ∇µ ρ 2 + φ 2 H 2 y 2 |I 4 | = |(ξ∇µ ξ , y)| ≤ ξ L 4 ∇µ ξ y L 4 ≤ C ξ 2 L 4 ∇µ ξ 2 + ν 8 ∇y 2 |I 5 | ≤ ∇µφ ρ L 4 y L 4 ≤ C µφ 2 ρ 2 H 1 + ν 8 ∇y 2 . Substituting the estimates of I 1 to I 5 in (4.26), 1 2 d dt y 2 + ν 2 ∇y 2 ≤ C( ∇û 2 + ∇φ 2 ) y 2 + C ∇µφ 2 ρ 2 H 1 + 1 4 ∇µ ρ 2 + C( z 4 H 1 + ξ 2 H 1 ∇µ ξ 2 ) ≤ C( ∇û 2 + ∇φ 2 ) y 2 + C ∇µφ 2 ρ 2 H 1 + 1 4 ∇µ ρ 2 + C η 4 U . (4.28) Now we will estimate the right-hand side of (4.27) as follows, |J 1 | ≤ 1 2 y 2 + 1 2 φ 2 H 3 ρ 2 |J 2 | ≤ z L 4 ∇ξ ρ L 4 ≤ 1 2 z 2 H 1 ∇ξ 2 + 1 2 ρ 2 H 1 |J 3 | ≤ y ∇φ L ∞ ∆ρ ≤ C φ 2 H 3 y 2 + 1 8 ∆ρ 2 |J 4 | ≤ C û 2 H 2 ∇ρ 2 + 1 8 ∆ρ 2 |J 5 | ≤ z L 4 ∇ξ L 4 ∆ρ ≤ C z 2 H 1 ξ 2 H 2 + 1 8 ∆ρ 2 To estimate the sixth term of the right-hand side of (4.27), we first note from the Taylor series expansion of F , F ′ (ϕ) − F ′ (φ) − F ′′ (φ)ψ = 1 2 F ′′′ (θϕ + (1 − θ)φ)ξ 2 + F ′′ (φ)ρ, for some θ ∈ (0, 1). Then, we can estimate the sixth term as |J 6 | ≤ C F ′′′ (θϕ + (1 − θ)φ) L ∞ ξ 2 ∆ρ + F ′′ (φ) L ∞ ρ ∆ρ ≤ C ξ 4 L 4 + 1 8 ∆ρ 2 + C ρ 2 . The last term of (4.27) can be written as, (F ′ (ϕ) − F ′ (φ) − F ′′ (φ)ψ, −∆µ ρ ) = 1 2 F (4) (θϕ + (1 − θ)φ)∇(θϕ + (1 − θ)φ)ξ 2 + F ′′′ (θϕ + (1 − θ)φ)ξ∇ξ+ F ′′′ (φ)∇φρ + F ′′ (φ)∇ρ, ∇µ ρ =: J 1 7 + J 2 7 + J 3 7 + J 4 7 ,(4.29) where J 1 7 , J 2 7 , J 3 7 , J 4 7 can be estimated as follows |J 1 7 | ≤ C F (4) (θϕ + (1 − θ)φ) L ∞ ( ∇φ L 4 + ∇ϕ L 4 ) ξ 2 L 4 ∇µ ρ ≤ 1 8 ∇µ ρ 2 + C( ∇φ 2 L 4 + ∇ϕ 2 L 4 ) ξ 4 L 8 ≤ 1 8 ∇µ ρ 2 + C( ∇φ 2 L 4 + ∇ϕ 2 L 4 ) ξ 4 H 1 , |J 2 7 | ≤ F ′′′ (θϕ + (1 − θ)φ) L ∞ ξ L 4 ∇ξ L 4 ∇µ ρ ≤ 1 8 ∇µ ρ 2 + C ξ 2 L 4 ∇ξ 2 L 4 , |J 3 7 | ≤ F ′′′ (φ) L ∞ ∇φ L 4 ρ L 4 ∇µ ρ ≤ C ∇φ 2 L 4 ρ 2 H 1 + 1 8 ∇µ ρ 2 , |J 4 7 | ≤ F ′′ (φ) L ∞ ∇ρ ∇µ ρ ≤ C ∇ρ 2 + 1 8 ∇µ ρ 2 . Substituting J 1 to J 7 in (4.27) we get 1 2 d dt ρ 2 H 1 + ∇µ ρ 2 + 1 2 ∆ρ 2 ≤ 1 2 + C + C φ 2 H 3 + û 2 H 2 ρ 2 H 1 + C( 1 2 + φ 2 H 3 ) y 2 + 1 2 ∇µ ρ 2 + 1 2 z 2 H 1 ∇ξ 2 + C z 2 H 1 ξ 2 H 2 + C ξ 4 L 4 + +C ξ 2 L 4 ∇ξ 2 L 4 + C( ∇φ 2 L 4 + ∇ϕ 2 L 4 ) ξ 4 H 1 ≤ 1 2 + C + C φ 2 H 3 + û 2 H 2 ρ 2 H 1 + C( 1 2 + φ 2 H 3 ) y 2 + 1 2 ∇µ ρ + C( ∇φ 2 L 4 + ∇ϕ 2 L 4 + 3 + 1 2C ) η 4 U . (4.30) Adding the inequalities (4.28) and (4.30) we obtain 1 2 d dt ( y 2 + ρ 2 H 1 ) + ν 2 ∇y 2 + 1 2 ∆ρ 2 + 1 4 ∇µ ρ 2 ≤ C ∇û 2 + ∇φ 2 + 2 φ 2 H 3 + ∇µφ 2 + ∇û 2 H 2 + 2 ( y 2 + ρ 2 H 1 ) + C ∇φ) 2 L 4 + ∇ϕ 2 L 4 + 1) η 4 U . (4.31) As (û,φ) and (u, ϕ) are strong solution of (1.1) with boundariesĥ andĥ + η respectively, so the pairs (û,φ) and (u, ϕ) are uniformly bounded in V from Theorem 2.7, i.e. (û,φ) V ≤ M 1 , (u, ϕ) V ≤ M 2 . Therefore we can write (4.31) as 1 2 d dt ( y 2 + ρ 2 H 1 ) + ν 2 ) ∇y 2 + 1 2 ∆ρ 2 + 1 4 ∇µ ρ 2 ≤C( y 2 + ρ 2 H 1 ) + C η 4 U ,(4.32) where C depends on M, M 1 , M 2 , Ω. Since the initial condition (y(0), ϕ(0)) = (0, 0), applying Gronwall lemma on (4.32) yields (y, ρ) 2 W ≤ C T η 4 U ,(4.33) Where C T depends on M, M 1 , M 2 , Ω, T . Thus lim η U →0 S(ĥ + η) − S(ĥ) − S ′ (ĥ)(η) W η U = lim η U →0 (y, ρ) W η U ≤ C T η U → 0 as η U → 0. This completes the proof. The First Order Necessary Optimality Condition Once we have shown that the control to state operator S is Frechet differentiable, the next step is to derive the first-order necessary optimality condition. In this section, we establish the first-order necessary condition of optimality satisfied by the optimal solution and the solution of the linearised system (4.1). Theorem 5.1. Let the Assumpstion 2.1 be satisfied and (u 0 , ϕ 0 ) ∈ V div × H 2 . Also suppose (u * , ϕ * , h * ) be the optimal triplet where h * ∈ U ad and S(h * ) = (u * , ϕ * ). Let (w, ψ) be the unique weak solution of linearized problem (4.1) with boundary data h − h * . Then following variational inequality holds: Q (u * − u Q ) · wdxdt + Q (ϕ * − ϕ Q ) · ψdxdt + Ω (u * (T ) − u Ω ) · w(T )dx + Ω (ϕ * (T ) − ϕ Ω ) · ψ(T ) + Σ h * (h − h * )dSdt ≥ 0. (5.1) Proof. Since U ad is a nonempty, convex subset of U, then from Lemma 2.21 of [38], we have the reformulated cost functionalJ satisfiesJ ′ h (u * , ϕ * , h * )(h − h * ) ≥ 0 ∀ h ∈ U ad ,(5.2) whereJ ′ h (u, ϕ, h) denotes the Gateaux derivative ofJ with respect to h. Now we will determineJ ′ h (u * , ϕ * , h * ). We have from (3.7) thatJ (h) =J (S(h), h), such that, S(h) = (u, ϕ) is the unique strong solution of (1.1) corresponding to control h. SinceJ is a quadratic functional, by chain rule, we can writẽ at (u * , ϕ * , h * ) in the direction of (y 1 , y 2 ) is given bỹ J ′ h (u, ϕ, h) =J ′ h (S(h), h) =J ′ S(h) (S(h), h) • S ′ (h) +J ′ h (S(h), h).(5.J ′ S(h) (S(h * ), h * )(y 1 , y 2 ) = Q (u * − u Q ) · y 1 dxdt + Q (ϕ * − ϕ Q )y 2 dxdt + Ω (u * (T ) − u Ω ) · y 1 (T )dx + Ω (ϕ(T ) − ϕ Ω )y 2 (T )dx,(5.4) for any (y 1 , y 2 ) ∈ W. Similarly, we calculate the Gateaux derivative ofJ h at (u * , ϕ * , h * ) in the direction of g asJ ′ h (S(h * ), h * )(g) = Γ h * · g dSdt,(5.5) for any g ∈ U. Also from the Theorem 4.3 we get that S ′ (h * )(h − h * ) = (w, ψ).J ′ h (u * , ϕ * , h * )(h − h * ) = Q (u * − u Q ) · wdxdt + Q (ϕ * − ϕ Q ) · ψdxdt + Ω (u * (T ) − u Ω ) · w(T )dx + Ω (ϕ * (T ) − ϕ Ω ) · ψ(T ) dx + Γ h * · (h − h * )dSdt. Therefore, we can conclude (5.1) from (5.2). First Order Necessary Optimality Condition Via Adjoint System. In this section, we would like to simplify the optimality condition (5.1) and write it in terms of optimal solution and adjoint variables. This optimality system can serve as the basis for computing approximations to optimal solutions numerically. Thus we will now derive the adjoint system corresponding to the system (1.1) by using the Lagrange multipliers method. We know that the adjoint system variables act as Lagrange multipliers corresponding to the state system variables (1.1). It is well-known that the necessary optimality conditions satisfied by the optimal control can be derived from the exact Lagrange multipliers method, which is also known as Karush-Kuhn-Tucker (KKT) theory for optimization problems in Banach spaces. This method is described for various elliptic and parabolic problems in [38,15]. The application of the method for nonlinear pde's is difficult, because it requires a lot of experience in matching the operators, functionals, and spaces involved. Despite the technical difficulty, KKT theory has been applied to study optimal boundary control problems for Navier-Stokes equations in [16,18,17], but it is not straight forward to use in our case for CHNS system (1.1) since our system is highly nonlinear coupled pde system. Thus we will use a formal Lagrange multiplier method [see section 2.10 in [38]] to derive the adjoint system, then prove the well-posedness of the adjoint system, and finally establish the necessary condition of optimality for our optimal control problem (OCP). For this purpose, we formally introduce the Lagrange functional for the control problem (OCP) as follows: L((u, ϕ),h, (p, ζ,P , p 1 , ζ 1 )) := J (u, ϕ, h) − Q [u t − ν∆u + (u · ∇)u + ∇π − (−∆ϕ + F ′ (ϕ))∇ϕ] · p dxdt − Q [ϕ t + u · ∇ϕ − ∆(−∆ϕ + F ′ (ϕ))] · ζ dxdt − Q (div u)P dxdt − Σ (u − h) · p 1 dSdt − Σ ∂ϕ ∂n ζ 1 dSdt (5.7) for any h ∈ U ad and (u, ϕ) = S(h). Here ζ, p,P , ζ 1 , p 1 are Lagrange multipliers corresponding to five state constraints in (1.1) respectively. Let (u * , ϕ * , h * ) be the optimal solution to the problem (OCP) such that (u * , ϕ * ) = S(h * ). Then by using the Lagrange principle, we conclude that (u * , ϕ * , h * ) together with Lagrange multipliers p, ζ,P , p 1 , ζ 1 , satisfies the first order optimality condition associated with the optimization problem related to the Lagrange functional L, defined as follows: min h∈U ad L((u, ϕ),h, (p, ζ,P , p 1 , ζ 1 )). Now, since (u, ϕ) has become formally unconstrained, the Fréchet derivative of L with respect to (u, ϕ) will vanish at the optimal point (u * , ϕ * , h * ), which implies L ′ (u,ϕ) ((u * , ϕ * ), h * , (p, ζ,P , p 1 , ζ 1 ))(u 1 , u 2 ) = 0 (5.8) for all smooth function (u 1 , u 2 ) such that u 1 (0, .) = 0, u 2 (0, .) = 0 in Ω. Furthermore, from the Lagrange principle, the constraints on h in (3.4) gives the following variational inequality L ′ h ((u * , ϕ * ), h * , (p, ζ,P , p 1 , ζ 1 ))(h − h * ) ≥ 0 (5.9) for all h ∈ U ad . Next, we want to determine the Lagrange multipliers p, ζ,P , p 1 , ζ 1 from (5.8), which are the adjoint states corresponding to state equations (1.1). For this purpose, we formally calculate (5.8), then perform integration by parts and take the terms together corresponding to (u 1 , u 2 , π) to derive the following linear system satisfied by (p, ζ,P ): Furthermore, on the boundary Σ, the two lagrange multipliers p 1 , ζ 1 can be uniquely determined in terms of (p, ζ,P ) and it satisfy the following equations:                              − ∂ t p − ν∆p + (u * · ∇)p − (p · ∇ T )u * + ζ∇ϕ * − ∇P = u * − u Q , in Q, − ∂ t ζ − u * · ∇ζ − p · ∇(∆ϕ * ) + div((∇p) · ∇ϕ * ) + div((∇ T (∇ϕ * )) · p) + ∆ 2 ζ − F ′′ (ϕ * )∆ζ = ϕ * − ϕ Q , in Q,p 1 +P n + ∂p ∂n = 0, on Σ (5.11) ∂ζ 1 ∂n + [(∇p) · ∇ϕ * + (∇ T (∇ϕ * )) · p]n = 0, on Σ. (5.12) We call the linear pde system (5.10) as the adjoint system corresponding to (1.1). (5.7) is not well-defined yet because we only have the regularity of (u, ϕ) ∈ V for the control h ∈ U. We do not know the regularity of Lagrange multipliers ζ, p,P , ζ 1 , p 1 yet. Thus the Lagrange multiplier method presented in this section has the sole purpose of identifying the correct form of the adjoint system. Now we establish the following existence result of the adjoint system (5.10). Remark 5.2. The expression of Lagrange functional L in Theorem 5.3. Let (u * , ϕ * ) ∈ V and assumption (2.1) on F be satisfied. Also, assume u Ω ∈ L 2 div (Ω), ϕ Ω ∈ H 1 (Ω). Then the linear problem (5.10) has a unique solution (p, ζ) such that p ∈L ∞ (0, T ; G div ) ∩ L 2 (0, T ; V div ) ∩ H 1 (0, T, V ′ div ), ζ ∈L ∞ (0, T ; H 1 ) ∩ L 2 (0, T ; H 2 ∩ H 3 ) ∩ H 1 (0, T ; (H 1 ) ′ ). for all t ∈ [0, T ]. Applying Gronwall's lemma to the inequality (5.18) 1 2 ( p(t) 2 + ζ(t) 2 H 1 ) ≤ p(T ) 2 + ζ(T ) 2 H 1 + 3 4ν T 0 u * (s) − u Q 2 ds + 1 2 T 0 ϕ * (s) − ϕ Q 2 ds ×exp T + T 0 F ′′ (ϕ * (s)) 2 L ∞ ds + T 0 u * (s) 2 H 2 ds ×exp T 0 6 ν ∇u * (s) 2 ds + 3 16Cν T 0 ϕ * (s) 4 L 4 ds + 1 4C T 0 ϕ * (s) 2 H 2 ds + C T 0 ϕ * (s) 2 H 4 ds + 2 T 0 ϕ * (s) 2 H 3 ds . (5.19) Since (u * , ϕ * ) is the strong solution of nonlinear system (1.1), the right-hand side of (5.19) is finite. Therefore using (5.18) and (5.19) we conclude that p ∈ L ∞ (0, T ; G div ) ∩ L 2 (0, T ; V div ) and ζ ∈ L ∞ (0, T ; H 1 ) ∩ L 2 (0, T, H 2 ∩ H 3 ). Now with this regularity of p, ζ and using the equations (5.10) 1 and (5.10) 2 we get a uniform estimates on time derivatives p t , ζ t so that p t ∈ L 2 (0, T ; V ′ div ), ζ t ∈ L 2 (0, T ; (H 1 ) ′ ). Hence we get a weak solution (p, ζ) as claimed. The uniqueness of weak solutions follows from the linearity of the system. Since (p, ζ) is the unique solution of the adjoint system, as in the Navier -Stokes equations (see [37]), we can determineP ∈ L 2 (0, T ; L 2 0 (Ω)), where L 2 0 (Ω) = {g ∈ L 2 (Ω) : Ω g(x)dx = 0}. Remark 5.4. Note that, if u Ω ∈ H 1 div then solution of the adjoint system, p satisfy p ∈ L ∞ (0, T ; V div ) ∩ L 2 (0, T ; H 2 ). This can be shown by multiplying the equation (5.10) 1 by Ap and proceeding similarly as in Theorem 5.3. Proof. Since (p, ζ,P ) is the unique solution of the adjoint system (5.10), we can determine (p 1 , ζ 1 ) from the equations (5.11) and (5.12) uniquely on the boundary Σ. We only need to show (5.20). From the Trace theorem we have ∂p ∂n ∈ L 2 (0, T ; H − 1 2 (∂Ω)) andP n ∈ L 2 (0, T ; H − 1 2 (∂Ω)). Therefore from (5.11), we have p ∈ L 2 (0, T ; H − 1 2 (∂Ω)). Moreover, using Gagliardo -Nirenberg interpolation inequality we obtain p · ∇ϕ * 2 H 1 = |α|≤1 Ω |D α (p · ∇ϕ * )| 2 ≤ |α|≤1 Ω |D α p · ∇ϕ * | 2 + Ω |p · D α (∇ϕ * )| 2 ≤ p 2 W 1,4 ∇ϕ * 2 L 4 + p 2 L 4 ϕ * 2 W 2,4 ≤ C p 3 2 H 2 p 1 2 L 2 ∇ϕ * L 4 + C ∇p 2 ϕ * 5 3 H 3 ϕ * 1 3 H 1 . As a consequence, using the fact (u * , ϕ * ) ∈ V we obtain p · ∇ϕ * ∈ L 2 (0, T ; H 1 (Ω)). Therefore, (∇p) · ∇ϕ * + (∇ T (∇ϕ * )) · p = ∇(p · ∇ϕ * ) ∈ L 2 (0, T ; L 2 (Ω)). Which implies [(∇p) · ∇ϕ * + (∇ T (∇ϕ * )) · p]n ∈ L 2 (0, T ; H − 1 2 (∂Ω)). Therefore, it follows from (5.12) that ∂ζ 1 ∂n ∈ L 2 (0, T ; H − 1 2 (∂Ω)), and hence from Trace theorem, ζ 1 ∈ L 2 (0, T ; H 1 2 (∂Ω)). Remark 5.6. Using the Theorem 5.3 and Corollary 5.5, we can now conclude that the Lagrange functional L defined in (5.7) is well-defined. In the following theorem, we will derive the first-order necessary optimality condition in terms of optimal solution of (OCP) and the adjoint variables p, ζ. Theorem 5.7. Let the Assumption 2.1 holds and (u 0 , ϕ 0 ) ∈ V div × H 2 . In addition, let (u * , ϕ * , h * ) be the optimal triplet, i.e., h * ∈ U ad be the optimal boundary control of the problem (OCP) and (u * , ϕ * ) be the strong solution of (1.1) corresponding to h * . Let (p, ζ) be the solution of the adjoint system (5.10). Then for any h ∈ U ad , the following variational inequality holds: Σ h * (h − h * ) dSdt − ΣP n · (h − h * ) dSdt − Σ ∂p ∂n · (h − h * ) dSdt ≥ 0. (5.21) Proof. We have from (5.22) that L ′ h ((u * , ϕ * ), h * , (p, ζ,P , p 1 , ζ 1 ))(h − h * ) ≥ 0 (5. 22) for all h ∈ U ad . A direct computation of (5.22) leads to the following inequality Σ (h * + p 1 )(h − h * ) dSdt ≥ 0. (5.23) Now substituting the value of p 1 from (5.11) in (5.23) we get Σ h * (h − h * ) dSdt − ΣP n · (h − h * ) dSdt − Σ ∂p ∂n · (h − h * ) dSdt ≥ 0, h ∈ U ad . That completes the proof. Remark 5.8. We can also prove optimality condition (5.21) in Theorem 5.7 from equation (5.1) in Theorem 5.1. For this, we take (w, ψ) to be the unique weak solution of linearised system (4.1) corresponding to boundary η = h− h * , for any h ∈ U ad . Then from the optimal condition (5.8), by taking (u 1 , u 2 ) as the linearised solution (w, ψ) and using the fact that w| Σ = h − h * , ∂ψ ∂n | Σ = 0, we can derive Q (u * − u Q ) · w dxdt + Q (ϕ * − ϕ Q ) · ψ dxdt + Ω (u * (T ) − u Ω ) · w(T ) dx + Ω (ϕ * (T ) − ϕ Ω ) · ψ(T ) dx = Σ p 1 · (h − h * ) dSdt, ∀h ∈ U ad . Now from the variational inequality (5.1), (5.21) follows easily. Finally, using the variational formula (5.21), we can interpret the optimal boundary control in terms of the adjoint variables. Corollary 5.9. Let h * ∈ U ad be a optimal boundary control associated with (OCP). Then h * and the adjoint ssystem (p, ζ) satisfy the projection formula h * = P U ad (−P n − ∂p ∂n ), (5.24) where P U ad is the orthogonal projector from L 2 (Σ) onto U ad . u t − ν∆u + (u · ∇)u + ∇π = µ∇ϕ, in Ω × (0, T ), div u = 0, in Ω × (0, T ), ∂ϕ ∂n = 0, ∂µ ∂n = 0, on Σ, u = h, on Σ, u(0) = u 0 , ϕ(0) = ϕ 0 , in Ω. H s (Ω) := H s (Ω; R). : Let e + ∇π = 0, in Q, div u e = 0, in Q, u e = h, on ∂Ω × (0, T ). . 2 . 2Suppose that h satisfies the conditions h ∈ L 2 (0, T ; H 3 2 3(∂Ω)) ∩ L ∞ (0, T ; H h ∈ L 2 (0, T ; H − 1 2 (∂Ω)). Then equation (2.1) admits a unique weak solution t=0 = 0, ψ| t=0 = 0, in Ω. . 1 . 1Existence of weak solution of linearize system. Now we state the existence result of the linearized system (4.1) and prove it by using a Faedo-Galerkin approximation scheme. y| ∂Ω = 0, on Σ, ∂ρ ∂n = 0 = ∂µ ρ ∂n , on Σ,y(0) = 0, ρ(0) = 0, in Ω. T ) = u * (T ) − u Ω , ζ(T ) = ϕ * (T ) − ϕ Ω , in Ω. Corollary 5. 5 . 5Let u Ω ∈ H 1 div and all the assumptions of Theorem 5.3 hold. Then the Lagrange multipliers (p 1 , ζ 1 ) are uniquely determined by the equations (5.11) and (5.12) such that p 1 ∈ L 2 (0, T ; H − 1 2 (∂Ω)), ζ 1 ∈ L 2 (0, Finally, we derive the first-order necessary optimality condition in terms of a variational inequality involving adjoint states. [See Theorem (5.7)]). [See Theorem (3.2)] (2) We show that the control to state operator S is Fréchet differentiable between suitable Banach spaces. [See Theorem (4.3)] (3) We derive the adjoint system corresponding to state problem (1.1) and establish the well-posedness of the adjoint system. [See Theorem (5.3)] (4) Proof. The proof follows from a similar argument as in Theorem 4.1 using the Faedo -Galerkin method. For the sake of simplicity, we omit the approximation scheme and do the apriori estimates only. We multiply the equation (5.10) 1 by p and the equation (5.10) 2 by ζ − ∆ζ we getNow we estimate each I i and J i one by one using Poincare, Young's, and Sobolev inequality. We estimate I 1 using Ladyzhanskya and Young's inequality asSimilarly, we can estimate I 2 asWhere C is a generic constant.Adding I 1 to I 3 in the equation (5.13) we getNow we will estimate J i .where we have used Agmon's inequality and Young's inequality. Similarly, J 2 can be estimated asSimilarly,Taking estimates of J 1 to J 4 into account, we write (5.14) asAdding the inequalities (5.15) and (5.16) we getIntegrating the inequality (5.17) from t to T yields On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Helmut Abels, Arch. Ration. Mech. Anal. 1942Helmut Abels. 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[ "A cold-atom Ramsey clock with a low volume physics package", "A cold-atom Ramsey clock with a low volume physics package" ]
[ "A Bregazzi \nDepartment of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n", "E Batori \nInstitute of Physics\nUniversity of Neuchâtel\nLaboratoire Temps-Fréquence\nAvenue de Bellevaux 512000NeuchâtelSwitzerland\n", "B Lewis \nDepartment of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n", "C Affolderbach \nInstitute of Physics\nUniversity of Neuchâtel\nLaboratoire Temps-Fréquence\nAvenue de Bellevaux 512000NeuchâtelSwitzerland\n", "G Mileti \nInstitute of Physics\nUniversity of Neuchâtel\nLaboratoire Temps-Fréquence\nAvenue de Bellevaux 512000NeuchâtelSwitzerland\n", "E Riis \nDepartment of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n", "P F Griffin \nDepartment of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n" ]
[ "Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom", "Institute of Physics\nUniversity of Neuchâtel\nLaboratoire Temps-Fréquence\nAvenue de Bellevaux 512000NeuchâtelSwitzerland", "Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom", "Institute of Physics\nUniversity of Neuchâtel\nLaboratoire Temps-Fréquence\nAvenue de Bellevaux 512000NeuchâtelSwitzerland", "Institute of Physics\nUniversity of Neuchâtel\nLaboratoire Temps-Fréquence\nAvenue de Bellevaux 512000NeuchâtelSwitzerland", "Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom", "Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom" ]
[]
We demonstrate a Ramsey-type microwave clock interrogating the 6.835 GHz ground-state transition in cold 87 Rb atoms loaded from a grating magneto-optical trap (GMOT) enclosed in an additively manufactured loop-gap resonator microwave cavity. A short-term stability of 1.5×10 −11 τ −1/2 is demonstrated, in reasonable agreement with predictions from the signal-to-noise ratio of the measured Ramsey fringes. The cavity-grating package has a volume of ≈67 cm 3 , ensuring an inherently compact system while the use of a GMOT drastically simplifies the optical requirements for laser cooled atoms. This work is another step towards the realisation of highly compact portable cold-atom frequency standards.Compact frequency standards based on the interrogation of both ions and neutral atoms continue to receive much interest, with many different schemes now reported in the literature 1-9 . While compact clocks based on thermal atomic vapours and coherent population trapping (CPT) remain unrivalled in terms of size, weight and power (SWaP) they typically exhibit stabilities that are limited in the medium to longterm due to light-shift effects 10 and the buffer gasses required to reduce atomic collisions 1 . In addition to this, CPT clocks struggle to reach atomic shot noise due to the limited signalto-noise ratio (SNR), a consequence of the low detected signal photon count per atom. This limitation often requires complex interrogation schemes to optimise the clock stability[11][12][13].The development in the last decade of pulsed optically pumped (POP) clocks in thermal vapours has driven new research, with state-of-the-art stabilities 2,14,15 . However, these clocks still suffer from buffer gas shifts, providing the ultimate limitation to their long-term stabilities. Achievable Ramsey times within these systems are also limited to a few ms by spin relaxation of the thermal atoms 2,16 , restricting the shortterm stability performance.In an effort to combat these limitations several groups have now developed compact cold atom microwave clocks based on spherical optical-integrating sphere cavities 17,18 , cylindrical cavities 19,20 and more recently loop-gap-resonator cavities21. Three examples of cold atom microwave clocks are now even commercially available 22-24 . Different laser cooling schemes with varying optical geometries have been used within these systems, with isotropic cooling of an optical molasses 17,18,20,22 , pyramid MOTs 25 and larger traditional 6-beam MOTs 19,21 all being utilised.In this letter we demonstrate operation and first stability measurement of a cold-atom atomic clock based on an additively manufactured loop-gap resonator (LGR) cavity with a volume of ≈ 67 cm 3 incorporating an integrated GMOT 26 . Although vapour-cell atomic clocks based on LGR structures with external diameters of order 1 cm have been demonstrated 16 , such small devices will have limited benefit when using cold atoms. Previous investigations with both a) Electronic mail: [email protected] 22 mm To LO PD Sig PD Ref NPBS B Grating chip MOT M MOT FIG. 1. Simplified schematic of the physics package. Trap light (red arrow) propagates parallel to the magnetic bias field (black dashed arrow), 90:10 non-polarising beam splitter (NPBS) splits the optical pumping (blue arrow) and readout (green arrow) light onto a reference photodiode (PD Ref ) and signal photodiode (PD Sig ) after being retro-reflected by a mirror (M). Local oscillator (LO) is connected via a SMA vacuum feedthrough.GMOTs 27 and 6-beam MOTs 28 have shown that by constraining the trap beam diameter the maximum trapped atom number is limited, therefore constricting the potential stability of the final clock.In the present study a sample of cold atoms is optically pumped into the clock state, probed in a double resonance Ramsey-type scheme and the resulting state populations read out through an absorptive method. The use of additive manufacturing allows for complex electrode geometries, while maintaining a highly scalable manufacturing process due to the lack of precision machining and assembly required 29,30 . While the main aim of this work has been to demonstrate the feasibility of integrating this cavity design and its fabrication technique with the GMOT architecture, we envisage that further substantial reduction in size is possible by adapting the cavity to form the bulk of the vacuum system. The minimal optical access required of this scheme also allows for a reduction in the number of apertures present in the cavity body which must be carefully considered during the design process so as not to degrade the cavity mode and field arXiv:2305.02944v1 [physics.atom-ph] 4 May 2023
null
[ "https://export.arxiv.org/pdf/2305.02944v1.pdf" ]
258,480,055
2305.02944
30bb36e7b98dac58447cfb21c4ed05c6ce5c2dc8
A cold-atom Ramsey clock with a low volume physics package A Bregazzi Department of Physics University of Strathclyde G4 0NGGlasgowUnited Kingdom E Batori Institute of Physics University of Neuchâtel Laboratoire Temps-Fréquence Avenue de Bellevaux 512000NeuchâtelSwitzerland B Lewis Department of Physics University of Strathclyde G4 0NGGlasgowUnited Kingdom C Affolderbach Institute of Physics University of Neuchâtel Laboratoire Temps-Fréquence Avenue de Bellevaux 512000NeuchâtelSwitzerland G Mileti Institute of Physics University of Neuchâtel Laboratoire Temps-Fréquence Avenue de Bellevaux 512000NeuchâtelSwitzerland E Riis Department of Physics University of Strathclyde G4 0NGGlasgowUnited Kingdom P F Griffin Department of Physics University of Strathclyde G4 0NGGlasgowUnited Kingdom A cold-atom Ramsey clock with a low volume physics package (Dated: 5 May 2023) We demonstrate a Ramsey-type microwave clock interrogating the 6.835 GHz ground-state transition in cold 87 Rb atoms loaded from a grating magneto-optical trap (GMOT) enclosed in an additively manufactured loop-gap resonator microwave cavity. A short-term stability of 1.5×10 −11 τ −1/2 is demonstrated, in reasonable agreement with predictions from the signal-to-noise ratio of the measured Ramsey fringes. The cavity-grating package has a volume of ≈67 cm 3 , ensuring an inherently compact system while the use of a GMOT drastically simplifies the optical requirements for laser cooled atoms. This work is another step towards the realisation of highly compact portable cold-atom frequency standards.Compact frequency standards based on the interrogation of both ions and neutral atoms continue to receive much interest, with many different schemes now reported in the literature 1-9 . While compact clocks based on thermal atomic vapours and coherent population trapping (CPT) remain unrivalled in terms of size, weight and power (SWaP) they typically exhibit stabilities that are limited in the medium to longterm due to light-shift effects 10 and the buffer gasses required to reduce atomic collisions 1 . In addition to this, CPT clocks struggle to reach atomic shot noise due to the limited signalto-noise ratio (SNR), a consequence of the low detected signal photon count per atom. This limitation often requires complex interrogation schemes to optimise the clock stability[11][12][13].The development in the last decade of pulsed optically pumped (POP) clocks in thermal vapours has driven new research, with state-of-the-art stabilities 2,14,15 . However, these clocks still suffer from buffer gas shifts, providing the ultimate limitation to their long-term stabilities. Achievable Ramsey times within these systems are also limited to a few ms by spin relaxation of the thermal atoms 2,16 , restricting the shortterm stability performance.In an effort to combat these limitations several groups have now developed compact cold atom microwave clocks based on spherical optical-integrating sphere cavities 17,18 , cylindrical cavities 19,20 and more recently loop-gap-resonator cavities21. Three examples of cold atom microwave clocks are now even commercially available 22-24 . Different laser cooling schemes with varying optical geometries have been used within these systems, with isotropic cooling of an optical molasses 17,18,20,22 , pyramid MOTs 25 and larger traditional 6-beam MOTs 19,21 all being utilised.In this letter we demonstrate operation and first stability measurement of a cold-atom atomic clock based on an additively manufactured loop-gap resonator (LGR) cavity with a volume of ≈ 67 cm 3 incorporating an integrated GMOT 26 . Although vapour-cell atomic clocks based on LGR structures with external diameters of order 1 cm have been demonstrated 16 , such small devices will have limited benefit when using cold atoms. Previous investigations with both a) Electronic mail: [email protected] 22 mm To LO PD Sig PD Ref NPBS B Grating chip MOT M MOT FIG. 1. Simplified schematic of the physics package. Trap light (red arrow) propagates parallel to the magnetic bias field (black dashed arrow), 90:10 non-polarising beam splitter (NPBS) splits the optical pumping (blue arrow) and readout (green arrow) light onto a reference photodiode (PD Ref ) and signal photodiode (PD Sig ) after being retro-reflected by a mirror (M). Local oscillator (LO) is connected via a SMA vacuum feedthrough.GMOTs 27 and 6-beam MOTs 28 have shown that by constraining the trap beam diameter the maximum trapped atom number is limited, therefore constricting the potential stability of the final clock.In the present study a sample of cold atoms is optically pumped into the clock state, probed in a double resonance Ramsey-type scheme and the resulting state populations read out through an absorptive method. The use of additive manufacturing allows for complex electrode geometries, while maintaining a highly scalable manufacturing process due to the lack of precision machining and assembly required 29,30 . While the main aim of this work has been to demonstrate the feasibility of integrating this cavity design and its fabrication technique with the GMOT architecture, we envisage that further substantial reduction in size is possible by adapting the cavity to form the bulk of the vacuum system. The minimal optical access required of this scheme also allows for a reduction in the number of apertures present in the cavity body which must be carefully considered during the design process so as not to degrade the cavity mode and field arXiv:2305.02944v1 [physics.atom-ph] 4 May 2023 We demonstrate a Ramsey-type microwave clock interrogating the 6.835 GHz ground-state transition in cold 87 Rb atoms loaded from a grating magneto-optical trap (GMOT) enclosed in an additively manufactured loop-gap resonator microwave cavity. A short-term stability of 1.5×10 −11 τ −1/2 is demonstrated, in reasonable agreement with predictions from the signal-to-noise ratio of the measured Ramsey fringes. The cavity-grating package has a volume of ≈67 cm 3 , ensuring an inherently compact system while the use of a GMOT drastically simplifies the optical requirements for laser cooled atoms. This work is another step towards the realisation of highly compact portable cold-atom frequency standards. Compact frequency standards based on the interrogation of both ions and neutral atoms continue to receive much interest, with many different schemes now reported in the literature [1][2][3][4][5][6][7][8][9] . While compact clocks based on thermal atomic vapours and coherent population trapping (CPT) remain unrivalled in terms of size, weight and power (SWaP) they typically exhibit stabilities that are limited in the medium to longterm due to light-shift effects 10 and the buffer gasses required to reduce atomic collisions 1 . In addition to this, CPT clocks struggle to reach atomic shot noise due to the limited signalto-noise ratio (SNR), a consequence of the low detected signal photon count per atom. This limitation often requires complex interrogation schemes to optimise the clock stability [11][12][13] . The development in the last decade of pulsed optically pumped (POP) clocks in thermal vapours has driven new research, with state-of-the-art stabilities 2,14,15 . However, these clocks still suffer from buffer gas shifts, providing the ultimate limitation to their long-term stabilities. Achievable Ramsey times within these systems are also limited to a few ms by spin relaxation of the thermal atoms 2,16 , restricting the shortterm stability performance. In an effort to combat these limitations several groups have now developed compact cold atom microwave clocks based on spherical optical-integrating sphere cavities 17,18 , cylindrical cavities 19,20 and more recently loop-gap-resonator cavities 21 . Three examples of cold atom microwave clocks are now even commercially available [22][23][24] . Different laser cooling schemes with varying optical geometries have been used within these systems, with isotropic cooling of an optical molasses 17,18,20,22 , pyramid MOTs 25 and larger traditional 6-beam MOTs 19,21 all being utilised. In this letter we demonstrate operation and first stability measurement of a cold-atom atomic clock based on an additively manufactured loop-gap resonator (LGR) cavity with a volume of ≈ 67 cm 3 incorporating an integrated GMOT 26 . Although vapour-cell atomic clocks based on LGR structures with external diameters of order 1 cm have been demonstrated 16 GMOTs 27 and 6-beam MOTs 28 have shown that by constraining the trap beam diameter the maximum trapped atom number is limited, therefore constricting the potential stability of the final clock. In the present study a sample of cold atoms is optically pumped into the clock state, probed in a double resonance Ramsey-type scheme and the resulting state populations read out through an absorptive method. The use of additive manufacturing allows for complex electrode geometries, while maintaining a highly scalable manufacturing process due to the lack of precision machining and assembly required 29,30 . While the main aim of this work has been to demonstrate the feasibility of integrating this cavity design and its fabrication technique with the GMOT architecture, we envisage that further substantial reduction in size is possible by adapting the cavity to form the bulk of the vacuum system. The minimal optical access required of this scheme also allows for a reduction in the number of apertures present in the cavity body which must be carefully considered during the design process so as not to degrade the cavity mode and field arXiv:2305.02944v1 [physics.atom-ph] 4 May 2023 homogeneity 21,30 . A commercial laser system (Muquans ILS) at 780 nm with an integrated electro-optic modulator (EOM) for the creation of optical sidebands on the carrier frequency is used throughout. The laser light is split into three distinct optical paths, each with a double passed acousto-optic modulator (AOM) for power and frequency control to enable trapping, optical pumping and state readout. The trapping light is coupled into a fibre to be passed to the physics package. The optical pumping and state readout beams are coupled into a single additional fibre and likewise passed to the physics package. The microwave cavity itself, described in detail in Ref 30 consists of a loop-gap structure with a four electrode geometry and has an outer radius of 44 mm and a height of roughly 44 mm. The cavity operates in a TE 011 -like mode, tuned to the ground-state-splitting of 87 Rb with a quality factor of Q≈360 and is mounted within a stainless steel vacuum chamber, maintained at ultra-high-vacuum (UHV) by an ion pump. Optical access is enabled via viewports in the vacuum chamber along two orthogonal axes. The first axis is parallel to the cylindrical symmetry axis of the cavity (in the following this is referred to as the cavity axis) and allows the trap light to be directed onto the grating chip after expansion and collimation from the fibre. Optical pumping and state readout light, expanded from the fibre to a 1/e 2 diameter of 7 mm, is directed onto the atoms through two 4 mm holes drilled in the side of the cavity body. A retro-reflecting mirror for this light is placed outside the vacuum chamber to decrease the acceleration experienced by the atoms when interacting with the optical pumping and probe beams and increase signal amplitudes. A simplified schematic of this is shown in Fig.1. No significant degradation of the cavity field is observed by the introduction of the holes in the cavity body 30 . A pair of anti-Helmholtz coils are mounted within the vacuum chamber, along the cavity axis in order to create the quadrupole magnetic field required for the trapping process. Three orthogonal pairs of Helmholtz shim coils, mounted externally to the vacuum chamber are used for the cancellation of external stray DC magnetic fields and to apply a magnetic bias along the cavity axis of ≈ 100 mG in order to lift the atomic degeneracy during optical molasses and clock interrogation. We note that the current demonstration is a proof of concept with no magnetic shielding of the experiment present, limiting the potential stability of the system. The experimental cycle is initiated by turning the trapping coils on with the trap light tuned to be approximately ∆=-2Γ red detuned (Γ/2π=6.07 MHz) from the 87 Rb D 2 F=2→F'=3 cycling transition, with re-pump light generated by the EOM operating at 6.57 GHz to produce 5% optical sidebands. A Rb vapour, maintained at the 1 × 10 −9 Torr level, is produced by resistively heating a pair of alkali metal dispensers. We then perform a 6 ms optical molasses by turning the trap coils off and linearly ramping the light detuning to ∆=-5Γ while simultaneously decreasing the trap light intensity. After molasses we measure atomic temperatures of ≈ 10 µK. In order to decrease the clock cycle time and mitigate experimental dead time, we employ atom recapture between experimental cycles 3,31 . In steady state this allows the trapping of > 3 × 10 6 atoms with a load time of 100 ms for a clock cycle operating at ≈7 Hz. Once the atoms have been trapped and cooled, the trap light is extinguished by an AOM and blocked by a mechanical shutter 32 to ensure no light leakage during microwave interrogation. After molasses, the atoms are assumed to be roughly evenly distributed between the five m F levels of the F = 2 hyperfine ground-state manifold. We therefore employ a 1 ms optical pumping stage with 10 µW total power of linearly polarised light, polarisation axis parallel to the quantization axis, tuned to the F=2→F'=2 and re-pump light tuned to the F=1→F'=2 transition. Due to selection rules this pumps > 95% of the atoms into the 5 2 S 1/2 , |F = 2, m F = 0 state 33 as seen by the almost complete elimination of the m F = 1 → m F = 1 transitions, when scanning the microwave detuning, increasing the contrast of the resulting clock signal. A Keysight E8257D microwave synthesizer is used as a local oscillator and microwave source throughout. Square microwave π/2 pulses with a duration of 200 µs and typical power levels of 0.04 mW are applied to the atoms. The specified phase noise performance of the local oscillator allows the Dick effect 34 to be reduced to the 6.12 × 10 −13 level at 1 s for the experimental cycle time. After the desired microwave pulses have been applied, the atomic states are read out by an absorption method using 30 µW of optical power, measured by recording the transmission on a photodiode of two subsequent probe pulses. First a short pulse of readout light tuned to the F=2→F'=3 transition is applied, giving an measure of the number of atoms in the F = 2 ground-state. We then apply a second readout pulse with re-pump light present. This measures the total number of atoms, providing a normalisation of the signal and reduced sensitivity to atom number fluctuations. Intensity noise in the readout pulses is reduced by the use of a reference photodiode. An example Ramsey fringe taken with a Ramsey time, T R =10 ms, corresponding to a fringe linewidth of 50 Hz is shown in Fig.2. This fringe exhibits an SNR of around 110, measured at the half-width points of the central fringe, where SNR is defined as the ratio of the peak amplitude divided by the amplitude noise. The predicted SNR limited short-term relative frequency instability of a local oscillator stabilised to an atomic transition in terms of Allan deviation is given by 35 : σ SNR (τ) = 1 πC ∆ f f 0 1 SNR T C τ (1) where C is the fringe contrast, ∆ f is the signal linewidth (≈ 1/2T R ), f 0 is the central frequency (6.8346... GHz), T C is the full experimental cycle time (T C = 140ms) and τ is the averaging time. We use this relationship as a basis to optimise our experimental cycle to maximise the potential stability of our clock. A plot of the predicted stability, using the measured SNR, as a function of the Ramsey time is shown in Fig.3. From this we find that as the Ramsey time is increased the predicted stability also improves up to the level of 8.95 × 10 −12 for a 10 ms Ramsey time. After this time however the stability begins to decrease as the SNR is degraded. This degradation in SNR is attributed to atomic losses due to both the thermal expansion of the cloud and the atoms falling out of the probe region under gravity. The extension of this Ramsey time should be possible by moving the holes in the cavity body lower down, introducing elliptical holes to maintain good probe-atom overlap along the path of gravity or by implementing a grating-chip atomic fountain 36 . This last option is particularly attractive because as with traditional atomic fountains it would be possible apply both π/2 pulses when the atoms are at the same vertical point of the cavity. This will allow the phase difference observed by the atoms between the two microwave pulses to be minimised, essential for high contrast Ramsey fringes in a relatively low-Q cavity such as used here. A grating-chip atomic fountain (using CPT interrogation) such as this has already demonstrated Ramsey times out to 100 ms, with a corresponding fringe linewidth of 5 Hz 36 . To assess the stability of our system we stabilise the local oscillator to the atomic signal. The signal is modulated and demodulated at the fringe's half maximum to construct the error signal which is used to feedback onto the local oscillator by voltage tuning its 10 MHz internal reference. Fig.4 shows an overlapping Allan deviation of the resulting frequency stability when compared to an oven controlled quartz crystal oscillator (Wenzel Associates 501-29647) disciplined to GPS (GPSDO). From Fig.4 we see that the clock stability averages down with a 1.5 × 10 −11 τ −1/2 dependence out to 10 s, at which point it deviates slightly from the ideal τ −1/2 dependence. This is in reasonable agreement with the theoretical stability obtained from Fig.3 at a 10 ms Ramsey time and is well above the ultimate stability limit set by the quantum projection noise (QPN) of 4.9 × 10 −13 τ −1/2 , calculated by replacing the SNR term in (1) by √ N, where N is the atom number. As the experiment is currently operated in a magnetically unshielded environment the instability contribution due to the second-order Zeeman shift was expected to limit the clock stability in the medium-term. This was confirmed by measuring the magnetic field stability via the |F = 1, m F = 1 → |F = 2, m F = 1 microwave transition, exhibiting a magnetic field sensitivity of β 1 =1.4 MHz/G 37 . The expected stability of the m F = 0 clock transition (β 0 =575 Hz/G 2 37 ) could then be calculated, shown as black points in Fig.4. This plot shows that the second-order Zeeman shift is indeed an important limitation to the clock stability at τ >10 s, due to a pronounced hump in the magnetic field stability around this time, after which the stability averages down slightly to the level of < 2 × 10 −12 . For future iterations of this set-up it will be imperative to introduce magnetic shielding for long-term stability performance. (1) Another limiting factor to the clock stability in the medium term is the GPSDO reference. Three-corner-hat stability measurements of this reference against two commercial Cs beam clocks (OSA 3235B and Microchip 5071A) indicates excellent short-term performance of the GPSDO < 1 × 10 −12 . At 700 s however, a peak in the instability is observed at ≈ 2 × 10 −12 . This peak is associated with the time constant of the tuning loop used to reference the oscillator to the GPS signal. When taken in conjunction with the stability limit due to the Zeeman effect, these two effects help explain the flattening of the clock stability for averaging times >100 s. To more fully characterise the short-term stability (τ=1 s) of the system a stability budget of the main instability contributions to the clock was constructed, shown in Table I. The short-term clock performance is found to be primarily limited by the Ramsey fringe SNR, discussed above. In future we expect to be able to improve the SNR contribution to shortterm stability towards the projection noise limit. This will be achieved by increasing the Ramsey time, optimisation of the optical detection process and suppression of intensity and frequency fluctuations in the probe beam. The next largest contributor to the 1 s clock stability after the fringe SNR is electronic noise on the voltage line used to tune the 10 MHz local oscillator. While this is not a limiting factor to the clock stability at present, careful minimisation of this noise source will be required when moving towards a magnetically shielded experiment where the expected clock stability should improve below the 1 × 10 −12 level. The total predicted short term stability from Table I is found to be 1.6 times better than the measured stability. Efforts are ongoing to identify the remaining instability contributions in order to minimise them in future iterations of the set-up. In conclusion, we have demonstrated a compact coldatom microwave clock based on an additively manufactured loop-gap-resonator cavity interacting with laser cooled atoms loaded from a GMOT chip. In the present system the experimentally optimised short-term clock stability is measured as σ y (τ) = 1.5 × 10 −11 τ −1/2 , primarily limited by the Ramsey fringe SNR. In the medium-term the clock stability is primarily limited by the second-order Zeeman shift due to operating in an unshilded environment. A secondary limit is placed on the medium-term stability performance by the reference oscillator itself exhibiting a stability bump at around 700 s. By addressing these issues, we expect to improve the short-term stability to be more in-line with other cold-atom cavity clock demonstrations 17,[21][22][23] . The current physics package is inherently compact and remains highly amenable to further miniaturisation. The use of additive manufacturing also maintains a highly scalable manufacturing process. Additively manufactured vacuum chambers have also recently demonstrated compatibility with UHV 38 pressure levels. This raises the enticing possibility of the cavity body itself simultaneously acting as a UHV chamber, allowing the entire physics package to be manufactured as a single bulk component, drastically reducing the size of the system. In addition to this, passively pumped vacuum chambers have now been shown to maintain the UHV levels required for atom trapping for extended periods 39,40 , allowing for a reduction in power consumption by negating the continuous use of an ion pump. While the stability of our clock is currently several orders of magnitude below the state-of-theart, the scope for improved performance and further miniaturisation is very good. We therefore believe this work represents a step towards highly compact and portable cold-atom frequency standards. ACKNOWLEDGMENTS The authors would like to thank R. Elvin and J. P. McGilligan for useful conversations and J. P. McGilligan for careful reading of the manuscript. A.B. was supported by a Ph.D. studentship from the Defence Science and Technology Laboratory (Dstl). E. B., C.A. and G. M. acknowledge funding from the European space Agency (ESA) and the Swiss Space Office (Swiss Confederation). We gratefully acknowledge funding from the International Network for Microfabrication of Atomic Quantum Sensors (EPSRC EP/W026929/1). DATA AVAILABILITY STATEMENT The data that support the findings of this study are available from https://doi.org/10.15129/bb0d3614-6394-4d0e-8649-b1bf66c71722 FIG. 2 . 2Typical Ramsey fringe taken with a 10 ms free evolution time. FIG. 3 . 3Predicted short-term stability as a function of Ramsey time. Inset: Typical central Ramsey fringe obtained at 10 ms free evolution time. Red points indicate measured normalised probe absorption while the black line shows a sinusoidal fit to the data. The fringe offset from 0 Hz detuning corresponds to the frequency shift expected from the second-order Zeeman shift at a 100 mG bias field. FIG. 4 . 4Red points: Overlapping Allan deviation of local oscillator's stability when locked to atomic signal. Blue solid line: stability predicted by , such small devices will have limited benefit when using cold atoms. Previous investigations with both a) Electronic mail: [email protected] mm To LO PD Sig PD Ref NPBS B Grating chip MOT M MOT FIG. 1. Simplified schematic of the physics package. Trap light (red arrow) propagates parallel to the magnetic bias field (black dashed arrow), 90:10 non-polarising beam splitter (NPBS) splits the optical pumping (blue arrow) and readout (green arrow) light onto a refer- ence photodiode (PD Ref ) and signal photodiode (PD Sig ) after being retro-reflected by a mirror (M). Local oscillator (LO) is connected via a SMA vacuum feedthrough. TABLE I . ITable of noise sources and their contributions to the 1 s Allan deviations stability. . S Knappe, V Shah, P D D Schwindt, L Hollberg, J Kitching, L.-A , S. Knappe, V. Shah, P. D. D. Schwindt, L. 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[ "Optical Signatures of Förster-induced energy transfer in organic/TMD heterostructures", "Optical Signatures of Förster-induced energy transfer in organic/TMD heterostructures" ]
[ "Joshua J P Thompson \nDepartment of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg\n", "Marina Gerhard \nDepartment of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg\n", "Gregor Witte \nDepartment of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg\n", "Ermin Malic \nDepartment of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg\n" ]
[ "Department of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg", "Department of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg", "Department of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg", "Department of Physics\nPhilipps-Universität Marburg\nRenthof 735032Marburg" ]
[]
Hybrid van der Waals heterostructures of organic semiconductors and transition metal dichalcogenides (TMDs) are promising candidates for various optoelectronic devices, such as solar cells and biosensors. Energy-transfer processes in these materials are crucial for the efficiency of such devices, yet they are poorly understood. In this work, we develop a fully microscopic theory describing the effect of the Förster interaction on exciton dynamics and optics in a WSe2/tetracene heterostack. We demonstrate that the differential absorption and time-resolved photoluminescence can be used to track the real-time evolution of excitons. We predict a strongly unidirectional energy transfer from the organic to the TMD layer. Furthermore, we explore the role temperature has in activating the Förster transfer and find a good agreement to previous experiments. Our results provide a blueprint to tune the light-harvesting efficiency through temperature, molecular orientation and interlayer separation in TMD/organic heterostructures. arXiv:2305.02853v1 [cond-mat.mes-hall] 4 May 2023
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[ "https://export.arxiv.org/pdf/2305.02853v1.pdf" ]
258,480,082
2305.02853
525cb0b7c6d144a2c0876644233c61ca41f4bfa2
Optical Signatures of Förster-induced energy transfer in organic/TMD heterostructures Joshua J P Thompson Department of Physics Philipps-Universität Marburg Renthof 735032Marburg Marina Gerhard Department of Physics Philipps-Universität Marburg Renthof 735032Marburg Gregor Witte Department of Physics Philipps-Universität Marburg Renthof 735032Marburg Ermin Malic Department of Physics Philipps-Universität Marburg Renthof 735032Marburg Optical Signatures of Förster-induced energy transfer in organic/TMD heterostructures Hybrid van der Waals heterostructures of organic semiconductors and transition metal dichalcogenides (TMDs) are promising candidates for various optoelectronic devices, such as solar cells and biosensors. Energy-transfer processes in these materials are crucial for the efficiency of such devices, yet they are poorly understood. In this work, we develop a fully microscopic theory describing the effect of the Förster interaction on exciton dynamics and optics in a WSe2/tetracene heterostack. We demonstrate that the differential absorption and time-resolved photoluminescence can be used to track the real-time evolution of excitons. We predict a strongly unidirectional energy transfer from the organic to the TMD layer. Furthermore, we explore the role temperature has in activating the Förster transfer and find a good agreement to previous experiments. Our results provide a blueprint to tune the light-harvesting efficiency through temperature, molecular orientation and interlayer separation in TMD/organic heterostructures. arXiv:2305.02853v1 [cond-mat.mes-hall] 4 May 2023 The controlled fabrication of two-dimensional heterostructures has paved the way for new cutting-edge materials and quantum technologies [1,2]. By combining materials with vastly different properties, novel device architectures can be designed to highlight and enhance desirable properties [3]. For typical, covalently bound monolayers, such as graphene, hBN and transition metal dichalcogenides (TMDs), the construction of van der Waals heterostructures has exciting physical and technological ramifications. From transistors [4,5] and superconductivity [6] to single photon emitters [7,8] and sensors [9], these structures are at the forefront of scientific development. Furthermore, more exotic combinations of materials including TMD/organic [10][11][12][13] and TMD/perovskite [14,15] heterostructures, are areas of significant interest, not just in the 2D materials community but also as a way to enhance the properties of the partner material [16]. Organic semiconductors (OSCs) have emerged as a leading candidate for a host of technological applications [17,18]. Owing to their low-cost, scalable and environmentally-friendly fabrication as well as their excellent light-harvesting properties, they are promising materials for photovoltaics [19] and sensor applications [20]. The main drawback of these materials is their low efficiency, driven, in part, by their poor mobility [21]. In a light-harvesting device, such as a solar cell, generated excitons (bound electron-hole pairs) can either recombine or separate into free charges at donor/acceptor interfaces or at the electrodes [18]. Charge separation is maximised when the mobility of the generated excitons is large allowing the charge carriers to be efficiently collected. One interesting approach to achieve this is to combine TMDs with OSCs [22]. The atomically flat interface of the TMDs acts as a good template on which to grow these crystals, and eliminates the effect of charge inhomogeneity [11]. Furthermore, depending on the band * [email protected] alignment of the chosen TMD and OSC, efficient charge and/or energy transfer can occur between the constituent layers. In a type-II heterostructure, charge transfer facilitates the formation of long-lived interlayer excitons [12,13,[23][24][25]. Occurring in both type-I and -II heterostructures, energy transfer could allow excitons generated in the high absorption efficiency OSC to be transferred into the high mobility TMD [26][27][28]. Both these schemes represent ways to use TMDs in order to enhance the light-harvesting efficiency of an OSC. In addition to light-harvesting, TMDs have been used as detectors of organic molecules. While this can be achieved by measuring a change in the electrical signal [29,30], significant performance has been achieved using optical methods [31]. Organic molecules have been shown to modify the photoluminescence (PL) spectrum of a TMD [28,32]. An exciting proposal is to use a layer of molecular aptasensors on the TMD as a selective probe of biological and infectious compounds [33,34], such as those involved in malaria [35], cancer [36] and liver function [37]. In the absence of these compounds, energy transfer between the molecular layer and the neighbouring TMD quenches the signal of the aptasensor. At high enough concentration, the biological compound binds to the aptasensor triggering it to detach from the TMD surface and leading to a notable fluorescence/PL signal. These devices represent selective probes of target biological molecules, which could be used as ultrathin, cheap, biological sensors, not only in the laboratory but also in the home and clinical settings. The fundamental mechanism describing the exciton energy transfer in these photovoltaic and biosensor systems is the Förster interaction [26,28], with the energy transfer commonly described as Förster-induced resonant energy transfer (FRET) [33]. Previous studies on two-dimensional systems found that the Förster interaction dominates over Dexter-mediated charge transfer processes, in spite of the sub nm separation d. This can be attributed to the deviation from the d −6 scaling law observed in localised molecular acceptor-donor systems, instead decaying more slowly at e −d for small distances and then d −4 at larger distances. This was observed in graphene-TMD systems [38], TMDsingle molecule [39] and molecule-graphene [40,41]. Thus, in this study we focus on the Förster-mediated energy transfer. We develop a fully microscopic model describing the Förster interaction between an exemplary OSC/TMD heterostructure comprised of WSe 2 and tetracene. Crucially, we describe the effect of the Förster interaction on optical spectra. We show pronounced Förster-induced signatures in differential absorption and time-resolved PL spectra. We predict a strongly unidirectional energy transfer from the organic to the TMD layer. Furthermore, we find that the transfer rate drastically increases with temperature due to the larger population of hot excitons. Our results provide a recipe with which to enhance the Förster interaction in a real heterostructure, through tuning the temperature, the interlayer distance, and the molecular orientation. By boosting the FRET in these heterostructures, radiative losses e.g. in solar cells could be diminished, while the sensitivity of molecular aptasensors at different concentrations or orientations could be controlled. I. RESULTS Tuning of the Förster transfer rate: The Förster interaction has already been intensively studied on a microscopic footing in quantum dots [42], TMD-graphene [38] and molecule-graphene [40] systems, whereas the effect of the Förster energy transfer has not been as well understood in the technologically promising TMD/OSC heterostructures. A real space illustration of the Förster interaction is shown in Fig 1(a) for a tetracene/WSe 2 heterostructure. The tetracene (Tc) in its bulk polymorph can be grown on top of a WSe 2 monolayer [12,28,43]. Using the semiconductor Bloch equations and the Markov approximation [44,45], we can derive an expression for the exciton transfer rate due to the Förster interaction [38] γ F = 2 1 Z ηQ Γ η Q exp(−E Tc Q,η /k B T ).(1) Here, Z is the Boltzmann partition function, such that Eq. 1 is a thermal average of the Förster dephasing rate Γ η Q = π µ |V µη (Q, z)| 2 δ(E Tc Q,η − E WSe2 Q,µ ) with V µη (Q, z) = A ηµ (Q, z)ϕ WSe2 * µ (r = 0)ϕ Tc ,η (r = 0) (c.f Methods for more details). This quantity describes exciton dephasing (and exciton linewidth broadening for Q = 0) due to the interaction of an exciton in state η in the Tc layer with momentum Q coupled to all states of the same momentum in state µ in the WSe 2 layer. The delta function ensures energy conservation, and coupled with the momentum and distance dependence contained in V µη (Q, z) fully determines the exciton dephasing. The reverse process (WSe 2 to Tc) can be obtained by taking the thermal average over the TMD states. The relatively flatter bands in the Tc crystal lead to a broader momentum occupation and hence higher population of Tc excitons at larger Q. As a result, the Förster-mediated exciton transfer is much larger from the Tc to the WSe 2 layer than the other way round. We therefore predict a strong net transfer of excitons from the organic layer to the TMD. In Fig. 2 (a), the exciton dephasing of the lowest Davydov state is shown as a function of the center-of-mass momentum magnitude Q and angle. The anisotropy of the Tc band structure gives rise to the characteristic shape of the intense region in (a), while the intensity varies in part due to the dipole orientation. The optical dipole of the lower Davydov state (LDS) is y-polarised (90 • ), aligned along the a-axis of the tetracene [46], and hence the strongest dephasing is observed in this orientation. This dipole dependence also explains the vanishing intensity at orthogonal angles (0 • and 180 • ). The two distinct bands appearing in Fig. 2 (a) originate from the coupling between the LDS in Tc and the A and B excitons in the TMD layer. The stronger emission stems from the cou- Förster-induced transfer rate from the Tc to the WSe2 layer as a function of temperature for a direct Tc/WSe2 interface and with an hBN spacer layer, respectively. The inset shows the temperature-dependent scattering rate both from the Tc to the WSe2 layer (red) and from the WSe2 to the Tc layer (green). pling with the B exciton, which occurs at lower momenta as energy conservation occurs at lower Q for this pair of bands. This can be seen in the inset of Fig 2(a), where the intersection of the excitonic dispersion (representing momentum and energy conservation) determines the shape of the features in Fig 2(a). In other many-particle processes, such as exciton-phonon scattering, the dephasing rate Γ leads to a broadening of the exciton linewidth in PL and absorption spectra. In Förster processes, however, this dephasing vanishes at Q = 0 corresponding to the light-cone. Previous experimental studies have estimated the Förster transfer rate, demonstrating a net transfer of excitons from a Tc layer into a WSe 2 layer [28] with a transfer time of approximately 3 ps corresponding to a transfer rate of ∼ 0.29 ps −1 . This occurs on a timescale much slower than typical exciton phonon-relaxation times in the material [47], but also quicker than singlet fission and charge transfer [28]. Evaluating Eq. (1), we calculate the temperature dependence of the exciton transfer rate γ F from the Tc to the WSe 2 layer, cf. Fig. 2 (b). We distinguish two cases: Tc directly on WSe 2 (light blue) and including an hBN spacer layer (dark blue). As the temperature increases, the population of higher momentum excitons increases. Since it is precisely these excitons that participate in the Förster transfer process (note the linear Q dependence in Eq. (3)), the scattering rate also increases with temperature. Adding additional hBN or organic spacer layers allows to tune the interlayer distance z. We recover a z −4 dependence at large distances, characteristic of 2D systems [38,40], while at shorter distances a more rapid exponential decay is found. As a result, we find a lower Förster rate (∼ 5× lower) in presence of a hBN spacer layer, cf. Fig 2. (b). This finding suggests that even in a multilayer Tc system, the main Förster interaction occurs between the WSe 2 and the Tc layer at the interface. This and the fact that Tc excitons are strongly confined to individual Herringbone Tc layers [46] suggests that our results can be extended to multi-layer Tc on WSe 2 . Förster visualized in differential absorption: The most convenient way to track the exciton transfer due to the Förster interaction is by employing optical spectroscopy techniques. The spectroscopy of choice should allow the population in one or both layers to be tracked. In differential absorption we can directly visualize the population change in either layer [48][49][50]. The exciton absorption can be described using the wellestablished Elliot formula [44,45] I ω (t) = µ (1 − B µ (t))|M µ | 2 (E WSe2 0,µ − ω) 2 + (γ µ + Γ phon µ ) 2(2) where γ µ and Γ phon µ are the radiative and non-radiative (phonon) decay rates, respectively. The excitonphoton light-matter interaction M µ , is reduced by the bleaching term B µ = K,µ ,q N µ K (|ϕ µ q+β µ K | 2 + |ϕ µ q−α µ K | 2 )|ϕ µ (r = 0)| −1 ϕ µ q where α µ = m µ c /M and β µ = m µ v /M . This term crucially depends on the timedependent excitonic population N µ K . We assume the optical excitation to be weak, such that the Coulomb renormalization of the excitonic resonance [48][49][50] is negligible and bleaching dominates. At much larger excitation fluences, clear resonance shifts should be observed due to population-induced screening and band renormalisation, while the bleaching term can become larger than 1, leading to optical gain. We derive a set of coupled semiconductor Bloch equations for the exciton polarization and incoherent exciton density, to describe this process (see SI for more details). We take explicitly into account in-and out-scattering between the incoherent excitons in the two layers due to both Förster and exciton-phonon scattering. The excitonic properties, calculated using the Wannier equation, in addition to the phonon properties, allow us to determine not only the scattering dynamics but also the exciton broadening appearing in Eq. (2). The differential absorption ∆I ω (t) = I ω (t) − I ω (0) is defined as the difference in the absorption at time t after excitation, and before the excitation t = 0. It is shown for a WSe 2 /Tc heterostructure in Fig 3(a). At time t = 0, the differential absorption is 0 by definition. However, following an excitation pulse resonantly exciting the Tc layer at t = 0, the generated excitons transfer via the Förster coupling to the WSe 2 layer. The phonon-driven intraband relaxation processes are, however, much faster than the Förster interaction [38,47], such that excitons in each layer remain thermalised. Using thermal averages, we find a simpler model for the exciton dynamics in the TMD (l = 0) and Tc (l = 1) layer ∂ t N l = −γ l loss N l − γ l F N l + γ 1−l F N 1−l(3) such that in the bleaching term described above we define N µl Q as N µl Q = N l Z exp(−E l Q,µ /k B T ). The first term in Eq. (3) describes the radiative and non-radiative loss [45] scaling with the decay rate γ loss . The second and third term describe the Förster-induced out-and in-scattering, respectively. These equations are accurate when the Förster transfer occurs on a longer timescale than the phonon-scattering rate. In Fig. 3(a), we find a negative differential absorption of WSe 2 for times t > 0, which is a result of the exciton transfer from the Tc layer. Time cuts show a characteristic dip, cf. Fig 3(b). The differential absorption decreases between t = 0 ps to around t = 2 ps owing to the build up of excitons on the TMD layer. After around 2 ps, a steady state is reached between the exciton population in the WSe 2 and the one in the Tc layer, such that the decrease in the exciton population in both layers is driven primarily by the radiative and non-radiative decay. The temporal evolution of the exciton populations in the TMD and molecular layer are shown in Fig 3 (c) and (d), respectively. We find that the population in the TMD layer initially increases before it drops, while the population of the Tc layer monotonously decreases. For the population in the TMD we find an interplay of Förster induced charger transfer from the TC layer (population increases) and the radiative exciton recombination (population decreases). This results in a maximum population around 2 ps, cf. Fig. 3(c). In the absence of radiative recombination, we find a complete exciton transfer from the Tc to the WSe 2 layer (dashed lines). The pronounced dip in the differential absorption is a clear indication of the Förster-induced energy transfer. The specific distance dependence of the Förster interaction (z −4 as explicitly shown in the SI) compared to other processes, such as tunneling or Dexter transfer [40], allows us to identify the microscopic origin of the differential absorption signatures. Förster visualized in time-resolved PL: An additional probe of the Förster-induced exciton dynamics can be achieved by using time-resolved photoluminescence [13,45]. As in the previous case, the LDS exciton in the Tc layer is resonantly excited leading to a thermalised exciton distribution on a short time-scale (< 1 ps) [47]. The dynamics of the relative populations follow the results outlined in Fig 3 (c) and (d). The time-resolved PL is proportional to the exciton population within the light cone, N µ l l 0 , and is described using the Elliot formula for PL [45] I PL ω (t) = µ l ,l N µ l l 0 |M l µ l | 2 (E l 0,µ l − ω) 2 + (γ µ l + Γ phon µ l ) 2(4) summing the contributions from both layers l. This equation differs from the aborption as it direcly depends on the exciton population N µ l l . The PL spectra are shown in Fig. 4(a) as a surface plot displaying the time and energy dependence. At time t = 0, the exciton population is primarily in the Tc layer, so a clear signature of the lower Davydov state is observed. After around 1 ps, the Förster energy transfer occurs, depleting the signature from the LDS and leading to a pronounced increase in the PL of the A exciton in the WSe 2 layer. Time snapshots are shown in Fig. 4(b). We estimate an energy transfer time of around 1-2 ps, which can be observed in the PL emission originating from the WSe 2 reaching a maximum at around this time. We observe again how the PL signal begins to decrease after around 2 ps due to the exciton loss due to radiative and non-radiative decay. Differences in the decay rates will not change the qualtitative predictions and will only alter the time at which the strongest signal is observed, both in the PL and the differential absorption. II. DISCUSSION Depending on the structural geometry, namely the separation between the organic and TMD layers, different spectroscopic methods may be better suited to observe the Förster-mediated energy transfer. When the interlayer separation is small, the rate is larger and higher-resolution methods, such as differential absorption, will be more appropriate. Similarly, in the case one or more spacer layers or lower temperatures, time-resolved PL may be better suited as higher timeresolution is less important. The Förster coupling could be technologically exploited. The observed strong quenching of the PL signal from the molecular crystal is the crucial mechanism behind many optical biosensors [33,35]. Here, in the absence of the target molecule, the signature from the aptasensor adsorbed on the TMD is efficiently quenched. When a target molecule is present, the sensing molecules forming the molecular crystal, detach from the surface. This increases the molecular distance from the TMD, causing the Förster coupling to quickly decrease, such that the molecular PL or fluorescence signal is no longer quenched. The coverage of molecules and the energetic/spatial separation between the molecular and TMD excitons allow the sensitivity of a biosensor to be tuned. Closer energetic alignment between exciton resonances will lead to even stronger exciton quenching. In our case, we used a well-known, exemplary organic crystal, however, our model can be easily extended to other organic molecules. Furthermore, an organic crystal is not strictly necessary, and similar results should be expected with a more sparse covering of molecules on the TMD. The competition with temperature should also allow for a thermally-activated biosensor. Anisotropic materials, such as the Rhenium TMDs [51], could also allow the orientation of the organic molecules relative to the TMD to be detected, due to the requirement that the TMD and organic dipoles should be aligned [27]. In the same vein, the "handedness" of chiral molecules could also be distinguished in the optical spectra [52], provided the Förster interaction ocurrs on a shorter timescale than the intervalley relaxation in the TMD. In this work, we presented microscopic insights into the Förster interaction between TMDs and organic crystals. Focusing on the exemplary WSe 2 /Tc heterostructure, we show a characteristic temperature and distance dependence of the Förster-induced dephasing and exciton transfer. We estimate the energy transfer time and find a good agreement with previous experimental studies. The main message of this work lies in the question of how Förster processes manifest in the optical spectra of an organic/TMD heterostructure. We demonstrate that the time-resolved differential absorption and photoluminescence can capture well the Förster-driven exciton dynamics, by tracking the change in the exciton population in the constituent layers. Our work represents a significant advance in the microscopic understanding of organic/TMD interfaces and energy-transfer processes therein, important for fundamental physics but also for the design of new optoelectronic devices, such as biosensors. III. METHODS Microscopic Model for Förster Transfer: Following optical excitation, the generated electron-hole pairs interact via the screened Coulomb interaction forming excitons [53]. We use the Wannier equation [44] to calculate the exciton energy dispersion E . The Wannier equation has been effective in determining the excitonic binding energy in previous calculations on tetracene and pentacene, with a good agreement to experiments [47]. This is further justified by the fact that the excitonic wavefunction in oligoacenes, such as tetracene and pentacene, is dispersed over multiple neighbouring molecules and hence resembles a localised Wannier exciton [46,47,54,55]. In tetracene, the symmetrically distinct molecules in the unit cell gives rise to a Davydov splitting of the excitonic state [46]. After photoexcitation, the exciton can then recombine, thermalise via scattering with phonons [47], undergo the so-called singlet fission forming two long-lived triplet excitons [18,56] and in the heterostructure case, take part in charge and energy transfer processes between the Tc and WSe 2 layers. Charge transfer, mediated by tunnelling/hopping processes, [25] gives rise to the formation of interlayer excitons. However, this requires a significant wavefunction overlap, which is reduced in the considered herringbone structure, due to the relatively large interlayer separation of 0.9 nm [28]. Furthermore, in the considered heterostructure Tc/WSe 2 , the band alignment is such that charge transfer processes occur on a much slower time scale [28] and hence the optically darker interlayer excitons are not expected to significantly contribute to optics nor the dynamics [13]. On the other hand, the Förster interaction depends less strongly on the interlayer separation [38,40], involving energy transfer between intralayer excitons in the constituent layers (cf. Fig 1(a)). In momentum space, the Förster interaction involves the recombination (annihilation) of an exciton (electron and hole pair) in one layer and the creation of an exciton in other layer, mediated by the Coulomb interaction [38]. The Hamiltonian describing this process in the Tc/WSe 2 heterostructure can be written as, H F = λ,λ ,ν,ν k,k ,q,q V k,k ,q,q λ,λ ,ν,ν â † λ,kB † ν,qBν ,q â λ ,k + h.c,(5) whereâ ( †) andB ( †) are annihilation (creation) operators acting on the Tc and WSe 2 layer, respectively. The indices λ and ν describe the hybrid spin/valley/band index. The Coulomb matrix element V k,k ,q,q λ,λ ,ν,ν is screened by the surrounding dielectric, , of a SiO 2 substrate [28]. Using a real-space coordinate transformation the Coulomb matrix element is V k,k ,q,q λ,λ ,ν,ν = sQ e −iQ(s+z) δ Q,k−k δ Q,q −q I s,z k,q,Q , where I s,z k,q,Q takes the form of a dipole-dipole interaction (cf. the SI), separated into in-plane, s, and out-of-plane, z, components. Since the dominant Förster transfer processes occur between the lowest lying energy levels we consider only the highest valence/lowest conduction band of the WSe 2 and the HOMO/LUMO of the Tc. We only consider spin-like states, as the Coulomb matrix elements vanish for excitons comprised of opposite spins [57]. The same applies to momentum dark intervalley excitons in both layers [53]. In the tetracene layer, the upper and lower Davydov excitons originate from the intermolecular coupling between the symmetrically distinct molecules in the unit cell. The resulting lower and upper Davydov excitons form around the M and Γ point respectively [46]. Taking this into account, we arrive at the Förster Hamil-ton operator in the excitonic basis (see the SI for a detailed derivation) H F = η,µ,Q A ηµ (Q, z)ϕ WSe2 * µ ϕ Tc ηX WSe2 † QµX Tc Qη + h.c. (6) whereX l( †) Qµ are exciton annihilation (creation) operators with momentum Q, and energy level µ, on layer l. Here, we can express the excitonic wavefunction in real-space coordinates as √ Aϕ µ/η (r = 0) = k ϕ k,µ/η , were the hybrid indices λ, ν have been absorbed in the excitonic indices µ and η. The appearing matrix element A ηµ (Q, z) can be expressed analytically (cf. the SI). A ηµ (Q, z) = e −iQzz e −|z|Q 4 0 d WSe2 · Q Q |Q| + iẑ · d Tc .(7) Here, we have introduced the optical transition dipoles |d WSe2 | = 0.4 nm [38] and |d Tc | = 0.05 [58] and are assumed to be approximately constant around the WSe 2 and Tc valleys, respectively [38]. The out-of-plane unit vectorẑ and out-of plane momentum Q z also appear in the the dipole-dipole interaction, and become more relevant in structures with organic dipoles pointing perpendicularly to the layer plane. Encoded in this Hamiltonian is the conservation of the center-of-mass momentum between the excitons in the WSe 2 and Tc. This is shown schematically in Fig 1(b). Importantly, the matrix element A(Q, z) describes the dependence on the exciton momentum and interlayer separation z. We find that the Förster interaction vanishes as Q → 0, which is typical for dipole-dipole-type coupling. The transition dipole of the TMD is circularly polarised, with handedness depending on the valley, which give rise to the well-known optical valley selection rules in TMDs [59]. In contrast, the lowest Davydov excitons in Tc are linearly polarised [58,60]. As a result the Förster matrix element becomes anisotropic in Q, favouring the transition dipole of the initial or final Davydov exciton. Unless otherwise stated, we take a typical interlayer separation of 0.9 nm, defined as the distance between the W-atom layer in the middle of the WSe 2 , and the centre of Tc layer (c.f . Fig 1 (a)). Acknowledgements This project has received funding from Deutsche Forschungsgemeinschaft via CRC 1083 and the European Unions Horizon 2020 research and innovation programme under grant agreement no. 881603 (Graphene Flagship). Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request. FIG. 1 . 1(a) Illustration of exciton energy transfer between the tetracene (blue) and the TMD layer. (b) Single particle electronic band structure of tetracene (left) and WSe2 (right) around the band gap. The Förster interaction, conserving energy and centre-of-mass momentum, is shown by the pink arrows. FIG. 2 . 2(a) Förster-induced exciton dephasing Γ Q in the lower Davydov state of the Tc crystal as a function of momentum direction (angle) and magnitude (Q). Inset shows schematic of excitonic band structures from the A (dark blue) and B (light blue) exciton in the TMD and the lower Davydov state (red) in the Tc layer. (b) FIG. 3 . 3(a) Time-resolved differential absorption spectrum of the WSe2 layer at 300 K. The lowest 1s exciton resonance is shown by the vertical line. (b) Time cuts of the differential absorption. Temporal evolution of the exciton density in the (c) TMD and the (d) Tc layer, taking into account the full dynamics (solid) and without the radiative loss (dashed). FIG. 4 . 4(a) Time-and energy-resolved photoluminescence spectrum of the WSe2 and Tc layer at 300 K. The 1s exciton resonance in the TMD and the lower-Davydov state in the Tc layer are marked by vertical lines, respectively. (b) Snapshots of the PL at constant times, with an arrow marking the trend of exciton transfer due to the Förster interaction. Competing Interests The authors declare no competing interests. Author Contributions E.M conceived the project. J.J.P.T developed the theory and performed the calculations. E.M, J.J.P.T, M.G and G.W analyzed the obtained results. J.J.P.T wrote the paper with all the au- Van der Waals heterostructures. A K Geim, I V Grigorieva, Nature. 499Geim, A. K.; Grigorieva, I. V. Van der Waals heterostruc- tures. Nature 2013, 499, 419-425. Van der Waals heterostructures and devices. Y Liu, N O Weiss, X Duan, H.-C Cheng, Y Huang, X Duan, Nature Reviews Materials. 1Liu, Y.; Weiss, N. O.; Duan, X.; Cheng, H.-C.; Huang, Y.; Duan, X. 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[]
[ "Multiscale mobility patterns and the restriction of human movement", "Multiscale mobility patterns and the restriction of human movement" ]
[ "Dominik J Schindler \nDepartment of Mathematics\nImperial College London\nLondonUK\n", "Jonathan Clarke \nDepartment of Mathematics\nImperial College London\nLondonUK\n", "Mauricio Barahona \nDepartment of Mathematics\nImperial College London\nLondonUK\n" ]
[ "Department of Mathematics\nImperial College London\nLondonUK", "Department of Mathematics\nImperial College London\nLondonUK", "Department of Mathematics\nImperial College London\nLondonUK" ]
[]
From the perspective of human mobility, the COVID-19 pandemic constituted a natural experiment of enormous reach in space and time. Here, we analyse the inherent multiple scales of human mobility using Facebook Movement Maps collected before and during the first UK lockdown. First, we obtain the pre-lockdown UK mobility graph, and employ multiscale community detection to extract, in an unsupervised manner, a set of robust partitions into flow communities at different levels of coarseness. The partitions so obtained capture intrinsic mobility scales with better coverage than NUTS regions, which suffer from mismatches between human mobility and administrative divisions. We then examine the evolution of mobility under lockdown, and show that mobility first reverted towards fine scale flow communities already found in the pre-lockdown data, and then expanded back towards coarser flow communities as restrictions were lifted. The improved coverage induced by lockdown is well captured by a linear decay shock model, which allows us to quantify regional differences both in the strength of the effect and the recovery time from the lockdown shock.
null
[ "https://export.arxiv.org/pdf/2201.06323v3.pdf" ]
256,105,790
2201.06323
26f6077038937f61657bb677004909894a3961fe
Multiscale mobility patterns and the restriction of human movement Dominik J Schindler Department of Mathematics Imperial College London LondonUK Jonathan Clarke Department of Mathematics Imperial College London LondonUK Mauricio Barahona Department of Mathematics Imperial College London LondonUK Multiscale mobility patterns and the restriction of human movement network analysiscomputational social sciencescales of human mobilityCOVID-19 lock- downmultiscale community detection From the perspective of human mobility, the COVID-19 pandemic constituted a natural experiment of enormous reach in space and time. Here, we analyse the inherent multiple scales of human mobility using Facebook Movement Maps collected before and during the first UK lockdown. First, we obtain the pre-lockdown UK mobility graph, and employ multiscale community detection to extract, in an unsupervised manner, a set of robust partitions into flow communities at different levels of coarseness. The partitions so obtained capture intrinsic mobility scales with better coverage than NUTS regions, which suffer from mismatches between human mobility and administrative divisions. We then examine the evolution of mobility under lockdown, and show that mobility first reverted towards fine scale flow communities already found in the pre-lockdown data, and then expanded back towards coarser flow communities as restrictions were lifted. The improved coverage induced by lockdown is well captured by a linear decay shock model, which allows us to quantify regional differences both in the strength of the effect and the recovery time from the lockdown shock. Introduction Spatiotemporal patterns of population mobility reveal important aspects of human geography, such as social and economic activity [1], the evolution of cities and economic areas [2], the response to natural disasters [3], or the spread of human infectious diseases [4]. Whilst mobility patterns are linked to, and influenced by, both geographical and administrative boundaries [5], they are also a direct reflection of social behaviour, and thus provide additional insights into the natural evolution of socio-economic interactions at the population level. The increasing access to detailed and continuously updated mobility data sets from various sources (e.g., mobile devices [5], GPS location traces [6], Twitter data [3]) opens up the opportunity to develop further quantitative approaches to harness the richness of such data [7][8][9]. Access to mobility data has recently become more widespread due to the COVID-19 pandemic, which prompted governments across the world to impose a range of restrictions on the daily activities and movements of their citizens [10]. Such mobility data was of immediate use to refine and assess interventions targeting the spread of COVID-19 [11][12][13][14][15][16], and to evaluate the unequal effects of the pandemic across populations [12]. Yet, from the perspective of mobility, the pandemic also constituted a natural experiment of enormous reach in space and time that accelerated both the sharing of such data sets and the study of a severe mobility shock, in which human activities were curtailed to reduced areas for a sustained period [17][18][19][20][21][22]. An important aspect of mobility is the presence of inherent spatial and temporal scales as a result of, e.g., administrative divisions, patterns of social interactions, jobs and occupations, as well as diverse means of transportation [1]. Recent work [6,23] has shown that this multiscale, nested structure of human activities contributes to the scale-free behaviour that had been previously found empirically [7,8,24]. Here, we apply data-driven, unsupervised network methods to study the multiscale structure of UK mobility in data collected before and during the first COVID-19 lockdown. Data from user-enabled, anonymised 'Facebook Movement maps' between UK locations [25] is used to construct directed, weighted mobility graphs which are then analysed using unsupervised multiscale community detection [26][27][28][29] to extract inherent flow communities at different levels of coarseness. Hence the inherent mobility scales emerge directly as robust flow communities in the data, obtained here through a scale selection algorithm. Our results show that multiscale flow communities extracted from pre-lockdown data broadly agree with the hierarchy of NUTS administrative regions (where NUTS stands for Nomenclature of Territorial Units for Statistics), yet with distinctive features that result from commuting patterns cutting across administrative divisions. We then quantify the extent to which mobility patterns under lockdown conform to the flow communities found in pre-lockdown data using data collected during the first UK COVID-19 lockdown (March-June 2020). We find that the imposition of lockdown reverted mobility towards the local, fine scale pre-lockdown flow communities and, as restrictions were lifted, mobility patterns expanded back towards the coarser pre-lockdown flow communities, thus providing empirical evidence for the presence of a quasi-hierarchical intrinsic organisation of human mobility at different scales [6]. Finally, we find regional differences in the response to the lockdown, both in the strength of the mobility contraction and the time scale of recovery towards pre-lockdown mobility levels. Results We use mobility data provided by Facebook under the 'Data for Good' programme [25] to construct directed, weighted networks of human mobility in the UK. The anonymised data sets ('Facebook Movement maps' [25,30]), which are collected from user-enabled location tracking, quantify frequency of movement of individuals between locations over time, thus capturing temporal changes in population mobility before and during the COVID-19 pandemic [19,20]. For details of the network construction see Methods. The directed graph of baseline UK mobility: quasi-reversibility and commuting travel patterns Using pre-lockdown mobility data (average of 45 days before 10 March 2020), we construct a strongly connected directed graph G with weighted adjacency matrix A ‰ A T (Fig. 1A). The N " 3, 125 nodes of this graph correspond to geographic tiles (width between 4.8-6.1 km, see Supplementary Fig. S5A) and the directed edges have weights A ij corresponding to the average daily number of inter-tile trips from tile i to j (see Methods for definition of 'trip' within the Facebook data set). The total average number of daily inter-tile trips is 2, 475, 527, as compared to 10, 416, 968 intra-tile trips. The matrix A is very sparse, with 99.7% of its entries equal to zero, i.e., there are no direct trips registered between the overwhelming majority of tile pairs. Furthermore, the non-zero edge weights are highly heterogeneous, ranging from 1.4 to 6, 709 daily trips (average 72.3, coefficient of variation 2.8), underscoring the large variability in trip frequency across the UK. To assess the directionality of the baseline network, we first compute the pairwise relative asymmetry (PRA) for each pair of tiles ij: 0 ď PRA ij :" |A ij´Aji | A ij`Aji ď 1,(1) defined for pairs where A ij`Aji ą 0. The distribution of the PRA ij ( Supplementary Fig. S7) shows that 25% of the tile pairs have PRA ě 0.23, a substantial asymmetry, including 3,226 one-way connections (8.64% of the total) with PRA " 1. It is thus helpful to use network analysis tools that can deal with directed graphs [31]. A natural strategy for the analysis of directed graphs is to employ a diffusive process on the graph to reveal important properties of the network, such as node centrality [32,33] or graph substructures [28], while respecting edge directionality. We consider a discrete-time random walk on graph G defined in terms of the NˆN transition probability matrix M : M :" Dò ut A,(2) where Dò ut is the pseudo-inverse of D out " diag pd out q, the diagonal matrix of out-strengths d out " A 1 N . A key property of the random walk is its stationary distribution π, a 1ˆN node vector defined through the equation π " πM.(3) The component π i is a measure of centrality (or importance) of node i; a high value of π i means that the random walk on G is expected to visit node i often at stationarity [34]. This is equivalent to PageRank without teleportation [35]. As expected, the centralities π i are highly correlated (R 2 " 0.97) with another node centrality measure, the out-strengths d out,i . Interestingly, the centralities π i are also correlated with the intra-tile mobility (R 2 " 0.83), a measure that is not used in the computation of π, see Supplementary Fig. S6 for details. Therefore, urban areas display high centrality due to the concentration of human mobility in those areas (Fig. 1A). A random walk on a directed graph might still not display strong directionality at equilibrium. This is our finding here: the random walk defined by M fulfills approximately the detailed balance condition: ΠM » ΠM T with }ΠM´ΠM T } F }ΠM } F " 0.033, where Π " diagpπq and }¨} F denotes the Frobenius norm (see Fig. S7 and [28,36] for a more indepth discussion). The random walk for our mobility graph is therefore close to being time-reversible at equilibrium [34], so that the probability of following a particular trajectory from node i to j is almost equal to the probability of going back on the same trajectory from j to i. This property coincides with our intuition that most journeys in the mobility network are linked to commuting travel patterns. Unsupervised community detection reveals intrinsic multiscale structure in the baseline mobility data To extract the inherent scales in the UK mobility data, we apply multiscale community detection to the baseline network. We use Markov Stability (MS) [26], a methodology that reveals intrinsic, robust graph partitions across all scales through a random walk Laplacian that simulates individual travel on the network based on the observed average daily trips. As random walkers explore the network, they remain contained within small subgraphs (communities) at shorter times, and then spill over onto larger communities at longer times. MS uses an optimisation to identify graph communities in which the flow of random walkers is contained over extended periods, uncovering a sequence of robust graph partitions of increasing coarseness (regulated by the Markov scale s). In the context of our mobility network, this set of partitions captures intrinsic scales of human mobility present in the data. See Refs. [26-29, 37, 38] and Methods for details of the methodology. Figure 1B summarises the MS analysis for our network, which was carried out with the PyGenStability Python package [39]. We find nine robust MS partitions Hps i q at different levels of resolution (s 1 , . . . , s 9 ) from fine to coarse, which comprise flow communities at different scales of human mobility (see Supplementary Table S3 and Supplementary Fig. S8 for further statistics and visualisations). Figure 1C shows that these data-driven flow communities correspond to geographic areas, even though our data only contains relational mobility flows without explicit geographic information. Furthermore, the nine partitions have a strong quasi-hierarchical structure, which is not imposed a priori by our graph partitioning method (see Supplementary Fig. S8 and S9). The obtained partitions thus reflect an inherent multiscale structure in the patterns of UK human mobility. Figure 1: Multiscale structure in the baseline mobility network. A Using mobility data averaged over 45 days before 10 March 2020, we create a weighted directed graph (N " 3, 125 nodes, E " 37, 349 edges) with edge weights equal to the average daily number of trips between geographic tiles (nodes). The stationary distribution π (3) of the associated random walk indicates high centrality of urban areas. B Multiscale community detection on the baseline mobility network using Markov Stability analysis. We find optimised MS partitions that are robust both across scales (blocks of low NVIps, s 1 q), and within scale (dips in NVIpsq). To find robust optimal partitions, we first determine basins in Block NVIpsq (the pooled diagonal of NVIps, s 1 q) and then find the minima of NVIpsq for each basin (see Methods). This selection process leads to nine robust scales (s 1 , . . . , s 9 ), from finer to coarser. C The graph partitions for the six scales with lowest Block NVI are plotted on the UK map, with different colours indicating different communities. Comparing the intrinsic mobility scales at baseline to administrative NUTS regions Next, we compare the MS partitions to NUTS regions, administrative and geographic regions defined at three hierarchical levels: NUTS1 build upon NUTS2 in turn consisting of NUTS3 regions. In the UK, NUTS3 regions represent counties and groups of unitary authorities; NUTS2 regions are groups of counties; and NUTS1 regions correspond to England regions, plus Scotland, Wales and Northern Ireland as whole nations (see Supplementary Table S4 for further statistics). In Fig. 2 we use the Normalised Variation of Information (NVI) (7) to evaluate the similarity of each of the three NUTS levels to the MS partitions at all scales. The best match of each NUTS level (as given by the minimum of NVI) is close to one of the robust partitions: NUTS3 corresponds closely to Hps 1 q; NUTS2 to Hps 2 q; and NUTS1 to Hps 4 q. Hence these three MS partitions of the mobility network capture the fine, medium and coarse scales in the UK, yet with some significant deviations from the administrative NUTS divisions. For instance, Greater London is separated from the rest of the South East at the level of NUTS1 regions, whereas the whole South East of England forms one flow community in partition Hps 4 q. Similarly, the south of Wales is connected strongly via flows to the South West of England in partition Hps 4 q, which is not reflected in the NUTS1 regions. On the other hand, the correspondence between NUTS3 regions and the fine MS partition Hps 1 q is strongest (lower value of NVI), with fewer such discrepancies between administrative and flow communities. To evaluate further how the MS partitions capture the patterns of mobility, we compute two measures: the Coverage C (9) and the Nodal Containment NC (11), where high values of these measures (normalised between 0 and 100%) indicate that mobility flows are captured within the boundaries of the communities A B C D 2 1 0 1 2 3 log10(s) 0.3 0.4 0.5 0.6 0.7 NVI NUTS3 NUTS2 NUTS1 Fine H(s 1 ) & NUTS3 H(s 2 ) & NUTS2 H(s 4 ) & NUTS1 Medium Coarse 1.0 0.5 0.0 0.5 1.0 NC i [H(s 1 )] -NC i [NUTS3] NC i [H(s 2 )] -NC i [NUTS2] NC i [H(s 4 )] -NC i [NUTS1] s1 s2 s4 s3 s5 s6 s7 s8 s9 (Table 1), and the node measure NC i (Eq. 10) shows the regional improvements where administrative divisions do not conform naturally to mobility patterns. Fine scale: Hps 1 q captures boundary areas fragmented under NUTS3 (e.g., Cornwall-Devon boundary, North Wales, among many others). Medium scale: Hps 2 q captures densely connected areas in Central and East London, fragmented under NUTS2, as well as the commuter belt in Birmingham. Coarse scale: Hps 4 q naturally captures Greater London's commuter belt that is excluded from the London NUTS1 region. of the partition. Table 1 shows that MS partitions are substantially better at reflecting baseline mobility than NUTS divisions, since they have higher values for both average C and NC measures at all scales, and especially at the finer scales. We have also evaluated both measures at a local level. Fig. S10A shows that the median of the Coverage of individual communities C k (8) is significantly higher for MS partitions (as compared to NUTS) for the fine and medium scales (p ă 0.0001, Mann-Whitney). Fig. S10B shows that the median of the Nodal Containment of individual nodes NC i (10) is also significantly higher for MS partitions relative to NUTS regions at fine, medium and coarse levels (p ă 0.001, Mann-Whitney). Indeed, the maps in Fig. 2B-D show that NC i pMSq ą NC i pNUTSq in regions where the administrative NUTS boundaries cut through conurbations or closely connected towns or cities. A prominent example is Greater London, where the NUTS2 regions split areas in Central and East London that are tightly linked and thus captured better by the medium MS partition Hps 2 q, and, similarly, the NUTS1 region of Greater London does not include its wider commuter belt that is naturally captured by the coarse MS partition Hps 4 q. Similar commuter belt effects are observed, e.g., on on the medium level for Birmingham, and on the fine level for Plymouth, which has associated flows across the Cornwall-Devon boundary. The contraction of UK mobility under lockdown and its relation to the baseline mobility multiscale network The first nationwide COVID lockdown in the UK was imposed on 24 March 2020, instructing the British public to stay at home except for limited purposes. Over the following months, restrictions were gradually eased to allow pupils to return to school (1 June 2020 in England but 17 August 2020 in Scotland), businesses to reopen (non-essential shops reopened on 13 June 2020 in England but 13 July in Scotland), and people to travel more freely for leisure purposes (13 May 2020 in England but 8 July 2020 in Scotland) [40]. We have analysed the response to these restrictions using the time-dependent Facebook Movement maps [25] from 10 March 2020-18 July 2020 (131 days or 18 weeks). We construct mobility networks for each day, Gpdq, and week, Gpwq, defined on the same nodes (i.e., tiles) as the baseline network G (see Methods). Fig. 3A shows the temporal change of the number of trips (intra-tile, inter-tile and total) relative to 10 March 2020, the first day of our study period. It is interesting to note that the decrease in mobility was already taking hold rapidly from 10 March, two weeks before the official enforcement of the lockdown. We find that, whilst the total number of trips remained largely unchanged throughout the period, the number of inter-tile trips decreased sharply to "25% of the initial value, followed by a steady increase towards levels of around 50% at the end of the study period in mid-July 2020. Conversely, the number of intra-tile trips increased to a maximum of 130% after lockdown before decreasing steadily to around 105% by mid-July 2020. Therefore lockdown induced a redistribution from inter-tile to intra-tile trips as a result of a reduction in commuting and long-distance travel, with mobility reverting to local neighbourhoods. The observed contraction of human mobility towards local neighbourhoods is consistent with the multiscale structure that was already present in the baseline mobility network pre-lockdown. The Coverage C of all MS partitions increased over lockdown, with larger relative improvement for the finer scales ( Fig. 3B and Supplementary Fig. S11). The surge in Coverage induced by lockdown, which then decays towards its pre-lockdown value, can be modelled with a simple linear model under an external stimulus αe´λ t . The relative change from the initial value is then given by [41]: ∆Cptq :" Cptq´Cpt 0 q Cpt 0 q " α β´λ´e´λ t´e´βt¯,(4) from which we estimate the amplitude (α) and characteristic time (1{λ) of the external stimulus, as well as the characteristic recovery time (1{β) of the system towards its pre-stimulus value (see Methods for details). Fig. 3B shows the fits of ∆Cptq (dashed lines) with estimated parameters in Table 2. The fine MS partition exhibits the largest relative increase ∆Cptq peaking at around 6%. The medium and coarse partitions peak at around 1% and 0.1%, respectively. This is also captured by the values of α, and indicates that during lockdown people reverted to local mobility neighbourhoods already present in pre-lockdown patterns. The adaptation to the new COVID situation and the pre-announcement of lockdown occurs quickly (over a characteristic time of 1{λ " 2 weeks), signifying that adoption was fast and was already in progress before the official start date of lockdown. Mobility patterns then returned towards pre-lockdown values over longer time scales 1{β between 16.4 weeks (fine scale) and 20.9 weeks (coarse scale) reflecting a slow re-adaptation following the new situation and loosening of restrictions. To highlight the local differences in the temporal response to the lockdown, Fig. 4 show the parameters of the temporal fits for the community coverages ∆C k for all the communities in the fine scale MS partition. We observe that urban centers like London, Birmingham, Liverpool or Manchester experience the strongest changes in the fine scale Coverage (high values of α) yet with faster recovery times (low values of 1{β). Conversely, rural areas, which were already more constrained to local communities prelockdown, exhibit smaller but long-lived effects in the Coverage at the local level. Our method also captures divergent trends across the different nations of the UK. For example, Scottish regions show longer time scales of recovery than most regions in England, consistent with the fact that Scotland maintained more stringent lockdown restrictions for a longer time [40], e.g., domestic travel restrictions were eliminated in Scotland only on 8 July 2020 and schools reopened on 17 August 2020, in contrast to 13 May and 1 June 2020 in England, respectively. α (95% CI) 1{β (95% CI) 1{λ (95% CI) Discussion Taking advantage of recent data availability, we have studied here the intrinsic multiscale structure of human mobility using, as a motivating example, UK data collected before and during the first COVID-19 lockdown. Firstly, we generated a directed mobility graph from geospatial Facebook Movement maps collected before lockdown, and exploited multiscale graph clustering (MS) to identify inherent flow communities at different levels of resolution (or scales) in the baseline data in an unsupervised manner. Three of the MS partitions so identified are of similar granularity to the NUTS hierarchy, yet with improved mobility Coverage and Nodal Containment, also revealing areas of mismatch between human mobility and administrative divisions. We then analysed spatio-temporal mobility data collected during the first UK COVID-19 lockdown. We found increased mobility Coverage for MS partitions, especially at the fine scale, suggesting that the mobility contraction during this natural experiment reverted to scales already present in the pre-lockdown data. Indeed, given that our MS communities are found through a random walk on a graph weighted by pre-lockdown trip frequency of natural mobility, the fine scales capture frequent trips that were not suppressed during lockdown, whereas the coarse scales are associated with less frequent trips over longer geographic distances for leisure or business. The enhancement of Coverage induced by lockdown is well captured by a linear decay model, whose parameters allow us to quantify regional differences, including differing trends across urban and rural areas and across the UK nations consistent with distinct lockdown regulations. Our study has several limitations. Whilst the 'Facebook Movement map' data is aggregated from 16 million UK Facebook users who enable location sharing (over 20% of the total UK population), the observed sample might be biased and not representative of the general UK population [30]. Furthermore, inter-tile flows with fewer than 10 trips within an 8-hour period are suppressed to prevent individual identification. However, such low frequency connections account for a comparatively small number of the total trips, and are not expected to affect the obtained partitions. This study also assumes rates of utilisation and activation of location sharing within the Facebook app remain constant, yet this limitation is mitigated by the derivation of flow partitions from average baseline data, rather than post-pandemic mobility information. Our work contributes to the current interest in the study of intrinsic scales in human mobility [23]. A recent study identified 'spatial containers' from granular GPS traces [6] organised in a nested hierarchy specific to each individual. Similarly, we also reveal a multiscale organisation of human mobility, but instead obtain a data-driven, unsupervised quasi-hierarchical community structure at the population level. Because our MS community detection is based on a diffusion on a mobility graph, the flow communities at different scales provide insights into the importance of physical and political geographies, and reveal the scales at which lockdown introduced frictions by restricting natural mobility. Methods Mobility data and network construction Data: Facebook Movement maps [25,30] provide movement data between geographic tiles as codified by the Bing Maps Tile System [42]. For the UK, there are 5,436 geographic tiles with widths between 4.8-6.1 km (see Supplementary Fig. S5A). For users that enable location sharing, Facebook computes the dominant tiles, in which the user spends the most time over adjacent 8-hour time windows, The 'trips' correspond to movements between dominant tiles across adjacent time windows. The dataset then provides the number of trips within each tile and to any other tile at intervals of 8 hours for all users. The data is anonymised by Facebook prior to release using proprietary aggregation methods, including addition of small amounts of random noise, spatial smoothing, and dropping counts of less than 10 trips within an eight hour period to avoid identifiability. Our data further aggregates the three 8-hourly datasets for a given day. Network construction: Given a directed graph G, a weakly connected component (WCC) is a subgraph where each pair of nodes in the WCC is connected by an undirected path. Similarly, in a strongly connected component (SCC) each pair of nodes is connected by a directed path [43]. The largest weakly connected component is denoted as LWCC and the largest strongly connected component as LSCC. As baseline, we use pre-lockdown data consisting of mobility flows averaged over the 45 days before 10 March 2020. To obtain the baseline network G, we remove the self-loops (i.e., we do not include intra-tile trips) and we define G as the LSCC of the graph of flows. As shown in Supplementary Fig. S5, the LWCC and LSCC are similar and 98.8% of the WCCs are singletons, hence the LSCC captures the large majority of relevant flows while simplifying the mathematical interpretation of the results. We also use time series of mobility flows from 10 March 2020 to 18 July 2020 inclusive (131 days or 18 weeks) to build daily mobility networks Gpdq, d " 1, ..., 131 and weekly mobility networks Gpwq, w " 1, ..., 18, (by averaging the daily networks over calendar weeks). In all cases, the networks are defined on the same set of nodes as G, and we remove self-loops as above. Multiscale community detection with Markov Stability analysis Here we provide a brief outline of the Markov Stability (MS) framework. For a fuller description, see the Supplementary Information and in-depth treatments, including extensions to other types of graph processes, in Refs. [26][27][28]37]. Consider a weighted and directed graph G with adjacency matrix A. Let L " I´Dò ut A denote the random walk Laplacian matrix, where I is the identity matrix, and Dò ut denotes the pseudo-inverse of the diagonal out-degree matrix. The matrix L defines a continuous-time Markov process on G governed by a diffusive dynamics: dp dr "´p L, where pprq is a 1ˆN node vector of probabilities, and r is the Markov scale. The solution to this equation is given by pprq " pp0q expp´Lrq, and the matrix exponential defines transition probabilities of the Markov process (see Supplementary Information). This process converges to a stationary distribution π given by πL " 0. The goal of MS is to obtain partitions of the graph into cprq communities such that the probability flow described by (5) is optimally contained within the communities as a function of r. MS solves this problem by maximising the following function: Hprq " argmax H Tr " H T`Π expp´Lrq´π T π˘H ‰ ,(6) where Π " diag pπq, and the matrix Hprq is a Nˆcprq partition indicator matrix with Hprq ij " 1 if i is part of community j, and 0 otherwise. We thus obtain a series of optimised partitions over the Markov scales described by the matrices Hprq. The scales are more naturally described in log scale, so we redefine the Markov scale as s " log 10 prq. The optimisation (6) is carried out using the Louvain algorithm [44] through the implementation in the PyGenStability python package [39]. Comparing partitions with the Normalised Variation of Information To assess the quality of the partitions we use the Normalised Variation of Information (NVI), as a similarity measure for partitions [45,46]. Consider two partitions described by Hpsq and Hps 1 q with potentially different numbers of communities. The NVI is defined as: 0 ď NVIps, s 1 q :" VIps, s 1 q Hps, s 1 q ď 1, where VIps, s 1 q is the Variation of Information [47] and Hps, s 1 q is the Mutual Information between Hpsq and Hps 1 q. The NVI is a metric and low values indicate a high similarity between the partitions [46]. Using NVI has the advantage of being a universal similarity metric [48], i.e., if Hpsq and Hps 1 q are similar under any non-trivial metric, then they are also similar under NVI [46]. Scale selection algorithm After obtaining optimised partitions Hpsq for a sequence of m Markov scales S " ts 1 , s 2 , ..., s m u, we select partitions that describe the network structure robustly at different levels of resolution. Robust partitions are persistent across scales and reproducible under the non-convex Louvain optimisation for its particular scale [28]. We formalise these requirements using NVI as follows: (i) The persistence across scales is assessed by computing the pairwise NVI for partitions across different scales s and s 1 leading to an an mˆm symmetric matrix denoted by NVIps, s 1 q, where regions of low values indicate high persistence across scales. (ii) For each Markov scale s, the robustness is evaluated by repeating the Louvain optimisation (300 times in our study) with different random initialisation and computing the average pairwise NVI for the resulting ensemble of partitions, denoted by NVIpsq such that low values indicate indicate strong reproducibility of the optimisation. As an aid to scale selection we propose here an algorithm that processes the information contained in NVIps, s 1 q and NVIpsq sequentially. First, we use tools from image processing to evaluate the block structure of the NVIps, s 1 q matrix and apply average-pooling [49] with a kernel of size k (and padding) such that the pooled diagonal y NVIpsq quantifies the average pair-wise similarity of all partitions corresponding to scales in the neighbourhood B k psq " tu P S : 0 ă |u´s| ď ku of scale s. We then compute the smoothed version of y NVIpsq, denoted Block NVIpsq. Blocks of low values of NVIps, s 1 q correspond to basins around local minima of the Block NVIpsq. We then obtain the minimum of NVIpsq for each basin, and determine those as the robust scales of the network. Our scale selection algorithm is implemented in the PyGenStability package [39]. Measures of flow containment: Coverage and Nodal Containment Consider the adjacency matrix A of the mobility graph G and a Nˆc indicator matrix H for a partition of G into c communities. Let us also defineÃ, the adjacency matrix of the graph with self-loops that contains the intra-tile flows on the diagonal. Then F " H TÃ H is the cˆc lumped adjacency matrix where the element pH TÃ Hq kl corresponds to the mobility flow from community k to community l. The Coverage of community k, C k pHq, is defined as: 0 ď C k pHq :" pD`F q kk ď 1,(8) whereD`is the pseudo-inverse ofD " diagpdq whered " F 1 c . C k pHq can be interpreted as the probability of the lumped Markov process to remain in state k; hence high values of C k pHq indicate that community k covers well the flows emerging from the community. The Coverage of a partition CpHq is standard, and is defined as the ratio of flows contained within communities by the total amount of flow [50]. It is easy to see that this is given by the weighted average 0 ď CpHq " ř kd k C k pHq ř kd k ď 1.(9) High values of CpHq indicate that mobility flows are contained well within the communities of the partition and movement across different communities is limited. The Nodal Containment NC i of node i quantifies the proportion of flow emerging from i that is contained within its community in a partition H: 0 ď NC i pHq :" pAHq iC i d i ď 1,(10) where C i is the community of node i and d i " pA 1 N q i . Large values of NC i indicate that the mobility flows emerging from node i are largely contained within its assigned community, indicating a good node assignment. Hence NC i measures the containment of flows from a node-centered perspective. To obtain a partition-level measure, we define the average Nodal Containment NCpHq: 0 ď NCpHq :" 1 N N ÿ i"1 NC i pHq ď 1,(11) where N is the number of nodes. Response to an exponentially decaying shock The response of a variable xptq to a shock can be modelled as a linear ODE under a stimulus Rptq: dx dt "´βx`Rptq, xp0q " 0(12) where 1{β is the characteristic relaxation time of the system, and we assume here an exponentially decaying external stimulus Rptq :" αe´λ t , with amplitude α ě 0 and characteristic decay time 1{λ. The solution of (12) is given by [41]: xptq " α β´λ pe´λ t´e´βt q. We use the Levenberg-Marquardt algorithm [51] implemented in the LMFIT [52] python package to fit the activation response function xptq to a set of n data points pt i ,x i q by minimising the sum of squares χ 2 :" n ÿ i"1`x pt i q´x i˘2(13) to determine parameter estimatesα,β, andλ. Confidence intervals (CIs) are obtained from an F-test [53]. Data and code availability Data used in this study was accessed through Facebook's 'Data for Good' program: https://dataforgood. facebook.com/dfg/tools/movement-maps. Shape files for the NUTS (2018) regions in the UK are available from the Open Geography Portal https://geoportal.statistics.gov.uk/ under the Open Government Licence v.3.0 and contain OS data © Crown copyright and database right 2023. We host data of the UK mobility networks alongside code to reproduce all results and figures in our study on GitHub: https://github.com/barahona-research-group/MultiscaleMobilityPatterns. We use the PyGenStability python package [39] for Markov Stability analysis and scale selection. Code and documentation are hosted on GitHub under a GNU General Public License: https://github. com/barahona-research-group/PyGenStability. when X 0 is distributed according to the stationary distribution π. Here d " denotes that the two random variables pX 0 , X 1 , ..., X r q and pX r , X r´1 , ..., X 0 q have the same distribution functions. When, as in our case, the transition matrix M is obtained from a the adjacency matrix of a graph, the above conditions for the Markov chain can be translated into graph properties. In particular, it is well known that a necessary and sufficient condition for the irreducibility of the Markov chain defined by a graph-based transition matrix M (S.14) is that the graph G is strongly connected [36]. The assumption of aperiodicity is harder to interpret but requires that the graph is non-bipartite. Under these conditions, the Markov chain defined by equation (S.14) is ergodic, and its unique stationary distribution π is related to PageRank without teleportation [35]. There are continuous-time processes associated with the random walk (S.15) (see Refs. [27,28] for a detailed discussion). In particular, let us define the rate matrix Q " M´I, (S.20) where I is the NˆN identity matrix. Note that L "´Q is the so called random walk Laplacian. We can then define the continuous-time Markov process Xprq with semi-group P prq governed by the forward Kolmogorov equation dP dr " P Q, (S. 21) which has the solution P prq " e rQ . (S.22) When the Markov process is ergodic, P prq converges in total variation to its unique stationary distribution π, defined by Eq. (S.16), which also fulfills πL " 0 [56]. From the theory of Markov processes [57], it is known that for all r ą 0, the jump times τ " inftr ą 0 : Xprq ‰ Xp0qu (S.23) are distributed according to Prτ ě rs " e´q i r (S. 24) with jump rates q i "´Q ii . Furthermore, it is known that for all r ą 0 P i rτ ě r and D l " lpωq ą 0 : Xpuq " j @ pτ, τ`lqs " e´q i r q ij q i . (S. 25) with q ij " Q ij . Hence, the ij element of the jump matrix is given by q ij {q i and contains the jump probability of the Markov process Xprq from state i to j. Clearly, when G has no self-loops, q i "´Q ii " 1, @i. Consequently, the jump times (S.23) are exponentially distributed with rate 1, and M itself is the jump matrix so that the jump probabilities of Xprq are governed by M , as q ij " M ij for i ‰ j. These results show that the jump probabilities of the diffusion process are determined by the edge weights in the graph, whereas the self-loops of the graph affords the flexibility to choose different jump times reflecting distinct modelling assumptions [28]. Markov Stability as a cost function for clustering algorithms The dynamics of the Markov chain with transition matrix M defined on the nodes of the graph can be exploited to get insights into properties of the graph G itself. Let P prq be the semi-group of a Markov process as defined in (S.22) with stationary distribution π on a graph with adjacency matrix A. Following [28], each partition of the graph into c communities corresponds to a Nˆc indicator matrix H where H ij " 1 if node i is part of community j and H ij " 0 otherwise. Define now the clustered autocovariance matrix for partition H as K r pHq " H T " ΠP prq´π T π ‰ H. (S.26) The diagonal elements K r pHq ii correspond to the probabilities that the Markov process starting in one community i does not leave the community up to time r, whereas the off-diagonal elements correspond S15 to the probabilities that the process has left the community in which it started by time r. It is important to remark that r is an intrinsic time of the Markov process that is used to explore the graph structure, and is clearly distinct from the physical time of some applications, e.g., the days in the mobility data. To avoid confusion, it is customary in Markov Stability analysis to refer to r and/or s " log 10 prq as the Markov scale.Following these observations, we define the Markov Stability of a partition H by R r pHq " min 0ďlďr Tr rK l pHqs « Tr rK r pHqs . (S.27) The approximation in (S.27) is supported by numerical simulations that suggest that Tr rK r pHqs is monotonically decreasing in r. The Markov Stability R r pHq is thus a dynamical quality measure of the partition for each Markov scale r which can be maximised to determine optimal partitions for a given graph and each scale of the associated Markov process. The objective is therefore to find a partition Hpsq that maximises Markov Stability up to a time horizon (scale) s for the Markov process on the graph: R s pHpsqq " max H R s pHq. (S.28) The optimisation of this cost function is known to be NP-hard. However, Markov Stability can be reexpressed in terms of a generalised modularity [58] for another directed graph [28], thus allowing the use of efficient algorithms developed for modularity optimisation, e.g., the Louvain algorithm [44]. Briefly, the Louvain algorithm starts by assigning each node to a different community, followed by two steps. In the first step, one loops over the nodes in a random order and a node i is added to a neighbouring community if the modularity increases. After no further increase is possible, a new network is generated with the communities as 'super-nodes' and aggregated edges. The two steps are repeated iteratively until no further (or only marginal) increase in modularity is possible and the resulting partition corresponds to a local maximum of the modularity. An efficient implementation of Markov Stability optimisation across scales with the Louvain algorithm is provided by the PyGenStability python software [39]. Optimisation of Markov Stability for different Markov scales s then leads to a series of partitions Hpsq. For small Markov scales, the Markov process can only explore local neighbourhoods, which leads to a fine partition, whereas increasing the Markov scale widens the horizon of the Markov process so that larger areas of the graph are explored, which leads to coarser partitions [28]. Hence, the notion of a community as detected by Markov Stability analysis is strictly based on the spread of a diffusion on the network. In particular, Markov Stability analysis is able to identify non clique-like communities, which are likely present in heterogeneous mobility networks [38]. B Pre-lockdown Facebook UK data: mobility graph, flow partitions and administrative regions Geographic tiles and mobility graph: As described in the main text, we used 'Facebook Movement maps' [30] to construct our baseline UK mobility graph. See Fig. S5 for some illustrative feaures of the data and graph. Stationary distribution and other node centralities: Fig. S6 analyses the correlation between the stationary distribution π (3) and the out-degrees and intra-tile movement. Figure S6: Correlation of stationary distribution π with out-degrees and intra-tile movement. We observe that π correlates strongly with the out-degree d out (R 2 " 0.97) and slightly less so with the intra-tile movement n i within tile i (R 2 " 0.83). S17 Detailed balance and asymmetry of graph diffusion at stationarity: In analogy to the pairwise relative asymmetry (PRA) in the main text (1), we define the pairwise detailed balance (PDB) for each pair of tiles ij: 0 ď PDB ij " |pΠM´ΠM T q ij | pΠM`ΠM T q ij ď 1. (S.29) We find that the top 25% of tile pairs have PDB ě 0.18 (Fig. S7). Hence the diffusive process on the graph displays a reduced flow asymmetry at equilibrium, indicating the importance of (symmetric) commuting travel patterns. Figure S8 shows that the set of MS partitions Hps i q at Markov scales s i , i " 1, . . . , 9 are quasihierarchical, a feature that is not imposed by the clustering algorithm but emerges as an intrinsic feature of the data. The intrinsic quasi-hierarchical nature of the set of partitions can be quantified using the Normalised Conditional Entropy [37]Ĥps|s 1 q for two partitions Hpsq and Hps 1 q given by: 0 ďĤps|s 1 q :" Hps|s 1 q logpN q ď 1, (S. 30) where Hps|s 1 q is the standard conditional entropy. This asymmetric quantity is a measure of hierarchy and vanishes if each community of the partition Hpsq is a union of communities in the partition Hps 1 q. Figure S9 shows that the sequence of MS partitions displays a strong hierarchical structure with low values ofĤps|s 1 q. Figure S8: Quasi-hierarchical structure of Markov Stability partitions. Small Markov scales correspond to fine partitions and large Markov scales to coarse partitions. One can observe a quasihierarchical structure in the Sankey diagram, which is not imposed by the algorithm but a feature of the data. Note that there is no consistency of the colours of communities in the maps across Markov scales. Median Q3 NUTS3 170 4 13 27 NUTS2 42 40 72 102 NUTS1 12 230 254 324 Table S4: Statistics for NUTS regions (restricted to the geographic tiles in the baseline network). Statistics for the three NUTS levels, including the number of regions for each level and the three quartiles Q1, Q2 and Q3 for the number of nodes per region. Note NUTS regions where no movement was recorded are not present in the baseline data, hence the number of NUTS regions in the baseline network is smaller than the actual number of NUTS regions. Figure S10 shows that the MS partitions better describe the patterns of human mobility both in terms of the community Coverage C k and Nodal Containment NC i . (10). Statistical significance is determined using the Mann-Whitney test (with **** indicating p ă 0.0001, and *** indicating p ă 0.001). MS partitions perform significantly better for each measure and have lower variability, indicating a better description of human mobility than the administrative NUTS regions. S20 C Mobility response to lockdown restrictions Figure S11 shows the relative change of the Coverage C (9) for the nine robust MS partitions, together with their fits to a linear response function with an exponentially decaying shock. 11 13 15 17 19 21 23 25 27 Week Figure 2 : 2A posteriori comparison between Markov Stability (MS) partitions and NUTS regions. A The MS partitions across all scales are compared to the three levels of administrative NUTS regions: NUTS3 (fine), NUTS2 (medium), NUTS1 (coarse). As indicated by the minima of the NVI, the NUTS3 division is closely similar to Hps 1 q; NUTS2 to Hps 2 q; and NUTS3 to Hps 4 q. The maps show NUTS regions (lines) and MS partitions (coloured communities) for fine, medium and coarse levels. B-D The MS partitions display improved average Nodal Containment (NC) relative to NUTS regions Figure 3 : 3Temporal response of mobility networks to lockdown restrictions. The vertical red lines mark the official start of the first lockdown in the UK (24 March 2020). A Percent change in number of daily trips (intra-tile, inter-tile, total) relative to 10 March 2020. B Relative change of Coverage ∆C of MS partitions (fine, medium and coarse scales) measured with respect to the first week of the study period (Week 11). ∆C improves for all scales, especially for the finer scales. Solid lines correspond to the observed values; dashed lines are fits of the activation response function (4). Figure 4 : 4Regional differences in the temporal response to the lockdown. The maps show the fitting parameters of the activation response function for the weekly Coverage C k of the communities in the finest MS partition. A The shock amplitude α is high for urban centers (most notably London, Birmingham, Manchester and Glasgow) and low for rural areas. B The time scale of recovery 1{β is low for urban centers but high for rural areas, especially in Cornwall, Scotland or the Morecambe Bay area in North West England. Scotland shows longer recovery times due to different calendars for the lifting of restrictions relative to the rest of the UK. Figure S5 : S5The Facebook Movement map pre-lockdown. A The geographic 12-digit tiles for which movement data is available overlaid on the UK map. The tiles outside of the UK are not used. B The directed mobility network, with nodes in the LSCC coloured in green; the remaining nodes in the LWCC in blue; and all other nodes in grey. Most of the grey nodes are unconnected to the rest of the network. C The geographic tiles in the LSCC are spread over most NUTS3 regions in the UK (borders shown by lines). Figure S7 : S7Asymmetry of baseline mobility graph as measured by the pairwise relative asymmetry and the pairwise detailed balance.MS partitions of the baseline mobility network: Figure S9 : S9Normalised Conditional Entropy of the sequence of MS partitions. The robust MS scales are located in areas of low Normalised Conditional Entropy indicating a strong quasi-hierarchical structure. Figure S10 : S10Comparison of Coverage and Nodal Containment between MS partitions and NUTS regions. Fine (fine (Hps 1 q vs. NUTS3), medium (Hps 2 q vs. NUTS2) and coarse (Hps 4 q vs. NUTS1) levels of the MS partitions and NUTS regions are compared for A community Coverage (8) and B Nodal Containment Figure S11 : S11Temporal change of Coverage and Nodal Containment in response to lockdown restrictions. The vertical red lines marks the official onset of the first lockdown in the UK on 24 March 2020. A Relative change of Coverage C (9) of MS partitions at all scales. The solid lines show the data and the dashed lines are the optimal fits of the activation response function (4) to the weekly Coverage curves. Table 2 : 2Parameters of temporal response of ∆Cptq for the MS partitions.Estimated values Table S3provides additional statistics of the MS partitions.Markov scale Number of communities Nodes per community Q1 Median Q3 s 1 "´1.48 201 7 13 22 s 2 "´0.38 59 34 49 67 s 3 " 0.27 32 67 87 116 s 4 " 0.71 20 98 133 201 s 5 " 1.06 16 107 168 274 s 6 " 1.38 13 126 263 313 s 7 " 1.80 9 246 274 376 s 8 " 2.13 7 260 319 560 s 9 " 2.65 6 253 365 835 Table S3 : S3Statistics for MS partitions. The number of communities and the three quartiles Q1, Q2 and Q3 for the number of nodes per community for the partitions at Markov scales s i . Table S4 provides S4additional statistics for the NUTS regions. Following Brexit, International Terri-S19 torial Level (ITL) geography hierarchy boundaries replace NUTS in the UK, but currently ITL mirrors the NUTS European system.Level Number of regions Nodes per region Q1 Table S5 shows S5the parameters of the fits of ∆Cptq (4) to the activation response function for all MS partitions.Markov scale α (95% CI) 1{β (95% CI) 1{λ (95% CI) s 1 "´1.48 0.042 (0.036-0.050) 16.4 (12.5-21.5) 2.0 (1.6-2.7) s 2 "´0.38 0.0086 (0.0074-0.0101) 18.8 (14.6-24.3) 1.92 (1.52-2.46) s 3 " 0.27 0.0034 (0.0029-0.0040) 21.8 (16.8-28.7) 1.87 (1.49-2.38) s 4 " 0.71 0.0014 (0.0012-0.0016) 20.9 (15.7-28.0) 1.98 (1.55-2.57) s 5 " 1.06 0.00090 (0.00077-0.00106) 21.4 (16.2-28.7) 1.93 (1.52-2.49) s 6 " 1.38 0.00063 (0.00053-0.00075) 18.8 (13.8-25.5) 2.07 (1.60-2.77) s 7 " 1.8 0.00048 (0.00041-0.00057) 20.9 (15.6-28.2) 2.00 (1.57-2.61) s 8 " 2.13 0.00030 (0.00026-0.00035) 24.0 (18.7-31.6) 1.80 (1.45-2.25) s 9 " 2.65 0.00018 (0.00016-0.00021) 26.5 (21.1-33.9) 1.70 (1.41-2.06) Table S5 : S5Response parameters for Coverage. Parameter estimates and 95% Confidence Intervals for the shock amplitude α, time scale of recovery 1{β and shock decay time 1{λ obtained from fitting the activation response function (4) to ∆C. AcknowledgementsWe thank Robert Peach and Michael Schaub for valuable discussions. We also thank Alex Pompe for insights regarding data collection.FundingMB and JC acknowledge support from EPSRC grant EP/N014529/1 supporting the EPSRC Centre for Mathematics of Precision Healthcare. JC acknowledges support from the Wellcome Trust (215938/Z/19/Z). DS acknowledges support from the EPSRC (PhD studentship through the Department of Mathematics at Imperial College London).Competing interestsThe authors declare that they have no competing interests.Author contributionsThe computations in this study were performed by DS and JC. 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The transition probability matrix M of a discrete-time random walk on G is:where Dò ut denotes the pseudo-inverse of D out . The matrix M defines a discrete time Markov chain on the finite state space defined by the nodes of G [55]:where p r is a 1ˆN probability vector with components equal to the probability of the random walk hitting the respective node at (discrete) time r. Clearly, p r¨1N " 1, @r, because p r is a probability vector. While p r defines the distribution of the Markov chain at time r, we denote by X r the random process that follows this distribution. 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[ "https://github.com/barahona-research-group/MultiscaleMobilityPatterns." ]
[ "Optimal Checkpointing for Adjoint Multistage Time-Stepping Schemes", "Optimal Checkpointing for Adjoint Multistage Time-Stepping Schemes" ]
[ "Hong Zhang \nArgonne National Laboratory\nUSA\n", "Emil M Constantinescu \nArgonne National Laboratory\nUSA\n" ]
[ "Argonne National Laboratory\nUSA", "Argonne National Laboratory\nUSA" ]
[]
We consider checkpointing strategies that minimize the number of recomputations needed when performing discrete adjoint computations using multistage time-stepping schemes that require computing several substeps within one complete time step. Specifically, we propose two algorithms that can generate optimal checkpointing schedules under weak assumptions. The first is an extension of the seminal Revolve algorithm adapted to multistage schemes. The second algorithm, named CAMS, is developed based on dynamic programming, and it requires the least number of recomputations when compared with other algorithms. The CAMS algorithm is made publicly available in a library with bindings to C and Python. Numerical results show that the proposed algorithms can deliver up to two times the speedup compared with that of classical Revolve. Moreover, we discuss the utilization of the CAMS library in mature scientific computing libraries and demonstrate the ease of using it in an adjoint workflow. The proposed algorithms have been adopted by the PETSc TSAdjoint library. Their performance has been demonstrated with a large-scale PDE-constrained optimization problem on a leadership-class supercomputer.
10.1016/j.jocs.2022.101913
[ "https://arxiv.org/pdf/2106.13879v2.pdf" ]
235,658,493
2106.13879
f1786287e8ec4e0edea2143568804384313b8c6d
Optimal Checkpointing for Adjoint Multistage Time-Stepping Schemes Hong Zhang Argonne National Laboratory USA Emil M Constantinescu Argonne National Laboratory USA Optimal Checkpointing for Adjoint Multistage Time-Stepping Schemes CCS Concepts:Mathematics of computing → Automatic differentiation, Mathematical software;Computing methodologies → Modeling and simulation, Machine Learning Additional Key Words and Phrases: Checkpointing, adjoint method, multistage time-stepping schemes, dynamic programming, PETSc We consider checkpointing strategies that minimize the number of recomputations needed when performing discrete adjoint computations using multistage time-stepping schemes that require computing several substeps within one complete time step. Specifically, we propose two algorithms that can generate optimal checkpointing schedules under weak assumptions. The first is an extension of the seminal Revolve algorithm adapted to multistage schemes. The second algorithm, named CAMS, is developed based on dynamic programming, and it requires the least number of recomputations when compared with other algorithms. The CAMS algorithm is made publicly available in a library with bindings to C and Python. Numerical results show that the proposed algorithms can deliver up to two times the speedup compared with that of classical Revolve. Moreover, we discuss the utilization of the CAMS library in mature scientific computing libraries and demonstrate the ease of using it in an adjoint workflow. The proposed algorithms have been adopted by the PETSc TSAdjoint library. Their performance has been demonstrated with a large-scale PDE-constrained optimization problem on a leadership-class supercomputer. INTRODUCTION Adjoint computation is a key component for solving partial differential equation (PDE)-constrained optimization, uncertainty quantification problems, and inverse problems in a wide range of scientific and engineering fields. It has also been used for training neural networks, where it is called backpropagation in machine learning. The adjoint method is an efficient way to compute the derivatives of a scalar-valued function with respect to a large number of independent variables including system parameters and initial conditions. In this method, the function can be interpreted as a sequence of operations, and one can compute the derivatives by applying the chain rule of differentiation to these operations in reverse order. Therefore, the computational flow of the function is reversed, starting from the dependent variables (output) and propagating back to the independent variables (input). Furthermore, because of the similarity between stepwise function evaluation and a time-stepping algorithm for solving time-dependent dynamical systems, the stepwise function evaluation in any programming language can be interpreted as a time-dependent procedure, where the primitive operation propagates a state vector from one time step to another. This interpretation makes it easier to address the challenge of the checkpointing problem described below. A general nonlinear dynamical system = ( , ) , ( 0 ) = 0 , ∈ R , 0 ≤ ≤(1) can be solved by a time-stepping algorithm +1 = N ( ), = 0, . . . , − 1, (2) where N is a time-stepping operator that advances the solution from to +1 . An explicit timestepping scheme is discussed in Section 2 as an example of N. To compute the derivative of a scalar functional ( ) that depends on the final solution with respect to the system initial state 0 , one can solve the discrete adjoint equation = N ( ) +1 , ∈ R , = − 1, . . . , 0,(3) for the adjoint variable , which is initialized with = ( ) . Details of the derivation of this formula can be found in, for example, [20,21]. The adjoint variable carries the derivative information and propagates it backward in time from to 0 . At each time step in the backward run, the solution of the original system (2) must be available. It is natural to save the solution vectors when carrying out the forward run and reuse them in the reverse run. However, the required storage capacity grows linearly with the problem size and the number of time steps [10], making this approach intractable for large-scale long-time simulations. When there is a limited budget for storage, one resorts to checkpointing the solution at selected time steps and recomputing the missing solutions needed [8]. This selection complicates the algorithm implementation, however, and may hamper the overall performance. Therefore, an optimal checkpointing problem, which will be formally defined later, needs to be solved in order to minimize the recomputation cost with a budget constraint on storage capacity. Two decades ago Griewank and Walther introduced the first offline optimal checkpointing strategy that addresses this problem [8,9]. They established the theoretical relationship between the number of recomputations and the number of allowable checkpoints (solution vectors) based on a binomial algorithm. Their algorithm was implemented in a software library called Revolve and has been used by automatic differentiation tools such as ADOL-C, ADtool, and Tapenade and by algorithmic differentiation tools such as dolfin-adjoint [7]. Many follow-up studies have extended this foundational work to tackle online checkpointing problems, where the number of computation steps is unknown [12,18,19], and multistage or multilevel checkpointing [3,4,11,15] for heterogeneous storage systems. Note that multistage checkpointing refers to checkpointing on multiple storage devices (e.g., memory and disk) and is not related to multistage time integration. In developing these algorithms, one commonly assumes that memory is limited and the cost of storing/restoring checkpoints is negligible, as is still the case for many modern computing architectures. Despite theoretical advances in checkpointing strategies, implementing a checkpointing schedule poses a great challenge because of the workflow complexity, preventing widespread use. The checkpointing schedule requires the workflow to switch between partial forward sweeps and partial reverse sweeps, a requirement that is difficult to achieve. For this reason, many adjoint models implement suboptimal checkpointing schedules [14,16] or even no checkpointing schedules [2,21]. In order to mitigate the implementation difficulty, the classical Revolve library is designed to be a centralized controller that guides users through the implementation. However, its use requires a specific intrusive workflow. In contrast, our proposed approach is implemented as an external guide. Our contributions in this paper are as follows. (1) We demonstrate that the classical Revolve algorithm can become suboptimal when multistage time-stepping schemes such as Runge-Kutta methods are used to solve the dynamical system (1). (2) We present two new algorithms, namely, modified Revolve and CAMS, that are suited to general and stiffly accurate multistage time-stepping schemes. (3) We present a library that implements the CAMS algorithm, and we discuss its utilization in adjoint ordinary differential equation (ODE) solvers. (4) We demonstrate that our library can be easily incorporated into scientific libraries such as PETSc and be used for efficient large-scale adjoint calculation with limited storage budget. (5) We show the performance advantages of our algorithms over the classical approach with experimental results on a leadership-class supercomputer. OPTIMAL CHECKPOINTING PROBLEMS IN ADJOINT CALCULATION Number of recomputations as a performance metric In a conventional checkpointing strategy, system states at selective time steps are stored into checkpoints during a forward sweep and restored during a reverse sweep. Before an adjoint step (5) can be taken, the state is needed to recompute the time step → +1 to recover all the intermediate information. And if is not checkpointed, one has to recompute from the nearest checkpoint to recover it first. Often in the literature (such as [9]) authors implicitly assume that the execution cost of every step is constant and the cost of memory access/copy is insignificant compared with the calculation cost. Consequently, the number of recomputations can be used as a good metric that reflects the computational cost associated with a checkpointing strategy. In the rest of this paper we make the same assumption and use the same metric when developing our checkpointing strategies. Conventional checkpointing problem The conventional checkpointing problem, defined in Problem 1 below, has been solved by Griewank and Walther [9] with the Revolve algorithm. Revolve generates a checkpointing schedule based on the assumption that a checkpoint accommodates only the solution at a time step. For one-stage time-stepping schemes such as the Euler method, the solution is guaranteed to be optimal according to Proposition 2.1 proved in [9]. Problem 1 ( conventional ( , )). Assume the solution at a selective time step can be saved to a checkpoint. Given the number of time steps and the maximum allowable number of checkpoints , find a checkpointing schedule that minimizes the number of recomputations in the adjoint computation for (2). Proposition 2.1 (Griewank and Walther [9]). The optimal solution to conventional ( , ) requires a minimal number of recomputations ( , ) = − + − 1 ,(4) where is the unique integer (called repetition number in [9]) that satisfies + −1 −1 < ≤ + . As an example, we illustrate in Figure 1 the optimal reversal schedule generated by Revolve, which requires the minimal number of recomputations to reverse 10 time steps given 3 maximum allowed checkpoints. For the convenience of notation, we associate the solution of the ODE system (1) at each time step with an index. The initial condition corresponds to index 0, and the index increases by 1 for each successful time step . In the context of adaptive time integration, a successful 0 1 2 3 4 5 6 7 8 9 10 7 8 9 7 8 4 5 6 7 5 6 4 5 0 1 2 3 4 2 3 1 2 0 1 Fig. 1. Application of Revolve to reversing 10 time steps given 3 allowable checkpoints. The up arrow and down arrow stand for the "store" and "restore" operations, respectively. The down arrows with solid lines indicate that the checkpointing units can be discarded after being used. The down arrows with dashed lines mean to restore the unit without removing it from memory. time step means the last time step taken after potentially several attempted steps to determine a suitable step size. Note that the failed attempts are excluded and not indexed. Starting from the final time step, the adjoint computation decreases the index by 1 after each backward step until the index reaches 0. During the forward run, the solutions at indices 0, 4, and 7 are copied into the first three checkpoints. When the final step 9 → 10 is finished, the solution and stage values at this step are usually accessible in memory, so the adjoint computation for the last time step can be taken directly. To compute the next backward step 9 → 8, one can acquire the solution at index 9 and the stage values by restoring the second checkpoint and recomputing two steps forward in time. The second checkpoint can be discarded after the backward step 8 → 7 so that its storage can be reused in following steps. Throughout the entire process, 15 recomputations are taken, as shown Figure 1. One can verify using Proposition 2.1 that the repetition number for this case is = 2 and an optimal solution should have (10, 3) = 15. Minimizing the number of recomputations for multistage methods The conventional strategy, however, can be suboptimal for multistage time-stepping schemes if we relax the assumption to allow saving the intermediate stages together with the solution as a checkpoint. For example, consider the modified problem. Multistage schemes such as Runge-Kutta (RK) methods are popular for solving systems of ODEs; their adjoint counterparts are implemented in ODE solver libraries such as FATODE [21] and more recently by PETSc TSAdjoint [1,5,20]. An ℓ-stage explicit RK method is expressed as = + ℎ −1 ∑︁ =1 ( ), = 1, · · · , ℓ, +1 = + ℎ ℓ ∑︁ =1 ( ).(5) Its discrete adjoint is ℓ, = ℎ ( ) +1 + ℓ ∑︁ = +1 ℓ, , = ℓ, · · · , 1 = +1 + ℓ ∑︁ =1 ℓ, ,(6) where is the adjoint variable that carries the sensitivity information and is propagated in a backward step during a reverse sweep. The adjoint step of a Runge-Kutta scheme requires all the stage values (see the sensitivity equation (6), for example). These intermediate values are usually obtained by restoring a state saved during the forward sweep and recomputing a time step using this state, as illustrated in Figure 2. Revolve requires its users to implement a basic action (named ) that recomputes a forward step followed immediately by a backward step. This strategy has been a de facto standard in classical adjoint computation. Recompute Restore Restore For cases where memory is limited, checkpointing the stage values may still lead to fewer recomputations. Based on this observation, we extend the classical optimal checkpointing scheme in [9] to solve Problem 2. See Figure 2 for a schematic illustration. Although saving more information at each time step means that fewer checkpoints are available, we will show in Section 3 that in certain circumstances the extended scheme may still outperform the original scheme, with the gain depending on the total number of time steps to be reversed. Our goal is to minimize the number of recomputations under practical assumptions. Further relaxing the assumption, we will seek in Section 4 an optimal solution for the following problem that requires the fewest recomputations among the solutions to the three problems (Problem1, 2, 3). Problem 3 ( multistage ( , )). Assume that either the solution or an intermediate stage (which has the same size as the solution) can be saved to a checkpoint. Given the number of time steps and the maximum allowable number of checkpoints , find a checkpointing schedule that minimizes the number of recomputations in the adjoint computation for (2). MODIFIED CHECKPOINTING SCHEME BASED ON REVOLVE In this section we first describe the Revolve nomenclature in order to provide the necessary background. We then introduce our solution to modified ( , ) using a concrete example, discuss its optimality, and address the implementation aspects. Modification to the Revolve offline algorithm To generate the schedule in Figure 1, one needs to call the API routine revolve() repeatedly and implement the actions prescribed by the algorithm. The return value of revolve() tells the calling program to perform one of the actions among advance, takeshot(store), restore, firsturn, and youturn, which are briefly summarized in Table 1 and are explained in detail in [9]. advance advance the solution forward takeshot(store) copy the solution into a checkpoint restore copy a checkpoint back into the solution firsturn take one backward step directly (usually after one forward step) youturn take one forward step and then one backward step In our modification of the schedule, every checkpoint position is shifted by one, and the stage values are included so that the Jacobian can be computed directly from these. To be specific, if the original Revolve algorithm determines that one should checkpoint the solution at index , we will store a combined checkpoint at the end of the time step → + 1 including the solution at index + 1 and the stage values. We do so by mapping the actions prescribed by Revolve to a series of new but similar actions, while guaranteeing the optimality for the new checkpointing settings. Table 2 lists the mappings we conduct in the modified schedule. Figure 3 illustrates the checkpointing schedule generated by the modified Revolve algorithm. Revolve Our modification advance from to advance the solution from + 1 to + 1 store solution copy the solution at + 1 and the stages into a checkpoint restore to solution restore the solution at + 1 and the stages from a checkpoint youturn take one backward step directly (firsturn) Optimality of the modified algorithm As we observe from the optimal schedule for adjoint computation provided by Revolve, the intermediate information required by each backward step must be recomputed from the solution restored from a checkpoint, except for the last time step. Because every checkpoint is shifted to one time step later, the distance between the restored checkpoint and the current solution in the reverse sweep is reduced by one, thus saving exactly one recomputation for each adjoint step. This observation leads to the following proposition regarding optimality for the modified Revolve algorithm. ( , ) = ( − 1) − + − 1 + 1,(7) where is the unique integer that satisfies + −1 −1 < ≤ + . Proof. According to the observation mentioned above, one can further save − 1 additional forward steps with the modified scheme. We will prove by contradiction that no further savings are possible with other schedules. If a schedule exists that satisfies the assumption and takes fewer recomputations than (7), one can move all the checkpoints backward by one step and exclude the stage values so that the checkpoints are composed of solutions only. The resulting schedule will cost − 1 additional recomputations by construction. It is clearly a solution to the classical checkpointing problem for time steps given allowed checkpoints, and the total number of recomputations required is less than − + −1 . This contradicts the optimality result in Proposition 2.1. □ Remark 1. The same modification can also be applied to the online checkpointing algorithms in [12,18,19] and to the multistage checkpointing algorithms in [17]. Considering the same number of allowable checkpoints, the number of recomputations saved is always equal to the total number of steps minus one. An immediate question is whether to use the modified or the original algorithm (in other words, whether to save stage values) given the same amount of storage capacity. According to Proposition 3.1, the optimal option depends on multiple factors, including the number of steps, the number of stages of the time-stepping algorithm, and the memory capacity. Therefore, for a given time-stepping algorithm and a fixed amount of storage capacity, the best choice can be made based on only the total number of time steps. For example, we suppose there is storage space for 12 solutions when using the original algorithm. In the modified algorithm, the same space can be used to store 6 checkpoints if each checkpoint consists of one solution and one stage, and it can be used to store 4 checkpoints if each checkpoint consists of one solution and two stages. In Figure 4 we plot the number of recomputations for the two algorithms in two scenarios-saving one additional stage and saving two additional stages. As can be seen, saving the stages together with the solution is more favorable than saving only the solution until a crossover point is reached (41 and 13 steps, respectively, for the two illustrated scenarios). Furthermore, the number of stages, determined by the time-stepping scheme, has a critical impact on the location of the crossover point; for schemes with fewer stages, saving the stages benefits a wider range of time steps (compare 41 with 13). Remark 2. Some multistage time-stepping methods have a property that can be exploited to reduce the size of a checkpoint. In particular, if the last stage is equal to the solution at the end of a time step, one can skip the last stage when storing a checkpoint. Many classical implicit RK methods and the Crank-Nicolson method have this property and thus can be treated as having one less stage. TRULY OPTIMAL CHECKPOINTING FOR THE ADJOINT OF MULTISTAGE SCHEMES Both the Revolve algorithm and the modified Revolve we propose are proved to be optimal under some assumptions on the checkpointing strategy (see Figure 2); however, neither of them is ideal in practice when evaluated without these assumptions, as indicated in Figure 4. To obtain a truly optimal checkpointing schedule, we consider the solution to multistage ( , ) with a further relaxed assumption allowing the solution or the stage values or both at one time step to be checkpointed, a situation that is not difficult to achieve in most ODE solvers. For the fairness of comparison, we introduce the concept of a checkpointing unit. By definition, one checkpointing unit can store one solution vector or one stage vector since they have the same size; one checkpoint may contain one or more units, thus having different types. The total memory (in bytes) occupied by checkpoints is ℎ × ℎ (in bytes). Therefore, when comparing different algorithms, using the same number of checkpointing units indicates using the same amount of total memory. In addition, many multistage schemes are constructed to be stiffly accurate in order to solve stiff ODEs. This property typically requires that the solution at the end of each time step be equal to the last stage of the method. Taking into account this observation as well as the relaxed assumption, we develop the optimal checkpointing algorithm CAMS that includes two variants, one for stiffly accurate schemes and the other for general cases, denoted by CAMS-SA and CAMS-GEN, respectively. Both variants are developed by using a divide-and-conquer strategy. CAMS for stiffly accurate multistage schemes First, let us consider a subproblem of the checkpointing problem multistage ( , ). If the initial state is already checkpointed, we want to know how many additional forward steps are necessary to reverse a sequence of time steps with a given amount of checkpointing units. To solve this subproblem, we establish a recurrence equation as follows. Lemma 4.1. Given allowed checkpointing units in memory and the initial state stored in memory, the minimal number of additional forward steps needed for the adjoint computation of time steps using an ℓ-stage time integrator satisfies IS ( , ) = min min 1≤˜≤ (˜+ IS (˜, ) + IS ( −˜, − 1)) min 1≤˜≤ (˜− 1 + IS (˜− 1, ) + IS ( −˜, − ℓ)) .(8) Proof. Assume that the next checkpoint is taken after˜steps with 1 ≤˜≤ . We need to consider two cases based on the type of the next checkpoint. For case 2, the th time step can be reversed directly since the stage values can be restored from memory; therefore, the first subsequence consists of − 1 time steps. The second subsequence still consists of −˜time steps, but there are ℓ fewer checkpointing units available. This case corresponds to the second recurrence equation in 8. □ With the solution of the reduced problem, we can solve the original problem easily. Notice that to reverse the first time step, one can either restore the initial state from checkpoints and recompute the time step or restore the stage values directly from memory. The latter option transforms the problem into checkpointing for − 1 time steps given − ℓ checkpointing units and the first state already checkpointed, which is IS ( − 1, − ℓ + 1). Theorem 4.2. Given allowed checkpointing units in memory, the minimal number of additional forward steps needed for the adjoint computation of time steps using an ℓ-stage time integrator is ( , ) = min ( IS ( , ), IS ( − 1, − ℓ + 1)) .(9) Based on Lemma 4.1, we can design a dynamic programming algorithm to compute IS ( , ) and save the results in a table. Computing ( , ) requires querying only two values from the table. Given , , ℓ and the index and the type of the last checkpoint, the routine CAMS( , , , , ℓ) returns the index and the type of the next checkpoint. These two returned variables are determined internally according to the choice of minimum made in (8). In particular, depends on˜, and depends on which case of the two yields the minimum. The index gives the position of the next checkpoint. The type can be either solution or stage values. This information, often dubbed the path to reach the minimum, can also be stored in tables when computing IS ( , ) recursively. Tabulation of intermediate results is a standard procedure in dynamic programming; thus we do not detail it here. CAMS for general multistage schemes Depending on how the first checkpoint is created, we split the problem into two scenarios: (1) the initial state is checkpointed, and (2) SV ( , ) = min ( IS ( − 1, − ℓ), SV ( − 1, − ℓ)) .(11) Theorem 4.5. Given allowed checkpointing units in memory, the minimal number of additional forward steps needed for the adjoint computation of time steps using an ℓ-stage time integrator is ( , ) = min ( IS ( , ), SV ( , )) .(12) The correctness of Lemmas 4.3 and 4.4 is illustrated in Figure 5. For IS ( , ), we suppose that the second checkpoint is placed after˜th time step during the forward sweep. Depending on the type of the second checkpoint, the second subproblem, which reverses the sequence of time steps starting from the second checkpoint, can be addressed by IS ( −˜, − 1) and SV ( −− 1 + 1, − 1), respectively. During the reverse sweep, the second subproblem is solved first, and then the initial state is restored from the first checkpoint and integrated to the location of the second checkpoint. Thus,˜or˜− 1 additional time steps are needed for solving the first subproblem. A similar strategy can be applied for SV ( , ) as well. But note that the second checkpoint must be placed after the second time step; otherwise, the second time step cannot be reversed. Theorem 4.5 combines the solution of the two subproblems, and the proof is straightforward. Based on Lemmas 4.3 and 4.4, we can develop dynamic programming algorithms to compute IS ( , ) and SV ( , ), respectively, and tabulate the values and the path information for any input ≤ and ≤ . The resulting algorithm for the adjoint computation is almost the same as Algorithm 3 and thus is omitted for brevity-except that an additional type of the checkpoint is considered. Specifically, the choices of a checkpoint type include solution only, stage values only, and solution plus stage values. Note that Lemmas 4.3 and 4.4 distinguish between the first two choices. If the next checkpoint is stage values and the checkpoint after the next is a solution calculated directly from the stage values, it makes the implementation easier to fuse the two checkpoints into a new type of checkpoint because both of them are available at the end of a time step. Performance analysis for CAMS Proposition 4.6. Using the CAMS algorithm takes no more recomputations than using the Revolve algorithm and the modified Revolve algorithm. Proof. By construction, to reverse time steps with checkpointing units, the Revolve algorithm requires the number of recomputations to satisfy R ( , ) = min 1≤˜≤ (˜+ R (˜, ) + R ( −˜, − 1)) ,(13) as shown in [9]. Following the same methodology, we can derive a recurrence equation for the modified Revolve algorithm: MR ( , ) = min 1≤˜≤ (˜− 1 + MR (˜− 1, ) + MR ( −˜+ 1, − 1 − ℓ)) .(14) Now we compare R ( , ) and MR ( , ) with ( , ) obtained with our CAMS algorithm. According to (12), ( , ) is always less than or equal to IS ( , ). The first case in (10) Note that IS ( , ) is a convex function in ∈ 1, . . . , . Therefore, IS ( −˜, − 1 − ℓ) ≤ IS ( −˜+ 1, − 1 − ℓ). Then we obtain IS ( , ) ≤ MR ( , ). □ We can conclude with this proposition that the number of recomputations needed by CAMS is bounded by the number of recomputations needed by Revolve, which is given in (4). We note that the lower bound for Revolve is − 1 whereas CAMS takes zero recomputations if there is sufficient memory. Figure 6 illustrates the application of Revolve and CAMS for reversing 10 time steps given 6 allowable checkpointing units. Each application starts from a forward time integration, at the end of which all the checkpointing slots are filled. The distribution of the checkpoints is determined by the corresponding checkpointing algorithm (e.g., Alg. 3). In the backward integration, a solution can be restored from checkpoints and then used for recomputing the intermediate states; for CAMS, stage values can be restored and used for reversing the time step directly without any recomputation. As a result, revolve takes 12 recomputations whereas CAMS takes 8 recomputations, or 6 recomputations if the integration method is stiffly accurate. The up arrow and down arrow stand for the"store" operation and "restore" operation, respectively. The down arrows with solid lines indicate that the checkpointing units can be discarded after being used. The down arrows with dashed lines mean to restore the unit without removing it from memory. To further study the performance of our algorithms, we plot in Figure 7 the actual number of recomputations taken in the algorithm versus the number of time steps to be reversed, and we compare our algorithms with the classical Revolve algorithm. For a fair comparison, the same number of checkpointing units is considered so that the total amount of memory usage is the same for both algorithms. Figure 7 shows that the two CAMS variants outperform Revolve significantly. For 30 checkpointing units and 300 time steps, CAMS-GEN takes 210 fewer recomputations than Revolve does, and CAMS-SA takes 211 fewer recomputations than Revolve does. If we assume the computational cost of a forward step is constant, then the result implies an approximate speedup of 1.6 times in running time for the adjoint computation. For 60 checkpointing units and 300 time steps, CAMS-GEN and CAMS-SA result in 261 and 269 fewer recomputations, respectively, which can be translated into an estimated speedup of 2 times. As the number of time steps increases, the gap between Revolve and CAMS can be further enlarged. The modified Revolve also performs better than Revolve but will eventually be surpassed by Revolve. Furthermore, when the number of time steps is small, which usually means there is sufficient memory, no recomputation is needed by modified Revolve, whereas Revolve requires the number of recomputations to be at least as large as one less than the number of time steps. In the ideal scenario when there is sufficient memory, using modified Revolve or CAMS can avoid recomputation completely, making the reverse sweep two times faster than using Revolve, provided the cost of a forward step is comparable to the cost of a backward step. UTILIZATION OF REVOLVE AND CAMS IN A DISCRETE ADJOINT ODE SOLVER Using Revolve in a discrete adjoint ODE solver The Revolve library is designed to provide explicit control for conducting forward sweeps and reverse sweeps in adjoint computations. Its user must implement primitive operations such as performing a forward and backward step, saving/restoring a checkpoint, and executing these operations in the order guided by Revolve. Thus, it can be intrusive to incorporate Revolve in other simulation software such as PETSc; and the workflow can be difficult to manage, especially when the software has an established framework for time integration and adaptive time-step control. To mitigate the intrusive effects, we use Revolve differently so that its role becomes that of a "consultant" rather than a "controller." Algorithm 1 describes our workflow for the adjoint computation with checkpointing. Revolve relies on several key parameters-capo, fine, and check-and updates them internally. These parameters are described in Table 3. For ease of implementation, we use a counter stepstogo to track the number of steps to advance. The routine forwardSweep(ind,n,state) advances the solution steps from the th time point, which is easy to implement in ODE solvers. It can be used to perform a full forward run and can also be reused to perform the recomputations in the backward adjoint run. revolveForward() wraps calls to the API routine revolve() and is intended to guide the selection of checkpoints. Because each call to revolve() changes the internal states of revolve, we must carefully control the number of times revolve() is called in revolveForward() based on the counter stepstogo and the returned value of the previous calls to revolve(). Despite the convoluted manipulations of calls to revolve(), the resulting checkpointing schedule is equivalent to the original one generated by calling revolve() repeatedly (the "controller" mode). This is justified primarily by two observations. • When revolve() returns takeshot (which means storing a checkpoint), the next call to revolve() will return either advance or youturn or firsturn. • In the reverse sweep, every backward step is preceded by restoring a checkpoint and recomputing from this point. Table 3. Key Revolve parameters capo the starting index of the time-step range to be reversed fine the ending index of the time-step range to be reversed check the number of checkpoints in use snaps the maximum number of checkpoints allowed Algorithm 2 depicts the adjoint computation using the modified Revolve. Compared with Algorithm 1, Algorithm 2 shifts the positions of all the checkpoints so that the call to revolve() is lagged. We Algorithm 1 Proposed adjoint computation for a sequence of time steps using Revolve. Assert(whatodo=restore) ⊲ always start from restoring a checkpoint 9: , ← restore( ℎ ) 10: ← forwardSweep( , − ,) 11: ← adjointStep( ) 12: end for 13: function forwardSweep( , , ) ⊲ advance n steps from the ind-th point 14: for ← to + − 1 do 2 ). This design is suitable for preserving the established workflow of the ODE solvers so that the influence on other interacting components such as TSAdapt (time-step adaptivity class) and TSMonitor (time-step monitor class) is minimized. PETSc uses a redistributed package 1 that contains a C wrapper of the original C++ implementation of Revolve. Users can pass the parameters needed by Revolve through command line options at runtime. PETSc provides additional command line options that allow users to monitor the checkpointing process. By design, PETSc manages the manipulation of checkpoints. The core data structure is a stack with push and pop operations, which is used to conduct the actions decided by the checkpointing scheduler. Deep copy between the working data and the checkpoints is achieved with the PetscViewer class. The data can be encapsulated in either sequential or parallel distributed vectors. Besides the offline checkpointing scheme, PETSc supports online checkpointing and multistage checkpointing schemes, which are also provided by the Revolve package. The proposed modification has been applied to these schemes as well. Apart from memory, other storage media such as disk can also be considered for storing checkpoints in binary format. For parallel file systems, which are common on high-performance computing clusters, the PetscViewer class can use MPI-IO to achieve high-performance parallel I/O. Using CAMS in a discrete adjoint ODE solver Motivated by our experience with incorporating Revolve in PETSc and the difficulties in handling the workflow, we design the main interface function of CAMS to be idempotent and simplify its output for better usability. The CAMS library is publicly available at https://github.com/caidao22/pkgcams. It provides both C and Python APIs. To use CAMS, users first need to call the function offline_cams_create(int m,int s,int l,int stifflyaccurate); to create a CAMS object and specify the number of time steps, the number of allowable checkpointing units, the number of stages, and a flag that indicates whether the integration method is stiffly accurate. At creation time, the dynamic programming algorithms presented in Sections 4.1 and 4.2 are executed, generating tables for fast query access later on. A side benefit of dynamic programming is that when multistage ( , ) is solved, the solutions to the subproblems multistage ( , ) ∀0 < < , 0 < < become available. Then users can query for the position and the type of next checkpoint by calling offline_cams(int lastcheckpointstep,int lastcheckpointtype,int s,int m,int l, int *nextcheckpointstep,int *nextcheckpointtype); with negligible overhead. The output of this function provides the minimal information necessary to guide a checkpointing schedule. By design, it can be called repeatedly and return the same output if provided the same input. This idempotence feature, however, is not available in Revolve because the output of Revolve also depends on hidden internal states. Redundant calls to the Revolve function could be detrimental. In contrast, CAMS allows more flexible and robust design of the workflow for the adjoint computation. Figure 9 illustrates how CAMS can be used in a typical adjoint ODE solver. The core of the workflow is to determine whether a checkpoint should be stored at the end of a time step, either during the forward run or during the recomputation stage in the adjoint run. The other operations such as restoring a checkpoint, recomputing from a restored state, or deleting an unneeded checkpoint are intuitively straightforward. A detailed description of the algorithm for reversing a sequence of time steps with CAMS is given in Algorithm 3. This algorithm has also been implemented as an option in the TSTrajectory class in PETSc with evident ease (compare Algorithm 3 with Algorithm 1 and 2). EXPERIMENTAL RESULTS To evaluate our algorithms for their performance and applicability to practical applications at large scale, we ran experiments on the Cori supercomputer at NERSC. In all the cases, we used 32 Intel Knights Landing (KNL) nodes in cache mode, 64 cores per node, with each MPI process bound to one core. DDR memory was used for storing checkpoints. As a benchmark, we consider a PDE-constrained optimization problem for which an adjoint method is used to calculate the gradient. The objective is to minimize the discrepancy between the simulated result and observation data obtained from a reference solution: minimize 0 ∥ ( ) − ( ) ∥ 2(15) subject to the Gray-Scott equations [13] and set as observed data. Solving the optimization problem implies recovering the initial condition from the observations. This example is a simplified inverse problem but fully represents the computational complexities and sophistication in large-scale adjoint computations for time-dependent nonlinear problems. Following the method of lines, the PDE is discretized in space with a centered finite-difference scheme, generating a system of ODEs that is solved by using an adjoint-capable time integrator in PETSc with a fixed step size. The step size used is 1.0 for implicit time integration and 0.001 for explicit time integration due to stability restrictions. For spatial discretization, the computational domain Ω ∈ [0, 2] 2 is divided into a uniform grid of size 128 × 128. The nonlinear system that arises at each time step is solved with Newton's method. Both the linear and the transposed linear systems are solved by using GMRES and a geometric algebraic multigrid preconditioner following the same parameter settings described in [20]. u = 1 ∇ 2 u − uv 2 + (1 − u) v = 2 ∇ 2 v + uv 2 − ( + )v,(16) The base performance for adjoint computations with three selective time-stepping schemes is presented in Table 4. For implicit time-stepping schemes, the reverse sweep takes much less time than the forward sweep does, since only transposed linear systems need to be solved in the reverse sweep whereas nonlinear systems need to be solved in the forward sweep. One recomputation takes about 4 times more than a backward step. Therefore, savings in recomputations can lead to a dramatic reduction in total running time. For explicit time-stepping schemes, to avoid the expensive cost of forming the Jacobian matrix, we use a matrix-free technique to replace the transposed Jacobian-vector product with analytically derived expressions. As a result, the cost of one recomputation is less than the cost of a backward step. Figure 10 shows the theoretical predictions regarding the additional recomputations in the adjoint computation and the real recomputation overhead in terms of CPU wall time. We see that the runtime decreases as the allowable memory for checkpointing ( ℎ × ℎ ) increases for the two time-stepping schemes considered, backward Euler and Crank-Nicolson, both of which are stiffly accurate. This decrease in runtime is expected because the number of additional recomputations decreases as the number of allowable checkpointing units increases. The best performance achieved by modified Revolve is approximately 2.2 times better than that by Revolve. CAMS performs the best for all the cases. We note that despite being noisy, the experimental results match the theoretical predictions well. The timing variability is likely due to two aspects: one is that the cost of solving the implicit system varies across the steps, which is expected for nonlinear systems; the other is the run-to-run variations [6] on the KNL systems. Figure 11 shows the result for the Runge-Kutta 4 method. Modified Revolve outperforms Revolve when the allowable memory for checkpointing per node is larger than roughly 6 GB, which is close to the theoretical prediction. We note that the performance of CAMS-GEN slightly deviates from the theoretical prediction. This deviation is because the cost of the memory movement accounts for a significant part of the recomputation overhead, which can be partially explained In the experiments, the Crank-Nicolson method and backward Euler are used for time integration. Both methods can be viewed as a two-stage stiffly accurate method, where the second stage is the same as the solution. All the runs are executed on Cori with 2, 048 MPI processes (32 nodes) and are repeated three times to mitigate the influence of run-to-run variations. by the fact that a KNL node has a low DRAM bandwidth of 90 / but powerful cores with large caches and wide SIMD units. However, Revolve is designed to minimize the number of times saving a state to a checkpoint as a secondary objective [9] and thus is less affected by the noticeable memory overhead than is CAMS-GEN. CONCLUSION The classical Revolve algorithm provides an optimal solution to the checkpointing problem for adjoint computation in many scientific computations when the temporal integration is abstracted at the level of time steps. When directly applied to multistage time-stepping schemes, however, it may not yield optimal performance. In this paper we have presented new algorithms that minimize the number of recomputations under the assumption that the stage values of a multistage scheme can be stored and the stage vectors are of the same size as the solution vector. By extending from Revolve and redefining the checkpoint content, we derived the modified Revolve algorithm that provides better performance for a small number of time steps. Based on dynamic programming, we proposed the CAMS algorithm, which proves to be optimal for computing the discrete adjoint of multistage time-stepping schemes. We also developed a variant of CAMS that takes advantage of the special property of stiffly accurate time-stepping schemes. The performance has been studied both theoretically and numerically. The results on representative cases show that our algorithms can deliver up to 2 times speedup compared with Revolve and do not need recomputation when memory is sufficient. In addition, the usage of CAMS is tailored to the workflow of practical ODE solvers. We also propose a new approach for integrating the algorithms introduced in this work into existing scientific libraries. In our approach the solver does not require using our algorithms as a centralized controller over the entire workflow, as proposed in the design of classical Revolve; thus our approach is less intrusive to the application codes. The proposed algorithms have been successfully incorporated into the PETSc library. For future work we will extend CAMS for online checkpointing problems and investigate new algorithms that take memory access overhead into account. Problem 2 ( 2modified ( , )). Assume a checkpoint is composed of the solution and the stage values at a time step. Given the number of time steps and the maximum allowable number of checkpoints , find a checkpointing schedule that minimizes the number of recomputations in the adjoint computation for(2). Fig. 2 . 2Strategies for obtaining the stage values during the reverse sweep of a discrete adjoint solver based on a Runge-Kutta method. Left is the classical strategy used with Revolve. Right is the strategy considered in this work. Let us count the number of recomputations required for an ideal case where the memory is unlimited. To reverse time steps, − 1 recomputations would be required if one checkpointed the solution at every time step. No recomputation would be needed, however, if one checkpointed the stage values instead of the solution for all the time steps. Fig. 3 . 3Application of the modified Revolve to reversing 10 time steps given 3 allowable checkpoints. The up arrow and down arrow stand for the "store" and "restore" operations, respectively. The down arrows with solid lines indicate that the checkpointing units can be discarded after being used. The down arrows with dashed lines mean to restore the unit without removing it from memory.Proposition 3.1. Assume that a checkpoint is composed of the solution and the stage values at a time step. Given number of allowed checkpoints in memory, the minimal number of additional forward steps (recomputations) needed for the adjoint computation of time steps is Fig. 4 . 4Minimal recomputations required by Revolve and the modified Revolve with different numbers of stages (2 in the left plot and 3 in the right plot). Adapted from[20]. Case 1 : 1The state˜is checkpointed. Case 2: The state˜and stage values at time step˜are checkpointed. For case 1, the sequence of time steps can be split into two parts, one with˜time steps and the other with −˜. The second part will be reversed first, requiring IS ( −˜, − 1) additional forward steps. The first part needs a forward sweep over the time steps before the reverse run can be performed. Summing up all the additional forward steps leads to the first recurrence equation in 8. the stage values of the first step are checkpointed. The corresponding recurrence equations are established in Lemmas 4.3 and 4.4, respectively, and are used to generate the final result in Theorem 4.5. The proofs are similar to the results in Section 4.1.Note that the two subproblems are intertwined in the derivation, and they are solved with double dynamic programming. Lemma 4. 3 .. 4 . 34Given allowed checkpointing units in memory and the initial state stored in memory, the minimal number of additional forward steps needed for the adjoint computation of time steps using an ℓ-stage time integrator satisfies IS ( , ) = min min 1≤˜≤ −2 (˜+ IS (˜, ) + IS ( −˜, − 1)) min 2≤˜≤ −1 (˜− 1 + IS (˜− 1, ) + SV ( −˜+ 1, − 1)) Given allowed checkpointing units in memory and the stage values of the first time step stored in memory, the minimal number of additional forward steps needed for the adjoint computation of time steps using an ℓ-stage time integrator satisfies Fig. 5 . 5Strategy to solve the intertwined subproblems IS ( , ) and SV ( , ). Each subproblem is broken into smaller subproblems. A dot stands for a solution, and crosses stand for stage values; , , ℓ are the number of time steps, the number of checkpointing units, and the number of stages, respectively. − 1 + 1indicates IS ( , ) ≤ R ( , ). The second case in (10) indicates IS ( , ) ≤ min 2≤˜≤ −1 (˜− 1 + IS (˜− 1, ) + SV ( −˜+ 1, IS (˜− 1, ) + IS ( −˜, − 1 − ℓ)) . Fig. 6 . 6CAMS with general two-stage schemes (8 recomputations) Application of CAMS and Revolve to reversing 10 time steps given 6 allowable checkpointing units. Fig. 7 . 7Performance comparison of various checkpointing algorithms. The plotted data is generated for 2-stage time-stepping schemes (ℓ = 2). Figure 8 8shows how the needed recomputations vary with the number of allowable checkpointing units for a fixed number of time steps. This indicates the memory requirement of each algorithm for a particular time-to-solution budget. Fig. 8 . 8Performance comparison of various checkpointing algorithms. The plotted data is computed for time integration methods with two stages (ℓ = 2). decreasing the counter stepstogo by one in the backward time loop (Line 22) indicates that one less recomputation is needed for each backward step.We have implemented both Algorithms 1 and 2 under the TSTrajectory class in PETSc, which provides two critical API functions: TSTrajectorySet() and TSTrajectoryGet(). The former function wraps revolveForward() in forwardSweep(). The latter function wraps all the operations before adjointStep in the for loop (Lines 7-10 in Algorithms 1 and Lines 7-8 in Algorithms Algorithm 2 Proposed adjoint computation using the modified Revolve algorithm. 20: end function where = [u v] is the PDE solution vector. A reference solution is generated from the initial condition u 0 = 1 − 2v 0 , v 0 = sin 2 (4 ) cos 2 (4 )/4, ∀ , ∈ [1.0, 1.5] 0 otherwise(17) Fig. 10 . 10Theoretical predictions and performance for Revolve, modified Revolve, and CAMS-SA. Fig. 11 . 11Theoretical predictions and performance for Revolve, modified Revolve, and CAMS-GEN. In the experiments, the Runge-Kutta 4 method is used for time integration. All the runs are executed on Cori with 2, 048 MPI processes and are repeated three times to mitigate the influence of run-to-run variations. Table 1 . 1Revolve nomenclature. Table 2 . 2Mapping the Revolve output to new actions. Fig. 9. Workflow for the adjoint computation using CAMS.Algorithm 3 Proposed adjoint computation for multistage time integration using CAMS.Remove the checkpoint if no longer needed 11: end for 12: function forwardSweep( , , )Integrate tIme step Yes No Consult CAMS on whether to store a checkpoint Restore a state from the nearest checkpoint Integrate time step Consult CAMS on whether to store a checkpoint Take adjoint step Yes No Last time step? First time step? Reached state ? Yes Yes Yes Initial state Final solution No Initialize Gradient Delete the restored checkpoint if not needed Stack of checkpoints in memory Recompute Forward run Adjoint run 1: , ← camsAPI(−1, −1, , , ℓ) 2: Decide if the initial state should be stored according to 3: forwardSweep(0, , ) 4: ← adjointStep( ) 5: for − 1 to 1 do 6: Restore the closest checkpoint and get , 7: , ← camsAPI( , , , , ℓ) 8: forwardSweep( , − , ) 9: ← adjointStep( ) 10: 13: for to + − 1 do 14: ← forwardStep( ) 15: if = then 16: , ← camsAPI( , , , , ℓ) 17: Store the checkpoint and update 18: Table 4 . 4Base performance for one adjoint computation (one forward sweep followed by one reverse sweep). In the tests with implicit schemes, the grid size is 2, 048 × 2, 048. For explicit schemes, the grid size is 16, 384 × 16, 384. Time is in seconds.Time-stepping schemes Wall time for forward sweep Wall time for reverse sweep Total time Backward Euler 39.69 11.15 50.84 Crank-Nicolson 39.20 12.93 52.13 Runge-Kutta 4 1.56 2.10 3.66 https://bitbucket.org/caidao22/pkg-revolve.git ACKNOWLEDGMENTSWe thank Jed Brown and Mark Adams for providing allocations on the Cori supercomputer. Shrirang Abhyankar, Jed Brown, Emil M Constantinescu, Debojyoti Ghosh, Barry F Smith, Hong Zhang, arXiv:1806.01437PETSc/TS: A modern scalable ODE/DAE solver library. math.NAShrirang Abhyankar, Jed Brown, Emil M. Constantinescu, Debojyoti Ghosh, Barry F. Smith, and Hong Zhang. 2018. PETSc/TS: A modern scalable ODE/DAE solver library. arXiv:1806.01437 [math.NA] CasADi -A software framework for nonlinear optimization and optimal control. Joris Joel A E Andersson, Greg Gillis, Horn, B James, Moritz Rawlings, Diehl, Mathematical Programming Computation. 11Joel A E Andersson, Joris Gillis, Greg Horn, James B Rawlings, and Moritz Diehl. 2019. 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SIAM Journal on Scientific Computing 44, 1 (2022), C1-C24. https://doi.org/10.1137/21M140078X FATODE: a library for forward, adjoint, and tangent linear integration of ODEs. Hong Zhang, Adrian Sandu, 10.1137/130912335SIAM Journal on Scientific Computing. 365Hong Zhang and Adrian Sandu. 2014. FATODE: a library for forward, adjoint, and tangent linear integration of ODEs. SIAM Journal on Scientific Computing 36, 5 (oct 2014), C504-C523. https://doi.org/10.1137/130912335 Government License (will be removed at publication): The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory. DE-AC02-06CH11357Argonne"). Argonne, a U.S. Department of Energy Office of Science laboratoryGovernment License (will be removed at publication): The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory ("Argonne"). 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[ "https://github.com/caidao22/pkgcams." ]
[ "MODIFIED WATERSHED APPROACH FOR SEGMENTATION OF COMPLEX OPTICAL COHERENCE TOMOGRAPHIC IMAGES", "MODIFIED WATERSHED APPROACH FOR SEGMENTATION OF COMPLEX OPTICAL COHERENCE TOMOGRAPHIC IMAGES" ]
[ "Maryam Viqar \nInstitute of Optical Materials and Technologies\nAcad. G. Bonchev Str\nBulgarian Academy of Sciences\n109, 1113SofiaBulgaria\n", "Violeta Madjarova \nInstitute of Optical Materials and Technologies\nAcad. G. Bonchev Str\nBulgarian Academy of Sciences\n109, 1113SofiaBulgaria\n", "Elena Stoykova \nInstitute of Optical Materials and Technologies\nAcad. G. Bonchev Str\nBulgarian Academy of Sciences\n109, 1113SofiaBulgaria\n" ]
[ "Institute of Optical Materials and Technologies\nAcad. G. Bonchev Str\nBulgarian Academy of Sciences\n109, 1113SofiaBulgaria", "Institute of Optical Materials and Technologies\nAcad. G. Bonchev Str\nBulgarian Academy of Sciences\n109, 1113SofiaBulgaria", "Institute of Optical Materials and Technologies\nAcad. G. Bonchev Str\nBulgarian Academy of Sciences\n109, 1113SofiaBulgaria" ]
[ "Materials, Methods & Technologies ISSN" ]
Watershed segmentation method has been used in various applications. But many a times, due to its over-segmentation attributes, it underperforms in several tasks where noise is a dominant source. In this study, Optical Coherence Tomography images have been acquired, and segmentation has been performed to analyse the different regions of fluid filled sacs in a lemon. A modified watershed algorithm has been proposed which gives promising results for segmentation of internal lemon structures.
10.48550/arxiv.2303.16609
[ "https://export.arxiv.org/pdf/2303.16609v1.pdf" ]
257,805,213
2303.16609
d32e0405060b7c9f620f3a7599d959bd9bc02399
MODIFIED WATERSHED APPROACH FOR SEGMENTATION OF COMPLEX OPTICAL COHERENCE TOMOGRAPHIC IMAGES Maryam Viqar Institute of Optical Materials and Technologies Acad. G. Bonchev Str Bulgarian Academy of Sciences 109, 1113SofiaBulgaria Violeta Madjarova Institute of Optical Materials and Technologies Acad. G. Bonchev Str Bulgarian Academy of Sciences 109, 1113SofiaBulgaria Elena Stoykova Institute of Optical Materials and Technologies Acad. G. Bonchev Str Bulgarian Academy of Sciences 109, 1113SofiaBulgaria MODIFIED WATERSHED APPROACH FOR SEGMENTATION OF COMPLEX OPTICAL COHERENCE TOMOGRAPHIC IMAGES Materials, Methods & Technologies ISSN 162022Journal of International Scientific Publications www.scientific-publications.net Page 180watershedsegmentationoptical coherence tomographyHann windowing Watershed segmentation method has been used in various applications. But many a times, due to its over-segmentation attributes, it underperforms in several tasks where noise is a dominant source. In this study, Optical Coherence Tomography images have been acquired, and segmentation has been performed to analyse the different regions of fluid filled sacs in a lemon. A modified watershed algorithm has been proposed which gives promising results for segmentation of internal lemon structures. OCT is a high-resolution, contactless, cross-sectional imaging technique capable of acquiring depth information in a transparent object within a range of few millimeters. It is based on the principle of low coherence interferometry. Most commonly used interferometers are fiber-based Michelson type and Mach-Zehnder type interferometers. In Michelson type, the light field is directed to a beam splitter that splits the light into two beams: one illuminates the sample and the other illuminates the reference arm. Then, the reflections from the sample arm and the reference arm propagate back to the beam splitter and get combined to form an interference pattern. The interference will only appear when the two optical path lengths are matched to within the coherence length of the light source. Based on the methods for detection of the interference pattern and post processing to provide information of the in-depth structures having varying refractive indices, the OCT is generally divided into time domain OCT (TD-OCT) and Fourier domain OCT (FD-OCT). In TD OCT each depth scans are acquired by movement of the reference mirror. In FD-OCT, the interference signal is either spectrally resolved at the detector, or a light source that sweeps through a given wavelength range. The obtained spectrally resolved interference signal is then Fourier transformed to generate information of the in-depth structures having varying refractive indices. The three-dimensional scans in OCT are obtained by incorporating multiple depth and lateral X-Y scans. Generally, the resolution of the OCT system is in the range of few micrometers whereas the system can support a depth penetration of a few millimeters [2]. The OCT based images can be used to extract additional information, which is not visible to a naked eye. The images can also be processed to perform segmentation and classification operations for detection/extraction of complex structures which otherwise would consume lot of time and money due to the requirement of trained professionals. To obtain morphological information, several Artificial Intelligence (AI) based methods have been proposed. These models are designed to extract specific layers and structures, to classify diseases in animals, to distinguish age, variety, specie, quality of fruits and vegetables. In this work, volume dataset is acquired for complex internal structures of lemon. The analysis of the internal structure of lemon requires a highly accurate segmentation method which can separate the fluid-filled sacs so that these structures can be used to estimate the age and quality of the lemon fruit in a non-destructive manner. To perform this task, a segmentation method is put forward for segmentation of inner sections of lemon called carpels which are fluid filled (juice) sacs. Several segmentation techniques have been proposed with application in fields of bio-medical imaging, bio-technology, agriculture, biology, etc. Image segmentation methods usually segregate the regions of interest (objects) in an image based on lines, curves, boundaries, etc. Generally, the techniques are application specific and universal segmentation techniques are yet a challenging task. Commonly used segmentation algorithms are based on following approaches: (i) region expanding methods as in [3]- [11], (ii) thresholding [12]- [13], (iii) graph-based methods [14], etc. One of the commonly used techniques is Watershed segmentation [3] which is based on topographical features. The image is considered as a surface having varying topographic features relative to the intensities in the image. This surface is filled with water starting with basins created due to higher areas (mountains) surrounding lower ones. The water is made to flow from the lower points in basins (local minima) until it starts merging with the water from different basins, where the watershed lines or dams are built. The water-flow ends when the water reaches to the highest point on the surface. The watershed transform suffers from over-segmentation problem as huge number of unwanted segments are generated due to noise present and inherent texture details of the image. Hence, several models have been proposed with advancements to avoid the over-segmentation problem. Belaid et. al. in [4] proposed a watershed-based model with adaptive topological gradient method. The method results in attenuation of over-segmentation problem along with less computational complexity making it suitable even for real time applications. A marker-based method presented in [5] uses basic morphological operations by segmenting a skeleton for the edge map. The aim is to keep the fragments of same region together. For the connected segments in the skeleton, the seeds are evaluated by finding out the local minima, to which the dilation process is applied. In this manner, the overlapping regions are combined together and used for segmentation, allowing lesser cancellation or addition of false edges apart from original edge map. Another computationally efficient modified watershed algorithm was introduced by Chien et. al. in [6] for segmentation in videos. This method exploits the temporal coherence and only modifies watersheds for each next frame rather than searching for watershed in all the frames. Furthermore, it uses Intra-inter watershed scheme wherein the I-watershed frames are inserted to maintain accuracy and avoid error propagation. An alternative to watershed for brain tumours is proposed by Aslam et. al. [7] which follows the motion of a falling ball on the surface having varying topography. The method is a region growing segmentation technique which uses Fuzzy logic for decision making. These segmentation methods suffer from several problems, such as application specific nature, computational complexity, over or under segmentation, etc. Hence, depending on the image, the amount of noise present, the textural details, the above-mentioned methods fail to give optimum performance. Typical structures like that of lemon fluid sacs are complex in nature and moreover noise is commonly experienced phenomenon in the images acquired using the OCT set-up due to back-scattering of light. Hence, in this work a method is proposed to overcome the problem of unwanted segmentation due to noisy and inherent textural details of an image employing the watershed transform as the back-bone. The novelty of the work lies in its segmentation accuracy even in the presence of noise and the ability to limit over-segmentation. The proposed model firstly pre-processes the image in three stages: Stage I where Hann transform is applied on raw A-scan to smoothen the effect of edges and corners; Stage II wherein pre-processing using the contours followed by gradient and marker-based method is used to segregate the foreground and background using the morphological operations; in Stage III the watershed transform is applied to segment into regions based on topography. The pre-processing stages involving the Hann operator and marker-based technique respectively, enhance the performance of the conventional watershed method. It identifies the real edges efficiently and henceforth, helps to avoid over-segmentation. The remaining section of the paper contains the following information: Section II describes the watershed transform and the proposed modified watershed transform model, Section III gives the results Section IV provides discussion and comparison of the results, Section V gives concluding remarks., followed by list of references. MATERIALS AND METHODS OCT Set-up and lemon The used OCT setup is produced by Optores in Munich, Germany [15]. It is a FD OCT system with Fourier Domain Mode Lock (FDML) laser [16] which operates at 1.6 MHz sweeping rate with 100 nm sweeping range at central wavelength 1309 nm. The depth resolution is calculated to 17 micro-meters and a lateral resolution to 40 micro-meters. The schematics of the acquisition set-up is given in Fig. 1 (a). The area scanned is of size 7 mm x 7 mm (1024 by 1024 points). The sample arm of the fiber-based interferometer is connected to a scanning head via an optical fiber. The scanning head performs the X-Y scanning. In this FD-OCT, the source sweeps through an optical bandwidth, while the detector acquires signal at each evenly distributed wavelength. This spectrally resolved signal is Fourier transformed to get depth-resolved profile. The line scan acquisition is done at several points by laterally scanning on the sample in X direction to get the complete 2D depth-image (B-scan). The volume scan is acquired by performing additional scan in Y direction. A fresh lemon was brought from Local Vegetable and Fruits market in Sofia, Bulgaria. It was peeled, placed on a Petri-dish on a height-adjustable table. The volume dataset was acquired using the OCT setup as shown in Fig.1 (a). while the lemon structure can be seen in Fig. 1 (b) and can be described as Journal of International Scientific Publications www.scientific-publications.net follows: (i) the outer layers of the lemon are exocarp and mesocarp which are followed by an inner part called the endocarp; (ii) the inner endocarp is segmented into main lobes by carpel which contains the juice filled fluid sacs. The fluid filled sacs contain the juice which is the most important part of the lemon as afar as edibility of the lemon is concerned. These sacs are of irregular shape and size making an overall complex structure to segment. Methodology The used watershed algorithm is based on the topological representation of an image. The image is assumed as surface flooded with water, wherein the water flows towards the lower areas as a result of gravitational force. Three regions are identified this way: (i) Minima, (ii) Watersheds and (iii) Catchment basins. The altitudes of the topological surface are designated using the gradient of the image. When water flows, firstly it goes towards the lowest points of the catchment basin. Furthermore, as the water is allowed to flow, the water reaches the top of catchment basins and may start to merge with neighbouring flows. This identifies the watershed, which corresponds to details of the image having high gradient information, which can be employed to segregate the regions of an image. The process of water-flow is immersion type in which the water emerges out of the spots at the minima of the basins and it over-flows the height of surrounding boundaries. The points of merger of this over-flow water are the lines where the dam is to be built. Upon flooding the complete surface with water, these dams are the watershed lines of the image. There are many implementations of the watershed algorithm [3]- [8] amongst which Vincent and Soille's algorithm [3] is computationally less complex and is described as below: • Firstly, the gradient of the complete image is calculated. • These gradient values are used to sort the pixels values of the image, starting with the lower gradient values. Special tags are given to pixels of connected areas. • First the lower gradient values are processed following the higher ones. On each of the gradient levels, pixels with tagged neighbours are inserted into a priority pipeline first. Initial tag of the pixel is decided by its neighbours in the pipeline. After processing of the tagged neighbouring pixels, untagged ones are also tagged. • After the tagging of all the pixels, the outlines separating the regions with different tags are the watershed lines of the image. The level of R increases from its minimum to maximum value, which results in expansion of basins. The union of sets of basins calculated at step R, are considered as " . The union of all the minima corresponding to different regions for height R, the watershed can be defined using the following recursion: * " !"# = " !"# , ∀ ∈ [ #$% , #&' − 1], "() = "() ∪ * "() ( " )9(1) In eq.1, * "() = ⋃ * "() % $+) ( " $ ), where n is number of minima of gray-level image I and the * $%& ( $ ) is defined in eq.2 as follows: Ω ( $ ) = { Ω, ∀ ≠ , Ω ( , $ ) ≤ Ω ( , % )} (2) For a grey-level image I, set " !'( is equivalent to set of catchment basins. The compliment of the " !'( in Ω gives the watershed of the image [4]. The watershed algorithm offers many benefits, firstly it results in to closed segments which are important in segmentation tasks and secondly it requires lesser time for processing compared to many of the state-of-art methods. But in addition to these advantages, there are also disadvantages associated with this method, over-segmentation being the most problematic one. The marker-based segmentation technique is an example of over-segmentation avoiding technique. In this work, we propose a pre-processing along with employing the marker model for the watershed transform to solve the problem more effectively. The pre-processing model makes use of Hann windowing which is a window function used for tapering the high-frequency features in an image. It is applied after the acquisition of A-scan prior to Fourier Transform. The OCT scans are pre-processed using the thresholding with a value of 245 (chosen heuristically), filling of holes and use of Chan-Vese and Edge based active contouring to get a binary image. The marker-based foreground and background demarcation followed by the watershed algorithm is used to segment the internal structures of lemon. The block diagram of the proposed scheme is presented in Fig. 2. The marker model works by identifying the foreground and background objects. The foreground objects are marked as connected sets of pixels covering the region of an individual object. One of such techniques is to use morphological operations namely reconstruction-based opening and closing. While using this operation, the markers reach even up to the level of the edges of the objects. Hence to erase the edges from the marked fore-ground regions, closing operation is followed by an erosion process. Furthermore, to mark the background regions basic thresholding process is done. But this process, marks down the whole background region whereas the desired result includes only the markers for the background region. To get these markers, the "skeleton by influence zones" technique is used. Lastly, minima are computed for regions of objects marked as foreground and background. This sums up the process of the marker, which is followed by the watershed segmentation described above [5]. The watershed along with the marker-based foreground and background pre-processing is known as the marker-based watershed transform. Though it aims to avoid the over-segmentation problem it does not address the problem efficiently. Hence, in this work more pre-processing techniques are used along the marker-based segmentation to attain the desired result for the segmentation of complex structures as seen in a lemon. The over-segmentation in the watershed algorithm is present many times due to edge-based artefacts which are very common in OCT images. In such cases, to supress these edge effects the use of the Hann window function prior to segmentation followed by pre-processing using markers and contours is proposed in this work. The Hann window function is one such window which reduces the impact of the edge and corner by tapering the image-based signal and can be expressed as follows: -&%% ( , ) = (1 − cos( !/$ 01) ))(1 − cos G !/2 31) H)(3) In eq.3, i, j, P and Q represent the current horizontal and vertical positions, width and height of a block respectively. This function is a 2-dimensional, which increases the efficiency of the segmentation method by prioritizing the central regions of an object (to be segmented) rather than their edges and corners [15]. RESULTS The results of the proposed and original watershed model along with inter-mediate steps are presented in Fig. 4. The original 3D OCT volume data are given in Fig. 3 and one slice of the original image having size 1024 × 900 pixels (7 mm by 5 mm) is shown in Fig. 4a as captured by the OCT system. The volume dataset consists of several B-scans, where each B-scan was extracted as an individual 2-D image and modified watershed processing was performed on these scans. Each single image is a grey-scale image afflicted with noise. The binary image after the pre-processing contour-based operation is represented in Fig. 4b, In summary, the segmented area are blobs of pixels marked with outliners (marked with white) to separate foreground from background region. Initial pre-processed image works as a foreground marker but it cannot separate out the fluid containing areas from the boundaries of the sacs. For extracting the edge-map of the fluid filled regions, further marker-based watershed transform is applied. It is able to extract the edges of the fluid filled sacs leaving behind the fibrous structures and skin area of lemon which do not have useful fluid content. Generally, most of image acquisition techniques like OCT in this work are afflicted with different kinds of distortions due to surrounding environment, capture mechanism, transmissions, mathematical processing algorithms. It is highly difficult to obtain distortion-less images. The time required for watershed transform is 0.18 seconds, whereas the time for the proposed modified watershed is 2.13 seconds. The time complexity has increased due to additional pre-processing algorithms but the segmentation outputs are far more practically applicable. Hence, methods must be such that even in presence of noise, they should be capable enough to segment accurately. DISCUSSION The modified approach which in addition to watershed segmentation, includes gradient, contour-marker and Hann windowing approaches is quite capable to segment the fluid filled sacs even in the presence of false edges and corners generated due to noise. The watershed approach is good when the image has only true edge and no effects due to external sources. The Hann-window works well to taper the effect of edges, and results into an image with dominant edges, as usually the edges due to noise do not have pronounced effect. The marker-based method further helps in marking the regions of interest using the foreground and background technique effectively. Lastly, the watershed technique when applied to such pre-processed images, results into appropriate segmented areas. The accurate segmentation results come at the cost of higher computational complexity as the time for processing increased by several folds. CONCLUSIONS The watershed technique is powerful for segmentation, but has certain limitations due to presence of artificially created points, corners, boundaries, edges, etc. The technique is not well-suited for images afflicted to distortion of types speckle noise, blocking artifacts, Gaussian noise, etc. The use of Hann windowing helps to smoothen and supress the effect of edges. This pre-processing step helped significantly to segment the lemon sacs more accurately avoiding the presence of undesired segmented areas arising due to distortions. These methods can in future be used to analyse the citrus fruits like lemon to estimate the quality, life or juice content. More processing would be required to relate the properties with the segmented regions as many times large segmented fluid sacs will not depict proportionality to juice content. The skin structures can be examined and automated methods can be developed for biotechnology to estimate the ageing in the lemon fruit. Fig. 1 1(a) Schematics of the acquisition set-up for image acquisition of lemon (b) Structure of lemon showing the external and interior portions in lemon. Internal structure consisting of fluid filled sacs Fig. 2 2Block diagram of the proposed modified Hann windowing and markerthe watershed can expressed as follows: Let the image be w(x,y), with (x,y) ∈ ℝ ! , Ω is the open-bounded domain of ℝ ! . The pixels c of Ω represent the pixels in catchment basin D(M) having M as the minima. The threshold set of w for a height R be " = { ∈ Ω, w(c)≤R}, and #$% and #&' are the minimum and maximum values of image w. Fig. 3 33D Fig. 4 4(a) Original OCT image of lemon (b) Pre-processed binary image (c) Gradient Magnitude (d) Watershed lines (e) Original watershed segmentation image ( where we can observe a binary image with showing different regions. Furthermore, Fig. 4c shows gradient estimation prior to marking of foreground and background regions. The watershed lines which separate regions based on their topographic characteristics are shown in Fig. 4d. The regions of fluid filled sacs detected by the modified watershed algorithm are shown in Fig.4f. It can be visualised that the modified approach can efficiently segment the sacs filled with lemon fluid demonstrating clearly the edges of each sac. In contrast to this, the results of original watershed segmentation in Fig. 4e exhibit over-segmentation. In Figs. 4e and 4f, the colours are for better visualization of the labelled regions; the original watershed's outcomes incorporate the noise into false edge and produce regions for original edges, noise generated edge and combination of both. ) . . Materials, Methods & Technologies Optical coherence tomography's current clinical medical and dental applications: a review. S Ali, S B S Gilani, J Shabbir, K S Almulhim, A Bugshan, I Farooq, Ali, S., Gilani, S.B.S., Shabbir, J., Almulhim, K.S., Bugshan, A. and Farooq, I., 2021. Optical coherence tomography's current clinical medical and dental applications: a review. F1000Research, 10. W Drexler, Fujimoto, Optical coherence tomography: technology and applications. BerlinSpringer2Drexler, W. and Fujimoto, J.G. eds., 2015. Optical coherence tomography: technology and applications (Vol. 2). Berlin: Springer. Watersheds in digital spaces: an efficient algorithm based on immersion simulations. L Vincent, P Soille, IEEE Transactions on Pattern Analysis & Machine Intelligence. 1306Vincent, L. and Soille, P., 1991. Watersheds in digital spaces: an efficient algorithm based on immersion simulations. IEEE Transactions on Pattern Analysis & Machine Intelligence, 13(06), pp.583-598. Image segmentation: a watershed transformation algorithm. Lamia Belaid, Walid Jaafar, Mourou, Image Analysis & Stereology. 282Belaid, Lamia Jaafar, and Walid Mourou. "Image segmentation: a watershed transformation algorithm." Image Analysis & Stereology 28, no. 2 (2009): 93-102. 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A novel falling-ball algorithm for image segmentation. arXiv preprint arXiv:2105.02615. Watershed segmentation brain tumor detection. P Dhage, M R Phegade, S K Shah, 1314-72692015 International Conference on Pervasive Computing (ICPC) (pp. 1-5). IEEE. Materials, Methods & Technologies ISSN. 16Dhage, P., Phegade, M.R. and Shah, S.K., 2015, January. Watershed segmentation brain tumor detection. In 2015 International Conference on Pervasive Computing (ICPC) (pp. 1-5). IEEE. Materials, Methods & Technologies ISSN 1314-7269, Volume 16, 2022 Journal of International Scientific Publications www.scientific-publications.net Region growing: a new approach. S A Hojjatoleslami, J Kittler, IEEE Transactions on Image processing. 77Hojjatoleslami, S.A. and Kittler, J., 1998. Region growing: a new approach. IEEE Transactions on Image processing, 7(7), pp.1079-1084. A new approach to combining region growing and edge detection. J P Gambotto, Pattern recognition letters. 1411Gambotto, J.P., 1993. A new approach to combining region growing and edge detection. Pattern recognition letters, 14(11), pp.869-875. Seeded region growing. R Adams, L Bischof, IEEE Transactions. 166Adams, R. and Bischof, L., 1994. Seeded region growing. IEEE Transactions on pattern analysis and machine intelligence, 16(6), pp.641-647. Local threshold and boolean function based edge detection. M B Ahmad, T S Choi, IEEE Transactions on Consumer Electronics. 453Ahmad, M.B. and Choi, T.S., 1999. Local threshold and boolean function based edge detection. IEEE Transactions on Consumer Electronics, 45(3), pp.674-679. Brain tumor detection by various image segmentation techniques with introduction to non negative matrix factorization. S J Prajapati, K R Jadhav, Brain. 43Prajapati, S.J. and Jadhav, K.R., 2015. Brain tumor detection by various image segmentation techniques with introduction to non negative matrix factorization. Brain, 4(3), pp.600-603. Medical image segmentation and its application in cardiac MRI. D Wei, C Li, Y Sun, Biomedical Image Understanding, Methods and Applications. Wei, D., Li, C. and Sun, Y., 2015. Medical image segmentation and its application in cardiac MRI. Biomedical Image Understanding, Methods and Applications, pp.47-89. High-speed OCT light sources and systems. T Klein, R Huber, Biomedical optics express. 82Klein, T. and Huber, R., 2017. High-speed OCT light sources and systems. Biomedical optics express, 8(2), pp.828-859. Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography. R Huber, M Wojtkowski, J G Fujimoto, Optics express. 148Huber, R., Wojtkowski, M. and Fujimoto, J.G., 2006. Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography. Optics express, 14(8), pp.3225-3237. Introducing Hann windows for reducing edge-effects in patchbased image segmentation. N Pielawski, C Wählby, PloS one. 153229839Pielawski, N. and Wählby, C., 2020. 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[ "A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming", "A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming", "A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming", "A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming" ]
[ "Olli Herrala \nDepartment of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland\n", "Tommi Ekholm \nFinnish Meteorological Institute\nHelsinkiFinland\n", "Fabricio Oliveira \nDepartment of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland\n", "Olli Herrala \nDepartment of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland\n", "Tommi Ekholm \nFinnish Meteorological Institute\nHelsinkiFinland\n", "Fabricio Oliveira \nDepartment of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland\n" ]
[ "Department of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland", "Finnish Meteorological Institute\nHelsinkiFinland", "Department of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland", "Department of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland", "Finnish Meteorological Institute\nHelsinkiFinland", "Department of Mathematics and Systems Analysis\nSchool of Science\nAalto University\nFI-00076AaltoFinland" ]
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Despite methodological advances for modeling decision problems under uncertainty, faithfully representing endogenous uncertainty still proves challenging, both in terms of modeling capabilities and computational requirements. A novel framework called Decision Programming provides an approach for solving such decision problems using off-the-shelf mathematical optimization solvers. This is made possible by using influence diagrams to represent a given decision problem, which is then formulated as a mixed-integer linear programming problem.In this paper, we focus on the type of endogenous uncertainty that received less attention in the introduction of Decision Programming: conditionally observed information. Multistage stochastic programming (MSSP) models use conditional non-anticipativity constraints (C-NACs) to represent such uncertainties, and we show how such constraints can be incorporated into Decision Programming models. This allows us to consider the two main types of endogenous uncertainty simultaneously, namely decision-dependent information structure and decision-dependent probability distribution. Additionally, we present a decomposition approach that provides significant computational savings and also enables considering continuous decision variables in certain parts of the problem, whereas the original formulation was restricted to discrete variables only.The extended framework is illustrated with two example problems. The first considers an illustrative multiperiod game and the second is a large-scale cost-benefit problem regarding climate change mitigation. Neither of these example problems could be solved with existing frameworks.
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[ "https://export.arxiv.org/pdf/2304.02338v1.pdf" ]
257,952,259
2304.02338
f9d7b49a8c428c00e797d3abf4a7709016cc39c0
A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming Olli Herrala Department of Mathematics and Systems Analysis School of Science Aalto University FI-00076AaltoFinland Tommi Ekholm Finnish Meteorological Institute HelsinkiFinland Fabricio Oliveira Department of Mathematics and Systems Analysis School of Science Aalto University FI-00076AaltoFinland A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming endogenous uncertaintystochastic programmingdecision programmingclimate change mitigation Despite methodological advances for modeling decision problems under uncertainty, faithfully representing endogenous uncertainty still proves challenging, both in terms of modeling capabilities and computational requirements. A novel framework called Decision Programming provides an approach for solving such decision problems using off-the-shelf mathematical optimization solvers. This is made possible by using influence diagrams to represent a given decision problem, which is then formulated as a mixed-integer linear programming problem.In this paper, we focus on the type of endogenous uncertainty that received less attention in the introduction of Decision Programming: conditionally observed information. Multistage stochastic programming (MSSP) models use conditional non-anticipativity constraints (C-NACs) to represent such uncertainties, and we show how such constraints can be incorporated into Decision Programming models. This allows us to consider the two main types of endogenous uncertainty simultaneously, namely decision-dependent information structure and decision-dependent probability distribution. Additionally, we present a decomposition approach that provides significant computational savings and also enables considering continuous decision variables in certain parts of the problem, whereas the original formulation was restricted to discrete variables only.The extended framework is illustrated with two example problems. The first considers an illustrative multiperiod game and the second is a large-scale cost-benefit problem regarding climate change mitigation. Neither of these example problems could be solved with existing frameworks. Introduction Stochastic programming (SP) is one of the most widespread mathematical programmingbased frameworks for decision-making under uncertainty. In general, SP casts decision problems subject to parametric uncertainty as deterministic equivalents in the form of large-scale linear or mixed-integer programming (LP/MIP) problems that can be solved with standard optimization techniques. One important feature related to SP models is the scenario tree. A scenario tree is a structural description of the dynamics between the random variables that model the uncertain elements. In this setting, the random variables are typically assumed to have discrete finite support, for which each possible realization and respective probability are known. The literature has seen a wealth of applications that rely on SP and associated developments (Ruszczyński and Shapiro, 2003;Wallace and Ziemba, 2005). The most widespread form of SP problems is two-stage stochastic programming (2SSP), emerging from problems with only two stages in the scenario tree. Settings with more than two stages are referred to as multi-stage problems. Multi-stage stochastic programming (MSSP) has likewise seen intensive developments in the last few decades, mostly motivated by developments in closely related fields such as optimal control and reinforcement learning. MSSP models arise at the intersection between dynamic programming and SP. Dynamic programming (DP) is a framework specialized for problems in which dynamics play a central role (Bellman, 1966), and its extension to problems involving stochastic processes is known as stochastic dynamic programming (SDP). MSSP problems are essentially SDP problems and can therefore utilize the sophisticated solution techniques available for SDP problems. A significant limitation in dynamic programming, however, is the no-forgetting assumption stating that at each decision stage, the full history of the stochastic decision process must be remembered. Dynamic programming circumvents this issue by considering Markov decision problems satisfying the Markov property, that is, given the current state and action, the next state is conditionally independent of the previous states and actions (Bellman, 1961). While the Markov property makes it possible to solve large decision problems, it has two main disadvantages. First, many problems, including ones with multiple cooperating decision-makers (Lauritzen and Nilsson, 2001), do not satisfy the Markov property. Second, the assumption of conditional independence makes considering risk measures such as value-at-risk generally challenging for approaches based on dynamic programming because the optimal strategy in such problems depends on not only the history of the process but also the decisions that would be made if the history was different. A notable framework combining the advantages of MSSP and DP is stochastic dual dynamic programming, presented in Pereira and Pinto (1991) and Zou et al. (2019), which can consider conditional value-at-risk while assuming the Markov property. A common assumption in SP models is that the stochastic processes, particularly the state probabilities and/or observed values, are not influenced by the previously made decisions. The uncertainty is thus exogenous. This is methodologically convenient, for the deterministic equivalent model has the same nature as its stochastic counterpart, retaining important characteristics such as linearity, or more generally, convexity. In this paper, we focus on the much less explored class of stochastic problems presenting endogenous uncertainty. In this more general setting, the decisions made at previous stages can affect the uncertainty faced in later stages. This endogenous nature is present in several important problems that have been addressed in the literature, such as disaster resilience in road networks (Peeta et al., 2010) and oilfield planning (Apap and Grossmann, 2017). Nonetheless, the literature considering endogenously uncertain SP problems is still scarce, particularly in comparison to the body of SP literature, likely due to the much more challenging nature of these problems. It is common to classify SP problems according to the nature of the endogenous structure arising in the decision problem. Hellemo et al. (2018) proposes a taxonomy of such problems, classifying the endogenous uncertainties into two distinct types. In Type 1 problems, earlier decisions influence the later events' probability distribution (i.e., realizations and/or the probabilities associated with each realization). For example, deciding to perform maintenance on a car engine influences the probability of it breaking in the future. In Type 2 problems, the information structure is influenced by the decision-making. Specifically, the decisions influence when information about the stochastic process becomes available without influencing the probabilities or observable outcomes associated with the random variables. Continuing with the car example, deciding to examine the engine does not affect the probability of it breaking, but provides information that enables a better-informed maintenance decision. It is noteworthy that Type 2 is more common in the literature on decision-making under endogenous uncertainty. Hellemo et al. (2018) also introduce Type 3, which combines the Type 1 and Type 2 endogenous uncertainties. Our main contribution is to provide a framework general enough to address Type 3 endogenously uncertain SP (T3ESP) problems, while still retaining prior computationally favorable properties of the mathematical model such as linearity or convexity. Our proposed framework builds upon Decision Programming (Salo et al., 2022) and expands its capabilities to consider Type 2 endogenous uncertainty. Decision Programming was designed to incorporate Type 1 endogenous uncertainties in the decision process in an intuitive way by using an influence diagram representation of the problem. This representation is then converted to a mixed-integer linear programming (MILP) problem, for which powerful off-the-shelf solvers exist. Salo et al. (2022) presents a simple problem with Type 2 endogenous uncertainty but does not discuss this class of endogenous uncertainty in detail. This paper explores Type 2 endogenous uncertainty further, showing explicitly how Decision Programming can be enhanced to become a suitable framework for T3ESP problems. The formulation presented in Salo et al. (2022) is designed for discrete decisions, and the authors acknowledge its limitations regarding accommodating continuous decisions. In addition to discussing T3ESP problems, we present a decomposition approach that not only provides significant computational savings but also allows for including continuous decision variables in the resulting subproblems, taking a step in the direction of incorporating continuous decision variables more generally in the Decision Programming framework. In this paper, two example problems are presented to demonstrate these developments. The first showcases the modelling of Type 3 endogenous uncertainty and the computational benefits of the decomposition. After this, we employ our developments in a climate change mitigation case study that originally motivated our developments. In this case study, we consider uncertain climate parameters and technological progress. This paper is structured as follows. In Section 2, we present an overview of multi-stage stochastic programming. In Section 3, our methodological contributions are described in detail, starting from the formulation in Salo et al. (2022) and continuing with conditionally observed information and the decomposition. In Section 4 we illustrate the use of the framework by considering a larger-scale problem of climate change cost-benefit analysis. Section 5 concludes and provides directions for further development. Modeling problems with endogenous and exogenous uncertainties Problems with exogenous uncertainty have been researched since the introduction of stochastic programming by Dantzig (1955). Since then, research on stochastic programming has introduced recourse problems and the notion of multi-stage stochastic programming (MSSP). Walkup and Wets (1967) first introduce the term stochastic program with recourse, denoting problems where decisions and random elements alternate sequentially. After making the firststage decision, a random event occurs and the second-stage recourse action is chosen accordingly. Ruszczyński and Shapiro (2003) and Birge and Louveaux (2011) present comprehensive overviews of the stochastic programming methodology. Solution approaches for MSSP are often based on formulating the deterministic equivalent problem using a scenario tree, as described in, e.g., Ruszczyński (1997). A scenario tree represents the structure of the uncertain decision process, and non-anticipativity constraints (NACs) (Rockafellar and Wets, 1991) are employed to enforce the information structure in the formulation. NACs state that a decision must be the same for two scenarios if those scenarios are indistinguishable when making the decision. Notable applications of MSSP include financial portfolio management (Yin and Han, 2020) and energy systems planning (Lu et al., 2019). Additionally, supply chain (Alonso-Ayuso et al., 2003) and production (Goel and Grossmann, 2006) planning have an inherent structure that can be formulated as an MSSP problem, with multiple stages connected through, e.g., inventory constraints, resulting in large-scale problems. To alleviate the computational challenges, numerous heuristics and approximation methods have been proposed. These include knapsack-based methods (Huang and Ahmed, 2009) and (Lagrangian) dual decomposition (Carøe and Schultz, 1999). A recent review on multi-stage optimization under uncertainty can be found in Bakker et al. (2020). In endogenously uncertain problems, the decisions can affect the timing, event probabilities, or outcomes of uncertain events further in the process. As previously discussed, endogenous uncertainty is often divided into decision-dependent probabilities (Type 1) and decision-dependent information structure (Type 2) (Hellemo et al., 2018). In this context, information structure often refers to when the realization of each uncertain event is observed, if ever. In contrast, exogenously uncertain problems have a fixed information structure with the timing of observations known a priori. Type 2 uncertainty has been more widely addressed in the SP literature, perhaps due to one of its subclasses having a strong connection to exogenously uncertain problems. In the taxonomy presented in Hellemo et al. (2018), this specific type of endogenous uncertainty is called conditional information revelation. In this subclass, the decisions only affect the time at which the (exogenous) uncertainty is revealed to the decision maker. One of the earliest publications on such uncertainty is Jonsbråten et al. (1998), where the authors describe a branching algorithm for solving a subcontracting problem. Goel and Grossmann (2006) consider a process network problem where the yield of a new process is uncertain prior to installation. Other applications include open pit mining (Boland et al., 2008), clinical trial planning for drug development (Colvin and Maravelias, 2010) and technology project portfolio management (Solak et al., 2008). Similar solution methods are employed in problems with exogenous and Type 2 endogenous uncertainties. The main difference is that conditional information revelation requires the use of conditional non-anticipativity constraints (C-NACs), as the distinguishability is dependent on earlier decisions. This conditional dependency results in disjunctive constraints that require specific reformulation techniques (Apap and Grossmann, 2017). The main challenge arising from this approach is that the number of constraints rapidly increases with problem size, resulting in computational intractability for large problems. Apap and Grossmann (2017) also propose omitting redundant constraints, in an attempt to mitigate the tractability issues. In their example production planning problem, this results in roughly a 99% decrease in the problem size. Despite these improvements, the reduced model is still very large and cannot be solved to the optimum within a reasonable computation time under their experimental setting, illustrating how challenging such problems are. In addition to information revelation, Type 2 uncertainty also encompasses problems where the information structure is altered by modifying the support of decision variables and changing the objective or constraint coefficients. Hellemo et al. (2018) report a 2SSP problem where the recourse costs depend on first stage decisions (Ntaimo et al., 2012). Gustafsson and Salo (2005) present a model for contingent portfolio programming, a project scheduling problem where projects can be expanded or terminated before they are finished. The decisions thus affect the decision spaces of future decision variables, resulting in additional consistency constraints, e.g., that a project can be continued in period t only if it was ongoing in period t − 1. Type 1 endogenous uncertainty is more challenging from a mathematical modeling standpoint because the uncertain events depend on earlier decisions, which, in turn, precludes a scenario tree-based representation as the scenario probabilities in a tree cannot depend on decisions. Therefore, the well-established solution techniques for MSSP cannot be directly applied to these problems. Despite these challenges, some discussion on Type 1 endogenous uncertainty is found in the literature. Peeta et al. (2010) discuss the fortification of a structure in a network, where the probability of failure depends on the fortification decision. Lauritzen and Nilsson (2001) present the "pig farm problem", where the health of a pig depends on the treatment decisions, introducing the concept of limited memory influence diagrams (LIMID), thus relaxing the noforgetting assumption in influence diagrams. The no-forgetting assumption states that when making a decision, all prior decisions and outcomes of uncertain events are known. This assumption is a significant limitation in distributed decision-making and leads to computational challenges in multi-period problems where the decisions towards the end are conditional on the full history of the problem. Dupačová (2006) presents a summary of problems with Type 1 uncertainties and Escudero et al. (2020) presents solution approaches to multi-stage problems where the first-stage decisions influence the scenario probabilities in later stages. Examples of such problems can be found in Zhou et al. (2022) and Li and Liu (2023). Finally, reformulations and custom algorithms for various Type 1 problems are further summarized in Hellemo et al. (2018). However, none of these approaches is easily generalizable to different problems. In this paper, we present a general solution framework for problems with both types of endogenous uncertainty discussed here. Modeling Type 1 endogenous uncertainty has previously relied on specific problem structures and reformulations, and the ability to model these uncertainties in a general setting makes the proposed framework versatile compared to these earlier methods. Additionally, we present two alternative approaches for incorporating Type 2 endogenous uncertainty in the decision models formulated with our framework. The ease of considering any combination of the two types of endogenous uncertainty also makes the framework more generally applicable to problems with endogenous uncertainty than the approaches discussed earlier. Decision programming with conditionally observed information and subdiagram decomposition Decision Programming Decision Programming (Salo et al., 2022) relies on a graph representation of the problem, in which nodes represent decisions, the observation of chance events, and the consequences of these decisions and chance events. In specific, let G(N, A) be an acyclic graph formed by nodes j ∈ N = C ∪ D ∪ V and arcs a ∈ A = {(i, j) | i, j ∈ N }, where C and D are the sets of chance and decision nodes, respectively, and V is a collection of value nodes representing the consequences incurred from the decisions made at nodes D and the chance events realized at nodes C. In Fig. 1, the decision nodes are represented by squares, the chance nodes by circles, and the value nodes by diamonds. Each decision and chance node j ∈ C ∪ D can assume a state s j from a discrete and finite set of states S j . For a decision node j ∈ D, S j represents the available choices; for a chance node j ∈ C, S j is the set of possible realizations. The set A = {(i, j) | i, j ∈ N } contains the arcs (i, j), which represent the influence between nodes. In Fig. 1, the arrows between the nodes represent the arcs. Before defining this notion of influence further, let us first define a few necessary concepts. The information set comprises the immediate predecessors of a given node j ∈ N and is defined as I(j) = {i ∈ N | (i, j) ∈ A}. In the graphical representation, this corresponds to the set of nodes that have an arrow pointing directly to node j. For example, in Fig. 1, the information set of C 2 consists of D 1 and C 1 . The decision s j ∈ S j made in each decision node j ∈ D and the conditional probabilities of the states s j ∈ S j in each chance node j ∈ C depend on their information state s I(j) ∈ S I(j) , where S I(j) = i∈I(j) S i is the set of all possible information states for node j. Continuing with our example, the probabilities of different outcomes in C 2 are conditional on the decision in D 1 and the random outcome in C 1 . C 1 D 1 C 2 D 2 V 1 V 2 V 3 Let us define X j ∈ S j as the realized state at a chance node j ∈ C. For a decision node j ∈ D, let Z j : S I(j) → S j be a mapping between each information state s I(j) ∈ S I(j) and decision s j ∈ S j . That is, Z j (s I(j) ) defines a local decision strategy, which represents the choice of some s j ∈ S j in j ∈ D, given the information s I(j) . Such a mapping can be represented by an indicator function I : S I(j) × S j → {0, 1} defined so that I(s I(j) , s j ) = 1, if Z j maps s I(j) to s j , i.e., Z j (s I(j) ) = s j ; 0, otherwise.(1) A (global) decision strategy is the collection of local decision strategies in all decision nodes: Z = (Z j ) j∈D , selected from the set of all possible strategies Z. A path is a sequence of states s = (s i ) i=1,...,n , with n = |C| + |D|; and S = {(s i ) i=1,...,n | s i ∈ S i , i = 1, . . . , n}(2) is the set of all possible paths. Moreover, the information states s I(j) and states s j can be seen as subpaths derived from a given path s and we say that strategy Z and path s are compatible if Z j (s I(j) ) = s j for all j ∈ D. Using the notion of information states, the conditional probability of observing a given state s j for j ∈ C is P(X j = s j | X I(j) = s I(j) ). The probability associated with a path s being observed given a strategy Z can then be expressed as P(s | Z) =   j∈C P(X j = s j | X I(j) = s I(j) )     j∈D I(s I(j) , s j )   .(3) Notice that the term j∈D I(s I(j) , s j ) in equation (3) takes value one if the strategy Z is compatible with the path s, and zero otherwise. Thus, one can pre-calculate the probability p(s) =   j∈C P(X j = s j | X I(j) = s I(j) )   of a path s being observed, given that the selected strategy is compatible with s. At the value node v ∈ V , a real-valued utility function U v : S I(v) → R maps the information state s I(v) of v to a utility value U v . The default objective is to maximize the expected utility of a strategy, which can be expressed as v∈V s∈S P(s | Z)U v (s I(v) ) = s∈S P(s | Z)U (s),(4) where we denote the total path utility as U (s) = v∈V U v (s I(v) ). Note that this value can be precalculated for each path, similarly to the probability p(s). A significant advantage of using Decision Programming is that other objectives such as conditional Value-at-Risk can also be used, as discussed in Salo et al. (2022). This problem can be formulated as a mixed-integer linear programming (MILP) model, which allows for employing standard techniques widely available in off-the-shelf solvers. For that, let us define the binary variable z(s j | s I(j) ) that takes value 1 if Z j (s I(j) ) = s j , and 0 otherwise, for all j ∈ D, s j ∈ S j , and s I(j) ∈ S I(j) . These variables correspond to the indicator function (1), representing local decision strategies at each decision node j ∈ D. Using these variables directly, the conditional path probability (3) would become P(s | Z) = p(s) j∈D z(s j | s I(j) ), which contains a product of |D| binary variables. To linearize this formulation, we instead define a continuous variable π(s) representing the conditional path probability P(s | Z) and use the z-variables to model the conditional behavior of π(s). Then, the MILP problem can be stated as (5)-(10). max Z∈Z s∈S π(s)U (s)(5) subject to s j ∈S j z(s j | s I(j) ) = 1, ∀j ∈ D, s I(j) ∈ S I(j) (6) 0 ≤ π(s) ≤ p(s), ∀s ∈ S (7) π(s) ≤ z(s j | s I(j) ), ∀j ∈ D, s ∈ S (8) π(s) ≥ p(s) + j∈D z(s j | s I(j) ) − |D|, ∀s ∈ S (9) z(s j | s I(j) ) ∈ {0, 1}, ∀j ∈ D, s j ∈ S j , s I(j) ∈ S I(j) .(10)Constraints (7)-(9) enforce π(s) = P(s | Z) = p(s) j∈D z(s j | s I(j) ). With these constraints, π(s) = p(s) if z(s j | s I(j) ) = 1 for all j ∈ D, and π(s) = 0 otherwise. Constraints (8) and (9) state that π(s) ≤ 0 if z(s j | s I(j) ) = 0 for any j ∈ D and π(s) ≥ p(s) if z(s j | s I(j) ) = 1 for all j ∈ D, respectively. We note that constraint (9) can be excluded from the model if we assume (w.l.o.g) that the path utilities U (s) are positive, as the expected utility maximization would then drive the values of π(s) to their upper bounds given by constraints (7) and (8). The objective function (5) corresponds to the expected utility (4). Finally, constraint (6) states that the z-variables must represent a mapping S I(j) → S j with each s I(j) ∈ S I(j) mapping to exactly one s j ∈ S j , for each decision node j ∈ D. For a further discussion on the dimensions of the model (5)-(10), as well as its benefits against similar solution approaches, we refer the reader to Salo et al. (2022). Conditionally observed information As originally proposed, Decision Programming focuses on problems with Type 1 endogenous uncertainty, i.e., decision-dependent probabilities. However, many MSSP problems involve conditionally observed information (also called conditional information revelation). Before introducing further extensions, let us illustrate the notion of conditionally observed information with a capacity expansion problem described in Goel and Grossmann (2006). The setting is a chemical process network where a product is made and then delivered to customers. The decision maker (DM) can increase production by installing a new process with an uncertain efficiency that can be observed only by installing the machine. The observation of the efficiency is thus conditional on the decision to install the process. After introducing some relevant concepts, we will connect them to this example using Fig. 2. A key concept with conditionally observed information is distinguishability. Suppose we represent the potential efficiency of the new process as a chance node. The outcome of this node is independent of the DM's choice to install the process and is observed after installing the process. If the potential efficiency differs between two paths s and s , the paths are clearly different. However, from the DM's perspective, there is no difference between s and s if the new process is not installed and the efficiency is thus not observed. In this case, such paths are indistinguishable. In conditionally observed information, the two key elements are the decisions or random events that the observation is conditional on, and the condition that the observation depends on. We refer to these as the distinguishability set T i,j ⊂ C∪D and the distinguishability condition F i,j : S T i,j → {0, 1}, respectively. Here, i ∈ C ∪ D denotes the conditionally observed node, usually a chance node, and j ∈ D is the decision node where that information is available if the distinguishability condition is fulfilled. In some cases, the conditionally observed node could also be a decision node. Such cases might arise in the context of distributed decision-making, where observing the decision of another agent does not happen automatically, or problems in which previous decisions are not remembered by default and the decision maker must instead decide if they want to, e.g., find the decision history in an archive. Using the notion of distinguishability sets and conditions, we can define conditional arcs a c ∈ A c = {(i, j) | i ∈ C ∪ D, j ∈ D, i < j, T i,j = ∅, F i,j (s T i,j ) = 1}(11) to describe conditionally observed information in influence diagrams. Specifically, we say that a conditional arc a c from node i ∈ C ∪ D to node j ∈ D is active (i.e., node i is observed when making the decision corresponding to node j) if the condition F i,j is fulfilled by the states of the nodes k ∈ C ∪ D in the distinguishability set T i,j . If T i,j is empty, there is no conditional observation of i in j and, thus, no conditional arc between these nodes exists. The concept is illustrated in Fig. 2. i k j Figure 2: An illustration of a conditional arc (i, j). Nodes i and j correspond to the earlier notation where the realization of i is conditionally observed in j. The distinguishability set is Ti,j = {k} and the middle part of the arc represents that the flow of information from i to j is conditional on k. In the capacity expansion example from Goel and Grossmann (2006), the chance node i in Fig. 2 corresponds to the yield of the new process, decision node k to the installation decision, and j to operational decisions such as which processes and raw materials to use for production. The distinguishability set is T i,j = {k}, and the distinguishability condition is F i,j (s T i,j ) = I(s k = "install new process"), where the indicator function I( · ) is defined as I(x = x * ) = 1 (true), if x = x * , 0 (f alse), otherwise. If the distinguishability set includes more than one node, alternative functions F i,j might be employed for modeling the conditional dependencies between the nodes. For example, if there are several projects that reveal the same information in node i ∈ N and completing any of these projects is sufficient for information revelation, F i,j (s T i,j ) = k∈T i,j I(s k = s * k ) can be used; or if all of the projects are required for the information revelation, F i,j (s T i,j ) = k∈T i,j I(s k = s * k ) is appropriate. An example of such conditions is found in Tarhan et al. (2009), where different uncertainties in oil field development are gradually revealed, and the uncertainty in the amount of recoverable oil in a reservoir can be resolved in two different ways, namely drilling a sufficient number of wells or using the reservoir for production for long enough. Incorporating conditionally observed information in Decision Programming The conditional arcs are designed to describe conditionally observed information in influence diagrams. However, they are a general representation of the concept, not a modeling solution. In what follows, we present two alternative approaches for incorporating this concept into the Decision Programming framework, which ultimately enables solving Type 3 endogenously uncertain stochastic problems. The first approach employs observation nodes, used in decision analysis problems such as the used car buyer problem (Howard and Matheson, 2005). The second approach utilizes conditional non-anticipativity constraints, which are used in stochastic programming for modeling the decision-dependent information structure. Observation nodes portray how the decision maker observes the information, see Herrala (2020) for details. Earlier decisions affect the probability distribution of the observations, and thereby Type 2 uncertainty is effectively transformed into Type 1 uncertainty, making it directly amenable to a Decision Programming formulation. This approach is also used in Salo et al. (2022). In effect, each conditional arc is replaced with an observation node, as illustrated in Fig. 3. The information set of the observation node is the union of the chance node i and the distinguishability set T i,j (T i,j = {k} in the example from Fig. 2), and the state space is S i ∪ "no observation". Then, the observation node replaces the node i ∈ C in the information set of j ∈ D, controlling whether or not the information in i is available in j. A benefit of this approach is that wrong or imperfect observations can also be modeled, e.g., if an expert could be consulted in the chemical production example to give an educated but nevertheless uncertain guess on the efficiency of the new process. One can also utilize the ideas of Ruszczyński (1997) and Apap and Grossmann (2017) for modeling conditional information revelation, in which the information structure can be connected to the decisions by using disjunctive constraints called conditional non-anticipativity constraints (C-NACs). These constraints are similar to the more traditional non-anticipativity constraints (Rockafellar and Wets, 1991) in stochastic programming, but the constraints are instead toggled "on and off" by the distinguishability conditions between the corresponding paths. To integrate C-NACs into Decision Programming, the conditional arcs introduced earlier need to be supplemented with the conditional information set I c (j) = {i ∈ C ∪ D | (i, j) ∈ A c } to represent the conditionally available information at node j ∈ D. The C-NACs then control whether or not this information is available when making the decision in node j. We observe that two paths s and s are always distinguishable at node j ∈ D if the non-conditional information states s I(j) and s I(j) are different. Instead, C-NACs are needed when the conditional information states differ. For notational clarity, we separate the conditional and non-conditional information sets I c (j) and I(j). From a formulation standpoint, I c (j) is simply a part of the full information set of a node. C-NACs are used to enforce conditional non-anticipativity when only the conditional information states s Ic(j) differ between two paths, and we use this notation to emphasize the idea that C-NACs are only introduced for such pairs of paths. If the conditional information states differ (i.e., node i ∈ I c (j) has different states in paths s and s ), distinguishability is dependent on the corresponding distinguishability condition(s) F i,j (s T i,j ). This distinguishability of two paths at node j ∈ D can be formulated as a Boolean variable f s,s j , defined as f s,s j = i∈Ic(j)|s i =s i F i,j (s T i,j ). The value of f s,s j is True (i.e., 1) if the conditionally revealed information makes scenarios s and s distinguishable at node j, and False (i.e., 0) otherwise. Finally, we extend the definition of the local decision strategy Z j (s I(j) ) and the corresponding binary variables z(s j | s I(j) ) to include the conditional information set I c (j). Combining these ideas, we can define C-NACs in the context of Decision Programming as ¬f s,s j =⇒ Z j (s I(j) , s Ic(j) ) = Z j (s I(j) , s Ic(j) ), ∀s I(j) ∈ S I(j) , s Ic(j) , s Ic(j) ∈ S Ic(j) =⇒ z(s j | s I(j) , s Ic(j) ) = z(s j | s I(j) , s Ic(j) ), ∀s j ∈ S j , ∀s I(j) ∈ S I(j) , s Ic(j) , s Ic(j) ∈ S Ic(j) . In light of the above, the C-NACs for binary variables z can also be conveniently written as |z(s j | s I(j) , s Ic(j) ) − z(s j | s I(j) , s Ic(j) )| ≤ f s,s j , ∀s j ∈ S j , ∀s I(j) ∈ S I(j) , s Ic(j) , s Ic(j) ∈ S Ic(j) .(14) Notice that the absolute value function used in the left-hand side of (14) can be trivially linearized without compromising the model complexity. This constraint states that for each decision node j ∈ D, if the non-conditional information states s I(j) are the same for two paths, and conditionally revealed information does not make these paths distinguishable either, the local decision strategy represented by the z-variables must be the same. In practice, the main challenge with using C-NACs is that the number of constraints (14) quickly becomes overwhelmingly large. With this in mind, Apap and Grossmann (2017) presents a thorough description of C-NACs, followed by a number of C-NAC reduction properties that can be exploited to decrease the number of such constraints. By making use of these C-NAC reduction properties, representing the decision-dependent information structure within Decision Programming might be more compact with C-NACs than the corresponding model using observation nodes. However, C-NACs lack the ability to model Type 1 endogenous uncertainty. Observation nodes are more versatile, allowing for modeling both types of endogenous uncertainty within one observation node, making it possible to model and solve T3ESP problems. In Section 3.5, we consider both modeling approaches and compare their applicability and computational requirements in different settings, and in Section 4, we present a large-scale example problem involving different endogenous uncertainties. Subdiagram decomposition A well-known challenge in decision analysis and stochastic programming is the curse of dimensionality (Bellman, 1961), which describes the characteristic exponential growth of problems as the number of nodes or stages increases. To mitigate this exponential growth in Decision Programming problems, we propose a decomposition approach splitting the problem into a main problem and subproblem(s). Methods like Benders (Benders, 1962) and Dantzig-Wolfe (Dantzig and Wolfe, 1960) decomposition use this idea to make large problems computationally tractable. Both methods require a specific problem structure, which is then exploited in a divide-and-conquer strategy. The subproblems are smaller, independent parts of the full problem. If the problem is separable, the solutions of the subproblems can be utilized in the main problem, which recombines them into a solution for the full problem. In what follows, we first present the decomposition, then a condition that exploits the memory encoded in the decision process to show separability in Decision Programming problems, and finally present an illustrative example problem along with computational experiments in Section 3.5. The proposed subdiagram decomposition is based on splitting the influence diagram into two subdiagrams, which represent the main problem and subproblem. We split the set of nodes N into main problem nodes N main and subproblem nodes N sub , and consequently, the paths s are split into subpaths s main and s sub . The main problem formulation is identical to (5)-(10), with the full path s replaced by a main problem subpath s main , and the full decision strategies Z replaced by Z main , consisting of local decision strategies Z j (s I(j) ) for each main problem decision node j ∈ D main = {j | j ∈ N main , j ∈ D}. We assume an expected value objective function for the full problem, and the main problem path utility U (s main ) is the optimal solution to the subproblem (15)-(20) corresponding to the main problem path s main . U (s main ) = max Z sub ∈Z sub s sub ∈S sub π(s sub | s main )U (s main , s sub )(15) subject to sj ∈Sj z(s j | s I(j) ) = 1, ∀j ∈ D sub , s I(j) ∈ S I(j) 0 ≤ π(s sub | s main ) ≤ p(s sub | s main ), ∀s sub ∈ S sub (17) π(s sub | s main ) ≤ z(s j | s I(j) ), ∀j ∈ D sub , s sub ∈ S sub (18) π(s sub | s main ) ≥ p(s sub | s main ) + j∈D sub z(s j | s I(j) ) − | D sub |, ∀s sub ∈ S sub (19) z(s j | s I(j) ) ∈ {0, 1}, ∀j ∈ D sub , s j ∈ S j , s I(j) ∈ S I(j) ,(16) where U (s main , s sub ) represents the utility of the path formed by the juxtaposition of the subpaths s main and s sub . If we multiply all the conditional probabilities together appropriately, we can see that the main problem objective function max Z main ∈Z main s main ∈S main π(s main )U (s main ) is equivalent to the objective function (5). We observe that because we solve a subproblem (15)-(20) for each main problem subpath s main , the main problem nodes N main are effectively added to the information set of all decision nodes in the subproblem. That is, the states of the main problem nodes N main are known when making the decision in each subproblem decision node j ∈ D sub , implying that the local decision strategies for each decision node j ∈ D sub in the subproblem can be different for each subpath s main . This suggests that, in order for the information structure of the decomposed problem to be equivalent to that of the full problem, the decomposition would have to be such that all main problem nodes would be in the information set of each subproblem decision node. This would rather defeat the purpose of considering limited memory influence diagrams where the no-forgetting assumption is not required. However, the developments in Lauritzen and Nilsson (2001) can be used to relax this requirement. The node i is said to be requisite for j if including i in the information set I(j) can affect the optimal strategy in j, and nonrequisite otherwise. In practice, requisiteness is defined using the concept of d-separation, and we refer the reader to Lauritzen and Nilsson (2001) for a thorough discussion on the topic. Consider a node i ∈ N main not in the information set I(j) of a subproblem decision node j ∈ D sub . If i is nonrequisite for j, implicitly adding i to I(j) does not affect the strategies Z j and the missing arc between the main problem and subproblem does not preclude partitioning the diagram into two subdiagrams as proposed. To ensure separability and the correct flow of information, we impose the following partitioning condition on the problem structure for the decomposed problem to be equivalent to the original problem: for all pairs of nodes i ∈ N main and j ∈ D sub for which the arc (i, j) does not exist in the original problem, node i must be nonrequisite for node j. The condition can also be stated as (i, j) ∈ A ∨ i nonrequisite for j, ∀i ∈ N main , j ∈ D sub . After calculating the subproblem utilities U (s main ) for all s main ∈ S main and solving the resulting main problem, we obtain a main problem strategy that maximizes the expected subproblem utilities. When deciding the optimal strategies in the subproblem, the partitioning condition allows for exploiting the memory of the decision maker(s) to represent the main problem strategies. Constraints (16)-(20) ensure that the z-variables in the subproblem represent local strategies Z j (s I(j) ) for all j ∈ D sub and the π-variables are the probabilities of each subproblem subpath given the main problem subpath and subproblem decision strategy. Lastly, the partitioning condition guarantees that when making the subproblem decisions, all the requisite information from the main problem is remembered and the information structure of the decomposed problem is equivalent to that of the original problem. This decomposition can also be exploited in a parallelized fashion as solving the subproblems greatly benefits from the employment of parallelized solution strategies. This creates a possibility to trade off the computational demand between evaluating the subproblems and solving the main problem. For example, once a node j ∈ D ∪ C is made part of the subproblem, the total number of paths |S main | in the main problem is reduced by a factor of |S j |, shifting some of the computational burden from solving the main problem to solving a collection of subproblems, which are in turn more demanding than calculating the path utilities U (s) in the corresponding full problem (5)-(10). In Section 4, we demonstrate how exploiting this trade-off makes a complex problem computationally tractable by employing the proposed decomposition approach. Furthermore, the proposed decomposition approach provides additional methodological insight. While being an otherwise versatile framework, Decision Programming is strictly limited to problems in which both the decision spaces S j of decision nodes j ∈ D and the sets of possible outcomes S j for the chance nodes j ∈ C are discrete and finite. This imposes a limitation on practical applications that would benefit from the employment of mathematical programming as a modeling framework. Recall that the main advantage of Decision Programming over the traditional stochastic programming approach of formulating the deterministic equivalent problem is being able to easily represent problems with Type 1 endogenous uncertainty. The subdiagram decomposition provides a way to relax the requirement of discrete and finite state spaces in Decision Programming and allow the subproblem decision nodes to represent continuous decisions. We note that if there is no Type 1 endogenous uncertainty in the subproblems, they can be represented using the scenario tree based deterministic equivalent formulation, thereby allowing for continuous decision variables. This is demonstrated in Section 4. Computational experiments To demonstrate the employment of the subdiagram decomposition, we introduce a multiperiod "unfair coins" game presented in Fig. 4. In every period, the player makes a decision of whether play or skip the turn, represented by nodes D i . If the player decides to play, they might win or lose money. If they skip, no money is gained or lost. This outcome is represented by nodes C i and the associated value by V i . In each period, a possibly unfair coin is randomly chosen from three alternatives in P i . The probabilities of picking different coins, i.e., the conditional probability distributions in P i , depend on the previous period outcome C i−1 . The arcs from P i to D i represent the assumption that the player always observes which coin was chosen before making the decisions, thus knowing the probability of winning each round. If the player wins a given round, the highest winning probability coin is more likely to be chosen in the next Figure 4: A four-period version of the unfair coins problem. D 1 C 1 P 2 V 1 D 2 C 2 P 3 V 2 D 3 C 3 P 4 V 3 D 4 C 4 V 4 period, and losing similarly makes the lower winning probability coin more likely. Skipping a turn makes the "middle" coin the most likely. Adding a stage to the game in Fig. 4 introduces two chance nodes and a decision node, rapidly increasing the number of paths in the full problem. To combat the associated solution time increase, we use the subdiagram decomposition and split the problem by periods so that all nodes in, for example, periods 3 and onwards, are moved into the subproblem. We note that the nodes on the left-and right-hand sides of C i are d-separated by C i , showing that the only requisite nodes for D i (with i > 1) are C i−1 and P i . The partitioning condition is thus fulfilled when the problem is split in this way. As an example of when this is not the case, if the state of the node P 3 was not known when making the decision D 3 (i.e., the arc between these nodes did not exist), a decomposition where P 3 remained in the main problem and D 3 was moved to the subproblem would implicitly add P 3 to I(D 3 ), the decision maker would gain new information compared to the full problem and the decomposed problem would no longer be equivalent to the full problem. The computational tractability of the framework is first tested with this game. Information about the computational setup and available source code for all computational experiments can be found in Appendix A. We use a five-period version of the "unfair coins" game and decompose the problem so that the first n main periods are the main problem and the rest is moved into the subproblem. We solve 20 replications of the problem with random winning probabilities and present the average solution times for different values of n main in Fig. 5. When n main = 5, the main problem becomes the full problem and the top right corner of the figure represents the solution time without decomposition. The main problem solution time increases exponentially with the number of stages, and the subproblem solution times decrease similarly. The solution time of a single subproblem is similar to that of a main problem with the same number of periods, but the number of subproblems increases exponentially with the number of periods in the main problem and this makes the total subproblem solution time higher. However, the subproblems could also be solved in parallel and, with sufficient parallel computing units, the total solution time would approach the sum of the main problem solution time and the solution time of the computationally most demanding subproblem, as shown in Fig. 5b. The total solution time with or without parallelization is the lowest with three stages in the main problem and the last two stages in the subproblem. This supports the intuitive idea that the optimal split would have the main problem only slightly larger, and consequently more computationally demanding, than each subproblem, thus compensating for the possibly large number of subproblems. The difference in total solution time between the full problem and a decomposed problem with three periods in the main problem is two orders of magnitude with no parallelization in solving the subproblems. If all subproblems were solved in parallel, the difference would be three orders of magnitude instead. We now introduce a modified example with conditionally observed information in Fig. 6. Instead of always observing the coin used in each round (represented by nodes P i ), the decision maker can choose to observe it, but with a cost. With this modification, the uncertainty in what coin will be picked in P i still depends on the previous decision D i−1 through C i−1 and observing this uncertainty is now conditional on the decision D o i , making this a Type-3 endogenously uncertain SP (T3ESP) problem. Figure 6: The game from Section 3.4 with conditionally observed information. D 1 C 1 P 2 V 1 D 2 D o 2 V o 2 C 2 P 3 V 2 D 3 D o 3 V o 3 C 3 P 4 V 3 D 4 D o 4 V o 4 C 4 V 4 The additional conditional arcs from P i to D i can be modeled using observation nodes or conditional NACs as described in Section 3.3. The mean solution times for different problem sizes over 20 replications are presented in Fig. 7. First, we observe that making the information conditionally observed instead of always observed significantly increases solution times. This is explained by the additional decision of whether to observe the coin in each stage, which increases the problem size. There is practically no difference between solution times for observation nodes and C-NACs. Upon closer inspection of the solution process, we find that the two models are the same after the presolve process of the Gurobi solver (Gurobi Optimization, LLC, 2022). This is an interesting observation since it suggests that the solver is able to detect the structure in conditionally observed information. Additional experiments show that without presolve, solving the models takes significantly longer, but there are still no considerable differences between the solution times for observation nodes and C-NACs. A major disadvantage of observation nodes is that by the definition in Eq. (2), adding nodes to the diagram increases the number of paths s ∈ S. However, this does not necessarily increase the size of the decision model. To show this, we start with the definition that an observation node j is a deterministic mapping from an information state s I(j) to a state s j , meaning that P(X j = s j | X I(j) = s I(j) ) = 1 for one of the states s j ∈ S j and zero for others. From this, we observe that adding an observation node does not increase the number of paths for which the upper bound p(s) in Eq. (7) is nonzero. For any path s with p(s) = 0, constraint (7) forces π(s) = 0 and the constraints (8)-(9) become redundant. It is therefore not necessary to consider such paths in the model and the problem size does not increase with added observation nodes. Nevertheless, the increased number of paths might prove challenging in implementing the model, since basic operations such as iterating over all paths can become computationally infeasible. For example, the number total of paths in the diagram in Fig. 6 becomes 17.9 million if we assume three coins in each period. Cost-benefit analysis for climate change mitigation Model description We now consider the cost-benefit analysis on mitigating climate change under uncertainty (see, e.g., Ekholm, 2018). Climate change is driven by greenhouse gas (GHG) emissions and can be mitigated by reducing these emissions, which incurs costs. However, mitigation reduces the negative impacts of climate change, referred to as climate damage. In cost-benefit analysis, the objective is to minimize the discounted sum of mitigation costs and climate damage over a long time horizon. Here, we consider three salient uncertainties involving both decision-dependent probabilities (Type 1) and conditionally observed information (Type 2). Moreover, some decision nodes involve continuous variables. The resulting problem is a multi-stage mixed-integer nonlinear problem (MINLP) with Type 3 endogenous uncertainty, thus demonstrating all the proposed novel features to the decision programming framework described in this paper, and which could not be solved with earlier frameworks for endogenously uncertain problems. The considered uncertainties concern 1) the sensitivity of climate to GHG emissions, 2) the severity of climate change damages to society, and 3) the cost of reducing emissions. Decisions can be made to first conduct costly research and development (R&D) efforts towards each source of uncertainty. For the uncertainties regarding climate sensitivity and damages, a successful R&D effort results in an earlier revelation of the parametrization, whereas an unsuccessful or no R&D effort reveals this information later. Similar models of R&D pipeline optimization under (Type 2) endogenous uncertainty are considered in Colvin and Maravelias (2011). For the mitigation costs, technological R&D for decreasing the costs of bioenergy with carbon capture and storage (BECCS) can be conducted with three levels in two stages. The first stage is a choice between low or medium R&D effort. The low-effort choice represents a businessas-usual perspective, which carries throughout the decision process. If the medium effort is chosen, one observes whether the R&D looks promising or not, and can then decide whether to continue with the medium or switch to a higher R&D effort. The three R&D effort levels and whether the development seems promising or not all affect the probabilities for achieving either low, medium or high mitigation costs later during the century. For a related discussion, Rathi and Zhang (2022) consider endogenous technology learning on power generation. The presented model is an extension from Ekholm and Baker (2022), which in turn is a simplification from the SCORE model (Ekholm, 2018). Compared to the formulation proposed here, these earlier analyses have assumed that the uncertainties are resolved exogenously over time and dealt with the mitigation cost uncertainty through separate scenarios. The details of the model structure and parametrization are described in Appendix B. We assume that all prior decisions and uncertainty realizations apart from the conditionally observed parameters are remembered when making decisions, but omit the arcs for clarity. D Dmg D T 1 D CS D E1 D T 2 D E2 D E3 C Dmg C T 1 C CS C T 2 O Dmg O CS The influence diagram for the problem is presented in Fig. 8. The first stage involves R&D decisions towards climate sensitivity (D CS ), damages (D Dmg ) and technology (D T 1 ). If successful, the climate parameter (climate sensitivity and damage exponent) R&D efforts modify the information structure so that the parametrization is partially revealed in 2050 instead of 2070. This is represented by the observation nodes O Dmg and O CS and the outcome (success/failure) of the projects by C Dmg and C CS . Decisions over emission reductions (D Ei , i ∈ {1, 2, 3}) are made in three stages: in 2030, 2050 and 2070, which represent the medium-term and long-term climate actions. The technological R&D potentially lowers the costs of emission reductions in 2050 and 2070. We connect our example to the discussion on the feasibility of large-scale deployment of bioenergy with carbon capture and storage (BECCS), which has been a crucial but contested result of many mitigation scenarios (Calvin et al., 2021). The R&D costs and probabilities for the three levels of BECCS costs are parametrized using expert-elicited estimates in Baker et al. (2015). These are then reflected in the overall emission reduction costs. It is worth noting the major challenges in longterm technological foresight, which is manifested in the wide spectrum of responses from the experts; but elicitation data is nevertheless useful for illustrating the importance of technological progress. After 2070, the level of climate change is observed based on the chosen emission reductions and the observed branch of climate sensitivity, which then determines the climate damages along with the observed branch of climate damages. Modifying the influence diagram As discussed in Section 3, the original formulation of Decision Programming was limited to problems with discrete and finite state spaces for all nodes. However, discretizing the emission levels D Ei , i ∈ {1, 2, 3} would result in suboptimal solutions and we would thus prefer having continuous decision variables for the emission levels. To achieve this, we employ the decomposition presented in Section 3.4 and move the nodes D Ei , i ∈ {1, 2, 3} to the subproblem. Additionally, we move into the subproblem all nodes that are not in the information set of each of the emission level decisions. The influence diagram resulting from this decomposition is presented in Fig. 9. We assume that the previous decisions and chance node realizations are remembered in each decision node, but omit these arcs in Fig. 8 for clarity. As a result, there is no Type 1 endogenous uncertainty in the subproblem and it can be modeled as a nonlinear three-stage stochastic programming problem with continuous decision variables. The nonlinearity comes from the SCORE model described in Appendix B. The impact of the decomposition on computational performance is discussed in Section 4.4. Model results The optimal R&D strategy for this problem is to carry out all R&D projects. We next discuss the endogenous impacts of the R&D outcomes in this optimal strategy. Fig. 10 presents the emission levels of the optimal mitigation strategy. The effect of the technology R&D on optimal emission pathways is shown with the three subfigures corresponding to the final R&D outcome after 2030. Intuitively, successful research and therefore cheaper abatement leads to more abatement. This has also a major impact on the total costs of the optimal strategy: with the low cost curves, the total expected cost is roughly 30% lower than with medium costs, and with high abatement costs 30% higher than with the medium cost curve. The climatic parameter R&D, the other endogenous effect in this model has a magnitude smaller effect on the expected costs than the technology R&D, but the effect on abatement levels is remarkable. The branches after 2030 represent different realizations of the technology research and partial learning of the climate parameters. If the research efforts for both parameters fail (blue lines in Fig. 10), the 2050 abatement decisions are made knowing only the outcome of the technology projects, and the four scenarios of partial learning only occur after 2050. Learning the parameters before 2050 results in more dispersed abatement strategies. Finally, the underlying parameter branching is represented with shapes in 2070. It can be seen that the impact of climate sensitivity is considerably smaller than that of the damage exponent. This is in line with the results in Ekholm and Baker (2022) and shows that the proposed framework could be applied in planning for optimal R&D pathways. Computational aspects Using the decomposed model presented in Fig. 9, the Decision Programming problem is solved in a tenth of a second, while solving all of the subproblems takes less than a minute. If we instead solve a problem corresponding to Fig. 8 without decomposition, the Decision Programming framework requires discretizing the abatement levels. Using discrete decision spaces to approximate a continuous variable leads to suboptimal solutions. Furthermore, even with only three levels for each abatement decision, the problem becomes remarkably more computationally demanding. The path utilities are calculated in one second, but solving the Decision Programming model takes over an hour. With more abatement levels, the discretized problem quickly becomes intractable. The decomposed problem instead uses continuous decision variables to represent the abatement decisions, precluding the need for a discretization. As a consequence, it is much faster to solve than the rudimentary approximation with three abatement levels. The discretized model has 144876 constraints and 16662 variables, of which 138 are binary; and the decomposed main problem has 158 constraints and 36 variables, of which 12 are binary. The discretized model is thus three orders of magnitude larger than the decomposed model and the difference in solution times is even larger, but the path utilities in the discretized model are calculated almost instantaneously. However, the trade-off of moving some of the computational burden into the subproblems makes it possible to solve the decomposed problem to optimality. The discretized model can technically also be solved to optimality, but it requires a very rough discretization of the abatement levels, which are continuous decisions by nature. Even with a low number of abatement options, the solution times are much larger than those for the decomposed model. In addition, such a rough discretization is likely to result in solutions that are far from the optimal values. While discretizing the decision spaces may seem like an obvious approach, as it enables direct implementation using the Decision Programming formulation (5)-(10), it is not computationally viable in all cases. On the other hand, our subdiagram decomposition, when applicable, can provide more accurate solutions within much smaller solution times. The computational performance of this case study thus supports our findings in Section 3.5, providing an illustrative example of an application where the developments in this paper make it possible to consider settings that are challenging both from modelling and computational standpoints. Conclusion In this paper, we propose a framework for Type 3 endogenously uncertain stochastic programming (T3ESP) problems. Our contributions consist of two different modelling approaches to consider conditionally observed information, making the framework more generally applicable to decision-making problems in contexts such as capacity expansion, and a subdiagram decomposition approach to improve computational efficiency and allow for the inclusion of continuous decision variables. The proposed framework is based on Decision Programming (Salo et al., 2022), originally developed for solving decision problems with decision-dependent uncertainties by converting an influence diagram representation of the problem to a mixed-integer linear programming (MILP) problem. According to Hellemo et al. (2018), Type 3 endogenously uncertain stochastic programming (T3ESP) problems have not been discussed in the literature before. To make Decision Programming applicable to such problems, we show how Type 2 endogenous uncertainties can be modeled by either adding observation nodes to the influence diagram or by adding conditional non-anticipativity constraints (C-NACs) to the Decision Programming model. The computational experiments do not indicate clear superiority for either of these approaches, as the solution times observed were similar for both approaches in the example considered. However, in the presence of decision-dependent probability distributions, for example, the presence of imperfect conditional observations, one cannot employ C-NAC constraints. An example of such a problem would be a version of the climate CBA problem in Section 4 where the climate parameter research does not reveal the correct branch, that is, remove one of the extreme parameter values, but gives a probability distribution that provides better information than the original. In practice, both approaches have their advantages and disadvantages. Adding observation nodes only requires modifying the influence diagram by adding new nodes, while C-NACs are additional constraints that must be added to the decision model (5)-(10). One of the ideas behind Decision Programming is to provide a methodology for converting an influence diagram representation of the problem into a mixed-integer linear programming model, and from a usability perspective, it might be beneficial to not require explicit modification of the decision model. However, if one starts modifying the decision problem by using C-NACs, it might be possible to also represent other parts of the decision process explicitly as added variables or constraints. Further research in this direction is thus relevant. As with mixed-integer programming in general, the models rapidly grow in complexity when the model size increases. In our computational examples, this is clearly seen as solution times grow exponentially with the number of stages in the unfair coins game. Similar challenges can be observed in the climate change mitigation problem, where the full problem with discretized abatement decisions is computationally demanding. We observe that some problems may satisfy a partitioning condition, which in turn reveals them to be separable in a way that the influence diagram can be separated into a main and a subproblem. We call this method the subdiagram decomposition. This decomposition approach allows for modeling a wider variety of problems as we are no longer strictly limited to discrete and finite decision spaces when the subproblem structure allows for utilizing an alternative solution method. For example, subproblems without Type 1 endogenous uncertainty can be cast as deterministic equivalent stochastic programming formulations, which can contain continuous decision variables, as shown in the climate change mitigation example in Section 4. Furthermore, the subdiagram decomposition can reduce the total solution time by multiple orders of magnitude. In the climate change mitigation example, the subdiagram decomposition makes the problem tractable and enables obtaining more accurate solutions than with discretized abatement decisions. In addition to the subdiagram decomposition approach proposed in this paper, the computational efficiency of Decision Programming could potentially be improved through other approaches. Ideas in this direction include decomposition methods for solving the MILP formulation (5)-(10), solution heuristics and tighter formulations. Incorporating new problem structures would make the framework applicable to a broader set of problems. Interesting examples of such research ideas are (distributionally) robust optimization and further examination of multi-objective decision-making. Despite the computational challenges and the number of possible improvements and extensions discussed, the framework is already general enough to model a challenging example problem with Type 3 endogenous uncertainty. It should also be noted that the problem formulation remains linear even with this type of uncertainty, guaranteeing global optimality of solutions. The idea in climate change cost-benefit analysis (CBA) is the minimization of costs from emission abatement and climate damages. The abatement cost calculation is based on marginal abatement cost curves, as presented in equation (B.1), using numerical estimates from the SCORE model (Ekholm, 2018). Coefficients α and β are the parameters of the cost curves, R is the total abatement level and c is the marginal cost of abatement. In (B.3), C is the total cost for abatement level R. The subscript t has been omitted for clarity, but parameters α and β change between stages due to assumed technological progress. R = αc β (B.1) =⇒ c = R α 1/β (B.2) =⇒ C = R 0 r α 1/β dr = β 1 + β 1 α 1/β R 1+1/β . (B.3) Departing from the predetermined technological progress that was assumed by Ekholm (2018), the parameter α in the model depends here on the result of technological R&D, as presented in Figure 8. We consider three levels of R&D effort in 2020 and 2030, which can then lead to three possible levels of MAC curves for years 2050 and 2070. For the effect of R&D efforts on bioenergy and carbon capture and storage (CCS) costs, we used expert elicited estimates from Baker et al. (2015). To convert these into emission reduction costs, we assume a coal power plant as a baseline and calculate the additional costs from bioenergy with carbon capture and storage (BECCS) relative to the amount of reduced emissions by switching from coal to BECCS. Both plants were assumed to have a lifetime of 30 years and operate at 80% capacity on average. The coal power plant was assumed to have a 40% efficiency and produce 885 tonnes of CO 2 per GWh of electricity. The generation cost was assumed to be 50$/MWh. Costs were discounted at 5% rate. Compared to coal, BECCS accrues additional costs per generated unit of electricity from the higher cost of biofuel and lower efficiency, and the additional investment to CCS and loss of efficiency from using some of the generated electricity in the carbon capture process. Baker et al. (2015) presented probability distributions for these parameters following three different levels of R&D efforts (low, medium and high). To calculate the cost differential to coal power plant, we performed a Monte Carlo sampling of these four parameters, separately for each R&D level, which was then compared to the amount of reduced emissions per generated unit of electricity. This yields a distribution of emission reduction costs for BECCS for each R&D level. We generalize the impact of R&D on BECCS's emission reduction cost to the overall marginal abatement cost (MAC) curve. This is obviously a simplification, but nevertheless reflects the major role that BECCS might have in decarbonizing the economy (e.g. Fuss et al., 2018;Rogelj et al., 2018). We take the high MAC from Ekholm (2018) as the starting point and define two MAC curves that are proportionally scaled down from the high MAC. The low R&D level yields an average emission reduction cost of around 100 $/t. We set three bins for three cost levels: high costs correspond to above 75 $/t, medium costs are between 25 and 75 $/t, and low costs are below 25 $/t. The probabilities of achieving high, medium or low emission reduction costs are then estimated from the Monte Carlo sampling for each R&D level, and presented in Table B.1. With medium and high R&D effort, the average cost in the low cost bin is around 20 $/t. Therefore we assign the reduction in the MAC as 50% for medium costs and 80% for low costs. The corresponding parameters are listed in Table B.2. These are still within the range of costs used in Ekholm (2018), where the low-cost MAC yielded the same emission reductions than the high-cost MAC with approximately 90% lower costs in 2050. The climate damage cost calculation is from DICE (Nordhaus, 2017). The damage function is presented in (B.4), where Y (t) is the world gross economic output at time t, a is a scaling parameter and b is the damage exponent. While climate change and the abatement decisions have an effect on the economic output, the effect is assumed small and Y (t) is defined exogenously in SCORE. D(t, ∆T ) = Y (t)a∆T b . (B.4) Finally, the temperature change ∆T is approximated with (B.5), where c is the climate sensitivity (the temperature increase from doubling of CO 2 emissions), M is the sum of emissions in 2030-2070 and k i are coefficients. ∆T = k 1 cM + k 2 c + k 3 M + k 4 . (B.5) In SCORE, both the DICE damage parameter and the climate sensitivity are uncertain with three options, low, medium and high, as presented in Table B.3, and the uncertainty is revealed in two steps in a binomial lattice. First, between 2050 and 2070, one of the extreme alternatives is removed from both uncertainties, that is, for both parameters, we know either that the value is not high or that it is not low. Then, after 2070, we learn the actual value. The implementation here combines Decision Programming and MSSP in a way that the underlying branching probabilities are used as the probabilities of the observations O Dmg and O CS in Fig. 8. For the damage exponent, all branching probabilities are 50%. The observation O Dmg thus has a 50% probability of removing either the high or low value. Depending on the branch, the low or high value then has a 50% probability in the later branching, with the other 50% for the medium value. This makes the medium branch have a 50% probability in total, while the two extreme values both have a 25% probability. For the climate sensitivity, the first branching is with a 50% probability for both branches. However, the second branching is different. If the high sensitivity is excluded in the first branch, there is a 21% conditional probability of the low branch in the second branching, meaning a 10.5% total probability for the low sensitivity. Similarly, there is a 23% conditional probability of high damages in the other branch, resulting in a 11.5% probability for the high sensitivity. The remaining 78% is the final probability of medium sensitivity. This uncertain process is modeled by means of a multi-stage stochastic programming problem, where new information is obtained gradually. It is possible to perform research on these parameters. If the research succeeds, one of the extreme values is excluded already before 2050, revealing the first branching in the observation process earlier than without or with failed research. The observation of the actual parameter value (the second branching) still happens after 2070, after all abatement decisions have been made. The total cost we aim to minimize is then a discounted sum of research costs, abatement costs (B.3) for years 2030, 2050 and 2070, and damage costs (B.4). Figure 1 : 1An influence diagram representation of a decision problem Figure 3 : 3Replacing a distinguishability arc with an observation node in the example fromFig. 2. Figure 5 : 5The solution times of the main and subproblems for a five-period unfair coins problem with different splits of the problem. Notice the logarithmic y-axis. Figure 7 : 7The solution times for different problem sizes using observation nodes or C-NACs. Notice the logarithmic y-axis. Figure 8 : 8Influence diagram of the climate change cost-benefit problem with endogenous probability due to R&D. Figure 9 : 9Influence diagram of the climate change cost-benefit main problem after decomposition. Figure 10 : 10Abatement levels for different outcomes of technology research and parameter observations in 2030. Table B . B1: Probabilities for different abatement costs (rows) with different levels of R&D effort (columns).Table B.2: Cost curve parametrizationyear α high α medium α lowLow R&D Medium R&D High R&D High costs 73 % 31 % 9 % Medium costs 27 % 64 % 73 % Low costs 0 % 5 % 18 % β 2030 3.57 3.57 3.57 0.340 2050 11.2 13.3 16.7 0.250 2070 21.1 24.3 29.3 0.203 Table B . 3 : B3Climate sensitivity and damage exponent valuesClimate sensitivity Damage exponent High 6 4 Medium 3 2 Low 1.5 1 Appendix A. Computational environmentAll problems are solved using a desktop computer with an Intel i5-4690K CPU at 3.9GHz and 16GB of RAM, and the problem code was implemented in Julia v1.7.2(Bezanson et al., 2017)using DecisionProgramming.jl v0.1.0(Oliveira et al., 2021)with the Gurobi solver v9.5.1 (Gurobi Optimization, LLC, 2022) and JuMP v0.22.3(Dunning et al., 2017). All the code used in the computational experiments is available and documented at a GitHub repository(Herrala, 2022).Appendix B. Cost-benefit model description An approach for strategic supply chain planning under uncertainty based on stochastic 0-1 programming. A Alonso-Ayuso, L F Escudero, A Garìn, M T Ortuño, G Pérez, 10.1023/A:1023071216923Journal of Global Optimization. 26Alonso-Ayuso, A., Escudero, L.F., Garìn, A., Ortuño, M.T., Pérez, G., 2003. An approach for strategic supply chain planning under uncertainty based on stochastic 0-1 programming. 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[ "Towards Generalisable Video Moment Retrieval: Visual-Dynamic Injection to Image-Text Pre-Training", "Towards Generalisable Video Moment Retrieval: Visual-Dynamic Injection to Image-Text Pre-Training" ]
[ "Dezhao Luo [email protected] \nQueen Mary University of London\n\n", "Jiabo Huang [email protected] \nQueen Mary University of London\n\n", "Shaogang Gong [email protected] \nQueen Mary University of London\n\n", "Hailin Jin [email protected] \nAdobe Research\n\n", "Yang Liu [email protected] \nWICT\nPeking University\n\n" ]
[ "Queen Mary University of London\n", "Queen Mary University of London\n", "Queen Mary University of London\n", "Adobe Research\n", "WICT\nPeking University\n" ]
[]
The correlation between the vision and text is essential for video moment retrieval (VMR), however, existing methods heavily rely on separate pre-training feature extractors for visual and textual understanding. Without sufficient temporal boundary annotations, it is non-trivial to learn universal video-text alignments. In this work, we explore multi-modal correlations derived from large-scale image-text data to facilitate generalisable VMR. To address the limitations of image-text pre-training models on capturing the video changes, we propose a generic method, referred to as Visual-Dynamic Injection (VDI), to empower the model's understanding of video moments. Whilst existing VMR methods are focusing on building temporalaware video features, being aware of the text descriptions about the temporal changes is also critical but originally overlooked in pre-training by matching static images with sentences. Therefore, we extract visual context and spatial dynamic information from video frames and explicitly enforce their alignments with the phrases describing video changes (e.g. verb). By doing so, the potentially relevant visual and motion patterns in videos are encoded in the corresponding text embeddings (injected) so to enable more accurate video-text alignments. We conduct extensive experiments on two VMR benchmark datasets (Charades-STA and ActivityNet-Captions) and achieve state-of-the-art performances. Especially, VDI yields notable advantages when being tested on the out-of-distribution splits where the testing samples involve novel scenes and vocabulary.
10.48550/arxiv.2303.00040
[ "https://export.arxiv.org/pdf/2303.00040v2.pdf" ]
257,255,395
2303.00040
848898eb190d96406c45edd4fe5fa4a1d6709881
Towards Generalisable Video Moment Retrieval: Visual-Dynamic Injection to Image-Text Pre-Training Dezhao Luo [email protected] Queen Mary University of London Jiabo Huang [email protected] Queen Mary University of London Shaogang Gong [email protected] Queen Mary University of London Hailin Jin [email protected] Adobe Research Yang Liu [email protected] WICT Peking University Towards Generalisable Video Moment Retrieval: Visual-Dynamic Injection to Image-Text Pre-Training The correlation between the vision and text is essential for video moment retrieval (VMR), however, existing methods heavily rely on separate pre-training feature extractors for visual and textual understanding. Without sufficient temporal boundary annotations, it is non-trivial to learn universal video-text alignments. In this work, we explore multi-modal correlations derived from large-scale image-text data to facilitate generalisable VMR. To address the limitations of image-text pre-training models on capturing the video changes, we propose a generic method, referred to as Visual-Dynamic Injection (VDI), to empower the model's understanding of video moments. Whilst existing VMR methods are focusing on building temporalaware video features, being aware of the text descriptions about the temporal changes is also critical but originally overlooked in pre-training by matching static images with sentences. Therefore, we extract visual context and spatial dynamic information from video frames and explicitly enforce their alignments with the phrases describing video changes (e.g. verb). By doing so, the potentially relevant visual and motion patterns in videos are encoded in the corresponding text embeddings (injected) so to enable more accurate video-text alignments. We conduct extensive experiments on two VMR benchmark datasets (Charades-STA and ActivityNet-Captions) and achieve state-of-the-art performances. Especially, VDI yields notable advantages when being tested on the out-of-distribution splits where the testing samples involve novel scenes and vocabulary. Introduction Video moment retrieval (VMR) aims at locating a video moment by its temporal boundary in a long and untrimmed video according to a natural language sentence [3,13]. It * Corresponding authors Our method takes the advantage of image-text pre-trained models and learns moment-text correlations by visual-dynamic injection. is a critical task which has been extensively studied in a variety of real-world applications including human-computer interaction [5], and intelligent surveillance [9]. In practice, raw videos are usually unscripted and unstructured, while the words being chosen for describing the same video moments can be varied from person to person [45,63]. To be generalisable to different scenes, VMR is fundamentally challenging as it requires the comprehension of arbitrary complex visual and motion patterns in videos and an unbounded vocabulary with their intricate relationships. For the fine-grained retrieval objective of VMR, the precise segment-wise temporal boundary labels are intuitively harder to be collected than conventional image/video-level annotations. In this case, rather than training from scratch with a limited number of temporally labelled videos, existing VMR solutions [3,13,14,62] heavily rely on singlemodal pre-training [8,48] for visual and textual understanding ( Fig. 1 (a)). By doing so, they focus on modelling the correlations between the pre-learned features of videos and sentences. Nonetheless, without sufficient training data, it is non-trivial to derive universal video-text alignments so to generalise to novel scenes and vocabulary. Separately, the recent successes achieved by joint visionlanguage pre-training in zero-shot learning [21,42] demonstrate the potential of adapting the multi-modal correlations derived from large-scale visual-textual data to facilitate generalisable VMR. Whilst it is intuitive to adopt the videotext pre-learned features [34,38,50] for moment retrieval ( Fig. 1 (b)), it has been shown that the models pre-trained with coarse-grained video-level labels can not transfer well to localisation-based tasks like VMR due to their unawareness of fine-grained alignments between text and frames or clips [2]. Such a misalignment problem is less likely to exist in pre-training by image-text matching. However, imagebased pre-training models [21,42] are less sensitive to the changes in videos and the words describing such dynamics in text [17]. This is inherent in matching sentences and images with static content but is significant in understanding video actions and activities ( Fig. 1 (c)). It is suboptimal to directly apply image-text pre-learned features on VMR. In this work, we propose a generic method for exploiting large-scale image-text pre-training models to benefit generalisable VMR by the universal visual-textual correlations derived in pre-training, dubbed as Visual-Dynamic Injection (VDI). The key idea is to explore the visual context and spatial dynamic information from videos and inject that into text embeddings to explicitly emphasise the phrases describing video changes (e.g. verb) in sentences ( Fig. 1 (d)). Such visual and dynamic information in text is critical for locating video moments composed of arbitrary evolving events but unavailable or overlooked in imagetext pre-training. Specifically, we consider it essential for VMR models to answer two questions: "what are the objects" and "how do the objects change". The visual context information indicates the content in the frames, e.g. backgrounds (scenes), appearances of objects, poses of subjects, etc. Meanwhile, the spatial dynamic is about the location changes of different salient entities in a video, which potentially implies the development of their interactions. VDI is a generic formulation, which can be integrated into any existing VMR model. The only refinement is to adapt the text encoder by visual-dynamic information injection during training. Hence, no additional computation costs are introduced in inference. Our contributions are three-folded: (1) To our best knowledge, this is the first attempt on injecting visual and dynamic information to image-text pre-training models to enable generalisable VMR. (2) We propose a novel method for VMR called Visual-Dynamic Injection (VDI). The VDI method is a generic formulation that can be integrated into existing VMR models and benefits them from the universal visual-textual alignments derived from large-scale imagetext data. (3) The VDI achieves the state-of-the-art performances on two standard VMR benchmark datasets. More importantly, it yields notable performance advantages when being tested on the out-of-distribution splits where the testing samples involve novel scenes and vocabulary. VDI's superior generalisation ability demonstrates its potential for adapting image-text pre-training to video understanding tasks requiring fine-grained visual-textual comprehensions. Related Work Video Moment Retrieval To tackle the VMR task and predict accurate moment boundaries, existing methods [19,20,32,55,61,62] first generated visual features and textual features from pretraining encoders [44,51], then they designed correlation models to align the two modalities. Ghosh et al. [14], Liu et al. [31] and Zeng et al. [60] proposed to select the starting and ending frames by leveraging cross-modal interactions between text and video. He et al. [15] and Wang et al. [54] proposed reinforcement learning methods for VMR. Gao et al. [13], Wang et al. [55], Zhang et al. [62] took a two-stage pipeline by generating proposals and ranking them relying on the similarity between proposal and query. Li et al. [27], Liu et al. [33], Yang et al. [58] focused on debias problems including the temporal location bias [33,58], or the word-composition bias [27]. Even though existing methods have demonstrated promising performance for VMR, we argue that models that take separate pre-training visual and textual feature extractors are suboptimal as they need to learn the alignment of the two modalities from scratch. It is demanding to learn from large-scale image-text datasets due to a lack of wellannotated moment-text datasets [38] resulting in poor generalisation [2]. Vision-Language Pre-Training Vision-language models have demonstrated great potential in learning generic visual representations and allowing transferring to a variety of downstream tasks [25,43]. Previously, Frome et al. [12], Mori et al. [40], Weston et al. [56] had explored the connection between images and words using paired text documents. As more and more data accessible from the Internet, image-text pre-training models including CLIP [42], ALIGN [21], ALBEF [28] and Florence [59] proposed to pre-train vision-language models with a contrastive loss. Benefiting from large-scale web data (400M for CLIP, 1.8B for ALIGN and 900M for Florence), image-text pre-training methods can learn powerful visual representation as well as their correlations. Similar ideas can be seen in video-text pre-training [2] with a largescale video-text dataset Howto100M [38]. Even though the pre-training image-text models can capture the object appearance from visual embeddings and their corresponding description (e.g. nouns) in the text embed- dings [49], we consider they are suboptimal to capture the change between frames as well as their corresponding descriptions (e.g. verbs) in the text. Video ( ) … $ 1 2 3 4 $ ∈ ℝ % × ( ! … … + $ ∈ ℝ % × )× *× ( ! Visual Encoder ( $ ) Transformer Decoder Visual Context Injection Correlation Transfer Learning Cao et al. [2] explored the video-text pre-training to learn the alignment between moment and text. However, it is suboptimal with a lack of fine-grained well-annotated video-text alignment samples (50% samples are not aligned in dataset Howto100M [38]). Meanwhile, image-text pretraining methods have shown promising generalisable ability with multiple downstream tasks, including image classification [6,35], action recognition [53], video retrieval [37]. Ju et al. [22] transferred CLIP models [42] to multiple video tasks by textual prompting and visual temporal modelling. Wang et al. [53] utilized CLIP models for video recognition by traditional end-to-end fine-tuning. Lin et al. [29] proposed to build temporal dependencies between images. Cheng et al. [4], Luo et al. [37] proposed to use correlation modelling ability from CLIP models and build their temporal relations by transformers [52] for video retrieval task. Even though existing methods [4,22,29,37] demonstrated promising results in transferring CLIP to video understanding task, a problem still remains: image-text pretraining models are less sensitive to actions [17]. In this work, we explore multi-modal correlations derived from large-scale image-text data for generalisable VMR. To enable the action understanding for image-text pre-training models, we extract visual context information and spatial dynamic information and enable the text encoder to understand the entities in video frames and their movements. VMR by Image-Text Pre-Training Given an untrimmed video V = {I i } L v i=1 composed of L v frames and a natural language sentence Q, the objective of video moment retrieval is to predict when the target video moment starts t s ∈ [1, L v ] and ends t e ∈ [t s , L v ] in the video V . For generalisable VMR, not only what happens in V and what is described in Q but also their alignments are supposed to be modelled. This is intrinsically challenging considering the free-form nature of both the unscripted videos and natural language sentences as well as their complex correlations. To retrieve a target video moment by a text sentence, the untrimmed video V and query sentence Q are first fed into a pre-training visual and a textual encoder in respective to obtain the video X v = {x v i } L v i=1 = {f v (I i )} L v i=1 ∈ R D×L v and sentence x q = f q (Q) ∈ R D features. Both the features will then be taken as the inputs to a video moment retrieval model f g (X v , x q ) to predict the temporal boundary of the target moment (t s ,t e ). As the visual and textual encoders are pre-trained on image-text data, they are unaware of temporal changes in videos or the words/phrases describing them in the text. Therefore, we additionally adapt the two encoders ( Fig. 2) to model the visual context and spatial dynamic in both modalities. Together with the pre-learned visual-textual correlations which are prone to be universal, the video change-sensitive features will enable f g to predict accurate temporal boundary even for video moments filmed in novel scenes or described by unseen vocabulary. Visual-Dynamic Injection We start with using a language parsing tool 1 to extract all the noun chunks (a noun plus the words describing the noun) in the query sentence Q, which are supposed to describe the entities of interest in the target video moment. We mask all the other words in the sentence Q and consider such a masked query can be matched with the corresponding video content without knowing the temporal dependencies among frames. In this case, we call the masked sentence about the static content in videos as static query Q s . In contrast, we construct another dynamic query Q d with all the noun chunks in Q being masked, which is critical for understanding complex video moments composed of arbitrary evolving events but missing in the image-text pretraining. Whilst the video change information can be captured by additional sequence analysis in the video moment retrieval model f g , it is impractical for f g to understand the phrases describing video changes in the text which are originally overlooked in x q . Therefore, we model video changes with visual context injection and spatial dynamic injection and enforce the text encoder to match the dynamic queries Q d with them. By doing so, the adapted text encoder is able to yield visual-dynamic sensitive representations for query sentences to ensure more accurate VMR by both visual and dynamic matching. Visual Context Injection. The visual context we discover in the videos is about how the frames display the entities related to the video changes. Such visual information is likely to encoding the presence of scenes (e.g. outdoor or indoor), the status of entities (e.g. boiling or cold water) and etc. It is important for recognising and locating video moments which involve specific objects and scenes. To that end, we apply a transformer decoder [52] and encode the visual context information into the static query Q s : x qs = f q (Q s ) ∈ R D , x qs = TransDecoder(x qs , X v , X v ) ∈ R D .(1) In Eq. (1), x qs is the D-dimensional textual feature of the static query Q s obtained by the text encoder f q , X v is the video frame features, andx qs is the visual context-aware feature of Q s computed by a transformer decoder whose three inputs correspond to the query, key and value, respectively. To inject such visual context information to the text encoder with a focus on the words describing the changes in videos, we compute the dynamic query feature Q d by the text encoder and encourage its consistency withx qs : x qd = f q (Q d ) ∈ R D L vc (V, Q) = ∥FC(x qd ) −x qs ∥ 2 2 ,(2) where FC(·) is a linear projection layer. The rationale behind this design is to probe the video frames by the static query in order to select the visual context engaging the entities potentially existing in the target moments. By doing so, the text encoder is updated to align the dynamic query with its visual context and avoid distractions from irrelevant video content. Spatial Dynamic Injection. Besides the visual context demonstrating the entity of interests in frames, another crucial information for video change is about how the spatial locations of different entities change in time order. However, the motion patterns encoding such dynamic information is hidden in the complex visual patterns in videos. It is non-trivial to discover and use them to raise the text encoder's attention on the corresponding descriptions. Therefore, we propose to extract the location changes of salient entities in videos and explicitly inject such spatial dynamics into the text encoder. To that end, we first obtain the per-frame spatial featuresX v i ∈ R D×H×W ∀ i ∈ [1, L v ] as the last feature maps produced by the convolutional neural networks [16,26,47] or the patch-wise features in Visual Transformers [10]. The H and W denote the height and width resolutions of concerns. We then adopt a transformer-like formulation to compute a heatmap for each video frame: M i = FC(x v i ) · FC(X v i )/ √ D ∈ R H×W .(3) The frame-wise heatmap is computed by the correlations between every spatial feature and the global image representation. Therefore, the salient entities whose visual information is encoded in the image feature will result in corresponding salient regions in the heatmap. We then flatten the heatmap and feed it into a linear projection layer to compute a D-dimensional vector as the holistic representation of the spatial feature for each frame. The spatial dynamics in the video can be given by any sequence analysis model: m i = FC(Flatten(M i )) ∈ R D m = SeqModel({m i } L v i=1 ) ∈ R D .(4) As the visual information is deprecated in the spatial features, we cannot probe them by the static query Q s . Hence, we choose a transformer encoder [52] to build their dependencies and take the averaged outputs as the spatial dynamic featurem of the video. To inject such dynamic information into the text encoder, we then enforce consistent correlations between the spatial dynamic features of different videos (V and V ′ ) and the corresponding descriptions in text (Q d and Q d ′ ) by ϵ v = cos(m,m ′ ) ϵ q = cos(FC(x qd ), FC(x qd ′ )) L sd (V, Q, V ′ , Q ′ ) = (ϵ v − ϵ q ) 2 .(5) In Eq. (5), the notation ϵ v stands for the cosine similarity between the spatial dynamic features of two videos while ϵ q is that of the two corresponding dynamic queries Q d and Q d ′ . In contrast to the visual context injection, since the spatial features used here lost all the visual cues in videos, we optimise their correlations consistency with that of the dynamic queries rather than directly pushing them closer to the matched textual features. By doing so, different sentences matched with the videos sharing similar motion patterns will be encouraged to focus on the common descriptions of such dynamic information in the text. Video Moment Retrieval. With the visual and textual features pre-learned from large-scale image-text datasets as well as our adaptation of textual features to be aware of temporal changes in videos, our VDI model is ready to benefit existing VMR models. Here, we take the state-of-the-art Mutual Matching Network (MMN) as an example. Specifically, given the frame-wise video features X v and the sentence feature x q , we first enumerate all the start-end frame pairs to generate L m = L v × L v video segments as the candidates of the target moment. We then construct a 2D feature map X m = Conv2D({x m i,j } L v i,j=1 ) ∈ R D×L v ×L v where x m i,j is the feature of the segment starting from the i-th frame and ending at the j-th frame. After that, both the features of video segments X m and query sentences x q will be linearly projected into a common space and their alignments are then measured by cosine similarities cos(·, ·): Y iou = cos(FC(x q ), Conv1D(X m )) ∈ R L v ×L v .(6) The predicted alignment scoresỸ iou between every segment and the query sentence will be supervised by the temporal IoU between it and the manually labelled temporal boundary (t s , t e ) of the target moment: y iou i,j = IoU((t s , t e ), (i, j)), Y iou = {y iou i,j } L v i,j=1 ∈ R L v ×L v , L iou (V, Q) = BCE(Y iou ,Ỹ iou ).(7) Besides learning to locate video moments by aligning positive segment-text pairs, we follow MMN to conduct mutual-modal contrastive learning among negative sample pairs. In particular, for a moment x m t s ,t e in video V and its description x q , we construct a negative video set X m− and a negative sentence set X q− . We then map the segments and queries features to another shared feature space by linear projections and conduct contrastive learning by: X m = {x m i,j } L v i,j=1 = Conv1D(X m ) ∈ R D×L v ×L v , x q = FC(x q ) ∈ R D , p m = exp(cos(x m t s ,t e ,x q )/τ ) In Eq. (8), the tilde on top of all the segments and queries features {x m t s ,t e , x q , x} denotes their linear projected counterparts. The variables p m and p q measure how likely the model identifies the target moment x m t s ,t e and the query sentence x q according to each other from the respective negative sets. In this multi-modal common space, we can compute another alignment scores between every candidate segment and the query sentence: Y cl = cos(x q ,X m ) ∈ R L v ,L v .(9) The two video-text alignment predictions will then be fused by the hadamard product and the temporal boundary predicted for the target moment can then be computed in a maximum likelihood manner: Y =Ỹ iou ⊙Ỹ cl t s = arg max(cmax(Ỹ )),t e = arg max(ỹ t s ).(10) By learning from L iou and L cl jointly, the model is trained to be aware of both the matched and unmatched video-text information. Algorithm 1 Visual-Dynamic Injection (VDI) Input: Untrimmed videos V , Query sentences Q, Temporal boundary labels (t s , t e ), A visual f v and a textual encoder f q from image-text pre-training. Output: An updated video moment retrieval model. Generates the static Q s and dynamic Q d query sentences; Computes the features of Q s and Q d by f q ; Computes the features of videos V by f v ; Computes the visual context L vc (Eq. (2)) and spatial dynamic L sd (Eq. (5)) losses; Adapts the textual encoder f q by minimising L vc and L sd ; Computes the features of the query Q by f q ; Feeds the features of video V and query Q to f g ; Computes the losses L iou (Eq. (7)) and L cl (Eq. (8)); Optimises the VMR model f g by minimising L iou and L cl . Model Training The VDI model is trained in a conventional batch-wise manner. A mini-batch is composed of n randomly sampled video-text pair (V, Q) as well as the temporal boundary labels (t s , t e ) of the target moments. The overall loss functions are computed by: L = 1 n n i=1 (λ iou L iou (V i , Q i ) + λ cl L cl (V i , Q i )+ λ vc L vc (V i , Q i )+ λ sd 1 n n j=1 L sd (V i , Q i , V j , Q j )).(11) The training objective function L in Eq. (11) is then be used to optimise both the VMR model f g and the text encoder f q from the pre-training model by any stochastic gradient descent algorithms. The overall training process of the VDI model is summarised in Alg. 1. Experiments To evaluate the importance of generalisable correlations between the visual and textual space, we conduct experiments on video moment retrieval (VMR) and compare with the SOTAs on both out-of-distribution (OOD) and independent and identically distributed (IID) data splits. In this section, we first explain the implementation details and then report our results in the comparison with recent methods. Finally, we evaluate the effectiveness of each component in our methods. Experimental Settings Dataset Charades-STA [13] is a benchmark dataset for VMR, which is built upon the Charades dataset [46]. The Charades dataset is collected for video action recognition and video captioning. Gao et al. [13] adapt the Charades dataset to VMR by collecting the query annotations. The Charades-STA dataset contains 6670 videos and involves 16124 queries, where 12404 pairs are used for training and 3720 for testing. The average duration of the videos is 30.59 seconds and each video contains 2.41 annotated moments, and the moment has an average duration of 8.09 seconds. ActivityNet-Captions [24] is collected for the dense video captioning task from ActivityNet [1] where the videos are associated with 200 activity classes, and the content is more diverse compared to Charades-STA. The ActivityNet-Captions dataset consists of 19811 videos with 71957 queries. The average duration of the videos is around 117.75 seconds and each video contains 3.63 annotated moments, and the moment has an average duration of 37.14 seconds. The public split of the ActivityNet-Captions dataset contains a training set and two validation sets val 1 and val 2, including 10009, 4917, 4885 videos separately. Evaluation Metrics We adopt "R@n, IoU = µ" and "mIoU" as the evaluation metrics, where "R@n, IoU = µ" denotes the percentage of language queries having at least one result whose intersection over union (IoU) with ground truth is larger than µ in top-n retrieved moments. "mIoU" is the average IoU over all testing samples. We report the results as n ∈ {1, 5} with µ ∈ {0.5, 0.7} for fair IID split comparison following MMN [55] and n∈ {1} with µ∈ {0.5, 0.7} and mIoU for fair OOD split comparison with [27]. Implementation Details We experiment with the MMN [55] as the VMR framework to evaluate our method. Specifically, we apply the pre-training visual extractor of the CLIP (ViT-B/32) [42] as the backbone, and directly feed to the VMR framework to generate proposals. Our VMR framework is similar to MMN, where we only delete the linear layer in the pooling module to maintain the feature structure of CLIP. During training, we only update the parameters of the text encoder to empower the understanding of video change, no additional computation cost is introduced in inference. We use AdamW [36] optimizer with a learning rate of 1 × 10 −4 and a batch size 48 for Charades-STA, a learning rate of 8 × 10 −4 and a batch size 48 for ActivityNet-Captions. Following MMN, we early stop the training when we observe the performance on the validation set start to drop. The learning rate of text encoder is always 1/10 of the main model. λ vc and λ sd is set to 0.5 and 0.01. Comparison with the SOTAs In this section, we compare the results of our methods under the VMR task with the baseline MMN [55] and existing SOTAs. To evaluate the importance of generalisability between the vision and text, we report the results under OOD testing and IID testing. Novel-Word OOD Testing To validate the generalisation ability of our method to capture unseen words and scenes, we conduct experiments on novel-word OOD testing. Specifically, the novel-word OOD split is recently released by Li et al. [27] where the testing split contains novel words which are not seen in the training split, and the corresponding scenes are not seen as well. We follow the settings in Li et al. [27] to report the performance of Charades-STA [13] and ActivityNet-Captions [24] under R@1. Novel-word OOD testing results are shown in Table 1. We collect the performance reported by Li et al. [27] and reproduce the baseline model [55] with CLIP features under the same settings. One can see that we outperform the SOTA method by a significant margin. Especially on Charades-STA dataset, we outperform the baseline model MMN [55] with a margin of 2.62%/4.46% under the IoU = 0.5/0.7. We outperform VISA [27] by a margin of 4.12%/7.75%. One can see that on ActivityNet-Captions, the margin is less obvious than Charades-STA (2.21% vs 4.12%), it is partially because ActivityNet-Captions display longer moments than Charades-STA (averaged 37.14s vs 8.09s), so it is more challenging to capture the video change. Also, compared with the 37 long moments (L mom /L vid ≥ 0.5) in Charades-STA, there are over Table 1. Novel-word testing comparison between our method with other state-of-the-art methods on Charades-STA [13] and ActivityNet-Captions [24]. The "Pre-train" column indicates the types of pre-trained models adopted. 15k in ActivityNet-Captions, which makes it trivial by predicting long predictions instead of capturing the semantics. Obtaining superior OOD performance over videobased (Kinetics [23]) pre-training models demonstrates that our method can take advantage of the image-text pretraining feature and obtain generalisable correlations to unseen words and scenes. Original Split Testing To further evaluate the effectiveness of our method, we also conduct extensive experiments on the standard testing, where the training and testing splits share independent and identical distribution. In Table 2, we include the recently reported results of SOTA methods as well as their visual pre-training data (Vis.P.T). One can see that when replacing the MMN from separated pre-training with joint pre-training features, the performance increases from 47.31% to 50.48%, indicating that the correlation between the vision and text is essential. With our proposed method to empower the model's understanding of actions, the performance further improves to 52.32%, outperforming the baseline model with a large margin of 5.01%. From the results of ActivityNet-Captions, we can see that even though there is a performance drop from video-based pre-training to image-based pre-training (48.59% vs 46.89%), our method can fill the gap by injecting the video change understanding into the model. To evaluate the learning of correlations, we compare with existing methods with a specific focus on image-based pre-training datasets [7]. As one can see from Table 2, not only can we outperform SOTAs by a large margin, and we can narrow down the gap between the image-based pretraining feature and video-based pre-training feature. Ablation Study In this section, we perform in-depth ablations to evaluate the effectiveness of each component in VDI on Charades-STA dataset [13] with novel-word splits [27]. We report the performance under R@1 for IoU ∈ {0.5, 0.7} and mIoU. Visual Context Injection. To evaluate that it is essential to inject visual context information into the text embeddings, we study different types of visual context generations as shown in Table 3. We take the MMN with CLIP pre-training features as our baseline. The superior performances yielded by the models with visual context injection over the baseline demonstrate the effectiveness of the design. Moreover, we observe that using static query to probe the videos (L vc w/ Q s ) produces better results than probing by the complete sentence (L vc w/ Q) and the pure visual context generation (L vc w/o Q) without text. This implies the importance to avoid correlating the text descriptions of video changes with irrelevant visual context. Spatial Dynamic Injection. To evaluate that spatial dynamic information is essential for the text encoder, we study two types of dynamic modelling, including LSTM [18] and Transformer Encoder [52]. As shown in Table 3, by introducing spatial dynamic information, both a recurrent network (L sd w/ LSTM) or a transformer encoder (L sd w/ Trans) is ready to contribute to a more precise VMR. With a combination of L vc and L sd , the performance improves to 46.47%, which further demonstrates the effectiveness of VDI. Dynamics Awareness. We further evaluate our baseline and VDI models by using either the complete sentences Q, the static Q s or the dynamic Q d queries to retrieve video moments on Charades-STA. As shown in Fig. 3 Table 2. Comparisons to the state-of-the-art methods on the standard splits of VMR benchmark datasets. The "Vis.P.T" column indicates that the video encoders adopted are pre-trained by only videos [23] ("Video"), only images [7] ("Image"), video-text pairs [38] ("Video-Text") and Image-text pairs [42] ("Image-Text"). Best performances among image-based pre-training methods are highlighted in bold. 50 Table 3. Ablation studies on visual context injection L vc with different text queries and spatial dynamic injection L sd with different sequence analysis models. moments by static queries than the dynamic ones while our VDI model is in opposed. This verifies the sensitivity of VDI to video changes by correlating visual context and spatial dynamics with text and explains its improvements to the baseline on all VMR tasks. Conclusion In this paper, we propose to learn universal visual-textual correlations for video moment retrieval (VMR). To address the limitation that the image-text pre-training methods are less sensitive to video changes, we design visual context and spatial dynamic injection to the text encoder with an emphasis on the words describing video changes. By doing so, the potentially relevant visual and motion patterns in videos are encoded in the corresponding text embeddings, enabling more accurate video-text alignments. Experiments on two important datasets (Charades-STA and ActivityNet-Captions) prove that VDI can learn both effective and generic visual-text correlations. Moreover, the comparison between the before and after visual-dynamic injection demonstrate the sensitivity of VDI to video changes. Figure 1 . 1Contemporary methods lack moment-text correlations. Figure 2 . 2The framework of VDI, in which the video V is fed into a visual encoder to generate image global features X v and image patch featuresX v . The sentence Q is parsed into static query Q s and dynamic query Q d . Visual Context Injection (L vc ) aligns the Q d with Q s guided visual context information. Spatial Dynamic Injection (L sd ) empowers the Q d with the awareness of the spatial dynamics. ,t e }∪X m− exp(cos(x,x q )/τ )p q = exp(cos(x m t s ,t e ,x q )/τ ) x∈{x q }∪X q− exp(cos(x m t e ,t e ,x)/τ ) L cl (V, Q) = − log p m − log p q . , the baseline MMN model yields better results when retrieving videoMethod Charades-STA ActivityNet-Captions Vis.P.T R@1, R@1, R@5, R@5, Vis.P.T R@1, R@1, R@5, R@5, IoU=0.5 IoU=0.7 IoU=0.5 IoU=0.7 IoU=0.5 IoU=0.7 IoU=0.5 IoU=0.7 VideoBert [50] Video-Text 32.70 19.50 68.10 46.20 Video-Text 37.20 21.00 66.70 53.60 MIL-NCE [39] 37.00 21.20 74.30 50.40 41.80 24.50 73.50 57.70 LocVTP [2] 43.60 26.30 81.90 55.30 48.20 30.50 80.10 64.70 MGSL [32] Video 63.98 41.03 93.21 63.85 Video 51.87 31.42 82.60 66.71 D-TSG [33] 65.05 42.77 94.42 65.16 54.29 33.64 86.58 69.36 2D-TAN [62] Image 39.70 23.31 80.32 51.26 44.51 26.54 77.13 61.96 VSL-Net [61] 39.20 20.80 - - 43.22 26.16 - - CBLN [31] 43.67 24.44 88.39 56.49 48.12 27.60 79.32 63.41 DCM [58] 47.80 28.00 - - 44.90 27.70 - - DRN [60] 42.90 23.68 87.80 54.87 45.45 24.36 77.97 50.30 MMN [55] 47.31 27.28 83.74 58.41 48.59 29.26 79.50 64.76 MMN [55] Image-Text 50.48 29.65 85.27 60.67 Image-Text 46.89 27.26 78.32 63.47 VDI (Ours) 52.32 31.37 87.03 62.30 48.09 28.76 79.69 64.88 Figure 3. Video Moment Retrieval by the complete Q, the static Q s or the dynamic Q d sentence descriptions..48 36.13 32.74 52.32 38.9 41.59 20 25 30 35 40 45 50 55 Q Nouns Verbs R1, IoU=0.5 29.65 21.67 20.35 31.37 23.06 25.91 20 22 24 26 28 30 32 Sentence Nouns Verbs R1, IoU=0.7 85.27 80.56 78.25 87.03 80.07 82.74 20 30 40 50 60 70 80 90 100 Sentence Nouns Verbs R5, IoU=0.5 60.67 52.47 49.68 62.30 54.38 56.67 20 25 30 35 40 45 50 55 60 65 Sentence Nouns Verbs R5, IoU=0.7 ! " ! " ! " ! " MMN VDI (Ours) Method R@1, R@1, mIoU IoU=0.5 IoU=0.7 Baseline 43.85 24.17 39.50 L vc w/o Q 43.12 25.32 39.43 L vc w/ Q 45.02 27.63 40.40 L vc w/ Q s 45.47 29.35 40.61 L sd w/ LSTM 45.32 25.76 40.07 L sd w/ Trans 44.60 26.06 40.09 L vc + L sd 46.47 28.63 41.60 spaCy: https://spacy.io/ AcknowledgementsThis work was supported by the China Scholarship Council, the Alan Turing Institute Turing Fellowship, Veritone, Adobe Research and Zhejiang Lab (NO. 2022NB0AB05). 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[ "\nSchool of Electronic and Information Engineering\nSchool of Electrical Engineering\nXi'an Jiaotong University\nXi'anChina\n", "\nDepartment of Electrical and Computer Engineering\nSouthwest Jiaotong University\nChengduChina\n", "\nSchool of Electronic and Information Engineering\nUniversity of Florida\n32611GainesvilleFLUSA\n", "\nXi'an Jiaotong University\nXi'anChina\n" ]
[ "School of Electronic and Information Engineering\nSchool of Electrical Engineering\nXi'an Jiaotong University\nXi'anChina", "Department of Electrical and Computer Engineering\nSouthwest Jiaotong University\nChengduChina", "School of Electronic and Information Engineering\nUniversity of Florida\n32611GainesvilleFLUSA", "Xi'an Jiaotong University\nXi'anChina" ]
[]
Traditional Kalman filter (KF) is derived under the well-known minimum mean square error (MMSE) criterion, which is optimal under Gaussian assumption. However, when the signals are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises, the performance of KF will deteriorate seriously. To improve the robustness of KF against impulsive noises, we propose in this work a new Kalman filter, called the maximum correntropy Kalman filter (MCKF), which adopts the robust maximum correntropy criterion (MCC) as the optimality criterion, instead of using the MMSE. Similar to the traditional KF, the state mean and covariance matrix propagation equations are used to give prior estimations of the state and covariance matrix in MCKF. A novel fixed-point algorithm is then used to update the posterior estimations. A sufficient condition that guarantees the convergence of the fixed-point algorithm is given. Illustration examples are presented to demonstrate the effectiveness and robustness of the new algorithm. Index Terms-Kalman Filter, Maximum Correntropy Criterion (MCC), Fixed-Point Algorithm.
10.1016/j.automatica.2016.10.004
[ "https://arxiv.org/pdf/1509.04580v1.pdf" ]
7,595,388
1509.04580
187b1ea75b52f3cc804ff2ec6d3652d9f0a09c67
School of Electronic and Information Engineering School of Electrical Engineering Xi'an Jiaotong University Xi'anChina Department of Electrical and Computer Engineering Southwest Jiaotong University ChengduChina School of Electronic and Information Engineering University of Florida 32611GainesvilleFLUSA Xi'an Jiaotong University Xi'anChina > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1  Traditional Kalman filter (KF) is derived under the well-known minimum mean square error (MMSE) criterion, which is optimal under Gaussian assumption. However, when the signals are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises, the performance of KF will deteriorate seriously. To improve the robustness of KF against impulsive noises, we propose in this work a new Kalman filter, called the maximum correntropy Kalman filter (MCKF), which adopts the robust maximum correntropy criterion (MCC) as the optimality criterion, instead of using the MMSE. Similar to the traditional KF, the state mean and covariance matrix propagation equations are used to give prior estimations of the state and covariance matrix in MCKF. A novel fixed-point algorithm is then used to update the posterior estimations. A sufficient condition that guarantees the convergence of the fixed-point algorithm is given. Illustration examples are presented to demonstrate the effectiveness and robustness of the new algorithm. Index Terms-Kalman Filter, Maximum Correntropy Criterion (MCC), Fixed-Point Algorithm. The optimization criteria in information theoretic learning (ITL) [6] [7] have gained increasing attention over the past few years, which uses the information theoretic quantities (e.g. entropy) estimated directly from the data instead of the usual second order statistical measures, such as variance and covariance, as the optimization costs. Information theoretic quantities can capture higher-order statistics and offer potentially significant performance improvement in machine learning and signal processing applications. The ITL links information theory, nonparametric estimators, and reproducing kernel Hilbert spaces (RKHS) in a simple and unconventional way. In particular, the correntropy as a nonlinear similarity measure in kernel space has its root in Renyi's entropy [8]- [12]. Since correntropy is also a local similarity measure (hence insensitive to outliers), it is naturally a robust cost for machine learning and signal processing [13]- [21]. In supervised learning, such as regression, the problem can be formulated as that of maximizing the correntropy between model output and desired response. This optimization criterion is called in ITL the maximum correntropy criterion (MCC) [6] [7]. Recently, the MCC has been successfully used in robust adaptive filtering in impulsive (heavy-tailed) noise environments [6] [9]- [11] [22]. The MCC solution cannot be obtained in closed form even for a simple linear regression problem, so one has to solve it using an iterative update algorithm such as the gradient based methods [9]- [11] [22]. The gradient based methods are simple and widely used. But they depend on a free parameter step-size and usually converge to an optimal solution slowly. The fixed-point iterative algorithm is an alternative efficient way to solve the MCC solution, which involves no step-size and may converge to the solution very fast [6] [24] [25]. A sufficient condition that guarantees the convergence of the fixed-point MCC algorithm was given in [26]. In the present paper, we develop a new Kalman filter, called the maximum correntropy Kalman filter (MCKF), based on the MCC and a fixed-point iterative algorithm. Similar to the traditional KF, the MCKF not only retains the state mean propagation process, but also preserves the covariance matrix propagation process. Especially, the new filter has a recursive solution structure and is suitable for online implementation. It is worth noting that in [23], the MCC has been used in hidden state estimation, but it involves no covariance propagation process and is in form not a Kalman filter. The rest of the paper is organized as follows. In Section II, we briefly introduce the maximum correntropy criterion and Kalman filter. In Section III, we derive the MCKF algorithm and give the computational complexity and convergence analysis. Simulation results are then provided in Section IV to show the excellent performance of the MCKF. Finally, conclusion is given in Section V. Maximum Correntropy Kalman Filter A. Maximum Correntropy Criterion Correntropy is a generalized similarity measure between two random variables. Given two random variables , XY   with joint distribution function   F, XY xy, correntropy is defined by         , , , F , XY V X Y X Y x y d x y       (1) where  denotes the expectation operator, and ( , )   is a shift-invariant Mercer Kernel. In this paper, without mentioned otherwise the kernel function is the Gaussian Kernel given by     2 2 , G exp 2 e x y e         (2) where e x y , and 0   stands for the kernel bandwidth. In most practical situations, however, only limited number of data are available and the joint distribution F XY is usually unknown. In these cases, one can estimate the correntropy using a sample mean estimator:      1 1 , G ( ) N i V X Y e i N     (3) where ( ) ( ) ( ) e i x i y i  ,   1 ( ), ( ) N i x i y i  are N samples drawn from F XY . Taking Taylor series expansion of the Gaussian kernel, we have       2 2 0 1 , 2! n n nn n V X Y X Y n          (4) As one can see, the correntropy is a weighted sum of all even order moments of the random variable XY  . The kernel bandwidth appears as a parameter weighting the second order and higher order moments. With a very large  (compared to the dynamic range of the data), the correntropy will be dominated by the second order moment. Given a sequence of error data   1 () N i ei  , the cost function of MCC is given by   1 1 G ( ) N MCC i J e i N     (5) Suppose the goal is to learn a parameter vector W of an adaptive model, and let () xi and () yi denote, respectively, the model output and the desired response. The MCC based learning can be formulated as the following optimization problem:   1 1 arg max G ( ) N W i W e i N     (6) where Ŵ denotes the optimal solution, and  denotes a feasible set of parameter. B. Kalman Filter Kalman filter provides a powerful tool to deal with state estimation of linear systems, which is an optimal estimator under linear and Gaussian assumptions. Consider a linear system described by the following state and measurement equations: ( ) ( 1) ( 1) ( 1), k k k k      x F x q (7) ( ) ( ) ( ) ( ). k k k k   y H x r (8) where () n k  x  denotes the n -dimensional state vector, () m k  y  represents the m -dimensional measurement vector at instant k . F and H stand for, respectively, the system matrix (or state transition matrix) and observation matrix. ( 1) k  q and () k r are mutually uncorrelated process noise and measurement noise, respectively, with zero mean and covariance matrices ( 1) ( 1) ( 1), T k k k          q q Q ( ) ( ) ( ) T k k k       r r R(9) In general, Kalman filter includes the following two steps: 1) Predict The prior mean and covariance matrix are given by   ( | 1) ( 1) ( 1| 1), k k k k k      x F x (10) ( | 1) ( 1) ( 1| 1) ( 1) ( 1). T k k k k k k k         P F P F Q(11) 2) Update The Kalman filter gain is computed as   1 ( ) ( | 1) ( ) ( ) ( | 1) ( ) ( ) T T k k k k k k k k k      K P H H P H R(12) The posterior state is equal to the prior state plus the innovation weighted by the KF gain,      ( | ) ( | 1) ( ) ( ) ( ) ( | 1) k k k k k k k k k      x x K y H x(13) Additionally, the posterior covariance is recursively updated as follows:     ( | ) ( ) ( ) ( | 1) ( ) ( ) + ( ) ( ) ( ) T T k k k k k k k k k k k     P I K H P I K H K R K(14) III. KALMAN A. Derivation of the Algorithm Traditional Kalman filter works well under Gaussian noises, but its performance may deteriorate significantly under non-Gaussian noises, especially when the underlying system is disturbed by impulsive noises. The main reason for this is that KF is developed based on the MMSE criterion, which captures only the second order statistics of the error signal and is sensitive to large outliers. To address this problem, we propose in this work to use the MCC criterion to derive a new Kalman filter, which may perform much better in non-Gaussian noise environments, since correntropy contains second and higher order moments of the error. For the linear model described in the previous section, we have  ( | 1) ( ) ( ) () () kk kk k k              I x x H y (15) where I is the nn  identity matrix, and () k  is    ( ) ( | 1) () () k k k k k              x x r , with ( ) ( ) ( | 1) 0 0 ( ) ( | 1) ( | 1) 0 0 ( ) ( ) ( ) ( ) T T p p T r r T k k kk k k k k k k k k k                         P R B B B B BB(16) where () k B can be obtained by Cholesky decomposition of ( ) ( ) T kk       . Left multiplying both sides of (15) by 1 () k  B , we get ( ) ( ) ( ) ( ) k k k k  D W x e(17) where  1 ( | 1) ( ) ( ) () kk k k k          x D B y , 1 ( ) ( ) () k k k         I W B H , 1 ( ) ( ) ( ) k k k    eB . Since ( ) ( ) T kk    e e I , the residual error () k e are white. Now we propose the following MCC based cost function:   1 1 ( ) G ( ) ( ) ( ) L L i i i J x d k k k L     wx (18) where () i dk is the i -th element of () k D , () i k w is the i -th row of () k W , and L n m  is the dimension of () k D . Then, under MCC criterion, the optimal estimate of () k x is (20) The optimal solution can thus be obtained by solving (21) It follows easily that (22) is actually a fixed-point equation of () k    1 ( ) arg max ( ) arg max G ( ) L L i x x i k J x e k      x (19) where () i ek is the i -th element of () k e : ( ) ( ) ( ) ( ) i i i e k d k k k  wx() 0 () L Jx k   x    1 1 1 ( ) G ( ) ( ) ( ) G ( ) ( ) ( ) L T i i i i L T i i i i k e k k k e k k d k                    x w w w (22) Since ( ) ( ) ( ) ( ) i i i e k d k k k  wx, the optimal solution x and can be rewritten as   ( ) f ( ) kk  xx (23) with       1 1 1 f ( ) G ( ) ( ) ( ) ( ) ( ) G ( ) ( ) ( ) ( ) ( ) L T i i i i i L T i i i i i k d k k k k k d k k k k d k                                x w x w w w x w A fixed-point iterative algorithm can be readily obtained as     1 ( ) f ( ) t t k k   x x(24) where  () t k x denotes the solution at the fixed-point iteration t . The fixed-point equation (22) can also be expressed as   1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) T T k k k k k k k   x W C W W C D (25) where ( ) 0 () 0 ( ) x y k k k     C C C , with       1 ( ) G ( ) ,..., G ( ) x n k diag e k e k   C ,       1 ( ) G ( ) ,..., G ( ) y n n m n m k diag e k e k      C . The equation (25) can be further expressed as follows (see the Appendix for a detailed derivation):     ( ) ( | 1) ( ) ( ) ( ) ( | 1) k k k k k k k k      x x K y H x(26) where > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4   1 1 1 ( ) ( | 1) ( ) ( ) ( | 1) ( ) ( ) ( | 1) ( | 1) ( ) ( | 1) ( ) ( ) ( ) ( ) T T T p x p T r y r k k k k k k k k k k k k k k k k k k k k                    K P H H P H R P B C B R B C B(27) Remark: Of course, the equation (26) is also a fixed-point equation of () k x because () k K depends on ( | 1) kk  P and () k R , both related to () k x via () x k C and () y k C , respectively. The optimal solution of the equation (26) depends also on the prior estimate  ( | 1) kk  x , which can be calculated by (10) using the latest estimate  ( 1| 1) k k   x . With the above derivations, we summarize the proposed MCKF algorithm as follows: 1) Choose a proper kernel bandwidth  and a small positive number  ; Set an initial estimate  (0 | 0) x and an initial covariance matrix (0 | 0) P ; Let 1 k  ; 2) Use equations (10) (11) to obtain  ( | 1) kk  x and ( | 1) kk  P , and use Cholesky decomposition to obtain ( | 1) p kk  B ; 3) Let 1 t  and   0 ( ( | ) | 1) k k k k   x x , where  ( | ) t kk x denotes the estimated state at the fixed-point iteration t ; 4) Use (28)-(34) to compute  ( |) t kk x ;       ( | ) ( | 1) ( ) ( ) ( ) ( | 1) t k k k k k k k k k      x x K y H x (28) with       1 ( ) ( | 1) ( ) ( ) ( | 1) ( ) ( ) , T T k k k k k k k k k      K P H H P H R (29)   1 ( | 1) ( | 1) ( ) ( | 1), T x p p k k k k k k k      P B C B (30)   1 ( ) ( ) ( ) ( ). T y r r k k k k   R B C B (31)        1 ( ) G ( ) ,..., G ( ) x n k diag e k e k    C   (32)        1 ( ) G ( ) ,..., G ( ) y n n m nm k diag e k e k      C   (33)  1 ( ) ( ) ( ) ( | ) i i i t e k d k k k k   wx (34) 5) Compare the estimation of the current step and the estimation of the last step. If (35) holds, set   ( | ) ( | ) t k k k k  x x and continue to 6). Otherwise, 1 tt  , and go back to 4).    1 1 ( | ) ( | ) ( | ) t t t k k k k kk      xx x (35) 6) Update the posterior covariance matrix by (36), 1 k k  and go back to 2).         ( | ) ( ) ( ) ( | 1) ( ) ( ) + ( ) ( ) ( ) T T k k k k k k k k k k k     P I K H P I K H K R K(36) Remark: As one can see, different from the traditional KF algorithm, the MCKF uses a fixed-point algorithm to update the posterior estimate of the state. The small positive number  provides a stop condition (or a threshold) for the fixed-point iteration. Since the initial value of the fixed-point iteration is set at the prior estimate  ( | 1) kk  x , the convergence to the optimal solution will be very fast (usually in several steps). The bandwidth  is a key parameter in MCKF. In general, a smaller bandwidth makes the algorithm more robust (with respect to outliers) but converge more slowly. On the other hand, when  becomes more and more larger, the MCKF will behave more and more like the ordinary KF algorithm. In particular, the following theorem holds. Theorem 1: When the kernel bandwidth   , the MCKF will reduce to the KF algorithm. Proof: see Appendix. B. Computational Complexity Next, we analyze the computational complexity in terms of the floating point operations for the proposed algorithm. The computational complexities of some basic equations are given in Table I.                (38) The fixed-point iteration number T is relatively small in general (see the simulation results in the next section). Thus the computational complexity of the MCKF is moderate compared with the traditional KF algorithm. C. Convergence Issue The rigorous convergence analysis of the proposed MCKF algorithm is very complicated. In the following, we present only a sufficient condition that guarantees the convergence of the fixed-point iterations in MCKF. The result is similar to that of [26] and hence, will not be proved here. Let . p denote an  denote the minimum eigenvalue of a matrix. According to the results of [26], the following theorem holds. Theorem 2: If 1 1 min 1 ( ) ( ) ( ) ( ) L T ii i L T ii i n k d k kk           w ww , and   † max ,      , in which  is the solution of the equation ()     , with     1 1 min 1 1 ( ) ( ) ( ) , 0, ( ) ( ) ( ) ( ) L T i i i L T i i i i i n k d k G k d k k k                     w w w w(39)( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) G ( ) ( ) ( ) ( ) L T T i i i i i i i i L T i i i i i n k d k k k k k d k k d k k k                          w w w w w w w w   0,   (40) then it holds that   1 f ( ) k   x , and   () 1 f ( ) k k    x x for all   1 ( ) ( ) : ( ) n k k k     x x x  , where   () f ( ) k k  x x denotes the nn  Jacobian matrix of   f ( ) k x with respect to () k x , that is       () 1 f ( ) f ( ) ... f ( ) ( ) ( ) k n k k k x k x k            x x x x (41) with         1 2 1 1 2 1 f ( ) () 1 ( ) ( ) ( ) ( ) ( ) f ( ) 1 ( ) ( ) ( ) ( ) ( ) j L j T i i i i i i L j T i i i i i i k xk e k w k G e k k k k e k w k G e k k d k                                   ww ww x N w w x N w where   1 G ( ) ( ) ( ) L T i i i i e k k k     ww N w w and () j i wk is the j -th element of () i k w . By Theorem 2, if the kernel bandwidth is larger than a certain value, we have     1 () 1 f ( ) f ( ) 1 k k k            x x x (42) By Banach Fixed-Point Theorem [24], given an initial state estimate satisfying   ). Theorem 2 implies that the kernel bandwidth has significant influence on the convergence behavior of MCKF. If the kernel bandwidth  is too small, the algorithm will diverge or converge very slowly. A larger kernel bandwidth ensures a faster converge speed but usually leads to a poor performance in impulsive noises. In practical applications, the bandwidth can be set manually or optimized by trial and error methods. IV. ILLUSTRATIVE EXAMPLES In this section, we present two illustrative examples to demonstrate the performance of the proposed MCKF algorithm, and compare it to the traditional KF algorithm. A. Example 1 Consider the following linear system: Table II shows the MSEs of 1 x and 2 x for different filters. Here the MSE is computed as an average over 100 independent Monte Carlo runs, and in each run, 1000 samples (time steps) are used to evaluate the MSE. Since all the noises are Gaussian, the Kalman filter performs very well and in this example, it achieves almost the best performance (that is, the smallest MSEs). One can also see that when the kernel bandwidth is too small, the MCKF may achieve a worse performance; while when the bandwidth becomes larger, its performance will approach that of the KF. Actually, it has been proved that when  , the MCKF will reduce to the traditional KF. In general, one should choose a larger kernel width under Gaussian noises. Second, we consider the case in which the process noises are still Gaussian but the observation noise is a heavy-tailed (impulsive) non-Gaussian noise, with a mixed-Gaussian distribution, that is As one can see, in impulsive noises, when kernel bandwidth is too small or too large, the performance of MCKF will be not good. In this case, however, with a proper kernel bandwidth (say 2.0   ), the MCKF can outperform the KF significantly, achieving a desirable error distribution with a higher peak and smaller dispersion. Again, when  is very large, MCKF achieves almost the same performance as the KF. Fig.3 shows the fixed-point iteration numbers at the time step (or instant) 1 k  for different kernel bandwidths. It is evident that the larger the kernel bandwidth, the faster the convergence speed. In particular, when the kernel bandwidth is large enough, the fixed-point algorithm in MCKF will converge to the optimal solution in just one or two iterations. In practical applications, to avoid slow convergence, the kernel bandwidth cannot be set at a very small value. Similar results can also be seen from Table III, where the average fixed-point iteration numbers of every time step for different filters are shown, which are computed as averages over 100 independent Monte Carlo runs, with each run containing 1000 time steps. The results clearly indicate that the MCKF can achieve much better tracking performance than the traditional KF algorithm. Now we consider a practical example about one-dimensional linear uniformly accelerated motion. The state vector is 1 1 1 2 2 2 ( ) ( 1) ( 1) cos( ) sin( ) ( ) ( 1) ( 1) sin( ) cos( ) x k x k q k x k x k q k                                    (43)   1 2 () ( ) 1 1 ( ) ()  1 2 3 ( ) ( ) ( ) ( ) T k x k x k x k  x , in which 1 () xk is the position, 2 () xk denotes the speed, and 3 () xk stands for the acceleration. We assume that there are certain noises in the system and only the speed can be observed, which is also affected by some measurement disturbances. T  represents the measurement time interval. Then, the state and measurement equations are given by 1 1 1 . First, the process noises are assumed to be Gaussian and the measurement noise is non-Gaussian with a mixed-Gaussian distribution, that is 2 2 2 3 3 3 ( ) 1 0 ( 1) ( 1) ( ) 0 1 ( 1)( 1)( ) 0 0 1 ( 1) ( 1) x k T x k q k x k T x k q k x k x k q k                                                  (  . Those results confirm again that the proposed MCKF can outperform the traditional KF significantly when the system is disturbed by Gaussian process noises and non-Gaussian measurement noise. q k N N q k N N q k N N r k N N            With the same initial values and parameters setting as before, the results are shown in Fig.9~11 and Table VII. As expected, the MCKF performs much better than the traditional KF when the system is disturbed by non-Gaussian process and measurement noises. V. CONCLUSION A new Kalman type filtering algorithm, called maximum correntropy Kalman filter (MCKF), has been proposed in this work. The MCKF is derived by using the maximum correntropy criterion (MCC) as the optimality criterion, instead of using the well-known minimum mean square error (MMSE) criterion. The propagation equations for the prior estimation of the state and covariance matrix in MCKF are the same as those in KF. However, different from the KF, the MCKF uses a novel fixed-point algorithm to update the posterior estimations. The computational complexity of the MCKF is not expensive and the convergence is ensured if the kernel bandwidth is larger than a certain value. When the kernel bandwidth is large enough, the MCKF will behave like the KF. With a proper kernel bandwidth, the MCKF can outperform the KF significantly, especially when the underlying system is disturbed by some impulsive non-Gaussian noises. This completes the proof. we consider the case in which the noises are all Gaussian, Fig.1 and Fig.2 illustrate the probability densities of the estimation errors of 1 x and 2 x . In the simulation, we set Fig. 1 .Fig. 2 . 12Probability densities of 1 x estimation errors with different filters Probability densities of 2 x estimation errors with different filters Fig. 3. Fixed-point iteration numbers at time step 1 k  for different kernel Fig. 4 and 4Fig.5 show the true and the estimated values of 1 () xk and 2 () xk with KF and MCKF ( Fig. 4 .Fig. 5 . 45The true and the estimated values of 1 x The true and the estimated values of 2 x B. Example 2 Fig the initial values of the true state, estimated state and covariance matrix are assumed to be: .6~Fig. 8demonstrate the probability densities of the Fig. 6 . 6Probability densities of 1 x estimation errors for KF and MCKF in Gaussian process noises and non-Gaussian measurement noise Fig. 7 . 7Probability densities of 2 x estimation errors for KF and MCKF in Gaussian process noises and non-Gaussian measurement noise Fig. 8. Probability densities of 3x estimation errors for KF and MCKF in Gaussian process noises and non-Gaussian measurement noise Next, we consider the situation where the process and measurement noises are all non-Gaussian with mixed- Fig. 9 . 9Probability densities of 1 x estimation errors for KF and MCKF in non-Gaussian process and measurement noises Fig. 10. Probability densities of 2 x estimation errors for KF and MCKF in non-Gaussian process and measurement noises Fig. 11. Probability densities of 3 x estimation errors for KF and MCKF in non-Gaussian process and measurement noises Badong Chen, Senior Member, IEEE, Xi Liu, Haiquan Zhao, José C. Prí ncipe, Fellow, IEEE E > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <2 II. PRELIMINARIES TABLE I COMPUTATIONAL ICOMPLEXITIES OF SOME EQUATIONS so it is very easy to obtain their inverse matrices. Assume that the average fixed-point iteration number is T . Then, according toTable I, the computational complexity of the MCKF is > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <equation multiplication and addition/subtraction division, matrix inversion, Cholesky decomposition and exponentiation (10) 2n 2 -n 0 (11) 4n 3 -n 2 0 (12) 4n 2 m+4nm 2 -3nm O(m 3 ) (13) 4nm 0 (14) 4n 3 +6n 2 m-2n 2 +2nm 2 -nm 0 (28) 4nm 0 (29) 4n 2 m+4nm 2 -3nm O(m 3 ) (30) 2n 3 n + O(n 3 ) (31) 2m 3 m + O(m 3 ) (32) 2n 2 n (33) 2nm m (34) 2n 0 (36) 4n 3 +6n 2 m-2n 2 +2nm 2 -nm 0 The traditional Kalman filter algorithm involves the equations (10)~(14). Thus from Table I，one can conclude that the computational complexity of Kalman filter is 3 2 2 2 3 8 10 6 ( ) KF S n n m n nm n O m       (37) The MCKF algorithm mainly involves the equations (10), (11), (28)~(34) and (36). Note that  () x k C and  () y k C are diagonal matrices, 5 3 2 2 2 3 3 3 (2 8) (6 4 ) (2 1) (4 2) (3 1) (4 1) 2 2 ( ) 2 ( ) MCKF S T n T Tn m T n T nm T nm T n Tm Tm TO n TO m TABLE II MSES IIOF 1 x AND 2 x IN GAUSSIAN NOISES Filter MSE of 1 x MSE of 2 x KF 0.035778 0.030052 MCKF   6 0.5, 10      0.131361 0.105729 MCKF   6 1.0, 10      0.103497 0.096126 MCKF   6 3.0, 10      0.035885 0.030139 MCKF   6 5.0, 10      0.035785 0.030047 MCKF   6 7.0, 10      0.035784 0.030051 TABLE III AVERAGE IIIITERATION NUMBERS FOR EVERY TIME STEP WITH DIFFERENT Filter Average Iteration Numbers MCKF   6 0.2, 10      3.89826 MCKF   6 0.5, 10      2.78835 MCKF   6 1.0, 10      2.36406 MCKF   6 2.0, 10      2.12967 MCKF   6 3.0, 10      2.01343 MCKF   6 10, 10      1.66423 We further investigate the influence of the threshold  on the performance. Table IV illustrates the MSEs of 1 x and 2 x with different  (where the kernel bandwidth is set at 2.0   ), and Table V presents the average fixed-point iteration numbers. One can see that a smaller  usually results in slightly lower MSEs but needs more iterations to converge. Obviously, the influence of  is not significant compared with the kernel bandwidth  . TABLE IV IVMSES OF 1 x AND 2 x WITH DIFFERENT  Filter MSE of 1 x MSE of 2 x MCKF   1 2.0, 10      0.221182 0.168721 MCKF   2 2.0, 10      0.220386 0.167958 MCKF   4 2.0, 10      0.220326 0.167900 MCKF   6 2.0, 10      0.220322 0.167899 MCKF   8 2.0, 10      0.220322 0.167899 TABLE V AVERAGE VITERATION NUMBERS FOR EVERY TIME STEP WITH DIFFERENT Filter Average Iteration Numbers MCKF   1 2.0, 10      1.03884 MCKF   2 2.0, 10      1.09134 MCKF   4 2.0, 10      1.44313 MCKF   6 2.0, 10      2.13328 MCKF   8 2.0, 10      2.78392 TABLE VI MSES VIOF 1 x , 2 x AND 3 x IN GAUSSIAN PROCESS NOISES AND NON-GAUSSIAN MEASUREMENT NOISE Filter MSE of 1 x MSE of 2 x MSE of 3 x KF 50.7874   2 m 0.8172   2 2 / m s 0.2719   2 4 / m s MCKF 10.1444   2 m 0.3133   2 2 / m s 0.2231   2 4 / m s TABLE VII VIIMSES OF 1 x , 2 x AND 3 x IN NON-GAUSSIAN PROCESS AND MEASUREMENT NOISES Filter MSE of 1 x MSE of 2 x MSE of 3 x KF 114.8233   2 m 1.6358   2 2 / m s 1.8149  2 4 / m s MCKF 44.1290   2 m 0.7229   2 2 / m s 1.5803  2 4 / m s Using the matrix inversion lemma with the identification:1`1 1 1 1 1 T T T T p x p r y r k k k               W C W B C B H B C B H (A.4) where we denote   |1 p kk  B by p B ,   r k B by r B ,   x k C by x C and   y k C by y C for simplicity.     1 1 1 1 , , , T T p x p T r y r         B C B A H B H C B C B D We arrive at           11 1 1 1 1 1 ( ) y r k k k k k k        W C D B C B x H B C B y (A.6) Combining (25), (A.5) and (A.6), we obtain (26). B. Proof of Theorem 1   1 2 1 2 () limG ( ) lim exp 1 2 i i ek ek                      (A.7) It follows easily that         1 1 1 1 lim ( ) lim G ( ) ,..., G ( ) (1,...,1) x n n k diag e k e k diag            C   (A.8)         1 1 1 1 lim ( ) lim G ( ) ,..., G ( ) (1,...,1) y n n m m k diag e k e k diag              C   (A.9)   1 (A.8) (16) lim ( | 1) lim ( | 1) ( ) ( | 1) ( | 1) ( | 1) ( | 1) T x p p T p p k k k k k k k k k k k kk               P B C B B B P (A.10)   1 (A.9) (16) lim ( ) lim ( ) ( ) ( ) ( ) ( ) ( ) T y r r T rr k k k k kk k         R B C B BB R (A.11)         k k k k k k k k k kk                     P I K H P I K H K R K I K H P I K H K R K P (A.14) (A.10)(A.11) 1(12) lim( ) lim ( | 1) ( ) ( ) ( | 1) ( ) ( ) ( | 1) ( ) ( ) ( | 1) ( ) ( ) ( ) T T T T k k k k k k k k k k k T T T kk k k k k k k k k k A new approach to linear filtering and prediction problems. 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[ "Dynamical System Analysis of a Three Fluid Cosmological Model : An Invariant Manifold approach", "Dynamical System Analysis of a Three Fluid Cosmological Model : An Invariant Manifold approach" ]
[ "Subhajyoti Pal ", "Subenoy Chakraborty ", "\nDepartment of Mathematics\nDepartment of Mathematics\nSister Nibedita Govt General Degree College For Girls\nKolkata-700027, West BengalIndia\n", "\nJadavpur University\nKolkata-700032, West BengalIndia\n" ]
[ "Department of Mathematics\nDepartment of Mathematics\nSister Nibedita Govt General Degree College For Girls\nKolkata-700027, West BengalIndia", "Jadavpur University\nKolkata-700032, West BengalIndia" ]
[]
The present paper considers a three-fluid cosmological model consisting of noninteracting dark matter, dark energy and baryonic matter in the background of the Friedman-Robertson-Walker-Lemaître flat spacetime. It has been assumed that the dark matter takes the form of dust whereas the dark energy is a quintessence (real) scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. The field equations for this model takes the form of an autonomous dynamical system after some suitable changes of variables. Then a complete stability analysis is done considering all possible parameter (the adiabatic index of the baryonic matter and the parameter arising from the dark energy potential) values and for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperbolic critical points, the invariant manifold theory (center manifold approach) is applied. Finally various topologically different phase planes and vector field diagrams are produced and the cosmological interpretation of this model is presented.
10.1140/epjc/s10052-019-6875-4
[ "https://arxiv.org/pdf/2002.07588v1.pdf" ]
149,835,086
2002.07588
d865955c442510760d1e51851ce63ebc30d2cc1f
Dynamical System Analysis of a Three Fluid Cosmological Model : An Invariant Manifold approach arXiv:2002.07588v1 [gr-qc] 17 Feb 2020 Subhajyoti Pal Subenoy Chakraborty Department of Mathematics Department of Mathematics Sister Nibedita Govt General Degree College For Girls Kolkata-700027, West BengalIndia Jadavpur University Kolkata-700032, West BengalIndia Dynamical System Analysis of a Three Fluid Cosmological Model : An Invariant Manifold approach arXiv:2002.07588v1 [gr-qc] 17 Feb 2020Non-hyperbolic pointCenter manifold theoryField equations PACS Numbers : 9880-k0545-a0240Sf0240Tt The present paper considers a three-fluid cosmological model consisting of noninteracting dark matter, dark energy and baryonic matter in the background of the Friedman-Robertson-Walker-Lemaître flat spacetime. It has been assumed that the dark matter takes the form of dust whereas the dark energy is a quintessence (real) scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. The field equations for this model takes the form of an autonomous dynamical system after some suitable changes of variables. Then a complete stability analysis is done considering all possible parameter (the adiabatic index of the baryonic matter and the parameter arising from the dark energy potential) values and for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperbolic critical points, the invariant manifold theory (center manifold approach) is applied. Finally various topologically different phase planes and vector field diagrams are produced and the cosmological interpretation of this model is presented. accelerated expansion phases, one occurred before the radiation dominated era and the other one started not too long ago. We are presently in this accelerated expansion phase. This present time accelerated expansion has been attributed to a unseen and unknown matter with very large negative pressure called the dark energy. Analysis of different cosmological data suggests that our universe is composed of around 70% dark energy, about 25% dark matter and the rest accounts for baryonic matter and radiation. Very few properties of the dark energy is known [9]. As far as the mathematical modelling of this exotic matter is concerned, the simplest choice for it has been the Cosmological Constant Λ [8,11,16]. Although these models are capable of explaining most of the observational data, all of them fails to explain the coincidence problem (why expansion is happening now and why is it accelerated) and the fine-tuning problem (why some of the parameters take exorbitantly high values where others are not). To resolve these issues, lots of dynamical dark energy models has been prescribed. There the dark energy has been modelled as a scalar field. Some scalar field models like quintessence [17,18], K -essence and Tachyonic models have attracted lots of attention. In this paper we consider a cosmological model consisting of non-interacting dark matter, dark energy and baryonic matter in the background of the Friedman-Robertson-Walker-Lemaître flat spacetime. The dark matter has been assumed to take the form of dust whereas the dark energy is assumed to be a scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. All these three fluids are assumed to be non-interacting and minimally coupled to gravity. In order to study this model qualitatively, we derive the Einstein's field equations and the Klein-Gordan equation for the scalar field. After some suitable changes of variables, these equations take the form of an autonomous dynamical system. Then we find the critical points and analyze the stability of each critical point. We note that in [10] the model like ours have been considered. But the authors did not consider the cases of non-hyperbolic critical points. Our results differ from them in various ways. Firstly, we apply some of the very rich theories for the dynamical systems, namely the Invariant Manifold and the Center Manifold Theory [13][14][15]20] to compute the center manifolds for all the non-hyperbolic critical points and then continue to do the stability analysis for them. The other one is that we have considered all theoretically possible values of the parameters to do a complete analysis here, whereas in their article, they chose only a few suitable values. Lastly, we have presented all possible topologically different phase plane diagrams here where they included only some of them in their article. The motivation to do stability analysis is that after considering all cosmological and observational constraints of data, the stable critical points in our model may depict our present universe as a global attractor. If they do fit with the data of present percentages of dark energy, dark matter and baryonic matter together with radiation in the universe then our model would successfully describe the universe. The organization of this article is as follows : The section II describes the Einstein field equations, Klein-Gordan scalar field equation and energy conservation relations for our model. Section III describes the formation of an autonomous system. The section IV is where we present our work on complete stability analysis. At the end of this section, we produce the phase plane diagrams for different topological cases. Finally section V presents the cosmological interpretations of our results and concludes our work. II. EQUATIONS The homogeneous and isotropic flat Friedman-Robertson-Walker-Lemaître spacetime is the background of our model. This universe is assumed to be filled up by non-interacting dark matter, dark energy and baryon. Dark matter is assumed to be dust with energy density ρ m and the dark energy is assumed to be a scalar field φ with the potential as V (φ). The density ρ d and the pressure p d of the scalar field follows the following equations : ρ d = 1 2φ 2 + V (φ)(1) and p d = 1 2φ 2 − V (φ).(2) Here˙denotes differentiation with respect to cosmic time t. The baryonic matter is assumed to be a perfect fluid with linear equation of state p b = (ν − 1)ρ b(3) where p b and ρ b are the density and the pressure of the fluid and ν is the adiabatic index of the fluid satisfying 2 3 < ν ≤ 2. In particular ν = 1 and ν = 4 3 corresponds to the dust and radiation respectively. Here we also assume that ν = 1 All three matter are non-interacting and minimally coupled to gravity. The Einstein field equations for this model is 3H 2 = k(ρ m + ρ d + ρ b ).(4) where H is the Hubble parameter and k = 8πG, where G is the gravitational constant, the speed of light has been scaled to 1. The Klein-Gordan equation of the scalar field is φ + 3Hφ + dV dφ = 0(5) The energy conservation relations take the following forṁ ρ m + 3Hρ m = 0 (6) ρ d + 3H(ρ d + p d ) = 0 (7) ρ b + 3H(ρ b + p b ) = 0(8) From (4),(5),(6), (7) and (8) we derive 2Ḣ = −k(ρ m + ρ b + p b +φ 2 ).(9) The equations (4), (9) and (5) are the evolution equations for this model. Next we find suitable coordinate changes such that these evolution equations form a system of autonomous dynamical system. This is done in the following section. III. THE AUTONOMOUS SYSTEM We introduce the following coordinate transformations of variables : x = k 6φ H ,(10)y = k 3 V (φ) H(11) and the density parameters Ω m = kρ m 3H 2 ,(12)Ω b = kρ b 3H 2 ,(13)Ω d = kρ d 3H 2 .(14) These coordinate changes transform the Friedmann equation (4) and the equation (9) respectively as the following : Ω d = x 2 + y 2 ,(15)Ω m + Ω b + x 2 + y 2 = 1 (16) andḢ = −3H 2 (x 2 + Ω m 2 + ν Ω b 2 ).(17) As Ω m and Ω b , the density parameters are non-negative real quantities, so from (16) 0 ≤ Ω m ≤ 1, 0 ≤ Ω b ≤ 1 and x and y satisfies x 2 + y 2 ≤ 1. The strict equality is only possible if the energy densities of the dark matter and the baryonic matter is zero. Differentiating the equations (10), (11) and (12) with respect to N where N = ln a (a(t) is the scale factor of the universe) and using (16), (17), (5) and (6), (7) and (8) we derive the following autonomous dynamical system : dx dN = 3x[x 2 − 1 + Ω m 2 + ν 2 (1 − Ω m − x 2 − y 2 )] − 3 2k 1 V dV dφ y 2(18)dy dN = y[3x 2 + 3 ν 2 (1 − Ω m − x 2 − y 2 ) + 3 Ω m 2 + 3 2k 1 V dV dφ x](19)dΩ m dN = −3Ω m [1 − Ω m − 2x 2 − ν(1 − Ω m − x 2 − y 2 )](20) We end this section by expressing the relevant cosmological parameters in terms of the above transformed variables as Ω d = x 2 + y 2 ,(21)ω d = p d ρ d = x 2 − y 2 x 2 + y 2 ,(22)ω ef f = p d + p b ρ m + ρ d + ρ b = −1 + 2x 2 + ν(1 − Ω m − x 2 − y 2 ) + Ω m(23) and q = −(1 +Ḣ H 2 ) = −[(1 − 3x 2 − 3ν 2 ) − 3Ω m 2 (1 − ν) + 3ν 2 (x 2 + y 2 )].(24) We note that for accelerated expansion, ω ef f ≤ − 1 3 and q ≤ 0 are two necessary conditions. IV. STABILITY ANALYSIS We start working on the stability analysis of the dynamical system (18) in this section. We assume that the potential of the scalar field representing the dark energy is exponential, ie dx dN = 3x[x 2 − 1 + Ω m 2 + ν 2 (1 − Ω m − x 2 − y 2 )] − αy 2 (25) dy dN = y[3x 2 + 3 ν 2 (1 − Ω m − x 2 − y 2 ) + 3 Ω m 2 + αx](26)dΩ m dN = −3Ω m [1 − Ω m − 2x 2 − ν(1 − Ω m − x 2 − y 2 )](27) There are ten critical points of this autonomous system. They are listed in the table below. C 1 (0, 0, 0) C 2 (1, 0, 0) C 3 (0, 0, 1) C 4 (−1, 0, 0) C 5 (− α 3 , 1 3 √ 9 − α 2 , 0) C 6 (− α 3 , − 1 3 √ 9 − α 2 , 0) C 7 (− 3ν 2α , 3 2α √ 2ν − ν 2 , 0) C 8 (− 3ν 2α , − 3 2α √ 2ν − ν 2 , 0) C 9 (− 3 2α , 3 2α , (1 − 9 2α 2 )) C 10 (− 3 2α , − 3 2α , (1 − 9 2α 2 )) The value of the relevant cosmological parameters for each of the critical points are given in the following table : ω d ω ef f Ω m Ω d Ω b q C 1 Undefined ν − 1 0 0 1 3ν 2 − 1 C 2 1 1 0 1 0 2 C 3 Undefined 0 1 0 0 1 2 C 4 1 1 0 1 0 2 C 5 2α 2 9 − 1 2α 2 9 − 1 0 1 0 α 2 3 − 1 C 6 2α 2 9 − 1 2α 2 9 − 1 0 1 0 α 2 3 − 1 C 7 ν − 1 ν − 1 0 9ν 2α 2 1 − 9ν 2α 2 3ν 2 − 1 C 8 ν − 1 ν − 1 0 9ν 2α 2 1 − 9ν 2α 2 3ν 2 − 1 C 9 0 0 1 − 9 2α 2 9 2α 2 0 1 2 C 10 0 0 1 − 9 2α 2 9 2α 2 0 1 2 For every critical point and for each of their subcases we will do two successive change of variables to bring them to a form with which the calculation of center manifold and dynamics of the reduced system will be easy. For each of the cases and subcases the transformations are as follows : If X =      x y Ω m      ,X =     x ȳ Ω m      andX =     x ȳ Ω m      , thenX = X − A andX = P −1X where A and P is some 3 × 1 and 3 × 3 matrices respectively with P being non-singular. The exact form of A and P will vary from case to case and we will mention them while studying each of the cases and subcases. A. Critical Point C 1 For critical point C 1 , A =      0 0 0      and the Jacobian matrix of the system (25) at this critical point has the characteristic polynomial λ 3 + (6 − 6ν)λ 2 + [ 3ν(3ν − 3) 2 + (3ν − 6)(9ν − 6) 4 ]λ − 3ν(3ν − 3)(3ν − 6) 4 = 0.(28) which has eigen values as 3ν − 3, 3ν 2 − 3 and 3ν 2 . The stability analysis when ν is not 0, 1 or 2 (hyperbolic cases) is an easy application of the stability analysis of the linear system and the Hartman-Gröbman Theorem. So now we will present a table containing the non-hyperbolic subcases and the result of their stability analysis. Later We will write the stability results for all possible subcases for C 1 . Before we write down the table, we introduce a notation. ′ represents derivative with respect to N. TABLE III: C 1 (Reduced System) ν Center Manifold Reduced System 0x = O(Ω 3 m ),ȳ = − α 3Ω 2 m + O(Ω 3 m )Ω ′ m = − α 2 3Ω 3 m + O(Ω 4 m ) 1ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 2x = O(ȳ 3 ),Ω m = O(ȳ 3 )ȳ ′ = 0 For all these subcases, P =      0 1 0 0 0 1 1 0 0      . We summarize our results for C 1 in the table below to end this subsection. a ν < 0 NA Stable b ν = 0 Stable Stable c 0 < ν < 1 NA Saddle d ν = 1 Center Center-Saddle e 1 < ν < 2 NA Saddle f ν = 2 Center Center-Unstable g 2 < ν NA Unstable Here RS stands for the Reduced System, DS stands for the whole system and NA stands for Not Applicable (Hyperbolic Cases). B. Critical Point C 2 For critical point C 2 , A =      1 0 0      and the Jacobian matrix of the system (25) at this critical point has the characteristic polynomial λ 3 + (3ν − α − 12)λ 2 + [18 − (3ν − 9)(α + 3) − 9ν]λ − (9ν − 18)(α + 3) = 0.(29) which has eigen values as 3, 6 − 3ν and α + 3. The stability analysis when ν is not 2 and α is not −3 (hyperbolic cases) is again an easy application of the stability analysis of the linear system and the Hartman-Gröbman Theorem. So we will present a table containing all the non-hyperbolic subcases and the result of their stability analysis. Later We will write the stability results for all possible subcases for C 2 . 2 −3Ω m = O( (x,ȳ) 3 )x ′ = −3xȳ 2 ,ȳ ′ = −3xȳ 2 = −3ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 = 2 −3ȳ = O(x 3 ),Ω m = − 1 2x 2 + O(x 3 )x ′ = − 3 2x 3 When ν = 2 and α = −3 the P is      1 0 − 1 2 0 1 0 0 0 1      . For ν = 2 and α = −3, P =      1 − 1 2 0 0 0 1 0 1 0      . When ν = 2 and α = −3 then P =      0 − 1 2 1 1 0 0 0 1 0      . We end this subsection by summarizing our results for C 2 in a table. Case ν α Stability(RS) Stability(DS) a < 2 < −3 NA Saddle b < 2 −3 Stable Saddle c < 2 > −3 NA Unstable d 2 > −3 Center Center-Unstable e 2 −3 Stable Saddle f 2 < −3 Center Center-Saddle g > 2 < −3 NA Saddle h > 2 −3 Stable Saddle i > 2 > −3 NA Saddle C. Critical Point C 3 For C 3 the A =      0 0 1      . The Jacobian Matrix at C 3 has the characteristic polynomial λ 3 + (3ν − 3)λ 2 − 9 4 λ + [ 27 4 − 27ν 4 ] = 0(30) which has the eigenvalues − 3 2 , 3 2 and 3 − 3ν. Hence the stability analysis is almost trivial if ν = 1. Because then it is an immediate application of the stability analysis of the linear cases and the Hartman-Gröbman theorem. Therefore we assume that ν = 1 and proceed with our stability analysis. 1ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 For this subcase P is      0 0 1 0 1 0 1 0 0      . At the end, here is the summary of the results for C 3 . For critical point C 4 , A =      −1 0 0      and the Jacobian matrix of the system (25) at this critical point has the characteristic polynomial λ 3 + (3ν + α − 12)λ 2 + [(3ν − 6)(α − 6) − 3α + 9]λ − (3ν − 6)(3α − 9) = 0.(31) which has eigen values as 3, 6 − 3ν and 3 − α. The stability analysis when ν is not 2 and α is not 3 (hyperbolic cases) is easy again as said before. Now we will present a table containing all the non-hyperbolic subcases and the result of their stability analysis. Later We will write the stability results for all possible subcases for C 4 . 2 3Ω m = O( (x,ȳ) 3 )x ′ = −3xȳ 2 ,ȳ ′ = 3xȳ 2 = 3ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 = 2 3ȳ = O(x 3 ),Ω m = 1 2x 2 + O(x 3 )x ′ = − 3 2x 3 When ν = 2 and α = 3 the P is 0 1 0      . We end this subsection by summarizing our results for C 4 in a table. Here it is necessary that | α |≤ 3. A=      − α 3 1 3 √ 9 − α 2 0      for this case. The Jacobian matrix of the system (25) at this critical point has the characteristic polynomial λ 3 + (3ν − 5 3 α 2 + 6)λ 2 + [18ν − 3να 2 − 7α 2 + 8 9 α 4 + 9]λ − (9ν − 2α 2 )(2α 4 − 27α 2 + 81) 27 = 0. (32) This characteristic polynomial has eigenvalues as 2α 3 3 − 3ν, α 2 3 − 3 and 2α 3 3 − 3. The stability analysis when ν = 2α 2 9 and α = ±3, ± 3 √ 2 (hyperbolic cases) is easy as said in the above subsections. Now we will present a table containing all the non-hyperbolic subcases and the result of their stability analysis. Since for C 5 , the total number of all possible subcases is too many, we would not present the results for the complete case in a table at the end as we did in the previous subsections. We would rather present a diagram in α − ν plane to show the stability analysis for all possible values of α's and ν's. 1 3 √ 2Ω m = 1 √ 2x 2 + O( (x,ȳ) 3 )x ′ = −3 √ 2x 2 ,ȳ ′ = −3 √ 2xȳ = 1 3 √ 2x = rȳ 2 + O(ȳ 3 ),Ω m = sȳ 2 + O(ȳ 3 )ȳ ′ = −3ȳ 2 1 − 3 √ 2Ω m = 1 √ 2x 2 + O( (x,ȳ) 3 )x ′ = 3 √ 2x 2 ,ȳ ′ = 3 √ 2xȳ = 1 − 3 √ 2x = rȳ 2 + O(ȳ 3 ),Ω m = sȳ 2 + O(ȳ 3 )ȳ ′ = −3ȳ 2 2α 2 9 αȳ = tx 2 + O(x 3 ),Ω m = O(x 3 )x ′ = −ux 2 + O(x 3 ) 0 0ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = O(x 3 ) Where "id" means "identical" and r = = − 3 √ 2 , P =      −1 − 1 √ 2 − 2−ν ν−1 1 0 1 0 1 0      . Also in the final two subcases P =      − 2α(9−α 2 ) 3/2 2α 4 −27α 2 +81 √ 9−α 2 α 3 2α 1 1 0 0 0 1      and P =      1 0 0 0 − 1 2 1 0 1 0      respectively. Since we will find that the stability diagram for C 5 and C 6 is identical, it will be presented at the end of the next subsection. We now proceed with the next subsection. F. Critical Point C 6 Here it is also necessary that | α |≤ 3. A=      − α 3 − 1 3 √ 9 − α 2 0      in this case. The Jacobian matrix of the system (25) at this critical point has the characteristic polynomial λ 3 + (3ν − 5 3 α 2 + 6)λ 2 + [18ν − 3να 2 − 7α 2 + 8 9 α 4 + 9]λ + (9ν − 2α 2 )(2α 4 − 27α 2 + 81) 27 = 0. (33) This characteristic polynomial has eigenvalues as 2α 3 3 − 3ν, α 2 3 − 3 and 2α 3 3 − 3. The stability analysis when ν = 2α 2 9 and α = ±3, ± 3 √ 2 (hyperbolic cases) is again easy. Now we will present a table containing all the non-hyperbolic subcases and the result of their stability analysis. For C 6 too the total number of all possible subcases is many, so we would not present the results for the complete case in a table. We would rather present a diagram in α − ν plane to show the stability analysis for all possible values of α's and ν's as we did for C 5 . = 2 −3 id to subcase (b) and (h) of C 2 id to subcase (b) and (h) of C 2 1 3 √ 2Ω m = − 1 √ 2x 2 + O( (x,ȳ) 3 )x ′ = −3 √ 2x 2 ,ȳ ′ = −3 √ 2xȳ = 1 3 √ 2x = −rȳ 2 + O(ȳ 3 ),Ω m = −sȳ 2 + O(ȳ 3 )ȳ ′ = −3ȳ 2 1 − 3 √ 2Ω m = − 1 √ 2x 2 + O( (x,ȳ) 3 )x ′ = 3 √ 2x 2 ,ȳ ′ = 3 √ 2xȳ = 1 − 3 √ 2x = −rȳ 2 + O(ȳ 3 ),Ω m = −sȳ 2 + O(ȳ 3 )ȳ ′ = −3ȳ 2 2α 2 9 αȳ = −tx 2 + O(x 3 ),Ω m = O(x 3 )x ′ = ux 2 + O(x 3 ) 0 0ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = O(x 3 ) Where r, s, t and u take usual values as defined in the subsection of C 5 . As like in the previous subsection here is also in the subcase before the last one in the table, the ordered pair (ν, α) is √ 2 , P =      1 − 1 √ 2 2−ν ν−1 1 0 1 0 1 0      . Also in the final two subcases P =      2α(9−α 2 ) 3/2 2α 4 −27α 2 +81 − √ 9−α 2 α 3 2α 1 1 0 0 0 1      and P =      1 0 0 0 − 1 2 1 0 1 0      respectively. We finish this subsection by producing the stability diagram for both C 5 and C 6 in α − ν plane in the following : Here the cyan shaded area ie the area bounded by α = 3 √ 2 , α = − 3 √ 2 and ν = 2α 2 9 represents the α − ν pairs for which (25) is stable. The yellow shaded region represents parameter values for which (25) is saddle. Lastly the origin is center-stable here. G. Critical Point C 7 In this case, it is necessary that 0 ≤ ν ≤ 2 and α = 0. A takes the form      − 3ν 2α 3 √ ν(2−ν) 2α 0      . The Jacobian matrix of the system (25) at C 7 has the following characteristic polynomial: λ 3 + (6 − 9ν 2 )λ 2 + [ 9(ν − 2)(9ν 2 − 2α 2 ) 4α 2 ]λ − 27ν(9ν − 2α 2 )(ν 2 − 3ν + 2) 4α 2 = 0.(34) It has eigenvalues as 3(ν − 1), 3 4α (να − 2α + (2 − ν)(36ν 2 − 9να 2 + 2α 2 )) and 3 4α (να − 2α − (2 − ν)(36ν 2 − 9να 2 + 2α 2 )). The stability analysis when ν = 1, ν = 2α 2 9 and ν = 2, ν = 0 (hyperbolic cases) are easy by application of linear stability analysis and reduction of non-linear case to linear case under some specific conditions. So we present a table containing all the non-hyperbolic subcases. For C 7 also, we will present a diagram in α − ν plane to show the stability analysis for all possible values of α's and ν's at the end. id to subcase (e) of C 5 id to subcase (e) of C 5 1 − 3 √ 2 id to subcase (g) of C 6 id to subcase (g) of C 6 = 1 ν = 2α 2 9 , α > 0 id to subcase (i) of C 5 id to subcase (i) of C 5 = 1 ν = 2α 2 9 , α < 0 id to subcase (i) of C 6 id to subcase (i) of C 6 1 ν = 2α 2 9ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 2 αx = O( (ȳ,Ω m ) 3 )ȳ ′ = αȳΩ m ,Ω ′ m = −vȳ 2 0 α id to subcase (a) of C 1 id to subcase (a) of C 1 where v = 4α 2 −36 4α . For the last two subcases P is      0 w −w 0 1 1 1 0 0      and      3 2α 1 0 0 0 1 1 0 0      respectively with w = α √ 36−α 2 −α 2 +9 2α 2 −9 . Again we will find that the stability diagram for C 7 is exactly same as C 8 . Hence it will be presented at the end of the next subsection. H. Critical Point C 8 In this case also, it is necessary that 0 ≤ ν ≤ 2 and α = 0 and A takes the form      − 3ν 2α − 3 √ ν(2−ν) 2α 0      . The Jacobian matrix of the system (25) at C 8 has the characteristic polynomial: λ 3 + (6 − 9ν 2 )λ 2 + [ 9(ν − 2)(9ν 2 − 2α 2 ) 4α 2 ]λ − 27ν(9ν − 2α 2 )(ν 2 − 3ν + 2) 4α 2 = 0.(35) This polynomial has eigenvalues as 3(ν − 1), 3 4α (να − 2α + (2 − ν)(36ν 2 − 9να 2 + 2α 2 )) and 3 4α (να − 2α − (2 − ν)(36ν 2 − 9να 2 + 2α 2 ) ). The stability analysis when ν = 1, ν = 2α 2 9 and ν = 2, ν = 0 (hyperbolic cases) are easy again. So we present a table containing all the non-hyperbolic subcases for C 8 . For this case also, at the end we will present a diagram in α − ν plane to show the stability analysis for all possible values of α's and ν's. id to subcase (e) of C 6 id to subcase (e) of C 6 1 − 3 √ 2 id to subcase (g) of C 5 id to subcase (g) of C 5 = 1 ν = 2α 2 9 , α > 0 id to subcase (i) of C 6 id to subcase (i) of C 6 = 1 ν = 2α 2 9 , α < 0 id to subcase (i) of C 5 id to subcase (i) of C 5 1 ν = 2α 2 9ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 2 αx = O( (ȳ,Ω m ) 3 )ȳ ′ = αȳΩ m ,Ω ′ m = −vȳ 2 0 α id to subcase (a) of C 1 id to subcase (a) of C 1 Where "v" takes it's value as defined for C 7 . For the last two subcases P is Here 0 ≤ ν ≤ 2 and α = 0. The cyan-shaded area ie the area bounded by ν = 0, ν = 1 and ν = 2α 2 9 represents stability. The yellow region represents saddle system (25). The 'o' marked lines are those parameter values for which the system (25) is center-stable. I. Critical Point C 9 In this case, it is necessary that α = 0. For this case, A =      − 3ν 2α 3ν 2α 2α 2 −9 2α 2      . The Jacobian matrix of the system (25) at C 9 has the characteristic polynomial as following : λ 3 + (3ν − 3 2 )λ 2 + [ 18να 2 − 81 4α 2 ]λ − 27(2α 2 − 9)(ν − 1) 4α 2 = 0.(36) This polynomial has eigenvalues as 3(1 − ν), −3α+3 √ 36−7α 2 4α and −3α−3 √ 36−7α 2 4α . The stability analysis when ν = 1 and α = − 3 √ 2 , 3 √ 2 (hyperbolic cases) are easy. Therefore we present a table containing all the non-hyperbolic subcases for C 9 . In the end we will present a diagram in α − ν plane to show the complete stability analysis as usual. id to subcase (a) of C 7 id to subcase (a) of C 7 1 − 3 √ 2 id to subcase (b) of C 7 id to subcase (b) of C 7 = 1 3 √ 2 id to subcase (f) of C 5 id to subcase (f) of C 5 = 1 − 3 √ 2 id to subcase (h) of C 6 id to subcase (h) of C 6 1 α = − 3 √ 2 , 3 √ 2ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 For the last subcase, P =      0 α 6 + α 2 (α− √ 36−7α 2 ) 12(2α 2 −9) α 6 + α 2 (α+ √ 36−7α 2 ) 12(2α 2 −9) 0 − α 6 + α 2 (α− √ 36−7α 2 ) 12(2α 2 −9) − α 6 + α 2 (α+ √ 36−7α 2 ) 12(2α 2 −9) 1 1 1      . We find again that the stability diagram for C 9 and C 10 is totally same. Hence it will be presented at the end of the next subsection of C 10 . J. Critical Point C 10 This is the last subcase. In this case too it is necessary that α = 0. Also, A =      − 3ν 2α 3ν 2α 2α 2 −9 2α 2      . The Jacobian matrix of the system (25) at C 10 has the characteristic polynomial λ 3 + (3ν − 3 2 )λ 2 + [ 18να 2 − 81 4α 2 ]λ + 27(2α 2 − 9)(ν − 1) 4α 2 = 0.(37) which has eigenvalues as 3(1 − ν), −3α+3 √ 36−7α 2 4α and −3α−3 √ 36−7α 2 4α . The stability analysis when ν = 1 and α = − 3 √ 2 , 3 √ 2 (hyperbolic cases) are easy again. Therefore we will present a table containing all the non-hyperbolic subcases for this critical point. At the end we will present the stability diagram in α − ν plane. id to subcase (a) of C 8 id to subcase (a) of C 8 1 − 3 √ 2 id to subcase (b) of C 8 id to subcase (b) of C 8 = 1 3 √ 2 id to subcase (f) of C 6 id to subcase (f) of C 6 = 1 − 3 √ 2 id to subcase (h) of C 5 id to subcase (h) of C 5 1 α = − 3 √ 2 , 3 √ 2ȳ = O(x 3 ),Ω m = O(x 3 )x ′ = 0 For the last subcase, P =      0 α 6 + α 2 (α− √ 36−7α 2 ) 12(2α 2 −9) α 6 + α 2 (α+ √ 36−7α 2 ) 12(2α 2 −9) 0 α 6 − α 2 (α− √ 36−7α 2 ) 12(2α 2 −9) α 6 − α 2 (α+ √ 36−7α 2 ) 12(2α 2 −9) 1 1 1      . Lastly, we provide the stability diagram in the α − ν plane for C 9 and C 10 and end this section : At the end of this section, we will present the phase plane diagrams of the autonomous system (25) for various values of the parameter ν and α in figures (4)- (7) The last diagram (8) depicts the vector fields of the autonomous system (25). The equilibrium point C 1 is completely dominated by the baryonic matter and as expected in standard cosmology, the model will be in decelerating phase if the baryonic fluid is normal (ie, non-exotic: ν > 2 3 ) and it will be in the accelerating era for exotic nature of the matter (ie, ν < 2 3 ). The equilibrium points C 2 and C 4 have identical nature and both of them are not interesting from the cosmological viewpoint as there is deceleration (of unusual magnitude) when matter is completely dominated by DE. The point C 3 describes a known result of standard cosmology: In the dust era there is deceleration with the value of the deceleration parameter 1 2 . The equilibrium point C 5 and C 6 correspond to cosmological era dominated by the dark energy(DE). The condition (α 2 < 3) restricts the DE to be exotic and there is acceleration. By choosing α appropriately it is possible to match the model with the recent observations. Both the equilibrium points C 7 and C 8 in the phase space have identical cosmological behaviors. For real points in the phase space as well as for realistic baryonic fluid, ν should be restricted as 0 < ν < 2. The cosmological model is dominated by both the DE and the baryonic matter. In fact, the cosmic phase described by these phase space points (ie C 7 and C 8 ) has dominance of baryon over DE if 2 3 < ν < 2 and there is deceleration. While if 0 < ν < 2 3 then DE dominates the cosmic evolution and there is acceleration. The scaling solutions represented by the equilibrium points C 9 and C 10 are dominated by both the dark matter (DM) and DE. But they are not of much physical interest as they always correspond to dust era of evolution. Thus, from the dynamical system analysis, the equilibrium points describe various cosmological eras and some of them are interesting from the cosmological viewpoints with recent observed data. Therefore, one may conclude that from complicated cosmological models one may get cosmological inferences without solving the evolution equations, rather by analyzing with dynamical system approach and the present work is an example of this inference. 2ν−3) , t = 3 √ 9−α 2 (9α 2 −4α 4 ) 4α 6 −72α 4 +405α 2 −729 and u = − 4α 2 √ 9−α 2 2α 2 −9 .We also note that in the subcase before the last one in the table, the ordered pair (ν, α) is not permitted to take the values ( FIG. 1: Stability Diagram (C 5 , C 6 ) with j = α √ 36−α 2 +α 2 −9 2α 2 −9 and q = −α √ 36−α 2 +α 2 −9 2α 2 −9 .This subsection concludes with the stability diagram in the α − ν plane for both the C 7 and C 8 FIG. 3 : 3Stability Diagram (C 9 , C 10 ) Here it is necessary that α = 0. The two squares in the top corners bounded by the lines ν = 1, α = 3 √ 2 and ν = 1, α = − 3 √ 2 respectively are cyan-shaded. They represent the parameter values for which the system (25) is stable. The rest of the region represents saddle system.The 'o' marked line represents center-stability. FIG. 8 : 8Vector field diagram of (25) V. COSMOLOGICAL INTERPRETATIONS AND CONCLUSION The present work deals with a cosmological model consisting of three non-interacting fluids namely the baryonic matter in the form of perfect fluid (p b = (ν − 1)ρ b ), dark matter in the form of dust and dark energy as a scalar field respectively. This cosmological model has been studied in the framework of dynamical system analysis by forming the evolution equations (Einstein's field equations) into an autonomous system with suitable transformation of the variables. In table I, it has been shown that there are 10 equilibrium points (C 1 -C 10 ) of the autonomous system. The values of the relevant cosmological parameters at the equilibrium points have also been presented in table II. TABLE I : ICritical PointsCritical Point Name Critical Point TABLE II : IIValues of the Different Cosmological Parameters at the Critical PointsCritical Points TABLE IV : IVSummary for the critical point C 1Case ν Stability(RS) Stability(DS) TABLE V : VC 2 (Reduced System)ν α Center Manifold Reduced System TABLE VI : VIA summary for the critical point C 2 TABLE VII : VIIC 3 (Reduced System)ν Center Manifold Reduced System TABLE VIII : VIIIThe critical point C 3 , a summaryCase ν Stability(RS) Stability(DS) a < 1 NA Saddle b 1 Center Center-Saddle c > 1 NA Saddle D. Critical Point C 4 TABLE IX : IXC 4 (reduced System) ν α Center Manifold Reduced System TABLE X : XC 4 , a stability analysisCase ν α Stability(RS) Stability(DS) a < 2 < 3 NA Unstable b < 2 3 Stable Saddle c < 2 > 3 NA Saddle d 2 > 3 Center Center-Saddle e 2 3 Stable Saddle f 2 < 3 Center Center-Unstable g > 2 < 3 NA Saddle h > 2 3 Stable Saddle i > 2 > 3 NA Saddle E. Critical Point C 5 TABLE XI : XIC 5 (Reduced System)ν α Center Manifold Reduced System 2 3 id to subcase (e) of C 4 id to subcase (e) of C 4 = 2 3 id to subcase (b) and (h) of C 4 id to subcase (b) and (h) of C 4 2 −3 id to subcase (e) of C 2 id to subcase (e) of C 2 = 2 −3 id to subcase (b) and (h) of C 2 id to subcase (b) and (h) of C 2 TABLE XII : XIIC 6 (Reduced System)ν α Center Manifold Reduced System 2 3 id to subcase (e) of C 4 id to subcase (e) of C 4 = 2 3 id to subcase (b) and (h) of C 4 id to subcase (b) and (h) of C 4 2 −3 id to subcase (e) of C 2 id to subcase (e) of C 2 TABLE XIII : XIIIC 7 (Reduced System)ν α Center Manifold Reduced System 1 3 √ 2 TABLE XIV : XIVC 8 (Reduced System) ν α Center Manifold Reduced System 1 3 √ 2 TABLE XV : XVC 9 (Reduced System)ν α Center Manifold Reduced System 1 3 √ 2 TABLE XVI : XVIC 10 (Reduced System)ν α Center Manifold Reduced System 1 3 √ 2 Measurements of Omega and Lambda from 42 high redshift supernovae. 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[]
[ "Back to the Future: On Potential Histories in NLP", "Back to the Future: On Potential Histories in NLP" ]
[ "Zeerak Talat [email protected] \nDigital Democracies Institute\nData Science Group\nSimon Fraser University Burnaby\nCanada\n", "Anne Lauscher [email protected] \nUniversity of Hamburg\nHamburgGermany\n" ]
[ "Digital Democracies Institute\nData Science Group\nSimon Fraser University Burnaby\nCanada", "University of Hamburg\nHamburgGermany" ]
[]
Machine learning and NLP require the construction of datasets to train and fine-tune models. In this context, previous work has demonstrated the sensitivity of these data sets. For instance, potential societal biases in this data are likely to be encoded and to be amplified in the models we deploy. In this work, we draw from developments in the field of history and take a novel perspective on these problems: considering datasets and models through the lens of historical fiction surfaces their political nature, and affords re-configuring how we view the past, such that marginalized discourses are surfaced. Building on such insights, we argue that contemporary methods for machine learning are prejudiced towards dominant and hegemonic histories. Employing the example of neopronouns, we show that by surfacing marginalized histories within contemporary conditions, we can create models that better represent the lived realities of traditionally marginalized and excluded communities.
10.48550/arxiv.2210.06245
[ "https://export.arxiv.org/pdf/2210.06245v1.pdf" ]
252,846,634
2210.06245
64fed8c2cbf75d20f27a450e2013a36d0fd8fc9a
Back to the Future: On Potential Histories in NLP Zeerak Talat [email protected] Digital Democracies Institute Data Science Group Simon Fraser University Burnaby Canada Anne Lauscher [email protected] University of Hamburg HamburgGermany Back to the Future: On Potential Histories in NLP Machine learning and NLP require the construction of datasets to train and fine-tune models. In this context, previous work has demonstrated the sensitivity of these data sets. For instance, potential societal biases in this data are likely to be encoded and to be amplified in the models we deploy. In this work, we draw from developments in the field of history and take a novel perspective on these problems: considering datasets and models through the lens of historical fiction surfaces their political nature, and affords re-configuring how we view the past, such that marginalized discourses are surfaced. Building on such insights, we argue that contemporary methods for machine learning are prejudiced towards dominant and hegemonic histories. Employing the example of neopronouns, we show that by surfacing marginalized histories within contemporary conditions, we can create models that better represent the lived realities of traditionally marginalized and excluded communities. Introduction The state-of-the art in NLP requires, among other steps, selecting, sampling, and annotating data sets which we can then use to train large machine learning (ML) models (e.g., Devlin et al., 2019;Liu et al., 2019). Previous work has shown that this is a sensitive process: for instance, potential societal biases present in the data are prone to be encoded and even amplified in our models and might jeopardize fairness (e.g., Blodgett et al., 2020). Researchers have thus argued that ML for NLP should be handled with care, and have proposed measures designed to counter potential ethical issues, e.g., via augmenting datasets (Zhao et al., 2018). In this work, we argue that all these steps along the ML pipeline are in fact acts of historical fiction. Historical fiction is a field of study in which history is constructed as a plurality rather than a singular entity or timeline (White, 2005). What the field of historical fiction affords is drawing out marginalized and minoritized histories that have otherwise been forgotten or suppressed (White, 2005). In contrast, traditional history creates histories from linear timelines and emphasizes the dominant norms (Foucault, 2013). Here we argue: if the act of creating a history then, is creating a fiction through which we can understand the past, then the creation of datasets for ML and training ML models similarly engage in acts of historical fiction. However, rather than highlighting marginalized narratives or histories, mainstream ML draws out the majoritarian histories. This occurs at the expense of marginalized narratives, giving rise to the marginalization that ML performs. In this way, current ML is a conservative practice, which polices and limits the expression of marginalized discourses, and thereby the existence of marginalized people. In this paper, we acknowledge the potential of historical fiction for fairer NLP. Strongly believing that our community can profit from this novel perspective, we a) introduce its theoretical background; b) review different possibilities of how NLP is currently performing acts of historical fiction; and c) demonstrate through case study how to construct histories for ML that are progressive by explicitly including the lived realities of groups that have otherwise been marginalized. We show that such constructions strongly impact the ways in which models come to embody information (Talat et al., 2021). Here, we resort to the case of neo-pronouns (novel and yet established pronouns), to showcase how a simple heuristic fiction process impacts how models embody these. Concretely, we replace gendered pronouns with a gender-neutral neo-pronoun and adopt existing model specialization methods (e.g., Lauscher et al., 2021), for injecting a potential history. Training a model on our fiction data shifts a marginalized pronoun from the edges of the vector space towards majoritarian pronouns. Using this example, we discuss how the underlying data influences the production and operationalization of socio-political constructs, e.g., gender, in ML systems. We hope that our work inspires more NLP researchers and practitioners to think about steps in ML as acts of historical fiction, leading to more plurality and thus, fairer and more inclusive NLP. Background Data-making has been conceptualized as fiction (e.g. Gitelman, 2013) and ML researchers have also begun to conceptualize ML, and data, as subjective (Talat et al., 2021) and value-laden (Birhane et al., 2022). Here, we lay the foundation for considering ML through the lens of historical fiction. Historical Fiction In his foundational text, "The Archaeology of Knowledge", Foucault (2013) argues that history as a field has been pre-occupied with the construction of linear timelines rather than constructing narratives, in efforts to describe the past. Describing this distinction, White (2005) notes that "historical discourse wages everything on the true, while fictional discourse is interested in the real." That is, through engaging with fiction, we are afforded knowledge and understanding of the realities of life in the period that is under investigation. Moreover, through purposefully engaging with historical fiction, histories that have otherwise been marginalized can be surfaced (White, 2005). Imagining histories in opposition to hegemony can provide space for viewing our contemporary conditions through the lens of values in our past that have been neglected. The resulting timelines are what Azoulay (2019) terms potential histories. Machine Learning and NLP ML has been critiqued for its discriminatory and hegemonic outcomes from multiple fields (Benjamin, 2019;Blodgett et al., 2021;Bolukbasi et al., 2016), which has lead to a number of methods that address the issue of discrimination by proposing to "debias" ML models (e.g. De-Arteaga et al., 2019;Dixon et al., 2018;Lauscher et al., 2020). Early efforts have however been complicated by notions of 'bias' being under-specified (for further detail see Blodgett et al., 2020). Zhao et al. (2018) perform data augmentation, with a goal of a less gender-biased co-reference solution system. Mov-ing a step further, Qian et al. (2022) collect data perturbed along demographic lines by humans, and train an automated perturber, and a language model trained on the perturbed data. Although such artifacts can be used towards efforts to debias, the artifacts can also be used to situate models within desired contexts. Other works provide critiques from theoretical perspectives. For instance, Talat et al. (2021) critique the disembodied view that the ML practice and practitioners take, arguing that "social bias is inherent" to data making and modeling practices. Rogers (2021) argues that through carefully curating data along desired values, ML can constitute a progressive practice. Finally, Solaiman and Dennison (2021) propose fine-tuning language models on curated data, which seeks to shift language models away from producing toxic, i.e. abusive content. Such work stands in contrast to a large body of literature, which uncritically collects and uses data, with the result of producing ML models that recreate discriminatory contemporaries (e.g. Green and Viljoen, 2020;Gitelman, 2013). Viewing ML through the lens of historical fiction, we argue that ML engages in creating fictions, without awareness. For instance, in the creation of data (Gitelman, 2013) and in the amplification of dominant discourses (Zhao et al., 2017). The predominant function of these fictions has been to imagine a single past that reflect hegemonic trends in our contemporary. Here, we provide a casestudy that illustrates the possibility of imagining pasts that reflect our current conditions, through constructing a fiction (i.e., a data set and a model which we train on this data) that is oppositional to hegemony. Through deploying these fictions of the past (i.e., data sets and corresponding ML models) in productive settings, we are, as a society, able to shape futures that are more aligned with our fictions of relalities that were formerly oppressed. Experiments: Neopronoun-Fiction We describe a showcase which demonstrates the idea behind historical fiction in NLP: we study the case of the neopronoun "xe". Neopronouns are not yet established pronouns (McGaughey, 2020). They are an important example of language change and are mostly used by individuals belonging to already marginalized groups, e.g., non-binary individuals (e.g., see the overview by . NLP has long been ignoring neopronouns, leading to exclusion of these individuals in lan-guage technology (Cao and Daumé III, 2021;Dev et al., 2021). We argue that we can write the potential history of "xe" being an established pronoun through simple data pertubation to change how pretrained language models (PLMs) "perceive" the past. We hypothesize that through deploying such anti-discrimination models, we can shift the hegemonic nature of ML. Note, that we could use a similar approach for other neo-pronouns, e.g., nounself pronouns (Miltersen, 2016), etc. Similarly, the general idea of selecting, augmenting, pertubating, and curating data to write potential histories can be used to create other historical fiction-models focused on larger ideological aspects beyond single words. History-Injection Methods We compare two straight-forward methods for the injection of the potential history of xe into PLMs, which have been used successfully for related cases of refinement of PLMs, e.g., domain specialization (e.g., Hung et al., 2022), and debiasing (e.g., Lauscher et al., 2021): (i) intermediate model training via standard full fine-tuning (e.g., Devlin et al., 2019), and (ii) adapter-based (Houlsby et al., 2019) history-injection. In (i), we run simple language modeling on fiction data, thereby fine-tuning the whole PLM. In contrast, in (ii), we inject lightweight bottleneck adapter-layers into the PLM. Here, we employ the architecture proposed by Pfeiffer et al. (2020b). During language modeling, we only adjust those parameters and keep the original parameters frozen. This increases the efficiency of our approach, as the adapter-layers are typically much smaller (in our case, we apply a reduction factor of 16), and we avoid the catastrophic forgetting of the already acquired knowledge of the PLM. Additionally, we modularize historical fiction: our adapters contain a potential history, which we can turn off and on on demand, and flexibily combine with other potential histories (Pfeiffer et al., 2021). Experimental Setup Data. We start from the English Wikipedia "wikitext-103-v1" data set (Merity et al., 2016) available on Huggingface Datasets. 1 It consists of a training, validation, and testing portion with 1, 801, 350 sequences, 3, 760 sequences, and 4, 358 sequences, respectively. Next, we perturb the data: to this end, we loop over each token in the data set. If the token is a singular gendered pronoun (i.e., he, she, and corresponding grammatical cases), we replace the pronoun with the corresponding case of the neopronoun xe. We take care to always replace with the right form using additional information from a part-of-speech tagging (POS) analysis. For instance, her can be the possessive dependent or accusative case. Through the POS-tag according to the Penn Treebank Project (Santorini, 1990), i.e., PRP for personal pronouns, and PRP$ for possessive pronouns, we can distinguish these cases and assign xem or xyr, respectively. Evaluation Measure. Lacking standard tests for the intrinsic evaluation of neopronoun knowledge in PLMs, we resort to the following evaluation regime: first, we build a set of gendered pronouns (V g ) consisting of each grammatical form of a gendered singular pronoun (i.e., she, her, etc. and he, him, etc.) and a set for our gender-neutral neopronoun (V n ) with the grammatical forms of xe, respectively. In addition, we consider the word person (p). For each of the tokens, we then extract static embeddings from the model using the same procedure as . To this end, we surround the word with the models' sequence start and end tokens and input the sequence into the model. For each token in the sequence, we compute a static representation x i as the average of the representations from layers m : n. To induce a word representation w, we average representations over all consecutive ranges [m : n], m ≤ n. Using the word representations w, we then compute the difference in average similarity between V g and V n towards p as d(p, Vg, Vn)= 1 |Vg| wg ∈Vg cos(p,wg)− 1 |Vn| wn∈Vn cos(p,wn) , (1) with cos(·, ·) as the cosine similarity. A higher value of d corresponds to gendered pronouns being more similar to person than the forms of xe. We couple this quantitative evaluation with a qualitative analysis of the topology of the space, using the same static embeddings extracted from the PLM. Model and Optimization. We use RoBERTa from Huggingface Transformers (Wolf et al., 2020) 2 in large configuration (24 layers, 16 heads, 1024 hidden size). For the adapter-based injection we use Adapter Transformers (Pfeiffer et al., 2020a). We train the models with a batch size of 32 and a learning rate of 1 · 10 −4 on our fiction Wiki using Adam (Kingma and Ba, 2015) for maximum 50 epochs. We apply early stopping based on the validation set perplexity (patience: 2 epochs). Results and Discussion. The results of our neopronoun-fiction showcase are depicted in Figures 1a-1c. Across almost all layer combinations, embeddings extracted from the original RoBERTa large are skewed towards gendered pronouns. In contrast, in our Xe-Fiction models, we were able to refine the Transformer representations towards forms of xe. The xe-embeddings from the full fine-tuning history-injection are closer to person than the gendered pronouns almost for any layer combination. For the adapter-based historyinjection, we can see a softer adjustment. The qualitative analysis of the topology of the static embedding space (Figures 2a-2c) yields a similar picture: in the original model ( Figure 2a) the grammatical forms of xe were pushed towards the edge of the embedding space. In contrast, in the Xe-Fiction models, xe-pronouns are closer to person. Conclusion The issue of socially discriminatory ML partially stems from the reliance on data and architectures that forefront discriminatory pasts and contemporaries. Here, we propose the deliberate use of historical fiction as a lens to understand potential issues in ML and to create data and models that narrate the real, rather than the hegemonic. By creating a simple dataset that fictionalizes a contemporary with greater social inclusion of neopronouns, we have shown how PLMs can come to more accurately represent the world. However, the scope of fictionalizing for NLP extends far beyond pronouns to other gendered and racialized inequities and wider social issues. Thus, there is ample space for future work to create fictions which seek to embed more equal representations of demographic groups and social issues. We conclude that historical fiction can address the difficult question of creating models that embody worlds more closely related to our own and provide NLP practitioners with methods that surface the real, rather than the factual. Limitations Energy Consumption and Environmental Impacts Our experiments highlight two modes of fictionalizing just futures in PLMs: adding a posthoc fine-tuning step and creating fictions within the optimization dataset. Choosing the former method adds another step in the machine learning pipeline, which will have negative costs for carbon emissions and the sustainability of developing machine learning models. We therefore advocate for the latter: by fictionalizing within the existing steps in the machine learning pipeline, researchers and practitioners can avoid incurring additional carbon costs (see Strubell et al., 2019;Dodge et al., 2022) of creating narratives within machine learning. Shifting Opinions While our work affords to more accurately describe the realities experienced in the world, i.e. more accurately describe pronoun use, our experiments and models are subject to the pronouns that are currently in use, that we are aware of. As gender is constantly in flux and conflict and subject to the experiences of individuals, the existence of current pronouns may cease while new may come to express a more fine-grained understanding of gendered and genderless existence. Dual Use Creating data which fictionalizes our contemporary can be used to create data and models that more accurately represent marginalized discourses. On the other hand, it can also be used to reinforce marginalizing discourses. Although a large body within machine learning does this, we believe that it is a by-product of data-driven machine learning being a relatively young field, rather than a product of malice. However, should a machine learning practitioner seek to erase certain histories and people, fictionalizing data which erases their existence could provide an avenue for such erasure. Limitations of data For our method, we are only using a very limited dataset, constructed for the explicit purposes of providing an example of how historical fiction can be used when applied purposefully to machine learning. Our data is likely to have constructions of pronouns that are not accurate with real-world application. For a more considerate dataset, we direct readers to the work of Qian et al. (2022), who performed in-depth analyses and corrections of incorrect and incoherent pronoun use. Further, our work serves as an illustration of the uses of historical fiction, and we suggest that readers deliberately consider the particular fictions that provide avenues for their objects of research. Figure 1 :Figure 2 : 12Xe-Fiction(Full Fine-tuning) (c) Xe-Fiction (Adapter-Fine-tuning) Results for our neopronoun-fiction experiments. We depict the difference in average similarity between gendered and gender-neutral pronouns towards the word person computed with static embeddings extracted from layers [m : n], m ≤ n. A positive value (red color) indicates gendered pronouns being closer to person. Topology of static embedding spaces extracted from layers 0-24 of (a) original RoBERTa large, (b) the Xe-Fiction model fully fine-tuned, and (c) the adapter-based Xe-Fiction model. We show the embeddings of the forms of gendered pronouns and of xe and of person projected in 2D-space via Principal Component Analysis. https://huggingface.co/datasets/ wikitext https://huggingface.com Potential history: unlearning imperialism. Ariella Azoulay, 1124512424Verso, London; Brooklyn, NY. OCLCAriella Azoulay. 2019. Potential history: unlearn- ing imperialism. Verso, London ; Brooklyn, NY. 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[]
[ "Tuning hyperparameters of doublet-detection methods for single- cell RNA sequencing data", "Tuning hyperparameters of doublet-detection methods for single- cell RNA sequencing data" ]
[ "Nan Miles Xi \nDepartment of Mathematics and Statistics\nLoyola University Chicago\n60660ChicagoILUSA\n", "Angelos Vasilopoulos \nDepartment of Mathematics and Statistics\nLoyola University Chicago\n60660ChicagoILUSA\n" ]
[ "Department of Mathematics and Statistics\nLoyola University Chicago\n60660ChicagoILUSA", "Department of Mathematics and Statistics\nLoyola University Chicago\n60660ChicagoILUSA" ]
[]
The existence of doublets in single-cell RNA sequencing (scRNA-seq) data poses a great challenge in downstream data analysis. Computational doublet-detection methods have been developed to remove doublets from scRNA-seq data. Yet, the default hyperparameter settings of those methods may not provide optimal performance. Here, we propose a strategy to tune hyperparameters for a cutting-edge doublet-detection method. We utilize a full factorial design to explore the relationship between hyperparameters and detection accuracy on 16 real scRNA-seq datasets. The optimal hyperparameters are obtained by a response surface model and convex optimization. We show that the optimal hyperparameters provide top performance across scRNA-seq datasets under various biological conditions. Our tuning strategy can be applied to other computational doublet-detection methods. It also offers insights into hyperparameter tuning for broader computational methods in scRNA-seq data analysis.
10.15302/j-qb-022-0324
[ "https://export.arxiv.org/pdf/2211.00772v2.pdf" ]
253,254,783
2211.00772
e95730e19bea84af3c8b6473c99ae3b5fd8c7251
Tuning hyperparameters of doublet-detection methods for single- cell RNA sequencing data Nan Miles Xi Department of Mathematics and Statistics Loyola University Chicago 60660ChicagoILUSA Angelos Vasilopoulos Department of Mathematics and Statistics Loyola University Chicago 60660ChicagoILUSA Tuning hyperparameters of doublet-detection methods for single- cell RNA sequencing data * Correspondence: Nan Miles Xi ([email protected])scRNA-seqdoublet detectionhyperparameter tuningexperimental designresponse surface model The existence of doublets in single-cell RNA sequencing (scRNA-seq) data poses a great challenge in downstream data analysis. Computational doublet-detection methods have been developed to remove doublets from scRNA-seq data. Yet, the default hyperparameter settings of those methods may not provide optimal performance. Here, we propose a strategy to tune hyperparameters for a cutting-edge doublet-detection method. We utilize a full factorial design to explore the relationship between hyperparameters and detection accuracy on 16 real scRNA-seq datasets. The optimal hyperparameters are obtained by a response surface model and convex optimization. We show that the optimal hyperparameters provide top performance across scRNA-seq datasets under various biological conditions. Our tuning strategy can be applied to other computational doublet-detection methods. It also offers insights into hyperparameter tuning for broader computational methods in scRNA-seq data analysis. Introduction Single-cell RNA sequencing (scRNA-seq) is a cutting-edge sequencing technology that can quantify genome-wide gene expression levels in a large number of cells 1,2 . Since its debut, scRNA-seq has been widely applied in various fields, including precision medicine 3 , drug discovery 4 , cancer therapy 5 , and vaccine development 6 . The successful application of scRNAseq relies on separating and labeling mRNA molecules from different cells. However, results may be confounded by the formation of doublets -when two cells are captured in one reaction volume by chance 7 . Because doublets appear as but are not real cells, they potentially bias downstream scRNA-seq data analysis. For example, doublets may be falsely identified as new cell types in cell clustering analysis 8 . To tackle this issue, the scRNA-seq community has developed computational methods to detect doublets from scRNA-seq data [7][8][9][10][11] . These methods utilize statistical and machine learning models, each with a set of default hyperparameters. Despite rapid development in methodology, one critical question remains untouched: whether default hyperparameters offer the best doublet-detection performance, especially for scRNA-seq datasets generated under various biological conditions. Here, we systematically explore the optimal hyperparameters of scDblFinder 9 , one cutting-edge computational doublet-detection method. We collect detection accuracy data for various hyperparameter combinations from 16 real scRNA-seq datasets with experimentally annotated doublets. Then, we fit a second-degree polynomial regression model with first-order, second-order, and interaction terms of three key hyperparameters. Convex optimization is performed to find the hyperparameters that maximize average detection accuracy across 16 datasets. We show that our optimal hyperparameters significantly improve doublet-detection accuracy over the method's default settings on most datasets. The detection accuracy of our optimal hyperparameters also ranks close to the best performance obtained by exhaustive searches in many datasets. We also apply our tuning strategy to scRNA-seq datasets under various biological conditions using different double-detection measurements. We find similar benefits from hyperparameter tuning, and the optimal hyperparameters vary depending on the biological conditions and accuracy measurements. Our exploratory strategy can be easily extended to other computational doubletdetection methods and provides hyperparameter tuning insights for broader computational methods in scRNA-seq data analysis. Datasets In this study, we utilize 16 public scRNA-seq datasets collected in a previous benchmark study 12 . All datasets contain ground-truth doublet labels annotated by experimental techniques. They are so far the most comprehensive scRNA-seq data collection with ground-truth doublet labels. The datasets cover a wide range of cell types, droplet numbers, gene numbers, and doublet rates, representing various difficulty levels in detecting doublets from scRNA-seq data. Table 1 summarizes the key characteristics of the 16 datasets used in this study. In scRNA-seq experiments, droplets refer to the reaction volumes that encapsulate the cell suspension. While most droplets contain one cell (singlets) as expected, others accidentally encapsulate two cells (doublets). Therefore, we will use "droplet" instead of "cell" to denote one data point in scRNA-seq datasets in the following text. Hyperparameter setting We choose scDblFinder as the target method for exploring optimal hyperparameter settings. The design of scDblFinder can be summarized in the following steps. First, it generates artificial doublets by combining gene expression profiles of two randomly selected droplets in the dataset. Second, it conducts PCA dimension reduction on the union of artificial doublets and true droplets using top highly variable genes. Third, scDblFinder constructs a nearest neighbor network on top of the low-dimensional representations from the dimension reduction. Fourth, it sets different neighborhood sizes to create multiple predictors that will be used for binary classification. Finally, it performs cross-validation to assign a doublet score to each droplet. In each iteration of crossvalidation, it trains a gradient boosting model to distinguish true droplets and artificial doublets in the training set, and then assigns each droplet in the test set a doublet probability (doublet score). The design of scDblFinder helps to reduce the impact of batch effects on doublet detection: since cross-validation randomly assigns droplets from different batches to training and test sets, the batch effects will not cause the domain shift problem 13 in the final classification step. scDblFinder has shown superior performance in previous benchmark studies 12,14 . We consider three key hyperparameters of scDblFinder, i.e., the number of top features, the number of top principal components, and the maximum depths of decision trees. We refer to them as , , and ℎ moving forward, respectively. These three hyperparameters are discrete numerical variables, and we set each of them to five different levels (Table 2). Therefore, there are 5 × 5 × 5 = 125 hyperparameter combinations in total. In experimental design literature, this is a Doublet detection We use the R package DoubletCollection 14 to execute scDblFinder on 16 real datasets with the 125 hyperparameter combinations listed in Table 2. Since doublet detection is essentially a binary classification task, we use the area under the precision-recall curve (AUPRC) to measure the overall doublet-detection accuracy. After execution, each dataset results in a 125 × 4 data matrix, in which the first three columns are , , and ℎ, and the last column is AUPRC. Each row in the data matrix represents one combination of three hyperparameters and corresponding AUPRC. The 16 scRNA-seq datasets generate 16 such data matrices. Finally, we merge the 16 data matrices by averaging their AUPRCs for each hyperparameter combination. The final data matrix is 125 × 4, which contains the relationship between hyperparameters and overall doubletdetection accuracy. We refer to this data matrix as detection accuracy data moving forward. Model setup and optimization We build a second-degree polynomial regression model to examine the relationship between hyperparameters and doublet-detection accuracy. Specifically, we set average AUPRC as the response variable and the first order of the three hyperparameters, the second order of the three hyperparameters, and their interactions as the independent variables. Model (1) shows the complete setup of this second-degree polynomial regression. = 0 + 1 + 2 + 3 ℎ + 4 2 + 5 2 + 6 ℎ 2 + 7 + 8 ℎ + 9 ℎ +(1) where , = 0, 1, . . . , 9, are the unknown model parameters, and is the random error. Second-degree polynomial regression is one classic model in the response surface methodology (RSM) 18 . It is commonly used to explore the relationship between several independent variables (hyperparameters) and one response variable (AUPRC) based on a full factorial design 19 . It can obtain an optimal response by estimating hyperparameters' main and quadratic effects and interactions between them. The second-degree polynomial regression balances model complexity and interpretation, while higher-degree models may cause overfitting and are harder to interpret. We fit this model by least square estimation using detection accuracy data. We perform a -test to assess the significance of estimated parameters ̂ and set 0.01 as the -value cutoff. We find that the first and second orders of and are significant. Equation (2) shows the estimated model (1) with significant independent variables (including the intercept). = 5.444 × 10 −1 + 1.016 × 10 −5 + 1.336 × 10 −3 − 3.760 × 10 −9 2 − 3.484 × 10 −5 2 (2) To obtain the and that maximize AUPRC, we take the partial derivative of AUPRC in respect of and in (2) and let the derivatives equal zero simultaneously. (4) Model diagnostics The 16 scRNA-seq datasets are generated by different sequencing protocols using various doublet annotation techniques. The error terms in model (1) Figure S1a shows that most residues have constant variance with no obvious patterns. The only concern is on the left, where the six residues may have a smaller variance. Second, we perform a Breusch-Pagan test 20 for heterogeneity. With the -value as 0.451, we fail to reject the null hypothesis that constant variance is present. Additionally, we perform a sensitivity analysis to examine the robustness of model (1). We conduct a natural log transformation and a square root transformation on the response variable AUPRC in the detection accuracy data. We then fit model (1) on the two transformed datasets and obtain the optimal hyperparameters using the same optimization method in (3). We find that the significant hyperparameters in model (1) and their optimal values are identical (after rounding to the nearest integers) to the results without transformation (Supplementary Table S1). The patterns of residue plots (Supplementary Figures S1b and S1c) are also similar to those before transformation (Supplementary Figure S1a). Log transformation and square root transformation on the response variable are common remedies for heterogeneity. If heterogeneity exists, then these two transformations would significantly change the model fitting, optimization, and residue plot. Similar results before and after transformations indicate that the heterogeneity is very mild, if any. We suspect that the heterogeneity is largely reduced or removed by averaging the AUPRCs of 16 datasets when creating the detection accuracy data (on which we fit model (1)). Optimal hyperparameter evaluation The optimal hyperparameters in equation (4) scRNA-seq datasets. Since hyperparameter ℎ is not significant in model (1), we set it to the default value in the execution. Table 3 compares the AUPRCs of the optimal hyperparameters in equation (4), the method's default hyperparameters, and the maximal AUPRCs achieved by one of 125 hyperparameter combinations. Our optimal hyperparameters outperform the method's performance with default settings on 12 out of 16 scRNA-seq datasets. Figure 1 summarizes the AUPRC improvement by hyperparameters tuning over the method's default settings. The most significant improvement is over 5% on dataset pbmc-1B-dm. There are eight datasets on which the improvement is over 2%. Figure 2 shows each dataset's AUPRC ranking under optimal hyperparameters among 125 hyperparameter combinations. We can see that the AUPRCs of optimal hyperparameters rank at or higher than the 20 th percentile on ten datasets. The highest ranking is 3 rd on dataset pdx-MULTI. The optimal hyperparameters also achieve the 50 th percentile or higher on all 16 datasets. Tuning hyperparameters for various biological conditions The previous analysis presents the optimal hyperparameters based on the average of 16 scRNAseq datasets. In practice, users mainly conduct doublet detection on datasets generated under specific biological or technical conditions. Those datasets need unique hyperparameter settings to achieve optimal performance. To demonstrate the generability of our tuning strategy to those applications, we replicate the hyperparameter optimization on two subsets of 16 datasets. The first is the pbmc-related subset, including six datasets: pbmc-1A-dm, pbmc-1B-dm, pbmc-1Cdm, pbmc-2ctrl-dm, pbmc-2stim-dm, and pbmc-ch. We find that the optimal is 18 (after rounding to the nearest integer), with the other two hyperparameters insignificant (Supplementary (Figures 1 and 2), the AUPRC improvement is greater with hyperparameters specifically tuned for pbmc-related datasets. The second subset includes the three HMEC-related datasets: HMEC-orig-MULTI, HMEC-rep-MULTI, and HEK-HMEC-MULTI. We find that the optimal is 1520 (after rounding to the nearest integer), with the other two hyperparameters insignificant (Supplementary Table S2). Supplementary Table S4 compares the AUPRCs of the optimal hyperparameters, the method's default hyperparameters, and the maximal AUPRCs achieved by one of 125 hyperparameter combinations on HMEC-related datasets. Supplementary Figure S2b shows each dataset's AUPRC improvement by hyperparameters optimization over the method's default settings, and AUPRC ranking under optimal hyperparameters among 125 hyperparameter combinations. Compared with the optimization on all 16 datasets (Figures 1 and 2), the AUPRC improvement is greater with hyperparameters specifically tuned for HMEC-related datasets. Tuning hyperparameters for various measurements The previous analyses use AUPRC, an overall accuracy measurement of doublet detection, as the optimization objective. In practice, users may also be interested in the method's capacity to identify doublets or singlets, i.e., the true positive or negative rate. Different from AUPRC, the calculation of true positive/negative rate requires a user-specified doublet rate. The true doublet rate is typically unknown to the users and needs to be estimated based on the sequencing platform, sequencing throughput, and prior knowledge 12,21 . Because optimization relies on the doublet rate, it is infeasible to find universal optimal hyperparameters for the true positive/negative rate. To provide hyperparameter guidance under this scenario, we set the doublet rates to their true values for each dataset (Table 1) and calculate the corresponding true positive/negative rates for 125 hyperparameter combinations. Then we replicate our tuning strategy using these two measurements as objectives. We find that the optimal maximum depths of decision trees ( ℎ) is 5 for both measurements (after rounding to the nearest integer), with the other two hyperparameters insignificant (Supplementary Table S2). Supplementary Figure S3a and Table S5 show each dataset's true positive rate improvement by hyperparameters optimization over the method's default settings and true positive rate ranking under optimal hyperparameters among 125 hyperparameter combinations. Most datasets exhibit similar improvement as AUPRC, except for J293t-dm, with a 29% increase, significantly larger than others. Such a difference indicates this dataset's unique biological and technical characteristics, which require stronger hyperparameter tuning efforts. Supplementary Figure S3b and Table S6 show each dataset's true negative rate improvement and ranking. Although most datasets still benefit from hyperparameter tuning, the improvement of the true negative rate is milder (below 1%) compared to other metrics. One reason is that the true negative rates under default hyperparameters are already high on many datasets (above 0.95, Supplementary Table S6), limiting the improvement space by hyperparameter optimization. It is worth noting that the optimal hyperparameters and corresponding true positive/negative rates are obtained using the true doublet rates. If users choose different doublet rates, the optimization results will be different. It is straightforward to generalize our tuning strategy in those cases. Discussion The existence of doublets is a key confounder in scRNA-seq data analysis. With the wide application of scRNA-seq technologies, much effort has been invested in developing computational doublet-detection methods. Such methods are primarily based on statistical and machine learning algorithms and are sensitive to hyperparameter configurations 22 . Although most methods provide a set of default hyperparameters, they cannot guarantee the best doublet-detection performance universally, especially when scRNA-seq data are generated under various biological conditions 23,24 . In this study, we utilize a full factorial design to build a model of hyperparameters and overall doublet-detection accuracy based on a leading method, scDblFinder, and 16 scRNA-seq datasets. The optimal hyperparameter combination obtained by convex optimization not only surpasses the default setting but also offers close-to-best detection accuracy on many datasets. We expand our optimization strategy to two subgroups of 16 datasets separately, providing optimal parameters for various biological conditions. We show that our method can also be applied to optimize different measurements of doublet-detection accuracy. The improved doublet-detection performance by hyperparameter tuning presents several insights regarding the data generalization and doublet annotation mechanisms. First, there are two datasets, using optimal hyperparameters than default hyperparameters (Figure 1). In contrast, most datasets generated by the other three doublet annotation techniques, i.e., cell hashing 26 , demuxlet 27 , and MULTI-seq 28 , benefit from hyperparameter tuning. One possible reason is due to their different doublet-annotation mechanisms. While species mixture can only annotate doublets from two species, the other three techniques utilize oligo-tagged antibody, SNP, or lipid-tagged index to label doublets from much broader sources. Consequently, the true doublets in hm-6k and hm-12k are likely undercounted, causing their inconsistent optimization results. Second, the hyperparameter tuning fails to improve the AUPRC for dataset pbmc-ch, even if the hyperparameters are optimized specifically for pbmc-related datasets (Supplementary Figure S2). In contrast, optimal hyperparameters consistently improve AUPRC for the other five pbmc-related datasets, and the improvements are greater with specifically tuned hyperparameters ( Table S2). If optimized on pbmc-related datasets, is no longer significant, and the optimal value of changes to 18. If optimized on HMEC-related datasets, is no longer significant, and the optimal value of changes to 1520. Depth is the only significant hyperparameter when optimizing the true positive and negative rate on all 16 datasets, with optimal values as 5 in both cases. This result indicates that existing and future doublet-detection methods need to fine-tune hyperparameters for a variety of biological conditions and accuracy measurements. In summary, doublet detection is one essential step in the quality control of scRNA-seq data analysis. The hyperparameter configuration significantly impacts the performance of computational doublet-detection methods. Our study is the first attempt to systematically explore the optimal hyperparameters under various biological conditions and optimization objectives. Our study provides much-needed guidance for hyperparameter tuning in computational doublet-detection methods. Future directions of our study include increasing the exploratory space by utilizing advanced experimental design strategies, e.g., space-filling design 29,30 and fractional factorial design 31,32 . Another direction is to optimize hyperparameters for other doublet-detection methods by our tuning strategy. More scRNA-seq datasets with experimentally annotated doublets could also be incorporated into the tuning process to enhance the generalizability of optimal hyperparameters. Data Availability The 16 scRNA-seq datasets used in this study are available at Zenodo repository https://doi.org/10.5281/zenodo.4562782 3 5 full factorial design15 . It allows investigation of the effects of individual hyperparameters, as well as the effects of interactions between different hyperparameters on the performance of doublet detection.We choose the five values for each hyperparameter according to the following rule. First, we start with the default values of each hyperparameter (Table 2). In general, the default values are selected by the developers based on extensive numerical experiments and thus are likely close to a local optimum in the hyperparameter space. Second, with the default value as the center, we increase or decrease each hyperparameter by one or several fixed step sizes, generating four extra alternative values. We determine the step size and the search space boundaries for each hyperparameter based on the common practice in scRNA-seq data analysis. The hyperparameter refers to the number of highly variable genes used in the downstream analysis. Its value is often set from several hundred to several thousand in many applications. For example, scDblFinder uses 1000 as the default value and Seurat 16 , a popular R package for scRNA-seq data analysis, chooses 2000. After including these two values in our search space, we insert 1500, the median value between 1000 and 2000, as the third search value, resulting in a step size of 500. We further expand the search space downward and upward by one step size separately. The final search space for is 500, 1000, 1500, 2000, and 2500. The hyperparameter is the number of principal components used in the downstream analysis after performing PCA dimension reduction on highly variable genes. Its value is often set from single digits to several dozen in practice. For example, the Seurat tutorial suggests exploring between 5 to 20 for various scenarios. We start with the default value of 10 and include 5 (the lower bound suggested by Seurat) in the search space, using a step size of 5. We further expand the search space by three step sizes up to 25. The final search space for is 5, 10, 15, 20, and 25. The hyperparameter ℎ is the maximum depth of decision trees in the gradient boosting model used in scDblFinder. The larger values indicate more complex gradient boosting models in binary classification (singlet vs. doublet). This hyperparameter is often set to below ten in ordinary classification tasks to avoid overfitting. For example, XGBoost 17 , a generic gradient boosting package, chooses 6 as the default value. We use scDblFinder's default value of 4 as the center of the search space. With 6 as the maximum and 1 as the step size, we create a final search space for ℎ including 2, 3, 4, 5, and 6. are obtained by maximizing the average AUPRC across 16 scRNA-seq datasets. To examine if these parameters can improve doublet-detection accuracy on individual datasets, we execute scDblFinder with and as in (4) on all 16 Figure 1 and 1Supplementary Figure S2). Such discrepancy is potentially due to the different doublet annotation techniques (cell hashing vs. demuxlet) and batch effects among those datasets. Further investigations, especially from the experimental perspective, are needed to reveal the impacts of these two factors on doublet detection. Third, the optimal hyperparameters vary depending on the biological conditions and optimization objectives. There are no universal hyperparameters adaptive to all scenarios. The significant hyperparameters when optimizing AUPRC across all 16 datasets are and , with optimal values as 19 and 1252, respectively (Supplementary may have non-constant variance, causing the heterogeneity issue. We conduct model diagnostics to examine the existence and severity of heterogeneity. First, we plot the residue against the fitted value of model(1). Supplementary Table S2 ) S2. Supplementary Table S3 compares the AUPRCs of the optimal hyperparameters, the method's default hyperparameters, and the maximal AUPRCs achieved by one of 125 hyperparameter combinations on pbmc-related datasets. Supplementary Figure S2a shows each dataset's AUPRC improvement by hyperparameters optimization over the method's default settings, and AUPRC ranking under optimal hyperparameters among 125 hyperparameter combinations. Compared with the optimization on all 16 datasets Table 2 . 2The three hyperparameters and their default and exploratory values in this study.Hyperparameter Default values Exploratory values 1000 500, 1000, 1500, 2000, 2500 10 5, 10, 15, 20, 25 ℎ 4 2, 3, 4, 5, 6 Table 3 . 3The AUPRC of doublet detection under optimal and default hyperparameters. The last column shows the highest AUPRC achieved by one of the 125 hyperparameter combinations. The highest AUPRC between the optimum and default of each dataset is underscored.SupplementaryTable S5. The true positive rate of doublet detection under optimal and default hyperparameters. The last column shows the highest true positive rate achieved by one of the 125 hyperparameter combinations. The larger true positive rate between optimum and default in each dataset is underscored.SupplementaryTable S6. The true negative rate of doublet detection under optimal and default hyperparameters. The last column shows the highest true negative rate achieved by one of the 125 hyperparameter combinations. The larger true negative rate between optimum and default in each dataset is underscored.Dataset Optimum Default Maximum cline-ch 0.4280 0.4202 0.4369 HEK-HMEC-MULTI 0.4723 0.4830 0.4966 HEK-orig-MULTI 0.4911 0.4873 0.5054 hm-12k 0.9281 0.9506 0.9850 hm-6k 0.9737 0.9896 0.9972 HMEC-rep-MULTI 0.6010 0.5964 0.6020 J293t-dm 0.2052 0.1999 0.2525 mkidney-ch 0.6125 0.6080 0.6183 nuc-MULTI 0.4600 0.4430 0.4704 pbmc-1A-dm 0.5454 0.5441 0.5693 pbmc-1B-dm 0.4375 0.4145 0.4818 pbmc-1C-dm 0.5953 0.5744 0.6082 pbmc-2ctrl-dm 0.6980 0.6749 0.7088 pbmc-2stim-dm 0.7003 0.6763 0.7124 pbmc-ch 0.6405 0.6472 0.6520 pdx-MULTI 0.4457 0.4268 0.4477 FiguresTablesSupplementary FiguresSupplementaryFigure S1. 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[ "Semi-supervised Development of ASR Systems for Multilingual Code-switched Speech in Under-resourced Languages", "Semi-supervised Development of ASR Systems for Multilingual Code-switched Speech in Under-resourced Languages" ]
[ "Astik Biswas \nDepartment of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa\n", "Emre Yılmaz \nDepartment of Electrical and Computer Engineering\nNational University of Singapore\nSingapore\n", "Febe De Wet \nDepartment of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa\n", "Ewald Van Der Westhuizen \nDepartment of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa\n", "Thomas Niesler \nDepartment of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa\n" ]
[ "Department of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa", "Department of Electrical and Computer Engineering\nNational University of Singapore\nSingapore", "Department of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa", "Department of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa", "Department of Electrical and Electronic Engineering\nStellenbosch University\nSouth Africa" ]
[]
This paper reports on the semi-supervised development of acoustic and language models for under-resourced, code-switched speech in five South African languages. Two approaches are considered. The first constructs four separate bilingual automatic speech recognisers (ASRs) corresponding to four different language pairs between which speakers switch frequently. The second uses a single, unified, five-lingual ASR system that represents all the languages (English, isiZulu, isiXhosa, Setswana and Sesotho). We evaluate the effectiveness of these two approaches when used to add additional data to our extremely sparse training sets. Results indicate that batch-wise semi-supervised training yields better results than a non-batch-wise approach. Furthermore, while the separate bilingual systems achieved better recognition performance than the unified system, they benefited more from pseudo-labels generated by the five-lingual system than from those generated by the bilingual systems.
null
[ "https://arxiv.org/pdf/2003.03135v1.pdf" ]
212,628,704
2003.03135
77f36b469958814187f02d0ed72be3a326ca2570
Semi-supervised Development of ASR Systems for Multilingual Code-switched Speech in Under-resourced Languages Astik Biswas Department of Electrical and Electronic Engineering Stellenbosch University South Africa Emre Yılmaz Department of Electrical and Computer Engineering National University of Singapore Singapore Febe De Wet Department of Electrical and Electronic Engineering Stellenbosch University South Africa Ewald Van Der Westhuizen Department of Electrical and Electronic Engineering Stellenbosch University South Africa Thomas Niesler Department of Electrical and Electronic Engineering Stellenbosch University South Africa Semi-supervised Development of ASR Systems for Multilingual Code-switched Speech in Under-resourced Languages Code-switched speechunder-resourced languagesmultilingual speechsemi-supervised trainingASR This paper reports on the semi-supervised development of acoustic and language models for under-resourced, code-switched speech in five South African languages. Two approaches are considered. The first constructs four separate bilingual automatic speech recognisers (ASRs) corresponding to four different language pairs between which speakers switch frequently. The second uses a single, unified, five-lingual ASR system that represents all the languages (English, isiZulu, isiXhosa, Setswana and Sesotho). We evaluate the effectiveness of these two approaches when used to add additional data to our extremely sparse training sets. Results indicate that batch-wise semi-supervised training yields better results than a non-batch-wise approach. Furthermore, while the separate bilingual systems achieved better recognition performance than the unified system, they benefited more from pseudo-labels generated by the five-lingual system than from those generated by the bilingual systems. Introduction Much research has already been dedicated to the development of automatic speech recognition (ASR) for codeswitching between various languages. English-Mandarin has probably been studied most extensively (Li and Fung, 2013a;Li and Fung, 2013b;Zeng et al., 2018;Vu et al., 2012;Taneja et al., 2019), but other language pairs such as Frisian-Dutch (Yılmaz et al., 2018c), Hindi-English (Pandey et al., 2018;Emond et al., 2018), English-Malay (Ahmed and Tan, 2012) and French-Arabic (Amazouz et al., 2017) have also received some attention. Developing models that are robust to the additional complexity associated with code-switching is challenging. The task becomes even more difficult when the languages in question are under-resourced since small text and acoustic data sets limit modelling capacity. South Africa has a multilingual population of 57 million people and 11 official languages, including English. Due to the variety of geographically co-located languages, codeswitching -the alternation between languages during communication -is a common phenomenon. Code-switching is most prevalent between English, a highly-resourced language, and the South African Bantu languages, which are all under-resourced. A corpus of code-switched speech originating from South African soap operas has recently been compiled to enable the development of ASR for this type of speech (van der Westhuizen and Niesler, 2018). Previous work demonstrated that multilingual training using in-domain soap opera code-switched speech (Biswas et al., 2018a;Yılmaz et al., 2018a) and poorly matched monolingual South African speech (Biswas et al., 2018b) improves the performance of both bilingual and five-lingual ASR systems when the additional training data is from a closely-related language. Specifically, isiZulu, isiXhosa, Sesotho and Setswana belong to the same Bantu language family and were found to complement each other when combined into a multilingual training set. It has also been shown that, in comparison with in-domain code-switched data, out-of-domain monolingual speech yields relatively little performance improvement in acoustic modelling (Biswas et al., 2018b). However, the in-domain training data that is currently available remains insufficient for robust ASR development and hence obtaining more indomain data remains key to improve the recognition accuracy of code-switched speech. Compiling a multilingual corpus of code-switched speech is an extremely labour intensive process, both in terms of effort and time, because manual segmentation and annotation of the data are required. In the absence of manually annotated material, automatically transcribed training material has been shown to be useful in under-resourced scenarios using semi-supervised training (Thomas et al., 2013;Yılmaz et al., 2018c;Guo et al., 2018). For example, this strategy was used successfully to obtain bilingual and five-lingual ASR systems using 11.5 hours of manually segmented but untranscribed soap-opera speech (Biswas et al., 2019). Furthermore, the bilingual systems trained with automatic transcriptions generated by the five-lingual transcription system achieved the best performance. Motivated by these results, we now investigate a batch-wise semi-supervised technique in which we incorporate additional batches of manually segmented but untranscribed soap opera data for acoustic and language modelling. Initial transcriptions were generated using our best systems trained on the manually transcribed speech. Given the multilingual nature of the data, the transcription systems must not only provide the orthography, but also the language(s) present at each location in the segment. Each utterance was therefore presented to the four individual code-switching systems as well as to the five-lingual system. In both cases two training configurations were considered, the first presenting all the data in one pass and the second presenting the data in smaller batches. Finally, we also present language modelling experiments that investigate the inclusion of the automatically generated transcriptions and artificially generated text as training material for English-isiZulu. Multilingual soap opera corpus This work uses a multilingual corpus including examples of code-switching between South African English and four Bantu languages. The corpus, which was compiled from South African soap opera episodes, contains 21 hours of annotated South African code-switched speech data divided into four language pairs: English-isiZulu (EZ), English-isiXhosa (EX), English-Setswana (ET), and English-Sesotho (ES). Of the Bantu languages, isiZulu and isiXhosa belong to the Nguni language family while Setswana and Sesotho are Sotho-Tswana languages. The speech in question is typically fast and often expresses emotion. These aspects of the data in combination with the high prevalence of code-switching makes it a challenging corpus for ASR experiments. The corpus is however still under construction. During the first phase of development, more than 600 South African soap opera episodes were manually segmented and a substantial portion of this also manually transcribed. The second phase is currently underway and has thus far contributed manually segmented but still untranscribed data to the corpus. Manually transcribed data (ManT) The version of the soap opera corpus we used to develop our first code-switching ASR systems consisted of 14.3 hours of speech divided into four language-balanced sets, as described in (van der Westhuizen and Niesler, 2018). In addition to the language-balanced sets, approximately 9 hours of manually transcribed monolingual English soap opera speech was also available. This data was initially excluded to avoid a bias toward English. However, pilot experiments indicated that, counter to expectations, its inclusion enhanced recognition performance. These 9 hours of English data were therefore merged with the balanced sets for the experiments described here. The composition of the unbalanced training speech is reported in Table 1. An overview of the statistics for the development (Dev) and test (Test) sets for each language pair is given in Manually segmented data: Batch 1 A set of approximately 11 hours of manually segmented speech representing 127 different speakers was produced in addition to the manually transcribed data introduced in the previous section. Segmentation was performed manually by experienced language practitioners. This data set (B1) was automatically transcribed during our initial investigations into semi-supervised acoustic model training (Biswas et al., 2019). Two sets of automatic transcriptions derived from B1 are considered: one obtained using four bilingual systems (AutoT B B1 ) and the other using a five-lingual system (AutoT F B1 ). Manually segmented data: Batches 2 & 3 A subsequent phase of corpus development, currently still underway, has produced two new batches of manually segmented data. Manual transcriptions of this data are not yet available. These data sets will be referred to as Batch 2 (B2) and Batch 3 (B3), respectively. In contrast to B1, the segmentation was done by trained assistants because no specialist language practitioners were available. Hence, the quality of the segments in B2 and B3 may differ from those in B1. Batch B2 includes approximately 24 hours of speech produced by 157 speakers, while B3 contains a further 30 hours of speech from 145 speakers. Most speakers occur in both batches and the languages spoken in the segments are not labelled. South African languages other than the five present in the transcribed data are known to occur in these batches, but to a limited extent. Acoustic Modelling All ASR experiments were performed using the Kaldi ASR toolkit (Povey et al., 2011) and the data described in Sec- tion 2. For multilingual training, the training sets of all the relevant languages were pooled. No phone merging was performed between languages and hence all acoustic models are language dependent. Context-dependent Gaussian mixture model -hidden Markov models (GMM-HMM) were trained to obtain the alignments required for neural network training. Threefold data augmentation (Ko et al., 2015) was applied prior to feature extraction for neural network training. The feature set used to train the neural network comprised MFCCs (40-dimensional, without derivatives), pitch features (3dimensional) and i-vectors for speaker adaptation (100dimensional) (Saon et al., 2013). The acoustic models of all ASR systems were trained according the standard Kaldi CNN-TDNN-F (Povey et al., 2018) Librispeech recipe (6 CNN layers and 10 time-delay layers followed by a rank reduction layer) using the default hyperparameters. For the bilingual experiments, the multilingual acoustic models were subsequently adapted to the four different target language pairs. Automatic transcription systems A recent study demonstrated that semi-supervised training can improve the performance of Frisian-Dutch codeswitched ASR (Yılmaz et al., 2018c). A similar approach was taken in this study, using the system configuration shown in Figure 1. The figure illustrates the two phases of semi-supervised training for the parallel bilingual as well as five-lingual configurations: automatic transcription followed by acoustic model retraining. Parallel bilingual transcription The first transcription system in Figure 1 consists of four subsystems, each corresponding to a code-switch language pair (4×CS in Figure 1). Acoustic models were trained on the manually transcribed soap opera data (ManT) described in Section 2.1 pooled with the automatically transcribed speech (AutoT B) introduced in Section 2.2. Because the languages in the untranscribed data were unknown, each utterance was decoded in parallel by each of the bilingual decoders. The output with the highest confidence score provided both the transcription and a language pair label for each segment. Five-lingual transcription The second transcription system was based on a single acoustic model trained on all five languages. The training data consisted of the manually transcribed soap opera speech (ManT) pooled with the transcriptions generated by a five-lingual system (AutoT F). Since the five-lingual system is not restricted to bilingual output, Bantu-to-Bantu language switching was possible and was observed in these transcriptions. Moreover, the automatically generated transcriptions sometimes contained more than two languages. Although the use of more than two languages within a single utterance is not common, our soap opera data does include a few such examples. Language Modelling Baseline language model The EZ, EX, ES, ET vocabularies contained 11 292, 8 805, 4 233, 4 957 word types respectively and were closed with respect to the training, development and test sets. The SRILM toolkit (Stolcke, 2002) was used to train and evaluate all language models (LMs). Four bilingual and one five-lingual trigram language model were used for the transcription systems as well as for semi-supervised training (Yılmaz et al., 2018c;Biswas et al., 2019). Table 3 summarises the development and test set perplexities. Details on the monolingual and code-switch perplexities are only provided for the test set (columns 3 to 6 in Table 3). Much more monolingual English text was available for language model development than text in the Bantu languages (471M vs 8M words). Therefore, the monolingual perplexity (MPP) is much higher for the Bantu languages than for English for each language pair. Code-switch perplexities (CPP) for language switches indicate the uncertainty of the first word following a language switch. EB corresponds to switches from English to a Bantu language and BE indicates a switch in the other direction. Table 3 shows that the CPP for switching from English to isiZulu and isiXhosa is much higher than switching from these languages to English. This can be ascribed to the much larger isiZulu and isiXhosa vocabularies, which are, in turn, due to the high degree of agglutination and the use of conjunctive orthography in these languages. The CPP values are even higher for the five-lingual language model. This is because the five-lingual trigrams allow language switches not permitted by the bilingual models. Semi-supervised language models Only bilingual transcriptions were considered for the semisupervised language model experiments. from the text in the transcriptions of the corresponding target language pair. The interpolation weights were optimised using the development data. A related study has shown that text data augmentation can be useful in under-resourced scenarios (Yılmaz et al., 2018b). This approach was evaluated on the EZ subset of our data by training a long short-term memory (LSTM) model on the manual and automatic transcriptions to generate additional artificial text data. The model was subsequently used to generate a data set of approximately 11.5 million words. The semi-supervised English-isiZulu language model described in the previous paragraph was interpolated with a trigram trained on this artificial data set. In a further attempt to strengthen the language model at code-switch points, 1 million of artificial code-switched trigrams were synthesised using the method described in (van der Westhuizen and Niesler, 2019). The perplexity values of the resulting language models are reported in Table 4. The first row in the table shows that adding the transcriptions of the automatically transcribed data to the LM training set reduces the test set perplexity of the EZ semisupervised language model by more than 50 relative to the baseline value in Table 3. The ET semi-supervised language model also achieved a significant perplexity reduction on the test set. However, the EX and ES semisupervised language models did not show any improvement compared to their respective baselines. This may be because there are far fewer isiXhosa and Sesotho segments in the automatically generated transcriptions than isiZulu and Setswana segments (cf. (Figure 2). Experiments Bilingual semi-supervised acoustic model training experiments were performed using batches B2 and B3 according to the two approaches illustrated in Figure 2. Similar configurations were used for the five-lingual experiments. In the first approach, both batches were first automatically transcribed using the baseline ASR (System A in Figure 2) followed by retraining using automatic transcriptions for both batches (System B). The second approach used batch-wise semi-supervised training. Using System A, B2 was transcribed first, followed by acoustic model retraining with the automatically transcribed B2 data included in the training set (System C). B3 was then transcribed using the updated models and the retraining process repeated, this time also including the transcriptions of B3 (System D). This order was also reversed i.e. B3 first (System E) followed by B2 (System F). The experimental procedure for the bilingual and fivelingual systems is summarised in Table 5). Type of ASR System AutoT B1 B2 B3 Bilingual A (Baseline) AutoT B - - B A A C A - D A C E - A F E A 5-lingual G (Baseline) AutoT F - - H G G I G - J G I K - G L K G Bilingual (5-lingual trans.) N AutoT F G I Table 5: Semi-supervised acoustic model configurations. Results and Discussion Automatic transcription The output of the transcription systems is summarised in Table 6. The first five rows in the table correspond to segments that were classified as monolingual while the last row shows the number of segments that contain codeswitching. The values in this row reveal a high number of code-switched segments in data sets B2 and B3. It should be kept in mind that transcriptions of mixed language utterances produced by configurations A, C and E can only contain code-switching between English and one Bantu language. In contrast, the utterances produced by G, I and K can contain examples of code-switching between two or more Bantu languages. Automatic speech recognition ASR quality was measured in terms of word error rate (WER) evaluated on the development and test sets for each language pair described in Table 2. Results for the different semi-supervised training configurations are reported in Table 7. Bilingual semi-supervised training The upper part of Table 7 shows that, on average, semisupervised training using 53 additional hours of speech data (B) yields an absolute improvement of 1.3% over the baseline (A) on the test set. It could be argued that this improvement is not large, given how much additional data was added to the training set. However, the segments were not created by language experts and may therefore not be accurate. Improving the quality of the segments might lead to Five-lingual semi-supervised training The lower half of Table 7 indicates that the five-lingual semi-supervised acoustic models also benefited from the additional data. As for the bilingual systems, the fivelingual system yielded better results when using batch-wise training. The best performing system (J) outperformed the baseline by 2.5% absolute on the test set. Although the WER achieved by the five-lingual system is higher than that achieved by the bilingual system, this remains a promising result. Five-lingual recognition is more difficult since it allows more freedom in terms of the permissible language switches. It does, however, more honestly reflect the large and undetermined mix of languages an ASR system may be confronted with when processing South African speech. Bilingual semi-supervised training with five-lingual transcriptions & semi-supervised language model The bilingual acoustic models retrained with transcriptions generated by the five-lingual system (N) achieved the best overall WER: 40.56% which is an absolute improvement of 0.7% over system D, its closest competitor. This improvement may be due to the five-lingual system's ability to transcribe in more than two languages, since the untranscribed soap opera speech is known to contain at least some utterances that do not conform to the four bilingual systems. System N was also evaluated in combination with the semi-supervised language model. The combination of the semi-supervised acoustic and language models (N LM 1 ) reduced the overall WER on the test set by another 0.5% absolute. Due to computational constraints, additional language model experiments were only conducted on the EZ data set. Table 8 shows that the use of language models derived from the additional text generated by the LSTM model, N LM 2 , and the synthesised trigrams including additional text generated by the LSTM model, N LM 3 , resulted in further small reductions in WER for both the development and test sets. Moreover, the corresponding code-switch bigram accuracy (row 2 in Table 8) also improved when the language model training data included the additional, artificially generated text and trigrams. , for which the Bantu language WERs are better. As already observed for the language model, the bias of the five-lingual system towards English is due to the much larger proportion of in-domain English training material available when pooling the four code-switched language pairs. The accuracy at the code-switch points is substantially better when using the bilingual semi-supervised system. This is probably due to the higher ambiguity encountered by the five-lingual system at code-switch points than the bilingual systems. The table also reveals that the improvements observed for systems using the semi-supervised language models are due mostly to improved English recognition (N LM 1 ). Conclusions In this study we evaluated semi-supervised acoustic and language model training with the aim of improving the ASR performance of under-resourced code-switched South African speech. Two batches (approximately 53 hours in total) of manually segmented but untranscribed soap opera speech, rich in code-switching, were used for experimentation. The new speech was processed both in a single step and in batches by two automatic transcription systems: one comprising four parallel bilingual recognisers and the other a single five-lingual system. The results indicate that the overall WER of both bilingual and five-lingual systems was reduced substantially and that batch-wise training was the better approach in both instances. However, the overall average performance of the bilingual systems remained better than that of the fivelingual system. This is probably because the five-lingual recognition task is inherently more complex. The five-lingual system exhibited a bias towards English while the four bilingual recognisers were more accurate for the Bantu languages. Despite the confusability inherent in decoding five languages, the five-lingual system achieved an error rate that was almost as good as that attained by the bilingual systems. Thus, it seems worthwhile developing bilingual and five-lingual code-switched ASR systems in parallel. The semi-supervised language model resulted in a significant reduction in perplexity that translated into a corresponding decrease in WER. Artificially generated codeswitched text and synthetic trigrams also showed potential to further improve the ASR performance. Future work will focus on incorporating automatic segmentation as well as speaker and language diarisation to extend the pool of training data. The effect of the number and size of training batches on system performance will also be studied more carefully. Figure 1 : 1Semi-supervised training framework for bilingual and five-lingual systems. Figure 2 : 2Semi-supervised training configurations for bilingual ASR. System names are given in parenthesis. (A: Baseline, B: Without batches, C to F: Batchwise) arXiv:2003.03135v1 [eess.AS] 6 Mar 2020Language Mono (m) CS (m) Total (h) Total (%) Word tokens Word types English 755.0 121.8 14.6 69.3 194 426 7 908 isiZulu 92.8 57.4 2.5 11.9 24 412 6 789 isiXhosa 65.1 23.8 1.5 7.0 13 825 5 630 Sesotho 44.7 34.0 1.3 6.2 22 226 2 321 Setswana 36.9 34.5 1.2 5.6 21 409 1 525 Total 994.5 271.5 21.1 100.0 276 290 24 170 Table 1 : 1Duration in minutes (min) and hours (h) as well as word type and token counts for the unbalanced training speech. Table 2 . 2The table includes values for the total duration as well as the duration of the monolin- gual and code-switched segments. The test sets contain no monolingual data. A total of approximately 4,000 language switches (English-to-Bantu and Bantu-to-English) are ob- served in the test set. The number of unique English, isiZulu, isiXhosa, Sesotho and Setswana words in the corpus are 8 275, 11 352, 6 169, Table 2 : 2Duration (minutes) of English, isiZulu, isiXhosa, Sesotho, Setswana monolingual (mdur) and code-switched (cdur) utterances in the code-switching development and test sets. 2 792, 1 902 respectively. IsiZulu and isiXhosa have rela- tively large vocabularies due to their agglutinative nature. This property adds to the challenge of developing accurate ASR systems in these languages. Table 3 : 3Development and test set perplexities. CPP: codeswitch perplexity. MPP: monolingual perplexity. Table 6 ). 6Resources Dev Test all CPP CPPEB CPPBE all MPP MPPE MPPZ EZ ManT + AutoT BB1 + AutoT BB2B3 392.2 547.4 2898.8 3297.6 2578.3 328.5 108.2 727.4 + 11.5M artificial text 362.8 507.5 2368.8 3005.8 1907.7 315.9 103.7 701.3 + 1M synthetic bigrams 358.3 501.9 2139.8 2613.5 1784.2 320.7 108.3 697.5 EX ManT + AutoT BB1 + AutoT BB2B3 345.4 787.4 5039.1 7176.3 3654.6 454.5 89.2 1411.4 ES ManT + AutoT BB1 + AutoT BB2B3 200.7 206.7 1074.3 347.0 3487.7 144.9 117.3 170.8 ET ManT + AutoT BB1 + AutoT BB2B3 138.4 164.8 938.0 214.6 3784.7 108.8 105.5 111.0 Table 4 : 4Development and test set perplexities for dif- ferent language model configurations. AutoT B B1 : B1 transcribed as described in (Biswas et al., 2019). Au- toT B B2B3 : B2 transcribed by system A and B3 transcribed by system C Table 4 4also shows that the additional text generated by the LSTM reduced the EZ development and test perplexity values substantially. The additional text also helped to bolster the language model at code-switch points. EZ CPP improved further after the 1M synthesized trigrams were added. Table 5 . 5The manual transcriptions introduced in Section 2.1 were always included in the training set. The composition of the automatic transcriptions included during training is shown inthe last three columns of the table, with the last two indi- cating which systems were used to generate transcriptions for B2 and B3 respectively. Preliminary experiments in- dicated that the bilingual ASR systems achieved best per- formance when trained on AutoT F transcriptions (Biswas et al., 2019). Thus, the bilingual systems considered here were also trained on the AutoT F transcriptions of B2 and B3 (System N in Table 6 : 6Number of segments per language for different transcription systems. Dev Test Dev Test Dev Test Dev Test Dev Test Dev TestBilingual ASR A (Baseline) B C D E F CS Pair EZ 34.5 40.8 33.4 39.3 32.7 39.6 32.8 38.6 34.6 39.9 32.8 39.0 EX 35.8 42.7 34.8 41.8 35.5 42.0 35.0 41.0 33.7 41.4 34.3 42.2 ES 51.6 48.7 49.3 47.3 49.0 46.5 48.7 45.8 49.5 47.4 49.5 46.4 ET 44.3 41.3 41.8 39.9 41.8 40.3 40.7 39.6 42.3 39.3 42.2 38.6 Overall 41.5 43.4 39.8 42.1 39.8 42.1 39.3 41.3 40.0 42.0 39.7 41.6 5-lingual ASR G (Baseline) H I J K L CS Pair Dev Test Dev Test Dev Test Dev Test Dev Test Dev Test EZ 37.6 43.6 36.3 41.0 35.7 42.5 34.3 42.1 35.7 42.5 35.0 42.0 EX 40.6 54.5 37.4 50.0 37.7 50.7 38.3 49.0 37.7 50.7 37.9 48.5 ES 54.5 49.3 51.5 48.1 52.8 46.9 51.5 47.8 52.8 49.9 51.6 48.0 ET 47.2 43.9 46.1 42.3 46.4 42.4 44.1 40.9 46.4 42.4 45.6 41.6 Overall 46.5 46.7 44.3 44.4 44.8 44.8 43.5 44.2 44.8 44.8 44.1 44.3 Table 7 : 7Mixed WERs (%) for the four code-switched lan- guage pairs evaluated using the baseline language model. better performance. The overall WER values for systems D and F show that batch-wise training results in better perfor- mance than processing all the untranscribed data in a single step. The best performing bilingual semi-supervised sys- tem (D) achieved an absolute overall WER improvement of 2.1% over the baseline on the test set. Table 8 : 8Mixed WER (%) and code-switched bigram accuracy (CSBA) (%) for EZ using artificial and synthetic text.For code-switched ASR, the performance of the recogniser at the code-switch points is of particular interest. Language specific WERs and code-switched bigram accuracy values for the different systems are presented inTable 9. All values are percentages.7.3. Language specific WER analysis System EZ EX ES ET E Z CSBA E X CSBA E S CSBA E T CSBA A (baseline) 37.9 48.7 33.3 37.8 54.5 25.8 43.7 61.4 25.2 36.2 51.8 35.6 D 32.1 43.6 39.8 31.3 48.3 33.9 33.0 55.9 34.8 27.6 47.6 42.3 G (baseline) 29.6 54.3 16.3 34.1 70.1 7.3 29.3 65.2 7.5 23.9 57.16 11.3 J 28.2 52.7 16.6 29.2 64.0 11.7 28.8 62.9 7.7 21.5 53.8 13.8 N 30.3 42.9 40.2 30.3 47.2 32.8 33.6 55.8 35.5 28.1 46.0 42.2 NLM1 29.0 42.9 40.1 29.9 47.0 32.7 32.2 55.8 35.8 26.3 46.1 43.6 Table 9 : 9Language specific WER (%) for English (E), isiZulu (Z), isiXhosa (X), Sesotho (S), Setswana (T) and code-switched bigram accuracy (CSBA) (%) for the test set.The table reveals that five-lingual ASR (systems G & J) is more biased towards English (lower WER) than the bilingual systems (systems A & D) AcknowledgementsWe would like to thank the Department of Arts & Culture (DAC) of the South African government for funding this research. 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InterspeechYılmaz, E., Heuvel, H. v. d., and van Leeuwen, D. (2018b). Acoustic and Textual Data Augmentation for Improved ASR of Code-Switching Speech. In Proc. Interspeech, pages 1933-1937. Semi-supervised acoustic model training for speech with code-switching. E Yılmaz, M Mclaren, H Van Den Heuvel, D Van Leeuwen, Speech Communication. 105Yılmaz, E., McLaren, M., van den Heuvel, H., and van Leeuwen, D. (2018c). Semi-supervised acoustic model training for speech with code-switching. Speech Com- munication, 105:12-22. On the end-to-end solution to Mandarin-English code-switching speech recognition. Z Zeng, Y Khassanov, V T Pham, H Xu, E S Chng, Li , H , arXiv:1811.00241arXiv preprintZeng, Z., Khassanov, Y., Pham, V. T., Xu, H., Chng, E. S., and Li, H. (2018). On the end-to-end solution to Mandarin-English code-switching speech recognition. arXiv preprint arXiv:1811.00241.
[]
[ "Transfer Functions of Generalized Bessel Polynomials", "Transfer Functions of Generalized Bessel Polynomials" ]
[ "Member, IEEEJosé R Martínez " ]
[]
[ "IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS" ]
The stability and approximation properties of transfer functions of generalized Bessel polynomials (GBP) are investigated. Sufficient conditions are established for the GBP to be Hurwitz. It is shown that the Padé approximants of e −s are related to the GBP. An infinite subset of stable Padé functions useful for approximating a constant time delay is defined and its approximation properties examined. The lowpass Padé functions are compared with an approximating function suggested by Budak. Basic limitations of Budak's approximation are derived.
10.1109/tcs.1977.1084347
[ "https://arxiv.org/pdf/1402.7092v1.pdf" ]
119,160,746
1402.7092
8b3e6c1dd937e13479032db429f5f4a6a4ed0333
Transfer Functions of Generalized Bessel Polynomials JUNE 1977 Member, IEEEJosé R Martínez Transfer Functions of Generalized Bessel Polynomials IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS 6325JUNE 1977 The stability and approximation properties of transfer functions of generalized Bessel polynomials (GBP) are investigated. Sufficient conditions are established for the GBP to be Hurwitz. It is shown that the Padé approximants of e −s are related to the GBP. An infinite subset of stable Padé functions useful for approximating a constant time delay is defined and its approximation properties examined. The lowpass Padé functions are compared with an approximating function suggested by Budak. Basic limitations of Budak's approximation are derived. I. INTRODUCTION S OME RECENT work has suggested the possibility of using ratios of generalized Bessel polynomials (GBP) to approximate the ideal delay function e −s . [1] However, the question of the stability of such rational functions was left unresolved since only a necessary, but not sufficient, condition for the stability of these polynomials was given. The purpose of this note is to discuss several useful properties of the GBP and associated rational functions. In particular, we establish a sufficient condition for the GBP to be Hurwitz. We also show that nonminimum phase rational functions of GBP yield simultaneous maximally flat approximations of a certain order of both constant time delay and magnitude; these rational functions are shown to be related to the Padé approximants of e −s . The Padé functions are compared with an approximation of e −s proposed by Budak. [2] II. GENERALIZED BESSEL POLYNOMIALS The GBP of degree n is defined by B n (s, α, β) = n k=1 n k (n + k + α − 2) (k) β k s n−k ,(1) where n k is a binomial coefficient, (q) (k) = q(q − 1)(q − 2) · · · (q − k + 1), q = n + k + α − 2, is the backward factorial function of order k, and α and β are real parameters, β = 0. [3], [4] From (1) it can be seen that the parameter β is a scaling factor since β n B n (s, α, β) = B n (βs, α, 1), hence the properties of B n will be determined by the parameter α. When α = β = 2, the GBP reduce to the classical Bessel polynomials of circuit theory. [5] Manuscript received March 1, 1976;revised February 4, 1977. The author is with Environmental Research and Technology, Inc., Santa Barbara, CA 93103. III. HURWITZ PROPERTY A necessary condition for B n (s, α, β) to be Hurwitz is that its coefficients be positive. This implies that α > 1 − n and β > 0, as can be seen from (1). This was essentially the condition given in [1]. It can be readily shown that the condition is not sufficient to guarantee the Hurwitz property. [6] Theorem 1 establishes a sufficient condition for B n (s, α, β) to be Hurwitz. It is recalled that the Hurwitz character of the special case B n (s, 2, β), β > 0, was established long ago by Storch. [5] Theorem 1 B n (s, α, β), n > 0, is Hurwitz for α 0, β > 0. The complete proof of Theorem 1 is found in [6]. That the conditions of Theorem 1 are not necessary is immediately seen by referring to B 2 (s, α, β). Additional information on the zeros of the GBP for α < 0 as well as other general properties of the zeros are found in [6]. We note, however, that for purposes of approximating e −s , the stable GBP with α < 0 generally yield poor approximations and thus are of little practical interest. [6] IV. RATIONAL FUNCTIONS In this section we show that the Padé approximants of e −s are rational functions of GBP for integral values of the parameter α and discuss several properties of this class of rational functions. The Padé functions are then compared with Budak's approximation of e −s [2]. We shall denote the Padé approximant by (n, m) = Q nm (s) P nm (s) ,(3) where Q nm and P nm are polynomials of degree m and n, respectively. For e −s , it is known [7] that Q nm and P nm are given by Q nm = n! (n + m)! m k=0 m k (n + k)! n! (−s) m−k (4) P nm = m! (n + m)! n k=0 n k (m + k)! m! s n−k .(5) Comparing B n (s, α, β) with (4) and (5), the following theorem is immediately evident. Theorem 2 The polynomials Q nm and P nm of the (n, m) Padé approximant of e −s are GBP. The relationship is given by Q nm = n! (n + m)! B n (−s, δ, 1), δ = n − m + 2,(6)P nm = m! (n + m)! B n (s, α, 1), α = m − n + 2.(7) Theorem 3 establishes the Hurwitz property of P nm . Its proof is evident upon applying Theorem 1 to (7). Theorem 3 The Padé approximant (n, m) of e −s is stable if m n − 2. It is readily ascertained that the condition of Theorem 3 is not necessary. [6], [8] Moreover, it can also be shown that not all the approximants are stable. [6] In fact, the number of stable as well as the number of unstable approximants is infinite. In order to preserve boundedness, the useful range of physically realizable approximants is defined by n − 2 m n, and we shall examine below the properties of the lowpass approximants (n, n − 1) and (n, n − 2). The approximation properties of the Padé functions are given in Theorems 4 and 5; the proofs are found in [6]. It is noted that the approximation of a unit amplitude or delay is said to be maximally flat of order k if, apart from the unit constant term, its Taylor expansion about the origin begins with the term ω 2k . Theorem 4 (Delay Approximation) The (n, m) Padé approximant of e −s yields a maximally flat delay approximation of order m + 1 for m = n − 1 and for m = n − 2. Theorem 5 (Amplitude Approximation) The (n, n − 1) and (n, n − 2) Padé approximants of e −s both yield a maximally flat approximation of order n of the ideal lowpass characteristic. Theorems 4 and 5 imply that the (n, m) approximant satisfies m phase and n − 1 amplitude flatness conditions at the origin for m = n−1 and m = n−2. This is the maximum number of simultaneous amplitude and phase conditions that may be imposed on a transfer function of degree m + n. [9] The usual all-pole Bessel transfer functions devote all the parameters to the delay approximation and none to approximating amplitude. By contrast, the Padé approximants allocate some parameters to phase and some to amplitude approximation. This brings us to comparing the Padé approximation of e −s and that suggested by Budak [2]. In Budak's approximation we have that e −s = e −γ s e −(γ−1) s , γ > 0,(8) and both numerator and denominator are approximated independently by all-pole Bessel transfer functions suitably scaled and not necessarily of the same degree. The end result is a transfer function of the form e −s ≈ G γ mn (s) = K B m [2(γ − 1) s, 2, 1] B n (2γ s, 2, 1) ,(9) where K is a constant defined by K = B n (0, 2, 1)/ B m (0, 2, 1). For γ = 1 or m = 0, we have the usual allpole transfer function. Some of the attributes of (9) have been recently discussed in [10]. In the case γ = 1/2 and m = n we have the well-known (n, n) Padé approximant. Marshak [10] has shown that the expansion of the time delay function, t d (ω), of (9) has the form t d (ω) = 1 − γ a 1 (γ ω) 2n + · · · − (1 − γ) b 1 (1 − γ) 2m ω 2m + · · · ,(10) where a 1 and b 1 are constants. Thus for m = 0, γ = 1, and m n, the series expansion of t d (ω)−1 begins with the term ω 2m . Hence the delay approximation obtained with (9) is maximally flat of order m. By contrast, the delay approximation given by the (n, m) Padé approximant is maximally flat of order m + 1. Thus (n, n − 1) and (n, n − 2) will yield better delay approximations than Budak's functions in the sense that they satisfy one more flatness condition at the origin. However, the Padé functions are always nonminimum phase functions, whereas Budak's approximation also includes minimum phase functions when γ 1. In the sequel we will show that, except in some special cases, the lowpass functions G γ mn (s), m < n, yield amplitude approximations that are maximally flat of unit order for arbitrary values of γ, which is no better than is obtained with all-pole transfer functions. Moreover, it will be shown that the amplitude approximation can never be better than maximally flat of order 2, regardless of the values of m and n. This largest order can be attained only for specific values of γ > 1/2; hence minimum-phase functions can achieve it. Using some results from [6] it can be shown that the squared-magnitude function |G γ mn (ω)| 2 is given by |G γ mn (ω)| 2 = (2n)! (2m)! 2 m! n! (11) × m i=0 m i (2i)! i! (m + i)! [2 (γ − 1) ω] 2(m−i) n k=0 n k (2k)! k! (n + k)! (2 γ ω) 2(n−k) In order for the first nonconstant term of the Taylor expansion of (11) to contain the factor ω 2q , q 1, it is necessary that the coefficients of the terms ω 2p , p = 1, 2, . . . , q − 1, be equal in numerator and denominator. If the coefficients of ω 2 are not equal, the approximation will be maximally flat of unit order. However, only certain values of γ will cause the appropriate pair of coefficients to be equal in (11). But as we shall see, with a given value of γ one can equate only a single pair of coefficients at a time. Equating the pair of coefficients of ω 2j , j = 1, 2, ..., m, in (11) we find that the equation for γ is given by γ γ − 1 2j = A j = [(2n)!] 2 [(n − j)!] 2 (n!) 2 [2 (n − j)]! (2n − j)! (12) × (m!) 2 [2 (m − j)]! (2m − j)! [(2m)!] 2 [(m − j)!] 2 , where j = 1, 2, . . . , m. Solving for γ we obtain γ = (A j ) 1/2j (A j ) 1/2j ± 1 ,(13) which implies that γ > 1/2. Therefore, minimum-phase functions can have any pair of coefficients of ω 2j in (11) rendered equal. It remains to show that coefficients can be equated only one pair at a time. To show this, suppose for the moment that a single value of γ satisfies (13) for more than one value of j, say for j and j + p, p > 0. Then we must require that γ 2 (γ − 1) 2 j = γ 2 (γ − 1) 2 j+p(14) and this can only be true when the quantity inside the brackets is unity. This implies, in turn, that γ = 1/2. But (13) only yields γ > 1/2, hence no value of γ obtained from (13) for a given j can satisfy (13) for any other j. The implication of this property of mutual exclusion is that We can eliminate from the Taylor expansion of (11) only the term containing ω 2 . In that case the first nonconstant term will contain ω 4 and the approximation will be maximally flat of order 2. This is the best that can be obtained. The value of γ that causes (11) to yield an approximation which is maximally flat of order 2 is given by γ = (2n − 1) ± (2n − 1) (2m − 1) 2(n − m) , m < n. (15) The preceding discussion indicates that the lowpass minimum-phase functions G γ mn (s) will at best satisfy a single flatness condition at the origin for values of γ given by (15). This is to be contrasted with the (n, n − 1) and (n, n − 2) approximants, both of which satisfy (n− 1) amplitude flatness conditions at the origin. It appears, therefore, that for a given n 2 and m = n − 1 or m = n − 2, the Padé approximants will yield a better amplitude approximation than G γ mn (s). V. EXAMPLES The following examples illustrate the properties of the two types of function discussed above. For illustrative purposes we will compare the Padé approximant (3, 2) with G γ 23 (s); these are shown below: (3, 2) = 3s 2 − 24s + 60 s 3 + 9s 2 + 36s + 60 , G γ 23 (s) = 5(γ − 1) 2 s 2 + 15(γ − 1)s + 15 γ 3 s 3 + 6γ 2 s 2 + 15γs + 15 . The delay functions for (16) and (17), respectively, are given by (3,2): t d (ω) = (18) 17 ω 8 + 592 ω 6 + 12384 ω 4 + 172800 ω 2 + 1440000 ω 10 + 33 ω 8 + 832 ω 6 + 12384 ω 4 + 172800 ω 2 + 1440000 and G γ 23 (s): t d (ω) = (19) a 4 ω 8 + a 3 ω 6 + a 2 ω 4 + a 1 ω 2 + 2025 b 5 ω 10 + b 4 ω 8 + b 3 ω 6 + b 2 ω 4 + b 1 ω 2 + 2025 , where a 4 = 3 (γ − 2)(γ − 1) 3 γ 5 a 3 = 9 (γ − 1) γ 3 4γ 3 − 13γ 2 + 13 γ − 5 a 2 = 9 γ 25γ 4 − 79γ 3 + 120γ 2 − 85 γ + 25 a 1 = b 1 = 135 8γ 2 − 10γ + 5 b 2 = 9 46γ 4 − 130γ 3 + 165γ 2 − 100 γ + 25 b 3 = 9 γ 2 8γ 4 − 24γ 3 + 32γ 2 − 20 γ + 5 b 4 = 3 (γ − 1) γ 4 3γ 2 − 4 γ + 2 b 5 = (γ − 1) 4 γ 6 Since a 1 = b 1 for all γ, then (19) is maximally flat of order 2, whereas (18) is flat of order 3. We cannot improve the approximation by setting a 2 = b 2 in (19) and solving for γ since this results in the polynomial (γ − 1) 5 . Hence γ = 1, which brings us back to the all-pole function. In general, one cannot improve the approximation without reverting to the allpole case, as can be seen from (10). The squared-magnitude functions for (16) and (17) are |(3, 2)| 2 = 9 ω 4 + 216 ω 2 + 3600 ω 6 + 9 ω 4 + 216 ω 2 + 3600 (20) |G γ mn (ω)| 2 = 25 (γ − 1) 4 ω 4 + 75 (γ − 1) 2 ω 2 + 225 γ 6 ω 6 + 6 γ 4 ω 4 + 45 γ 2 ω 2 + 225 It is clear that (20) yields an approximation that is maximally flat of order 3. Using (15), we find that γ = 5 + √ 15 /2 ≈ 4.436 causes (21) to have flatness of order 2, the best that we can do. VI. CONCLUDING REMARKS What has the inclusion of the parameter γ done for G γ mn (s), m < n, when compared to the all-pole Bessel transfer function of degree n? As is well known, the latter produces a delay approximation that is maximally flat of order n, and an amplitude approximation that is flat of unit order. On the other hand, G γ mn (s) yields a delay approximation of order m and, for the choice of γ given by (15), an amplitude approximation of order 2. Thus G γ mn (s) can exhibit an enhanced bandwidth at the expense of the delay approximation. Comparing the Padé approximants (n, n − 1) and (n, n − 2) with G γ mn (s) shows that the former yield superior approximations of amplitude and delay based on the criterion of maximal flatness. Thus for this type of lowpass function in which the degree of the numerator polynomial is lower than the denominator's by at most two units, there seems to be no reason to use G γ mn (s) in preference over the Padé approximants. However, this is not the case if one requires minimum-phase functions. Finally, we note that we showed in [6] that (n, n − 2) yields a delay approximation that always exhibits some time lag. This must be considered in selecting one function over another if one must choose between (n, n − 2) and G γ mn (s). On circuit-theory polynomial classes. D E Johnson, J R Johnson, IEEE Trans. Circuit Theory. 20D. E. Johnson and J. R. Johnson, "On circuit-theory polynomial classes," IEEE Trans. Circuit Theory, vol. CT-20, pp. 603-605, Sept. 1973. A maximally flat phase and controllable magnitude. A Budak, IEEE Trans. Circuit Theory. 279CT-12A. Budak, "A maximally flat phase and controllable magnitude," IEEE Trans. Circuit Theory, vol. CT-12, p. 279, June 1965. The Bessel Polynomials. J L Burchnall, Canadian J. Math. 3J. L. Burchnall, "The Bessel Polynomials," Canadian J. Math, vol. 3, pp. 62-68, 1951. A new class of orthogonal polynomials: The Bessel polynomials. H Krall, O Frink, Trans. Am. Math. Society. 65H. Krall and O. Frink, "A new class of orthogonal polynomials: The Bessel polynomials," Trans. Am. Math. Society, vol. 65, pp. 100-115, 1949. Synthesis of constant-time-delay ladder networks using Bessel polynomials. L Storch, Proc. IRE. IRE42L. Storch, "Synthesis of constant-time-delay ladder networks using Bessel polynomials," Proc. IRE, vol. 42, pp. 1666-1675, 1954. Generalized Bessel polynomials and their applications to linear phase approximation. J R Martínez, Las Cruces, NMNew Mexico State Universitydoctoral dissertationJ. R. Martínez, "Generalized Bessel polynomials and their applications to linear phase approximation," doctoral dissertation, New Mexico State University, Las Cruces, NM, Oct. 1970. Sur la representation approchée d'un fonction par des fractions rationelles. H Padé, 1892thesis, Ann. de l'Ecole NormaleH. Padé, "Sur la representation approchée d'un fonction par des fractions rationelles," thesis, Ann. de l'Ecole Normale, (3), v. 9, 1892. Padé approximants, generalized Bessel polynomials, and linear phase filters. J R Martínez, Proc. 14th Midwest Symposium on Circuit Theory. 14th Midwest Symposium on Circuit TheoryDenver, COJ. R. Martínez, "Padé approximants, generalized Bessel polynomials, and linear phase filters," in Proc. 14th Midwest Symposium on Circuit Theory, Denver, CO, 1971. The derivation of pole-zero patterns by derivative adjustment. F L Dennis, D A Linden, J. Franklin Institute. 268F. L. Dennis and D. A. Linden, "The derivation of pole-zero patterns by derivative adjustment," J. Franklin Institute, vol. 268, p. 283-293, 1959. A Bessel rational filter. A H Marshak, D E Johnson, J R Johnson, IEEE Trans. on Circuits and Systems. 21A. H. Marshak, D. E. Johnson, and J. R. Johnson, "A Bessel rational filter," IEEE Trans. on Circuits and Systems, vol. CT-21, pp. 797-799, Nov. 1974.
[]
[ "On the stability of the invariant probability measures of McKean-Vlasov equations", "On the stability of the invariant probability measures of McKean-Vlasov equations" ]
[ "Quentin Cormier " ]
[]
[]
We study the long-time behavior of some McKean-Vlasov stochastic differential equations used to model the evolution of large populations of interacting agents. We give conditions ensuring the local stability of an invariant probability measure. Lions derivatives are used in a novel way to obtain our stability criteria. We obtain results for non-local McKean-Vlasov equations on R d and for McKean-Vlasov equations on the torus where the interaction kernel is given by a convolution. On R d , we prove that the location of the roots of an analytic function determines the stability. On the torus, our stability criterion involves the Fourier coefficients of the interaction kernel. In both cases, we prove the convergence in the Wasserstein metric W1 with an exponential rate of convergence.
null
[ "https://export.arxiv.org/pdf/2201.11612v3.pdf" ]
258,987,609
2201.11612
8df93703fda572c50e8d6aacbb4166dea09a929e
On the stability of the invariant probability measures of McKean-Vlasov equations May 2023 Quentin Cormier On the stability of the invariant probability measures of McKean-Vlasov equations May 2023McKean-Vlasov SDELong-time behaviorMean-field interactionLions deriva- tive Mathematics Subject Classification Primary: 60H10 Secondary: 60K3545D0537A30 We study the long-time behavior of some McKean-Vlasov stochastic differential equations used to model the evolution of large populations of interacting agents. We give conditions ensuring the local stability of an invariant probability measure. Lions derivatives are used in a novel way to obtain our stability criteria. We obtain results for non-local McKean-Vlasov equations on R d and for McKean-Vlasov equations on the torus where the interaction kernel is given by a convolution. On R d , we prove that the location of the roots of an analytic function determines the stability. On the torus, our stability criterion involves the Fourier coefficients of the interaction kernel. In both cases, we prove the convergence in the Wasserstein metric W1 with an exponential rate of convergence. Introduction We are interested in the long-time behavior of the solutions of a class of McKean-Vlasov stochastic differential equations (SDE) of the form: dX ν t = V(X ν t , µt)dt + σdBt, (1.1) µt = Law(X ν t ), µ0 = ν. In this equation, (Bt) t≥0 is a standard R d -valued Brownian motion, σ is a deterministic matrix, and ν is the law of the initial condition X ν 0 , assumed to be independent of (Bt) t≥0 . McKean-Vlasov equations appear naturally as the limit N → ∞ of the following particle system (1.2) where µ N t is the empirical measure (X i,N t ) t≥0 , solution of dX i,N t = V(X i,N t , µ N t )dt + σdB i,N t , 1 ≤ i ≤ N,µ N t = 1 N ∑ N j=1 δ X j,N t and (B i,N t ) t≥0 are N independent standard Brownian motions. We refer to [45] for an introduction to this topic. Such particle systems and their mean-field counterparts are used in a wide range of applications such as plasma physics [22,32], fluid mechanics [26], astrophysics (particles are stars or galaxies [50]), bio-sciences (to understand the collective behavior of animals [8]), neuroscience (to model assemblies of neurons, such as integrate and fire neurons [21,25] or FitzHugh-Nagumo neurons [39]), opinion dynamics [18] and economics [10]. In these applications, one important question concerns the long-time behavior of the solutions. As such, the ergodic properties of McKean-Vlasov equations (1.1) have been studied in many different contexts and approaches. Two families of assumptions are known to ensure that (1.1) admits a unique, globally attractive invariant probability measure. The first type of assumption deals with kernels given by V(x, µ) = −∇V (x) − ∇W * µ(x), where V, W have suitable convexity properties. The first results in this direction were obtained in [3,4] in dimension one. In larger dimensions, [41,49] proved the convergence uniformly in time of a suitable particle system towards the mean-field equation. As such, they obtained the ergodicity of the McKean-Vlasov equation from the ergodicity of the particle system. These uniformly in time propagation of chaos arguments have been used wisely; see for instance [19,31] for recent results in this direction. These results have also been obtained by using functional inequalities [5,11]: the idea is to define a measure-valued functional (known as the entropy or free energy), which only decreases along the trajectories of the solution of (1.1). The second kind of assumption involves weak enough interactions. When the dependence of V with respect to the measure is sufficiently weak, one expects global stability because this situation can be seen as a perturbation of the case without interactions. As such, it is possible to extend techniques from ergodic Markov processes to the case of weak interactions. This includes, for instance, coupling techniques [27,9,1,23,24] or Picard iterations in suitable spaces [16]. It is also well-known that, in general, such global stability results cannot hold because (1.1) may have multiple invariant probability measures and periodic solutions [43,34,48]. These examples motivate the current question of the paper, namely the study of the local stability of a given invariant probability measure of (1.1). That is, being given ν∞ an invariant probability measure of (1.1), we address the following question: Is there exist an open neighborhood of ν∞ such that for all initial conditions ν within this neighborhood, the law of X ν t converges to ν∞, as t goes to infinity? If so, for which metric does the convergence hold, and what is the rate of convergence? Such local stability results can be obtained via partial differential equation (PDE) techniques, using that the marginals of the non-linear process solve a non-linear PDE (the Fokker-Planck equation). The strategy is to linearize the non-linear PDE around ν∞, to study the existence of a spectral gap for the linear equation in appropriate Banach spaces, and to use perturbation techniques to obtain the convergence for the non-linear PDE. We refer to [33,40] for an overview of these techniques. When the non-linear PDE admits a gradient flow structure, it is also possible to study the local stability of an invariant probability measure using functional inequalities; see [47,12]. Our approach differs from these two methods on several points. We do not rely on the non-linear Fokker-Planck PDE but instead use the stochastic version (1.1). Our strategy is to differentiate the interaction kernel with respect to the initial probability measure, in the neighborhood of ν∞. There are several notions of derivation with respect to probability measures (see [10]): we use here the Lions derivatives. We denote by P2(R d ) the set of probability measures on R d having a second moment. For all x ∈ R d and t ≥ 0, we consider the function P2(R d ) ∋ ν ↦ V(x, Law(X ν t )) =∶ v x t (ν) ∈ R d , where X ν t is the solution of (1.1) starting with law ν at time 0. We prove that under suitable assumptions, this function is Lions differentiable at ν∞, meaning that for all ν ∈ P2(R d ), we have V(x, Law(X ν t )) = V(x, ν∞) + E∂νv x t (ν∞)(Z0) ⋅ (Z − Z0) + o((E Z − Z0 2 ) 1 2 ). In this equation, Z, Z0 are any random variables defined on the same probability space, with laws equal to ν and ν∞. We write E(Z − Z0 Z0) = h(Z0), where h is a deterministic function from R d to R d . As such, the function h encodes the correlations between the initial conditions Z and Z0. It follows from the Cauchy-Schwarz inequality that E h(Z0) 2 ≤ E Z − Z0 2 < ∞. Therefore, h ∈ L 2 (ν∞). We define the linear operator Ωt ∶ L 2 (ν∞) → L 2 (ν∞) by Ωt(h) ∶= x ↦ E∂νv x t (ν∞)(Z0) ⋅ h(Z0). The fact that Ωt(h) ∈ L 2 (ν∞) for all h ∈ L 2 (ν∞) is not granted apriori and will follow from our assumptions on the function V. So we have (recall that ν = Law(Z) and E(Z − Z0 Z0) = h(Z0)) V(x, Law(X ν t )) = V(x, ν∞) + Ωt(h)(x) + o((E Z − Z0 2 ) 1 2 ). Our spectral conditions under which we prove that ν∞ is locally stable can be stated in terms of the decay of the function t ↦ Ωt, as t goes to infinity. We show that the integrability of this function on R+ implies the stability of ν∞. In addition, the decay of t ↦ Ωt as t goes to infinity gives precisely the rate of convergence of Law(X ν t ) towards ν∞, in Wasserstein metrics. Crucial to our analysis, we provide an explicit integral equation to compute this function Ωt. To do so, we consider the linear process (Y ν t ) t≥0 associated with (1.1) and ν∞, defined as the solution of dY ν t = V(Y ν t , ν∞)dt + σdBt, starting from Law(Y ν 0 ) = ν. We define similarly for x ∈ R d and t ≥ 0 the function P2(R d ) ∋ ν ↦ u x t (ν) ∶= V(x, Law(Y ν t )). Under non-restrictive assumptions, u x t is Lions differentiable at ν∞, and we can define ∀h ∈ L 2 (ν∞), Θt(h) ∶= x ↦ E∂νu x t (ν∞)(Z0) ⋅ h(Z0). We prove the following key relation between Θt and Ωt ∀t ≥ 0, Ωt(h) = Θt(h) + t 0 Θt−s(Ωs(h))ds. That is, Ω is a solution of a Volterra integral equation whose kernel is given by Θ: in the language of integral equations, Ω is the resolvent of Θ. This relation is helpful because it is easier to get estimates on u x t , which involves a linear Markov process, rather than getting estimates on v x t , which involves the solution of the McKean-Vlasov equation (1.1). In particular, this relation allows to deduce the decay properties of Ω from properties of Θ, using Laplace transform. We obtain our stability results for the Wasserstein W1 metric. The contributions of this work are the following. First, in Section 2, we consider dynamics of the form V(x, µ) = b(x) + ∫ R d F (x, y)µ(dy), for some smooth functions b ∶ R d → R d and F ∶ R d × R d → R d . The function b is assumed to be confining. Our main result, Theorem 2.5, states that the stability of an invariant probability measure is determined by the location of the roots of an explicit analytic function associated with the dynamics. Stability holds when all the roots lie on the left half-plane, and we prove convergence in Wassertein metric W1 with an exponential rate. Our result shows that the stability is completely determined by a discrete set, this set being given by the zeros of an analytic function associated to the underlying Markov process. Note that we do not require any structural assumption on b and F : in particular, we do not require any convexity assumption on the coefficients. Our stability criterion is analogous to the Jacobian stability criterion for ODE, for which the location of the zeros of the characteristic polynomial determines the stability. Second, in Section 3, we consider a McKean-Vlasov equation on the torus T d ∶= (R (2πZ)) d , with an interaction kernel given by a convolution: V(x, µ) = − ∫ T d ∇W (x − y)µ(dy), where W is a smooth function from T d to R. We assume that σ = 2β −1 I d for some β > 0, where I d is the identity matrix. This setting covers many interesting models; see [12]. We study the stability of the uniform probability measure U (dx) ∶= dx (2π) d . Our second main result, Theorem 3.2, states that when inf n∈Z d ∖{0} n 2 (β +W (n)) > 0,W (n) being the n-th Fourier coefficient of W , then U is locally stable for the W1 metric. Our result complements the results of [12], for which static bifurcations are studied: in particular, we exhibit the same critical parameter. In both parts, we use the strategy described above, using Lions derivatives and probabilistic tools. The criteria we obtain are optimal: violations of the criteria occur strictly at bifurcation points. The strategy presented in this work also applies to mean-field models of noisy integrateand-fire neurons. In an unpublished preliminary version of this work [15], we study the stability of the stationary solutions of such a mean-field model of noisy neurons. In addition, periodic solutions via Hopf bifurcations are studied in [17]. For the sake of clarity, we restrict here ourselves to a diffusive setting. Finally, we mention an important open problem concerning the long-time behavior of the particle system (1.2). On the one hand, general conditions are known to ensure that the particle system is ergodic. On the other hand, numerical studies show that this particle system can have metastable behavior in the sense that the convergence of the empirical measure µ N t towards its invariant state can be very slow when N is large. The locally stable invariant probability measures of the non-linear equation (1.1) are good candidates to be metastable states of the particle system (1.2). Characterizing those metastable states in quantitative terms is a challenging mathematical question. Recent partial results have been obtained in this direction [7,2,37,14,20], and we hope to progress on this question in future works. McKean-Vlasov equations on R d Main result Let P1(R d ) be the space of probability measures on R d with a finite first moment. We consider the following McKean-Vlasov equation on R d : dX ν t = b(X ν t )dt + R d F (X ν t , y)µt(dy)dt + σdBt with µt = Law(X ν t ). (2.1) Here, the initial condition X ν 0 has law ν ∈ P1(R d ). Here, (Bt) t≥0 is a d-dimensional standard Brownian motion, σ ∈ M d (R) is a constant d × d matrix with det σ > 0. Assumption 2.1. The functions b ∶ R d → R d and F ∶ R 2d → R d are C 2 , b∂x i F ∞ + ∂ 2 x i ,x j F ∞ < ∞. This ensures in particular that (2.1) has a unique path-wise solution. Let ν∞ ∈ P1(R d ) be an invariant probability measure of (2.1), that is: ∀t ≥ 0, Law(X ν∞ t ) = ν∞. Denote by α(x) the interaction term under ν∞: ∀x ∈ R d , α(x) ∶= R d F (x, y)ν∞(dy). (2.2) Each invariant probability measure of (2.1) is characterized by its associated function α, and we sometimes denote by ν α ∞ such invariant probability measure to emphasize the dependence on α. We assume: Assumption 2.2. There exists β > 0 and R ≥ 0 such that ∀x, x ′ ∈ R d , x − x ′ ≥ R ⇒ (x − x ′ ) ⋅ (b + α)(x) − (b + α)(x ′ ) ≤ −β x − x ′ 2 . Remark 2.3. In particular, this is satisfied provided that F is bounded (or independent of x) and there exists β > 0 and R ≥ 0 such that: x − x ′ ≥ R ⇒ (x − x ′ ) ⋅ (b(x) − b(x ′ )) ≤ −β x − x 2 . Let (Y α t ) t≥0 the solution of the linear SDE dY α t = b(Y α t )dt + α(Y α t )dt + σdBt. (2.3) Note that ν∞ = ν α ∞ is also an invariant probability measure of this linear SDE. In addition, a result of Eberle [23] (see (2.7) below) ensures that ν∞ is the unique invariant probability measure of (2.3). Consider H ∶= L 2 (ν∞) the Hilbert space of measurable functions h ∶ R d → R d satisfying: h 2 H ∶= h(y) 2 ν∞(dy) < ∞. We denote by L(H) the space of bounded linear operators from H to itself. Key to our analysis is the following family of bounded linear operators Θt ∈ L(H), t ≥ 0: ∀h ∈ H, Θt(h)(x) ∶= R d ∇yEyF (x, Y α t ) ⋅ h(y)ν α ∞ (dy). (2.4) The notation EyF (x, Y α t ) means that the initial condition of (Y α t ) is set to be y ∈ R d (that is Y α 0 = y). In addition, ∇yEyF (x, Y α t ) is the Jacobian matrix of y ↦ EyF (x, Y α t ). The result of Eberle [23] (see (2.8) below) implies that there exists κ * > 0 and C > 0 such that ∀h ∈ H, Θt(h) ∞ ≤ Ce −κ * t h H . (2.5) Denote by I ∈ L(H) the identity operator. Let D = {z ∈ C, R(z) > −κ * }. For all z ∈ D, consider the Laplace transform of Θt: Θ(z) = ∞ 0 e −zt Θtdt. We recall some properties of this operator-valued function. z ∈ D ∖ S, where S is a discrete subset of D. In addition, (I −Θ(z)) −1 is meromorphic in D, holomorphic in D ∖ S. (c) The function D ∋ z ↦ det(I −Θ(z)) ∈ C is analytic, where det is the Fredholm determi- nant. In addition, I −Θ(z) is invertible if and only if det(I −Θ(z)) ≠ 0. Proof. By Fubini, we haveΘ (z)(h) = x ↦ R d Kz(x, y)h(y)ν∞(dy), where Kz(x, y) ∶= ∫ ∞ 0 e −zt ∇yEyF (x, Y α t )dt. Using (2.8), it holds that sup x,y∈R d Kz(x, y) < ∞ for all z ∈ D. Therefore, Kz(⋅, ⋅) ∈ L 2 (R d × R d , ν∞ ⊗ ν∞) and so Theorem VI.23 in [42] applies and gives the first point. For R(z) large enough, it holds that Θ (z) H < 1. Therefore, I −Θ(z) is invertible provided that R(z) is large enough, the inverse is given by its Neumann series. So the second point follows from the analytic Fredholm theorem, see Theorem VI.14 in [42]. Finally, the third point is a fundamental result of the theory of Fredholm determinant, see for instance [44]. Our main result is Theorem 2.5. Consider ν∞ an invariant probability measure of (2.1) and let α be given by (2.2). Assume that Assumptions 2.1 and 2.2 hold. Define the "abscissa" of the rightmost zeros of det(I −Θ(z)): −λ ′ ∶= sup{R(z) z ∈ D, det(I −Θ(z)) = 0}. (2.6) Assume that λ ′ > 0. Then ν∞ is locally stable: there exists C, ǫ > 0 and λ ∈ (0, λ ′ ) such that for all ν ∈ P1(R d ) with W1(ν, ν∞) < ǫ, it holds that ∀t ≥ 0, W1(Law(X ν t ), ν∞) ≤ CW1(ν, ν∞)e −λt . Remarks and examples We now give explanations on Theorem 2.5, in particular on the spectral assumption involving (2.6). Gradients bounds We denote by (Y α,δx t ) the solution of (2.3) with initial condition Y α,δx 0 = x. Under assumptions 2.2, Theorem 1 in [23] applies: there exists κ * > 0 and C * > 1 such that for all x, y ∈ R d and all t ≥ 0, W1(Law(Y α,δx t ), Law(Y α,δy t )) ≤ C * e −κ * t x − y . (2.7) We deduce from this inequality the following gradient bound. For all f ∈ C 1 (R d ): ∀y ∈ R d , ∇yEyf (Y α t ) ≤ C * ∇f ∞e −κ * t ,(2.8) In particular, by choosing f = F (x, ⋅), we obtain the estimate (2.5). Note that it is possible to get gradient bounds similar to (2.8) under less restrictive assumptions on b and σ; see [46]. On the spectral condition We consider the following bounded linear operator Ωt ∈ L(H) by taking the Neumann series: ∀h ∈ H, Ωt(h) ∶= i≥1 Θ ⊗i t (h), (2.9) where the linear operators Θ ⊗(i) t are defined recursively by ∀t ≥ 0, Θ ⊗(i+1) t (h) = t 0 Θt−s(Θ ⊗i s (h))ds, and Θ ⊗1 t (h) = Θt(h). The series (2.9) converges uniformly on any compact [0, T ] for T > 0. The operators Ωt and Θt satisfy the following Volterra integral equation: ∀h ∈ H, Ωt(h) = Θt(h) + t 0 Θt−s(Ωs(h))ds (2.10) = Θt(h) + t 0 Ωt−s(Θs(h))ds. We denote by Ωt L(H) the operator norm of Ωt: Ωt L(H) ∶= sup h L 2 (H) ≤1 Ωt(h) H . Then the spectral condition λ ′ > 0 is equivalent to the exponential decay of t ↦ Ωt L(H) : Proposition 2.6. The two following statements are equivalent (a) λ ′ > 0, where λ ′ is given by (2.6). (b) ∃λ > 0 such that sup t≥0 e λt Ωt L(H) < ∞. Proof. We first show that (b) implies (a): by assumption, there exists λ ∈ (0, κ * ) such that z ↦Ω(z) is an analytic operator-valued function on R(z) > −λ. In view of (2.10), we have for R(z) > −λ,Ω(z) =Θ(z) +Θ(z) ⋅Ω(z) =Θ(z) +Ω(z) ⋅Θ(z), and so I = (I +Ω(z))(I −Θ(z)) = (I −Θ(z))(I +Ω(z)). Therefore, I −Θ(z) is invertible for all R(z) > −λ, and consequently λ ′ ≥ λ > 0. We then show that (a) implies (b): this follows from a Paley-Wiener theorem. Let λ ∈ (0, λ ′ ) and define Kt ∶= e λt Θt and Rt ∶= e λt Ωt. It holds that K ∈ L 1 (R+; L(H)) (because λ < κ * ) and I −K(z) is invertible for all R(z) ≥ 0 (because λ < λ ′ ). Therefore, by [30, Ch. 2, Th. 4.1] * , it holds that R ∈ L 1 (R+; L(H)). Using the estimate (2.5) and (2.10), we find that Ωt(h) ∞ ≤ Ce −κ * t h H + C t 0 e −κ * s Ωt−s L(H) h H ds ≤ C h H e −κ * t + e −λt t 0 e λ(t−s) Ωt−s L(H) ds ≤ C h H 1 + ∞ 0 Rs L(H) ds e −λt . This shows that sup t≥0 e λt Ωt L(H) < ∞. This ends the proof. * The result in [30] is stated and proved for matrix valued operators. The extension to Hilbert-Schmidt operators is straightforward by the exact same arguments. The spectral condition is necessary Let h ∈ H be fixed and let Z0 be a random variable of law ν∞. For ǫ ∈ R, let νǫ ∶= Law(Z0 + ǫh(Z0)). We will see that the operator Ωt admits the following probabilistic representation (see Remark 2.22 below): ∀t ≥ 0, Ωt(h)(x) = lim ǫ→0 1 ǫ R d F (x, y)(Law(X νǫ t ) − ν∞)(dy), x ∈ R d , (2.11) where (X νǫ t ) is the solution of the McKean-Vlasov equation (2.1) starting with law νǫ. We have by the Cauchy-Schwarz inequality W1(νǫ, ν∞) ≤ ǫE h(Z0) ≤ ǫ h H . Assume that the conclusion of Theorem 2.5 holds. Then there exists C, λ > 0 such that for all ǫ small enough, W1(Law(X νǫ t ), ν∞) ≤ Cǫe −λt h H . We deduce that: ∀x ∈ R d , R d F (x, y)(Law(X νǫ t ) − ν∞)(dy) ≤ Cǫ ∇yF ∞e −λt h H . Therefore, Ωt(h) ∞ ≤ C ∇yF ∞e −λt h H . Using Proposition 2.6, we deduce that λ ′ > 0, where λ ′ is given by (2.6). In other words, the spectral condition λ ′ > 0 is necessary to have stability in W1 norm. Structural stability with convex coefficients The classical situation where stability is known to hold globally is the following: assume that there exist functions V, F ∶ R d → R such that b(x) = −∇V (x) and F (x, y) = −∇W (x − y). Assume that W is convex and even and that V is strictly convex: ∃θ > 0, ∀x ∈ R d , W (x) = W (−x), ∇ 2 W (x) ≥ 0, and ∇ 2 V (x) ≥ θI d . (2.12) Then, by [13,Th. 7], it holds that ∀t ≥ 0, ∀ν, µ ∈ P2(R d ), W2(Law(X ν t ), Law(X µ t )) ≤ e −θt 2 W2(ν, µ) . Therefore, these structural conditions ensure that (2.1) has a unique invariant probability measure, which is globally stable in W2 norm. We show that our spectral condition is satisfied under these conditions. Lemma 2.7. Under the structural assumption (2.12), it holds that λ ′ > 0, λ ′ given by (2.6), and so Theorem 2.5 applies. Proof. Let ν∞ be the unique invariant probability measure of (2.1). Let h ∈ H, ǫ ∈ R and let νǫ = L(Z0 + ǫh(Z0)), where Law(Z0) = ν∞. We have R d ∇W (x − y)(Law(X νǫ t ) − ν∞)(dy) ≤ ∇ 2 W ∞W1(Law(X νǫ t ), ν∞) ≤ ∇ 2 W ∞W2(Law(X νǫ t ), ν∞) ≤ ∇ 2 W ∞e −θt 2 W2(νǫ, ν∞) ≤ Ce −θt 2 E ǫh(Z0) 2 = C ǫ e −θt 2 h H . Therefore, by (2.11), we have ∀h ∈ H, Ωt(h) ∞ ≤ Ce −θt 2 h H . So Proposition 2.6 applies and λ ′ > 0. Case of weak interactions One way to check that the spectral condition λ ′ > 0 holds, λ ′ given by ( Θt L(H) dt < 1. Then λ ′ > 0 and so ν∞ is locally stable. Proof. By assumption, there exists δ > 0 small enough such that ∞ 0 e δt Θt L(H) dt < 1. For R(z) ≥ −δ, it holds that Θ (z) L(H) ≤ ∫ ∞ 0 e −R(z)t Θt L(H) dt < 1. We deduce that I −Θ(z) is invertible for R(z) ≥ −δ, with inverse given by ∑ k≥0 (Θ(z)) k . So λ ′ ≥ δ > 0. This assumption is typically satisfied if the non-linear part in (2.1) is weak enough. Given M ∈ M d (C), let M 2 be the spectral norm of the matrix M . We let [F ] ∶= sup x,y∈R d ∇yF (x, y) 2. Then, from (2.7), we have Θt L(H) ≤ C * [F ]e −κ * t . Therefore, if [F ] < κ * C * , then λ ′ > 0, and so any invariant probability measure of (2.1) is locally stable. Case of "separable" interactions Assume that F is "separable", in the sense that there exist functions wi ∶ R d → R d and fi ∶ R d → R such that ∀x, y ∈ R d × R d , F (x, y) = p i=1 fi(y)wi(x). Let H0 be the following subspace of H of finite dimension H0 ∶= {h h = p i=1 βiwi, β ∈ R p }. For all h ∈ H and for all t ≥ 0, it holds that Θt(h) ∈ H0. The restriction of Θt to H0 can be represented by a p × p matrix, again denoted by Θt, and we have Θ i,j t = R d ∇yEyfi(Y α t ) ⋅ wj(y)ν∞(dy), 1 ≤ i, j ≤ p. In that case, the determinant in Theorem 2.5 is the standard determinant of matrices. This instance already covers a number of interesting examples. Case with no noise (σ ≡ 0) The case σ ≡ 0 would require special treatment and is not included in Theorem 2.5. It is however instructive to see that criterion involving (2.6) is equivalent to the classical stability criterion of deterministic dynamical systems in R d . When σ ≡ 0, the invariant measures are of the form δx * for some F (x, y). We have: Lemma 2.9. Assume that there exists κ > 0 such that sup t≥0 e κt e t∇xV(x * ,x * ) < ∞. Then, the criterion λ ′ > 0, λ ′ given by (2.6), is equivalent to the fact that the Jacobian matrix of the vector field x ↦ V(x, x) at the point x * has all its eigenvalues in the left half-plane R(z) < 0. x * ∈ R d . Let V(x, y) = b(x) + The proof is given in the Appendix. A simple explicit example We close this Section with a simple explicit example. Consider for J ∈ R * the following McKean-Vlasov SDE on R: dXt = −Xtdt + JE cos(Xt)dt + √ 2dBt. (2.13) The associated linear process (Y α t ) is the solution of the Ornstein-Uhlenbeck SDE dY α t = −Y α t dt + αdt + √ 2dBt. This linear process admits a unique invariant probability measure given by ν α ∞ = N (α, 1), such that if G is a standard Gaussian random variable, E cos(Y α,ν α ∞ t ) = E cos(α + G) = cos(α) √ e . We deduce that the invariant probability measures of (2.13) are {N (α, 1) α ∈ R, √ e J α = cos(α)}. Let α ∈ R such that √ e J α = cos(α). We have: ∀t ≥ 0, Θt = J R d dy Ey cos(Y α t )ν α ∞ (dy) = − J √ e e −t sin(α). So, for R(z) > −1,Θ(z) = − J z+1 e −1 2 sin(α) and the equationΘ(z) = 1 has a unique solution z = −Je −1 2 sin(α) − 1. This root is strictly negative if and only if J sin(α) > − √ e. We deduce by Theorem 2.5 that ν α ∞ is locally stable provided that J sin(α) > − √ e. Recall that α √ e = J cos(α). So among all the invariant probability measures of (2.13), the (locally) stable ones are the N (α, 1) with α tan(α) > −1. Notations For x ∈ R d ,= b(Y α+k,ν t )dt + α(Y α+k,ν t )dt + kt(Y α+k,ν t )dt + σdBt,(2.14) where the initial condition satisfies Law(Y α+k,ν 0 ) = ν. Note that Y α+k,ν t is a solution of (2.1) provided that k satisfies the following closure equation: ∀x ∈ R d , ∀t ≥ 0, α(x) + kt(x) = EF (x, Y α+k,ν t ). A key ingredient in the proof of Theorem 2.5 is the notion of Lions derivatives. A function u ∶ P2(R d ) → R is Lions differentiable at ν0 ∈ P2(R d ) if there exists a deterministic function ∂νu(ν0) ∶ R d → R d such that for every random variables Z0, H defined on the same probabilistic space (Ω, F, P) with Law(Z0) = ν0 and E H 2 < ∞, we have for ν = Law(Z0 +H): u(ν) = u(ν0) + E [∂νu(µ0)(Z0) ⋅ H] + o( E H 2 ), as E H 2 goes to zero. We refer to [10] for a detailed presentation of this theory. Let ψ ∈ C 1 (R d ) with ∇ψ ∞ < ∞. We recall that the linear functional u(ν) = R d ψ(y)ν(dy) is Lions differentiable at every ν0 ∈ P2(R d ), with a Lions derivative given by ∂µu(ν0)(y) = ∇ψ(y). An integrated sensibility formula Let α ∶ R d → R d satisfying (2.2). The goal of this Section is to prove the following "integrated sensitivity formula". Proposition 2.10. Let k ∈ C(R+; K) and ν ∈ P1(R d ). Let g ∈ C 2 (R d ) with ∇g ∞ + ∇ 2 g ∞ < ∞. It holds that for all t ≥ 0, Eg(Y α+k,ν t ) − Eg(Y α,ν t ) = t 0 R d [∇yEyg(Y α t−θ ) ⋅ k θ (y)] Law(Y α+k,ν θ )(dy)dθ. Without loss of generality, we can assume that the initial condition ν belongs to P2(R d ). We define for all 0 ≤ θ ≤ t: ∀ν ∈ P2(R d ), u g t,θ (ν) ∶= Eg(Y α,ν t−θ ). By the Markov property, u g t,θ is linear with respect to ν: u g t,θ (ν) = R d Eyg(Y α t−θ )ν(dy). By [29,Th. 7.18], the function y ↦ Eyg(Y α t−θ ) is continuously differentiable with a bounded derivative. So u g t,θ is Lions differentiable with ∂νu g t,θ (ν)(y) = ∇yEyg(Y α t−θ ). Therefore, it suffices to prove that Eg(Y α+k,ν t ) − Eg(Y α,ν t ) = t 0 E ∂νu g t,θ (Law(Y α+k,ν θ ))(Y α+k,ν θ ) ⋅ k θ (Y α+k,ν θ ) dθ. Given θ ≥ 0 and k ∈ C(R+; K), we write for all u ≥ 0: ∀y ∈ R d , k [θ] u (y) ∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ku(y) if u ≤ θ 0 if u > θ (2.16) The proof of Proposition 2.10 is deduced from the following Lemma and from the fundamental theorem of calculus. Lemma 2.11. The function θ ↦ Eg(Y α+k [θ] ,ν t ) is differentiable for all θ ∈ (0, t) and d dθ Eg(Y α+k [θ] ,ν t ) = E∂νu g t,θ (Law(Y α+k,ν θ ))(Y α+k,ν θ ) ⋅ k θ (Y α+k,ν θ ). (2.17) 0 θ θ + δ t Y θ ∼ µ θ Y 2 θ+δ ∼ µ 2 θ+δ Y 1 θ+δ ∼ µ 1 θ+δ time (Y α+k [θ] u ) u∈[0,t] (Y α+k [θ+δ] u ) u∈[0,t] Figure 1 Proof. Fix θ ∈ (0, t) and δ > 0 small enough such that θ + δ ∈ (0, t). We write Y θ ∶= Y α+k,ν θ , Y 1 θ+δ ∶= Y α+k [θ] ,ν θ+δ , Y 2 θ+δ ∶= Y α+k,ν θ+δ µ θ ∶= Law(Y θ ), µ 1 θ+δ ∶= Law(Y 1 θ+δ ), µ 2 θ+δ ∶= Law(Y 2 θ+δ ). The notations are illustrated on Figure 1. We have by the Markov property satisfied by Y at time θ + δ Eg(Y α+k [θ+δ] ,ν t ) − Eg(Y α+k [θ] ,ν t ) = Eg(Y α,µ 2 θ+δ t−(θ+δ) ) − Eg(Y α,µ 1 θ+δ t−(θ+δ) ). By definition of the Lions derivative at the point µ 1 θ+δ we have Eg(Y α,µ 2 θ+δ t−(θ+δ) ) − Eg(Y α,µ 1 θ+δ t−(θ+δ) ) = E∂νu g t,θ+δ (µ 1 θ+δ )(Y 1 θ+δ ) ⋅ (Y 2 θ+δ − Y 1 θ+δ ) + o( E Y 2 θ+δ − Y 1 θ+δ 2 ). (2.18) By Lemma 2.12 (a) below, it holds that o( E Y 2 θ+δ − Y 1 θ+δ 2 ) = o(δ) as δ goes to zero. We now approximate Y 1 θ+δ and Y 2 θ+δ by a one-step Euler scheme: Y 1 θ+δ ≈Ỹ 1 θ+δ ∶= Y θ + α(Y θ )δ + σ(W θ+δ − W θ ) (2.19) Y 2 θ+δ ≈Ỹ 2 θ+δ ∶= Y θ + α(Y θ )δ + k θ (Y θ )δ + σ(W θ+δ − W θ ) Note thatỸ 2 θ+δ −Ỹ 1 θ+δ = k θ (Y θ )δ. These one-step Euler schemes have an error in L 2 norm of size o(δ) (see Lemma 2.12 (c) below), therefore: E Y 1 θ+δ −Ỹ 1 θ+δ 2 + E Y 2 θ+δ −Ỹ 2 θ+δ 2 = o(δ), so (2.18) gives Eg(Y α,µ 2 θ+δ t−(θ+δ) ) − Eg(Y α,µ 1 θ+δ t−(θ+δ) ) = δE ∂νu g t,θ+δ (µ 1 θ+δ )(Y 1 θ+δ ) ⋅ k θ (Y θ ) + o(δ). Finally, one has E ∂νu g t,θ+δ (µ 1 θ+δ )(Y 1 θ+δ ) ⋅ k θ (Y θ ) − E ∂νu g t,θ (µ θ )(Y θ ) ⋅ k θ (Y θ ) ≤ E ∂νu g t,θ+δ (µ 1 θ+δ )(Y 1 θ+δ ) ⋅ k θ (Y θ ) − E ∂νu g t,θ+δ (µ θ )(Y θ ) ⋅ k θ (Y θ ) + E ∂νu g t,θ+δ (µ θ )(Y θ ) ⋅ k θ (Y θ ) − E∂νu g t,θ (µ θ )(Y θ ) ⋅ k θ (Y θ ) =∶ A1 + A2. By Lemma 2.13 (a) there exists a constant C(t) such that A1 ≤ C(t) k θ K E Y 1 θ+δ − Y θ 2 Lem. 2.12 (b) ≤ C(t) √ δ sup θ∈[0,t] k θ K . Let ǫ > 0 be fixed. Lemma 2.13 (b) yields for δ small enough: A2 ≤ ǫ sup θ∈[0,t] k θ K . Altogether, we find that Eg(Y a+k [θ+δ] ,ν t ) − Eg(Y a+k [θ] ,ν t ) = δE∂νu g t,θ (µ θ )(Y θ ) ⋅ k θ (Y θ ) + o(δ). This ends the proof. We used the following classical estimates (the constants depend on α and k): Lemma 2.12. We have, with the notations introduced in the proof of Lemma 2.11, (a) it holds that E Y 2 θ+δ − Y 1 θ+δ 2 ≤ C(t)δ 2 . (b) it holds that E Y 1 θ+δ − Y θ 2 ≤ C(t)δ. (c) the Euler scheme (2.19) satisfies E Y 1 θ+δ −Ỹ 1 θ+δ 2 + E Y 2 θ+δ −Ỹ 2 θ+δ 2 = o(δ 2 ) , as δ goes to zero. We also used the following regularity results on ∂νu g t,s (ν)(y) = ∇yEyg(Y α t−s ). The proofs follow easily from the stochastic representation of y ↦ ∇yEyg(Y α t−s ): in particular this function has a bounded derivative (because ∇g ∞ + ∇ 2 g ∞ < ∞, see [29,Th. 7.18]). Lemma 2.13. It holds that: (a) there exists a constant C(t) such that any square-integrable variables Z, Z ′ , sup 0≤s≤t E ∂νu g t,s (Law(Z))(Z) − ∂ν u g t,s (Law(Z ′ ))(Z ′ ) 2 ≤ C(t)E Z − Z ′ 2 . (b) the function s ↦ ∂νu g t,s (Law(Z))(Z) is continuous: for all ǫ > 0 there exists δ > 0: ∀s, s ′ ∈ [0, t], s − s ′ ≤ δ ⇒ E ∂νu g t,s ′ (Law(Z))(Z) − ∂νu g t,s (Law(Z))(Z) 2 < ǫ. Remark 2.14. It is also possible to prove Proposition 2.10 without using Lions derivatives. Fix t > 0 and define for s ∈ (0, t) (s, y) ↦ φ(s, y) ∶= Eyg(Y α t−s ). Given k ∈ C([0, t]; K), we denote by L α+k θ ψ ∶= (b + α + k θ ) ⋅ ∇ψ + 1 2 d i,j=1 (σσ * ) i,j ∂x i ∂x j ψ, θ ∈ [0, t], the infinitesimal generator associated to Y α+k . It holds that φ ∈ C 1,2 ([0, t) × R d ) with ∂ ∂s φ(s, y) = −L α s φ(s, y). So, by Itô's lemma, Eφ(s, Y α+k,ν s ) = Eφ(0, Y α+k,ν 0 ) − s 0 EL α θ φ(θ, Y α+k,ν θ )dθ + s 0 EL α+k θ φ(θ, Y α+k,ν θ )dθ = Eφ(0, Y α+k,ν 0 ) + s 0 E∇yφ(θ, Y α+k,ν θ ) ⋅ k θ (Y α+k,ν θ )dθ. We used that L α+k θ ψ − L α θ ψ = ∇ψ ⋅ k θ . Using the definition of φ, we find: Eφ(s, Y α+k,ν s ) = Eg(Y α,ν t ) + s 0 R d ∇yEyg(Y α t−θ ) ⋅ k θ (y)L(Y α+k,ν θ )(dy)dθ. Finally, we let s converges to t and find the stated formula. Note that by choosing g = F (x, ⋅), Proposition 2.10 gives: EF (x, Y α+k,ν t ) − EF (x, Y α,ν t ) = t 0 R d ∇yEyF (x, Y α t−θ ) ⋅ k θ (y)Law(Y α+k,ν θ )(dy)dθ. (2.20) Recall that Θt(k) is defined by (2.4). When ν = ν∞ and when k is small, we obtain: EF (x, Y α+k,ν∞ t ) − α(x) ≈ t 0 R d ∇yEyF (x, Y α t−θ ) ⋅ k θ (y)ν∞(dy)dθ = t 0 Θ t−θ (k θ )(x)dθ. This observation is crucially used in the next Section. Control of the non-linear interactions Recall that (X ν t ) denotes the solution of the McKean-Vlasov equation (2.1). For all t ≥ 0 and ν ∈ P1(R d ), we define ϕ ν t (x) ∶= EF (x, Y α,ν t ) − α(x), k ν t (x) ∶= EF (x, X ν t ) − α(x). Recall that Ωt(h) is defined by (2.9). In this Section, we prove that: Proposition 2.15. For all T > 0, there is a constant C T such that for all t ∈ [0, T ], for all x ∈ R d and for all ν ∈ P1(R d ): k ν t (x) − ϕ ν t (x) − t 0 Ωt−s(ϕ ν s )(x)ds ≤ C T (W1(ν, ν∞)) 2 . The first step is to show the following apriori estimate on k ν t : Lemma 2.16. Let T > 0. There exists a constant C T such that ∀ν ∈ P1(R d ), sup t∈[0,T ] k ν t K ≤ C T W1(ν, ν∞). In addition, k ν ∈ C([0, T ]; K). Proof. We have k ν t (x) = R d F (x, y)(Law(X ν t ) − ν∞)(dy). Using Lemma 2.18 below, we deduce that k ν t (x) ≤ C T ∇yF ∞W1(ν, ν∞). Similarly, ∇k ν t (x) ≤ C T ∇ 2 x,y F ∞W1(ν, ν∞). This shows the bound on sup t∈[0,T ] k ν t K . Moreover, k ν t (x) − k ν s (x) = R d F (x, y) (Law(X ν t ) − Law(X ν s )) . In addition, for all T > 0 and ν ∈ P1(R d ), there exists a constant C(T, ν) such that ∀0 ≤ s ≤ t ≤ T, W1(Law(X ν t ), Law(X ν s )) ≤ E X ν t − X ν s ≤ C(T, ν) √ t − s. We deduce that k ν ∈ C([0, T ]; H). Next, using Proposition 2.10, we show that: Lemma 2.17. For all T > 0, there is a constant C T such that for all t ∈ [0, T ], for all x ∈ R d and for all ν ∈ P1(R d ): k ν t (x) − ϕ ν t (x) − t 0 Θt−s(k ν s )(x)ds ≤ C T (W1(ν, ν∞)) 2 . Proof. Using the closure equation (2.15), we have: k ν t (x) = EF (x, Y α+k ν ,ν t ) − α(x) = EF (x, Y α+k ν ,ν t ) − EF (x, Y α,ν t ) + ϕ ν t (x). We apply Proposition 2.10 and obtain: k ν t (x) − ϕ ν t (x) = t 0 R d ∇yEyF (x, Y α t−θ ) ⋅ k ν θ (y)Law(X ν θ )(dy)dθ. We used that Law(Y α+k ν ,ν θ ) = Law(X ν θ ). Let G x t,θ (y) ∶= ∇yEyF (x, Y α t−θ ) ⋅ k ν θ (y). We deduce that: k ν t (x) − ϕ ν t (x) = t 0 Θ t−θ (k ν s )(x)ds + Rt(x), where Rt(x) ∶= t 0 EG x t,θ (X ν θ ) − EG x t,θ (X ν∞ θ ) dθ. Using Lemma 2.16, we deduce for all T > 0, there exists a constant C T , such that ∀x ∈ R d , ∀0 ≤ θ ≤ t ≤ T, ∇G x t,θ ∞ ≤ C T W1(ν, ν∞). Therefore, using Lemma 2.18 below, we deduce that Rt(x) ≤ C T (W1(ν, ν∞)) 2 . By iterating the estimate of Lemma 2.17, we deduce the proof of Proposition 2.15. Proof of Proposition 2.15. For i ∈ N, define: ψ i t (x) ∶= ϕ ν t (x) + t 0 Θ t−θ (ϕ ν θ )(x)dθ + ⋯ + t 0 Θ ⊗i t−θ (ϕ ν θ )(x)dθ. Let D T a constant such that for all k ∈ K, sup t∈[0,T ] Θt(k) ∞ ≤ D T k ∞. By induction, we have: ∀t ∈ [0, T ], Θ ⊗i t (k) ∞ ≤ (D T ) i t i−1 (i − 1)! k ∞. Let C T be the constant of Lemma 2.17. By induction, we deduce that: k ν t − ψ i t − t 0 Θ ⊗(i+1) t−θ (k ν θ )dθ ∞ ≤ C T (W1(ν, ν∞)) 2 1 + D T t + ⋯ + (D T t) i i! . To conclude, it suffices to take the limit i → ∞: ψ i t converges uniformly to ϕ ν t + ∫ t 0 Ω t−θ (ϕ ν s )ds, and ∫ t 0 Θ ⊗(i+1) t−θ (k ν θ ) dθ converges uniformly to zero. This ends the proof. We used the following apriori estimate on the solution of (2.1). Lemma 2.18. Let T > 0. There exists a constant C T such that for all µ1, µ2 ∈ P1(R d ), ∀t ∈ [0, T ], W1(Law(X µ 1 t ), Law(X µ 2 t )) ≤ C T W1(µ1, µ2). Proof. Consider (X µ 1 t , X µ 2 t ) the solutions of (2.1) coupled with the same Brownian motion. The initial conditions (X µ 1 0 , X µ 2 0 ) are chosen such that E X µ 1 0 − X µ 2 0 = W1(µ1, µ2). Let µ 1 t ∶= Law(X µ 1 t ) and µ 2 t ∶= Law(X µ 2 t ). From (2.1) and Assumption 2.1, we have E X µ 1 t − X µ 2 t ≤ E X µ 1 0 − X µ 2 0 + E t 0 b(X µ 1 s ) − b(X µ 2 s ) ds + t 0 R d EF (X µ 1 s , y) − EF (X µ 2 s , y) Law(X µ 1 s )(dy)ds + t 0 R d EF (X µ 2 s , y)(Law(X µ 1 s ) − Law(X µ 2 s ))(dy) ds. The functions F and b are Lipschitz, so there exists a constant L such that E X µ 1 t − X µ 2 t ≤ E X µ 1 0 − X µ 2 0 + L t 0 E X µ 1 s − X µ 2 s ds. By Grönwall's inequality, we deduce that W1(Law(X µ 1 t ), Law(X µ 2 t )) ≤ E X µ 1 t − X µ 2 t ≤ e Lt E X µ 1 0 − X µ 2 0 = e Lt W1(µ1, µ2). Proof of Theorem 2.5 Control of the Wasserstein distance The next step towards the proof of Theorem 2.5 is the following apriori control of the Wasserstein distance between two solutions of SDE driven by slightly different drifts. Recall that κ * > 0 is given by (2.7). Proposition 2.19. There is a constant C > 0 such that for all t ≥ 0 and k ∈ C([0, t]; K), it holds that W1(Law(Y α+k,ν t ), Law(Y α,ν t )) ≤ C t 0 e −κ * (t−θ) k θ ∞dθ. Proof. The proof uses a similar coupling argument that in Proposition 2.10. We consider ∀θ ∈ (0, t), Gt(θ) ∶= W1(Law(Y α+k [θ] t ), Law(Y α t )), where k [θ] is given by (2.16). As in the proof of Proposition 2.10, let µ θ ∶= Law(Y α+k θ ), µ 2 θ+δ ∶= Law(Y α+k θ+δ ), µ 1 θ+δ ∶= Law(Y α+k [θ] θ+δ ). The notations are summarized in Figure 1. Let δ > 0 small enough such that θ + δ ∈ (0, t). By the triangular inequality satisfied by W1, we have Gt(θ + δ) − Gt(θ) ≤ W1(Law(Y α+k [θ+δ] t ), Law(Y α+k [θ] t )) = W1(Law(Y α,µ 2 θ+δ t−(θ+δ) ), Law(Y α,µ 1 θ+δ t−(θ+δ) )) ≤ Ce −κ * (t−(θ+δ)) W1(µ 2 θ+δ , µ 1 θ+δ ). We used (2.7) to obtain the last inequality. By Grönwall's inequality, there exists a constant C such that for all δ < 1: W1(µ 2 θ+δ , µ 1 θ+δ ) ≤ E Y α+k,µ θ δ − Y α,µ θ δ ≤ C θ+δ θ ku ∞du. Therefore, we deduce that ∀θ ∈ (0, t), lim sup δ↓0 1 δ Gt(θ + δ) − Gt(θ) ≤ Ce −κ * (t−θ) k θ ∞. Using that θ ↦ k θ ∞ is uniformly continuous on [0, t], we deduce that the lim sup above is uniform with respect to θ, and so we have Gt(t) ≤ C ∫ t 0 e −κ * (t−θ) k θ ∞dθ, as stated. Proof of Theorem 2.5 We now give the proof of Theorem 2.5. By combining Proposition 2.15, Proposition 2.19, and Proposition 2.6, we obtain Lemma 2.20. Assume that λ ′ > 0, where λ ′ is given by (2.6). Let λ ∈ (0, λ ′ ). There exists a constant C λ such that for all T > 0, there is a constant C T such that for all ν ∈ P1(R d ) and for all t ∈ [0, T ]: W1(Law(X ν t ), ν∞) ≤ C λ e −λt W1(ν, ν∞) + C T (W1(ν, ν∞)) 2 . Importantly, the constant C λ above does not depend on T . Proof. We write W1(Law(X ν t ), ν∞) = W1(Law(Y α+k ν ,ν t ), ν∞) ≤ W1(Law(Y α+k ν ,ν t ), Law(Y α,ν t )) + W1(Law(Y α,ν t ), ν∞) = C t 0 e −κ * (t−θ) k ν θ ∞dθ + C * W1(ν, ν∞)e −κ * t . We used Proposition 2.19 and (2.7). Applying Proposition 2.15, we deduce that t 0 e −κ * (t−θ) k ν θ ∞dθ ≤ t 0 e −κ * (t−θ) φ ν θ ∞ + θ 0 Ω θ−u (φ ν u ) ∞du dθ + C T (W1(ν, ν∞)) 2 . The estimate (2.7) implies that φ ν θ ∞ ≤ C * ∇yF ∞e −κ * θ . Fix λ ∈ (0, λ ′ ). In view of (2.5), (2.10) and Proposition 2.6, the spectral assumption λ ′ > 0 implies that there exists a constant C such that ∀h ∈ H, Ωt(h) ∞ ≤ Ce −λt h H . Therefore, using that λ < κ * , we have W1(ν, ν∞). θ 0 Ω θ−u (φ ν u ) ∞du ≤ C θ 0 e −λ(θ−u) e −κ * u W1(ν, ν∞)du ≤ C λ e −λθ Altogether, we deduce the stated inequality. Finally, the proof of Theorem 2.5 is deduced from Lemma 2.20 by following the argument of [7,Proposition 5.2]. Proof of Theorem 2.5. We choose T large enough such that C λ e −λT ≤ 1 4 . We choose ǫ > 0 small enough such that W1(ν, ν∞) ≤ ǫ ⇒ C T (W1(ν, ν∞)) 2 ≤ 1 4 W1(ν, ν∞Θt(h)(x) = lim ǫ→0 u x t (Law(Z0 + ǫh(Z0))) − α(x) ǫ , Ωt(h)(x) = lim ǫ→0 v x t (Law(Z0 + ǫh(Z0))) − α(x) ǫ . Proof. The first two points follow from the fact that u x t (ν) depends linearly on ν: u x t (ν) = Eg(Z0 + H), with g(y) ∶= EyF (x, Y α t ) . This function g is C 2 and there exists a constant C T such that for all t ∈ [0, T ], for all x ∈ R d , ∇ 2 g ∞ ≤ C T . Therefore: u x t (ν) − α(x) = Eg(Z0 + H) − Eg(Z0) = E 1 0 ∇g(Z0 + θH) ⋅ Hdθ = E∇g(Z0) ⋅ H + E 1 0 (∇g(Z0 + θH) − ∇g(Z0)) ⋅ Hdθ = E∇g(Z0) ⋅ H + O T (E H 2 ). So u x t is Lions differentiable at ν∞ with ∂νu x t (ν∞)(y) = ∇g(y). In addition: E∇g(Z0) ⋅ H = E[∇g(Z0) ⋅ E(H Z0)] = E∇g(Z0) ⋅ h(Z0) = Θt(h)(x). The third point is a direct consequence of Proposition 2.15 and of the inequality: W1(ν, ν∞) 2 ≤ W2(ν, ν∞) 2 ≤ E H 2 . To check the last point, it suffices to show that E[∂νv x t (ν∞)(Z0) ⋅ h(Z0)] = Ωt(h), where ∂νv x t (ν∞)(y) is given by the right-hand side of (2.21). This equality follows by the linearity of h ↦ Ωt(h) and by Fubini's theorem. Static bifurcation analysis: a Green-Kubo formula Our result also provides some information on the number of invariant probability measures of (2.1). In this Section, in addition to Assumption 2.1, we assume that F is bounded sup x,y∈R d F (x, y) < ∞, and that the drift is confining in the sense that: ∃R, β > 0, x − y ≥ R ⇒ (b(x) − b(y)) ⋅ (x − y) ≤ −β x − y 2 . We consider the following open ball of K K0 ∶= {α ∈ K ∶ α K ≤ F ∞ + ∇xF ∞}. Let α ∈ K0. Then Assumption 2.2 holds with constants R, β independent of α. So the SDE (2.14) has a unique invariant probability measure, denoted by ν α ∞ . In addition, the bound (2.7) holds, for some constants C * and κ * which do not depend on α. We consider Ψ ∶ K0 → K0, defined by: ∀α ∈ K0, Ψ(α) ∶= x ↦ R d F (x, y)ν α ∞ (dy). Note that there is a one-to-one correspondence between the invariant probability measures of (2.1) and the fixed-points of Ψ in K0. The following result is a generalization of the Green-Kubo formula [36, Ch. 5]: Proposition 2.23. The function Ψ is Frechet differentiable at every α ∈ K0 and DαΨ(α) ⋅ ǫ = ∞ 0 Θt(ǫ)dt =Θ(ǫ)(0), ǫ, α ∈ K0, where Θt is given by (2.4). Proof. We have for all T ≥ 0, R d F (x, y)(ν α+ǫ ∞ − ν α ∞ )(dy) = EF (x, Y α+ǫ,ν α+ǫ ∞ T ) − EF (x, Y α+ǫ,ν α ∞ T ) + EF (x, Y α+ǫ,ν α ∞ T ) − EF (x, Y α,ν α ∞ T ) =∶ A(x) + B(x). By (2.7), there exist positive constants C, κ * such that A K ≤ Ce −κ * T W1(ν α+ǫ ∞ , ν α ∞ ): this term can be made arbitrarily small by choosing T sufficiently large. In addition, using Proposition 2.10, we have B(x) = T 0 R d [∇yEyF (x, Y α T −θ ) ⋅ ǫ(y)] Law(Y α+ǫ,ν α ∞ θ )(dy)dθ. It follows that B K ≤ C ǫ ∞. Letting T → ∞ proves that Ψ is continuous. By refining the previous argument, we show that this function is Frechet differentiable with the stated derivative. Define G x t (y) ∶= ∇yEyF (x, Y α t ) ⋅ ǫ(y). We have B(x) = ∫ T 0 EG x T −θ (Y α+ǫ,ν α ∞ θ )dθ and, by Girsanov's theorem, provided that ǫ 2 ∞ T < 1, we have: EG x T −θ (Y α+ǫ,ν α ∞ θ ) − EG x T −θ (Y α,ν α ∞ θ ) ≤ C G x T −θ ∞ √ θ ǫ ∞ ≤ Ce −κ * (T −θ) ǫ 2 ∞ √ T . Therefore, we deduce that B(x) − ∞ 0 R d ∇yEyF (x, Y α θ ) ⋅ ǫ(y)ν α ∞ (dy)dθ ≤ C( √ T ǫ 2 ∞ + e −κ * T ). Overall, there exists a constant C such that R d F (x, y)(ν α+ǫ ∞ − ν α ∞ )(dy) − ∞ 0 R d ∇yEyF (x, Y α θ ) ⋅ ǫ(y)ν α ∞ (dy)dθ ≤ C[e −κ * T + ǫ 2 ∞ √ T ]. We choose T = 1 ǫ ∞ and let ǫ ∞ goes to zero: the right-hand term is a o( ǫ ∞). The same estimate holds on the derivative of Ψ with respect to x: R d ∇xF (x, y)(ν α+ǫ ∞ − ν α ∞ )(dy) − ∞ 0 R d ∇yEy∇xF (x, Y α θ ) ⋅ ǫ(y)ν α ∞ (dy)dθ ≤ C[e −κ * T + ǫ 2 ∞ √ T ]. Altogether, we deduce that α ↦ ∫Rd F (⋅, y)ν α ∞ (dy) is Frechet differentiable with the stated derivative. The interpretation of this result is the following: static bifurcations, leading to a change of the number of invariant probability measures of (2.1), occur for parameters satisfying det(I −Θ(0)) = 0. On the other hand, we expect Hopf bifurcations to occur at parameters for which det(I −Θ(iω)) = 0, for some ω > 0. Altogether, this covers the two canonical ways to break the stability condition λ ′ > 0, where λ ′ is given by (2.6). The study of these bifurcations is left to future research. McKean-Vlasov of convolution type on the torus Let β > 0. We consider the following McKean-Vlasov equation on the torus T d ∶= (R 2πZ) d : dX ν t = − T ∇W (X ν t − y)µt(dy)dt + 2β −1 dBt with µt = Law(X ν t ). (3.1) with initial condition Law(X ν 0 ) = ν ∈ P(T d ). Here (Bt) is a Brownian motion on T d . This equation generalizes the Kuramoto model [1,6,38,28], for which d = 1 and W = −κ cos for some constant κ ≥ 0. We refer to [12] for a detailed presentation of examples that fit in the framework of (3.1), as well as a study of the static bifurcations of this equation. In this Section, we study the local stability of the uniform probability measure using the strategy and the tools introduced in Section 2. Main result Write the interaction kernel W ∶ T d → R in Fourier: W (x) = n∈Z dW (n)e in⋅x , x ∈ T d ,(3.2) where n ⋅ x = ∑ d i=1 nixi. Let n 2 = n ⋅ n. The Fourier coefficients of W are given bỹ W (n) = 1 (2π) d T d W (y)e −in⋅y dy, n ∈ Z d . Assumption 3.1. Assume that W ∈ C 3 (T d ) and that ∑ n∈Z d n 2 W (n) < ∞. The uniform probability measure U (dx) ∶= dx (2π) d is an invariant probability measure of (3.1) and Theorem 3.2. In addition to Assumption 3.1, assume that λ ′ ∶= inf n∈Z d ∖{0} n 2 β −1 + R(W (n)) > 0. (3.3) Then U (dx) = dx (2π) d is locally stable: there exists λ ∈ (0, λ ′ ), ǫ > 0 and C > 1 such that for all ν ∈ P(T d ) with W1(ν, U ) < ǫ, it holds that ∀t ≥ 0, W1(Law(X ν t ), U ) ≤ CW1(ν, U )e −λt . Remark 3.3. When the interaction kernel W is even, [12] studies the existence of bifurcations of the invariant probability measures of (3.1), provided that there exists n ∈ Z d ∖ {0} such that β −1 + R(W (n)) = 0. Therefore, criterion (3.3) is sharp: we prove that the uniform measure is stable up to the first bifurcation. Note that we do not require here W to be even. Lemma A.1. Let z ∈ C such that R(z) > −κ. Then det I d − A x * (z) = 0 if and only if z is an eigenvalue of ∇xV(x * , x * ) + ∇yV(x * , x * ). Proof. When det I d − A x * (z) = 0, there exists u ∈ R d ∖ {0} such that u = ∇yV(x * , x * )(zI d − ∇xV(x * , x * )) −1 u. Setting v = (zI d − ∇xV(x * , x * )) −1 u, we have v ≠ 0 and zv = (∇xV(x * , x * ) + ∇yV(x * , x * ))v. So z is an eigenvalue of ∇xV(x * , x * ) + ∇yV(x * , x * ). The converse statement is proved similarly. Now, assume that all the real parts of the eigenvalues of ∇xV(x * , x * ) + ∇yV(x * , x * ) are less than −λ ′ , for some λ ′ ∈ (0, κ). Then, for all R(z) ≥ −λ ′ , det(I − A x * t ) ≠ 0. Applying [30,Th. 4.1], we deduce that ∫ ) = B x * t ⋅ h(x * ) ≤ Ce −λ ′ t h(x * ) = Ce −λ ′ t h H . Therefore, λ ′ > 0. Conversely, if λ ′ > 0 where λ ′ is given by (2.6) holds, then for R(z) > −λ ′ , det(I d − A x * (z)) ≠ 0. So by Lemma A.1, z is not an eigenvalue of ∇xV(x * , x * ) + ∇yV(x * , x * ). We deduce that all the eigenvalues of this matrix have real parts less or equal to −λ ′ . Proposition 2. 4 . 4We have: (a) For all z ∈ D,Θ(z) ∈ L(H) is an Hilbert-Schmidt operator. In particular,Θ(z) is compact. (b) D ∋ z ↦Θ(z) is an analytic operator-valued function, and (I −Θ(z)) −1 exists for all 0 e λ ′ t B x * t dt < ∞. Using (A.1), one deduces that B x * t e λ ′ t < ∞, and so Ωt(h) H = Ωt(h)(x * 2.6), is to compute the L 1 norm of Θt. Recall that Θt L(H) = sup h H ≤1 Θt(h) H .Lemma 2.8. Assume that ∫ ∞ 0 we denote by x its Euclidean norm. Recall that H ∶= L 2 (ν∞) is the Hilbert space of measurable functions h ∶ R d → R d such that We denote by K ∶= W 1,∞ (R d ; R d ) the subspace of H consisting of all bounded and Lipschitz continuous functions k ∶ R d → R d . We equip K with:∀k ∈ K, k K ∶= k ∞ + ∇k ∞. map H to K.Given I a closed interval of R+, we denote by C(I; K) the space of continuous functions from I to K. Let α ∶ R d → R d satisfying (2.2). Let k ∈ C(R+; K).h 2 H ∶= h(x) 2 ν∞(dx) < ∞. Note that, using Assumptions 2.1 and 2.2, the operators Θt and Ωt, defined by (2.4) and (2.10), Consider Y α+k,ν t the solution of the following linear non-homogeneous R d -valued SDE dY α+k,ν t ) . )(d) The function v x t is Lions differentiable at ν∞, with a derivative given by Remark 2.22. In particular, from points 1 and 3, we have for all h ∈ H:∂νv x t (ν∞)(y) = ∇yEyF (x, Y α t ) + t 0 Ωt−s(∇yEyF (⋅, Y α s ))(x)ds. (2.21) Acknowledgments. The author would like to express his gratitude to Etienne Tanré for many suggestions at different steps of this work and hearty encouragement. He also thanks Romain Veltz and René Carmona for their valuable advice. This research has received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 945539 (Human Brain Project SGA3) and was supported by AFOSR FA9550-19-1-0291.Therefore we have, by induction, provided that W1(ν, ν∞) ≤ ǫ:We write t = kT + s for some s ∈ [0, T ). Using Lemma 2.18, there exists a constant C such that W1(Law(X ν t ), ν∞) ≤ C(1 2) k W1(ν, ν∞) ≤ Ce −ct W1(ν, ν∞), where c ∶= log(2)T . This ends the proof of Theorem 2.5.Connections with Lions derivativesIn this Section, we give a probabilistic interpretation of the linear maps Θt(h) and Ωt(h). Let (Ω, F, P) be a probability space. Let Z0, H ∈ L 2 (Ω, F, P) such that Law(Z0) = ν∞. We denote by h ∶ R d → R d a measurable function such that:P(dω)p.s. E[H Z0] = h(Z0).It follows from E H 2 < ∞ and from the Cauchy-Schwarz inequality that h ∈ H. Define for all x ∈ R d , t ≥ 0 and ν ∈ P2(R d ):The function u x t is Lions differentiable at ν∞, with a derivative given byProofTo simplify the notations, we first assume that d = 1. We discuss the case d > 1 afterward, most of the arguments being the same. We write σ ∶= 2β −1 . The proof is divided into the following steps.Step 1. Because ∇W is Lipschitz, the equation (3.1) has a unique path-wise solution satisfying the following apriori estimate:Step 2. We define for ν ∈ P(T), x ∈ T and t ≥ 0:Step 3. We now use that there exists C * > 1 and κ * > 0 such thatWe refer to[35,Prop. 4]. We define for all t ≥ 0, x ∈ T and ν ∈ P(T):where X ν 0 is independent of (Bt) t≥0 and has law ν. Because ∇ 2 W ∞ < ∞, by the preceding result, there exists a constant C > 0 such that:Step 4. Let K ∶= W 1,∞ (T) be the space of bounded and Lipschitz continuous functions from T to R. For k ∈ C(R+; K) and ν ∈ P(T), we consider (Y k,ν t ) the solution of the followingThe integrated sensibility formula of Section 2.4writes, in this context:Step 5. We let for h ∈ L 2 (T) and x ∈ T:Using that Ee inσB t = e − n 2 σ 2 2 t = e − n 2 t β , we find that the Fourier series of Θt(h)(x) is:So Θt is diagonal in the Fourier basis (e inx ) n∈Z and Θt(h)(n) = −n 2W (n)e − n 2 t βh (n). In addition, using h (n) ≤ h ∞ we have:where C0 ∶= ∑ n∈Z n 2 W (n) < ∞.Step 6. We then define Ωt(h) to be the unique solution of the Volterra integral equation:Θt−s(Ωs(h))ds.Again, Ωt is diagonal in the Fourier basis:Let λ ′ be given by (3.3). We have:So, under the condition λ ′ > 0, (Ωt) decays at an exponential rate towards zero.Step 7. Let x ∈ T be fixed. We now apply Step 4 with g(y) ∶= −∇W (x − y), and with kt(y) ∶= k ν t (y), where k ν t is defined in Step 2. Note that with this choice, Y k,ν t = X ν t and soStep 3. Therefore, we have:Using the apriori estimates of Step 2, we deduce that there exists a constant C T such that for all 0 ≤ θ ≤ t ≤ T : ∇yG x t,θ (y) ≤ C T W1(ν, U ). Using Step 1, we conclude that Rt(x) ≤ C T (W1(ν, U )) 2 . To summarize, we have proven that for all T > 0, there exists a constant C T such that for all ν ∈ P(T) and for all t ∈ [0, T ]:Step 8. By iterating the last inequality of Step 7, we obtain that for all T > 0, there exists a constant C T such thatStep 9. We prove that there exists a constant C > 0 such that for all t > 0, for all k ∈ C([0, t]; K) and for all ν ∈ P(T), it holds thatThe proof is similar to the proof of Proposition 2.19; it uses the estimates of Step 3.Step 10. We fix λ ∈ (0, min(κ * , λ ′ )). Using Step 8, Step 6, and Step 3, we deduce that there exists C λ > 0 such that for all T > 0, there is a constant C T such that for all t ∈ [0, T ] and ν ∈ P(T):we have:ByStep 3, we haveByStep 9, we haveAltogether, we deduce that there is a constant C λ such that for all T > 0, there exists C T > 0 such that for all t ∈ [0, T ], for all ν ∈ P(T), we have:The proof of Theorem 3.2 is deduced from this estimate, exactly as we did at the end of Section 2.6. This ends the proof for d = 1.The case d > 1 is similar; the only differences are in the expressions of Θt and Ωt of Steps 5 and 6. Given n ∈ Z d , we denote by P (n) the d × d matrix defined by P (n) = (ninj ) 1≤i,j≤d . We find that for all h ∈ L 2 (T d ; R d ) and for all x ∈ T d ,andThe eigenvalues of P (n) are n 2 (of order 1) and zero (of order d − 1). In addition, it holds that for θ ∈ R, (e θP (n) ) i,j = δ {i=j} + ninj n 2 (e θ n 2 − 1).Therefore, the estimates of Steps 5 and 6 still hold in dimension d > 1. 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[]
[ "Internal dynamics of the z ∼ 0.8 cluster RX J0152.7−1357 ⋆", "Internal dynamics of the z ∼ 0.8 cluster RX J0152.7−1357 ⋆" ]
[ "M Girardi \nDipartimento di Astronomia\nUniversità degli Studi di Trieste\nVia Tiepolo 11I-34100TriesteItaly\n", "R Demarco \nESO-European Observatory\nKarl-Schwarzschild-Str. 285748GarchingGermany\n\nDepartment of Physics & Astronomy\nJohns Hopkins University\n3400 N. Charles Street21218BaltimoreMDUSA\n", "P Rosati \nESO-European Observatory\nKarl-Schwarzschild-Str. 285748GarchingGermany\n", "S Borgani \nDipartimento di Astronomia\nUniversità degli Studi di Trieste\nVia Tiepolo 11I-34100TriesteItaly\n" ]
[ "Dipartimento di Astronomia\nUniversità degli Studi di Trieste\nVia Tiepolo 11I-34100TriesteItaly", "ESO-European Observatory\nKarl-Schwarzschild-Str. 285748GarchingGermany", "Department of Physics & Astronomy\nJohns Hopkins University\n3400 N. Charles Street21218BaltimoreMDUSA", "ESO-European Observatory\nKarl-Schwarzschild-Str. 285748GarchingGermany", "Dipartimento di Astronomia\nUniversità degli Studi di Trieste\nVia Tiepolo 11I-34100TriesteItaly" ]
[]
We present the results from the dynamical analysis of the cluster of galaxies RX J0152.7−1357, which shows a complex structure in its X-ray emission, with two major clumps in the central region, and a third clump in the Eastern region. Our analysis is based on redshift data for 187 galaxies. We find that RX J0152.7−1357 appears as a well isolated peak in the redshift space at z = 0.836, which includes 95 galaxies recognized as cluster members. We compute the line-of-sight velocity dispersion of galaxies, σ V = 1322 +74 −68 km s −1 , which is significantly larger than what is expected in the case of a relaxed cluster with an observed X-ray temperature of 5-6 keV. We find evidence that this cluster is far from dynamical equilibrium, as shown by the non Gaussianity of the velocity distribution, the presence of a velocity gradient and significant substructure. Our analysis shows that the high value of σ V is due to the complex structure of RX J0152.7−1357, i.e. to the presence of three galaxy clumps of different mean velocity. Using optical data we detect a low-velocity clump (with σ V = 300-500 km s −1 ) in the central South-West region and a high-velocity clump (with σ V ∼ 700 km s −1 ) in the Eastern region, well corresponding to the South-West and East peaks detected in the X-ray emission. The central North-East X-ray peak is associated to the main galaxy structure with a velocity which is intermediate between those of the other two clumps and σ V ∼ 900 km s −1 . The mass of the whole system within 2 Mpc is estimated to lie in the range (1.2-2.2)×10 15 M ⊙ , depending on the model adopted to describe the cluster dynamics. Such values are comparable to those of very massive clusters at lower redshifts. Analytic calculations based on the two-body model indicate that the system is most likely bound, currently undergoing merging. In particular, we suggest that the South-West clump is not a small group, but rather the dense cluster-core of a massive cluster, likely destined to survive tidal disruption during the merger.
10.1051/0004-6361:20053232
[ "https://export.arxiv.org/pdf/astro-ph/0506211v1.pdf" ]
17,101,199
astro-ph/0506211
b9f07b475e6df78f41b7f6b506fe1334bbfd7f47
Internal dynamics of the z ∼ 0.8 cluster RX J0152.7−1357 ⋆ March 20, 2022 M Girardi Dipartimento di Astronomia Università degli Studi di Trieste Via Tiepolo 11I-34100TriesteItaly R Demarco ESO-European Observatory Karl-Schwarzschild-Str. 285748GarchingGermany Department of Physics & Astronomy Johns Hopkins University 3400 N. Charles Street21218BaltimoreMDUSA P Rosati ESO-European Observatory Karl-Schwarzschild-Str. 285748GarchingGermany S Borgani Dipartimento di Astronomia Università degli Studi di Trieste Via Tiepolo 11I-34100TriesteItaly Internal dynamics of the z ∼ 0.8 cluster RX J0152.7−1357 ⋆ March 20, 2022Received ¡date¿ / accepted ¡date¿arXiv:astro-ph/0506211v1 9 Jun 2005 Astronomy & Astrophysics manuscript no. mgirardiprep1 (DOI: will be inserted by hand later)Galaxies: clusters: general -Galaxies: clusters: individual: RX J01527−1357 -Galaxies: distances and redshifts -Cosmology: observations We present the results from the dynamical analysis of the cluster of galaxies RX J0152.7−1357, which shows a complex structure in its X-ray emission, with two major clumps in the central region, and a third clump in the Eastern region. Our analysis is based on redshift data for 187 galaxies. We find that RX J0152.7−1357 appears as a well isolated peak in the redshift space at z = 0.836, which includes 95 galaxies recognized as cluster members. We compute the line-of-sight velocity dispersion of galaxies, σ V = 1322 +74 −68 km s −1 , which is significantly larger than what is expected in the case of a relaxed cluster with an observed X-ray temperature of 5-6 keV. We find evidence that this cluster is far from dynamical equilibrium, as shown by the non Gaussianity of the velocity distribution, the presence of a velocity gradient and significant substructure. Our analysis shows that the high value of σ V is due to the complex structure of RX J0152.7−1357, i.e. to the presence of three galaxy clumps of different mean velocity. Using optical data we detect a low-velocity clump (with σ V = 300-500 km s −1 ) in the central South-West region and a high-velocity clump (with σ V ∼ 700 km s −1 ) in the Eastern region, well corresponding to the South-West and East peaks detected in the X-ray emission. The central North-East X-ray peak is associated to the main galaxy structure with a velocity which is intermediate between those of the other two clumps and σ V ∼ 900 km s −1 . The mass of the whole system within 2 Mpc is estimated to lie in the range (1.2-2.2)×10 15 M ⊙ , depending on the model adopted to describe the cluster dynamics. Such values are comparable to those of very massive clusters at lower redshifts. Analytic calculations based on the two-body model indicate that the system is most likely bound, currently undergoing merging. In particular, we suggest that the South-West clump is not a small group, but rather the dense cluster-core of a massive cluster, likely destined to survive tidal disruption during the merger. Introduction Clusters of galaxies are visible tracers of the network of matter in the Universe, marking the high-density regions where filaments of dark matter join together. In the hierarchical scenario of large-scale structure, clusters form via merging of smaller clumps and accretion of material from large scale filaments (e.g., Borgani & Guzzo 2001;Evrard & Gioia 2002). From the observational side, signatures of past merging processes are found in cluster substructure and evidences for ongoing cluster mergers are rapidly accumulating (e.g., Böhringer & Schuecker 2002;Buote 2002;Girardi & Biviano 2002;Evrard 2004). Over the last few years significant progress has been made to extend the above studies from local to distant clusters. Pioneering analyses suggest that no evidence of dynamical evo-Send offprint requests to: M. Girardi ([email protected]) ⋆ Based in part on observations carried out at the European Southern Observatory using the ESO Very Large Telescope on Cerro Paranal (ESO programs 166.A-0701, 69.A-0683, and 72.A-0759) and the ESO New Technology Telescope on Cerro La Silla (ESO program 61.A-0676). lution is shown by the cluster population out to z ∼ 0.3-0.4 (Adami et al. 2000;Girardi & Mezzetti 2001;but cf. Plionis 2002). On the other hand, z > 0.5 clusters have more X-ray substructures than lower-z clusters (Jeltema et al. 2005) and most clusters identified at z ∼ > 0.8 show an elongated, clumpy, or possibly filamentary structure (e.g., Donahue et al. 1998;Gioia et al. 1999; thus suggesting that present observations are approaching the epoch of cluster formation. Our results on RX J0152.7−1357 at z ∼ 0.8 add further insights on this issue. The galaxy cluster RX J0152.7−1357 was discovered in the ROSAT Deep Cluster Survey (RDCS, Rosati et al. 1998) in the ROSAT PSPC field rp60000rn00 observed in January 1992. It was independently discovered in the Wide Angle ROSAT Pointed Survey (WARPS, Ebeling et al. 2000) and reported in the Bright SHARC survey (Romer et al. 2000). It appeared also in the list of X-ray extended sources obtained from Einstein IPC data by Oppenheimer et al. (1997). The BeppoSax observations were used to derive a cluster X-ray bolometric luminosity L X,bol = (22 ± 5) × 10 44 erg s −1 (h = 0.5 and q 0 = 0.5), and a gas temperature kT = 6.46 +1.74 −1.19 keV (Della Ceca et al. 2000). RX J0152.7−1357 is characterized by a complex morphology with at least two cores, both in the optical and X-ray data as recovered by Keck imaging and Beppo-SAX data (Della Ceca et al. 2000). Observations with Chandra also show a complex structure in the intra-cluster medium with the presence in the central cluster region of two peaks in the X-ray emission 95 ′′ apart (North-East: R . 42, Dec.=−13 • 58 ′ 5 ′′ . 52 [J2000.0]), see Maughan et al. (2003). The existence of an Eastern peak was confirmed by spectroscopic VLT data and an independent analysis of the Chandra data by Demarco et al. (2005), who detect it at the > 3σ c.l. in X-rays (see their Fig. 1). Chandra observations gave a gas temperature for the North-East and South-West central X-ray clumps of kT = 5.5 +0.9 −0.8 keV and kT = 5.2 +1.1 −0.9 keV, respectively (Maughan et al. 2003). A complex structure with several clumps is also shown by the gravitational lensing analysis of Jee et al. (2005): in particular, the mass clump A corresponds to the Eastern X-ray peak. A number of evidences suggest that RX J0152.7−1357 may be undergoing a merger: the displacement between peaks of gas distribution and of galaxy/dark matter distribution (Maughan et al. 2003;Jee et al. 2005); the possible presence of a shock front (Maughan et al. 2003); the presence of galaxies showing a very recent star formation episode (Jørgensen et al. 2005); the segregation of star-forming and non star-forming galaxies probably induced by the intra-cluster medium interaction (Homeier et al. 2005). Demarco et al. (2005) have performed an extensive spectroscopic survey of RX J0152.7−1357 based on observations carried out with FORS1 and FORS2 on the ESO Very Large Telescope, obtaining more than 200 redshifts in the cluster field. Their analysis shows that RX J0152.7−1357 is characterized by a large velocity dispersion, ∼ 1600 km s −1 , and indicates a very complex structure. In particular, the galaxy populations inhabiting the regions around the three main X-ray peaks are characterized by different kinematical behaviour, in agreement with a cluster merging scenario. On the basis of Demarco et al. data we further investigate the internal dynamics of RX J0152.7−1357. The spatial and kinematical analysis of member galaxies is a powerful way to detect and measure the amount of substructure, to identify and analyze possible pre-merging clumps or merger remnants (Girardi & Biviano 2002 and refs. therein). This optical information is complementary to X-ray information since galaxies and intra-cluster gas react on different time scales during a merger (see, e.g., numerical simulations by Roettiger et al. 1997;Ricker & Sarazin 2001;Schindler 2002). The paper is organized as follows. We describe member selection and present our results for global properties of RX J0152.7−1357 in Sect. 2. We present our analysis of internal dynamics in Sect. 3. We discuss our results suggesting a tentative picture of the dynamical status of RX J0152.7−1357 in Sect. 4. We summarize our results in Sect. 5. Unless otherwise stated, we give errors at the 68% confidence level (hereafter c.l.) Throughout the paper, we assume a flat cosmology with Ω m = 0.3, Ω Λ = 0.7 and H 0 = 70 km s −1 Mpc −1 . For this cosmological model 1 arcmin corresponds to 458 kpc at the cluster redshift. Member selection and global properties Our data sample consists of the spectroscopic survey of RX J0152.7−1357 presented by Demarco et al. (2005), i.e. 187 galaxies with available redshift (see their Tables 4 and 5). We assume a typical redshift error of 8 × 10 −4 according to the authors prescriptions. The identification of cluster members proceeds in two steps, following a procedure already used for nearby and medium-redshift clusters Girardi et al. 1996;Girardi & Mezzetti 2001). First, we perform the cluster-member selection in velocity space by using only redshift information. We apply the adaptive kernel method (Pisani 1993) to find the significant (> 99% c.l.) peaks in the velocity distribution. This procedure detects RX J0152.7−1357 as a well isolated peak at z = 0.836 assigning 103 galaxies considered as candidate cluster members (see Fig. 1). Out of non-member galaxies, 61 and 23 are foreground and background galaxies, respectively. In particular, a second significant peak of 31 galaxies is shown at z = 0.638 suggesting the presence of a foreground system. All the galaxies assigned to the RX J0152.7−1357 peak are analyzed in the second step, which uses the combination of position and velocity information. We apply the procedure of the "shifting gapper" by Fadda et al. (1996). This procedure rejects galaxies that are too far in velocity from the main body of galaxies and within a fixed bin that shifts along the distance from the cluster center. The procedure is iterated until the number of cluster members converges to a stable value. We use a gap of 1000 km s −1 -in the cluster rest-frame -and a bin of 0.6 Mpc, or large enough to include 15 galaxies. As for the center we consider the position of the biweight center, i.e. we perform the biweight mean-estimator (ROSTAT package; Beers et al. 1990) for ascension and declination separately: this center is positioned between the North-East and South-West X-ray peaks (see § 1). The choice of using either one of the two Xray peaks as cluster center does not affect the final results. The shifting-gapper procedure rejects eight galaxies to give 95 fiducial members. The list of selected members corresponds to that in Table 4 of Demarco et al. (2005), but excluding galaxies #306,509,557,650,895,1146#306,509,557,650,895, ,1239 shows the plot of rest-frame velocity V rf = (cz − cz )/(1 + z ) vs. clustercentric distance R of galaxies in the main redshift peak. Finally, we recompute the biweight center on the 95 cluster members obtaining: R.A.=1 h 52 m 41 s . 669, Dec.=−13 • 57 ′ 58 ′′ . 32 (J2000.0). Unless otherwise stated, we adopt this as cluster center. By applying the biweight estimator to cluster members (Beers et al. 1990), we compute a mean cluster redshift of z = 0.8357 ± 0.0005. We estimate the line-of-sight (LOS) velocity dispersion, σ V , by using the biweight estimator and The redshift galaxy density, as provided by the adaptive-kernel reconstruction method. Unit on the y-axis is normalized to the density of the highest peak applying the cosmological correction and the standard correction for velocity errors (Danese et al. 1980). We obtain σ V = 1322 +74 −68 km s −1 , where errors are estimated through a bootstrap technique. To evaluate the robustness of the σ V estimate we analyze the integral velocity dispersion profile (Fig. 3). The value of σ V (< R) sharply varies in the internal cluster region. A similar behaviour is shown by the mean velocity V(< R) suggesting that a mix of clumps at different redshifts is the likely cause for the high value of the velocity dispersion rather than individual contaminating field-galaxies. A robust value of σ V is reached in the external cluster regions where the profile flattens, as found for most nearby clusters (e.g., Fadda et al. 1996). The question of the presence of substructure is deferred to the following sections. Here we assume that the system is in dynamical equilibrium to compute virial global quantities. Following the prescriptions of Girardi & Mezzetti (2001), we assume for the radius of the quasi-virialized region R vir = 0.17 × σ V /H(z) = 2.0 Mpc (see their eq. 1 after introducing the scaling with H(z), see also eq. 8 of Carlberg et al. 1997 for R 200 ). Thus the cluster is sampled out to a significant region, i.e. R out = 0.82 × R vir . We compute the virial mass (Limber & Mathews 1960; see also, e.g., Girardi et al. 1998) using the data for the N g observed galaxies: M = 3π/2 · σ 2 V R PV /G − C,(1) where C is the surface term correction (The & White 1986), and R PV , equal to two times the (projected) harmonic radius, is: , where V and σ V at a given (projected) radius from the cluster center is estimated by considering all galaxies within that radius. The error bands at the 68% c.l. are shown. In the lower panel, the horizontal line represent X-ray temperature with the respective errors transformed in σ V imposing β spec = 1 (see Sect. 4). R PV = N g (N g − 1)/(Σ i j R −1 i j ),(2) where R i j is the projected distance between two galaxies. The estimate of σ V is generally robust when computed within a large cluster region (see Fig. 3 for RX J0152.7−1357 and Fadda et al. 1996 for other examples). The value of R PV depends on the size of the sampled region and possibly on the quality of the spatial sampling (e.g., whether the cluster is uniformly sampled or not). Here we obtain R PV = (1.45 ± 0.05) Mpc, where the error is obtained via a jacknife procedure. The value of C strongly depends on the radial component of the velocity dispersion at the radius of the sampled region and could be obtained by analyzing the velocity-dispersion profile, although this procedure would require several hundreds of galaxies. We apply the correction obtained in the literature by combining data of many clusters sampled out to about R vir (C/M V ∼ 20%, Carlberg et al. 1997;Girardi et al. 1998). We obtain M(< R out = 1.65 Mpc) = (2.2 ± 0.3) × 10 15 M ⊙ . Calling into question the quality of the spatial sampling, one could use an alternative estimate of R PV on the basis of the knowledge of the galaxy distribution. We assume a King-like distribution, with parameters typical of nearby/medium-redshift clusters: a core radius R C = 1/20 × R vir and a slope-parameter β fit = 0.8, i.e. the volume galaxy density at large radii goes as r −3β f it = r −2.4 (see G98 and Girardi & Mezzetti 2001). We obtain R PV = 1.25 Mpc, with a 25% error, thus in agreement with the above direct estimate. The mass is then M(< R out ) = (1.9 ± 0.5) × 10 15 M ⊙ , in good agreement with our first estimate. We can use the second of the above approaches to obtain the mass within the whole assumed virialized region, which is larger than that sampled by observations, M(< R vir = 2.0 Mpc) = (2.2 ± 0.6) × 10 15 M ⊙ . Dynamical analysis Velocity distribution We analyze the velocity distribution to look for possible deviations from Gaussianity that could provide important signatures of complex dynamics. For the following tests the null hypothesis is that the velocity distribution is a single Gaussian. We base our analysis on shape estimators, i.e. the kurtosis and the skewness. As for the kurtosis, we find K = 2.04±0.49, that indicates a ∼ 2σ departure from a Gaussian distribution (reference value K = 3). In addition, we compute the scaled tail index (S T I), which also measures the shape of a distribution, but is based on order statistics of the dataset instead of its moments (see, e.g., Beers et al. 1991). This estimator, S T I = 0.860, indicates that the tails are underpopulated if the parent population is really a single Gaussian with a c.l. between ∼ 90% and ∼ 95%, (see Table 2 of Bird & Beers 1993). Finally, also the W-test (Shapiro & Wilk 1965) rejects the null hypothesis of a Gaussian parent distribution at the 98% c.l.. Then we investigate the presence of gaps in the distribution, which can be the signature of subclustering. A weighted gap in the space of the ordered velocities is defined as the difference between two contiguous velocities, weighted by the location of these velocities with respect to the middle of the data. We obtain values for these gaps relative to their aver- Table 1. age size, precisely the midmean of the weighted-gap distribution. We look for normalized gaps larger than 2.25 since in random draws of a Gaussian distribution they arise at most in about 3% of the cases, independent of the sample size (Wainer and Schacht 1978; see also Beers et al. 1991). Three significant gaps (2.312, 2.366, 2.395) in the ordered velocity dataset are detected (see Fig. 4). From low to high velocities the dataset is divided in parts containing 39, 29, 3, and 24 galaxies: thus the gaps individuate substantially three main subsets. In order to detect the presence of groups within our velocity dataset we use the Kaye's mixture model (KMM) test (Ashman et al. 1994). The KMM algorithm fits a user-specified number of Gaussian distributions to a dataset and assesses the improvement of that fit over a single Gaussian. In addition, it provides the maximum-likelihood estimate of the unknown n-mode Gaussians and an assignment of objects into groups. KMM is most appropriate in situations where theoretical and/or empirical arguments indicate that a Gaussian model is reasonable. This is valid in the case of cluster velocity distributions, where gravitational interactions drive the system to-ward a Gaussian distribution. However, one of the major uncertainties of this method is the optimal choice of the number of groups for the partition. Moreover, only in mixture models with equal covariance matrices for all components the algorithm converges, while this is not always true for the heteroscedastic case (see Ashman et al. 1994, for further details). Our search for significant gaps suggests the presence of two Gaussians (separated by the two very close second and third gaps at ∼ 252 × 10 3 km s −1 ) or possibly three Gaussians (corresponding to the three main subsets). In the homoscedastic case the KMM algorithm fits a two-group partition by rejecting the single Gaussian at the 97.4% c.l. (as obtained from the likelihood ratio test). The three-group partition is fitted at the 97.9% c.l.. In the heteroscedastic case we use the results of the gap analysis to determine the first guess and we fit two velocity groups around the guess mean-velocities of 249 × 10 3 and 254 × 10 3 km s −1 . The algorithm fits a two-group partition at the 99.4% c.l. Similarly, we fit three velocity groups around the guess mean-velocities of 247 × 10 3 , 250 × 10 3 , and 254 × 10 3 km s −1 to obtain a three-group partition at the 97.2% c.l. The high probability value obtained in the heteroscedastic bimodal case suggests the presence of a main cluster of 76 galaxies (KMM1), with the presence of a high-velocity clump of 19 galaxies (KMM2). In turn, the main cluster can be subdivided in two clumps of 19 and 57 galaxies according to the heteroscedastic trimodal case (KMM1a and KMM1b, respectively). Table 1 lists the kinematical properties of these clumps: the three corresponding Gaussians are displayed in Fig. 4. Fig. 5 shows that the galaxies of the KMM1a group mainly populate the South-West central region of the cluster. This spatial segregation suggests to investigate the velocity field in more detail. Velocity field The cluster velocity field may be influenced by the presence of internal substructures, possible cluster rotation, and the presence of other structures on larger scales, such as nearby clusters, surrounding superclusters, and filaments. Each asymmetry effect could produce a velocity gradient in the cluster velocity field. To investigate the velocity field of RX J0152.7−1357 we divide galaxies in a low-and a high-velocity samples by using the median value of galaxy velocitiesV = 250626 km s −1 and check the difference between the spatial distributions of the two samples. High-and low-velocity galaxies appear segregated in the E-W direction (see Fig. 6). The corresponding spatial distributions are different at the 99.2% c.l. according to the twodimensional Kolmogorov-Smirnov test (hereafter 2DKS-test; see Fasano & Franceschini 1987, as implemented by Press et al. 1992). To estimate the direction of the velocity gradient we perform a multiple linear regression fit to the observed velocities with respect to the galaxy positions in the plane of the sky (see also den Hartog & Katgert 1996; Girardi et al. 1996). We find a position angle on the celestial sphere of PA = 97 • ± 16 • (measured counter-clock-wise from North), i.e. higher-velocity Open symbols and crosses indicate galaxies assigned to KMM1 and KMM2 groups, respectively (see text). Squares and triangles indicate KMM1a and KMM1b groups, respectively. The plot is centered on the cluster center defined in Sect. 2. Three circular regions, corresponding to regions of extended X-ray emission are indicated, too (see Fig. 1 by Demarco et al. 2005). galaxies lie in the East-South-East region of the cluster, in agreement with the visual impression of galaxy distribution in Fig. 6. To assess the significance of this velocity gradient we perform 1000 Monte Carlo simulations by randomly shuffling the galaxy velocities and for each simulation we determine the coefficient of multiple determination (RC 2 , see e.g., NAG Fortran Workstation Handbook 1986). We define the significance of the velocity gradient as the fraction of times in which the RC 2 of the simulated data is smaller than the observed RC 2 . We find that the velocity gradient is significant at the 98.3% c.l.. We also analyze the central cluster region using 22 galaxies within a radius of 0.4 Mpc. This choice allows us to include the position of both X-ray peaks and exclude the East region populated by higher-velocity galaxies only. We find a very significant (99.5%) position angle of PA = 59 •+28 • −25 • , i.e. highervelocity galaxies lie in the direction of the North-East X-ray clump. 3D substructure and detection of subclumps The existence of correlations between positions and velocities of cluster galaxies is a footprint of real substructures. Here we combine velocity and position information to compute the ∆statistics devised by Dressler & Schectman (1988). This test is sensitive to spatially compact subsystems that have either an average velocity that differs from the cluster mean, or a ve- Beers et al. (1990) for samples with N g ≥ 15 and with N g < 15 galaxies, respectively (see also Girardi et al. 1993). locity dispersion that differs from the global one, or both. We find ∆ = 154 for the value of the parameter which gives the cumulative deviation of the local kinematical parameters (velocity and velocity dispersion) from the global cluster parameters. To compute the significance of substructure we run 1000 Monte Carlo simulations, randomly shuffling the galaxy velocities, and obtain a value of > 99.9%. This technique also provides information on the positions of substructures. Fig. 7 shows the distribution on the sky of all galaxies, each marked by a circle: the larger the circle, the larger the deviation δ i of the local parameters from the global cluster parameters, i.e. the higher the evidence for substructure. A clump of galaxies with low velocity is the likely cause of large values of δ i in the region which lies closely at South-West of the cluster center, i.e. in correspondence of the South-West X-ray peak. The other possible substructure, populated by high-velocity galaxies, lies in the Eastern region. To assign galaxies to the 3D-subclumps, we resort to the technique developed by Biviano et al. (2002), who used the individual δ i -values of the Dressler & Schectman method. The critical point is to determine the value of δ i that optimally separates between internal and external substructures. To this aim we consider the δ i -values of all 1000 Monte Carlo simulations already used to determine the significance of the substructure (see above). The resulting distribution of δ i is compared to the observed one finding a difference of 99.8% c.l. according to the Fig. 6. Spatial distribution on the sky of the 95 member galaxies: the larger the triangle, the smaller is the radial velocity. Open and solid triangles indicate galaxies with velocity lower and higher than the median cluster velocity, respectively. The plot is centered on the cluster center. The big and the small arrows indicate the position angle of the cluster gradient as measured over the whole cluster and in internal regions, respectively. The three circles correspond to the regions of extended X-ray emission. KS-test. The "simulated" distribution is normalized to produce the observed number of galaxies and compared to the observed distribution in Fig. 8: the latter shows a tail at large values. This tail is populated by galaxies that presumably are in substructures. For the selection of galaxies within substructures we choose the value of δ * = 3.35, since only after the rejection of the values δ i > δ * , the observed and simulated distributions are no longer distinguishable according to the KS-test. With this choice, 14 galaxies of the cluster are assigned to substructures: six to the central South-West clump (DS-SW*) and eight to the East clump (DS-E*), see Fig. 9. The velocity dispersions computed for these structures, σ V ≃ 300 km s −1 and ≃ 650 km s −1 for DS-S* and DS-E* clumps, respectively, are likely to be considered as lower limits since our analysis does not guarantee the detection of all substructure members. We consider also a more relaxed criteria, by selecting galaxies with δ i > 3 as suggested by the histogram of Fig. 8: Table 1 shows that the results for the South-West clump (DS-SW vs. DS-SW*) are very robust, while only one additional galaxy in the Eastern clump (DS-E vs. DS-E*) leads to an increase of 200 km s −1 in the velocity dispersion. We also consider the remaining 76 galaxies of the main structure (DS-M). DS-M does not contain significant structure according to the Dressler-Schectman test. However, since we cannot exclude a residual contamination from substructure members, its value of velocity dispersion σ V ∼ 1300 km s −1 is likely to be an upper limit. The Dressler-Schectman results superseed those of the KMM test. Again, there is the presence of a low-velocity clump and now its South-West position is better defined by the detection of the DS-SW clump. The presence of a highvelocity clump is confirmed and located at the East by the detection of the DS-E clump. Moreover, the location of DS clumps well coincide with X-ray peaks of extended emissions. Analysis of X-ray -centered clumps The good spatial agreement between detected galaxy clumps and peaks of X-ray emission prompts us to analyze the profiles of mean velocity and velocity dispersion of galaxy systems corresponding to the South-West, East, and North-East X-ray peaks, i.e. using the position of the X-ray peaks as system-centers (see Figs. 10, Fig. 11, and Fig. 12, respectively). This allows an independent analysis of the individual galaxy clumps. An increase of the velocity-dispersion profile in their central regions might be due to dynamical friction and galaxy merging (e.g., Menci & Fusco-Femiano 1996;Girardi et al. 1998;Biviano & Katgert 2004), or simply induced by the presence of interlopers or of a secondary clump (e.g., Girardi et al. 1996). The latter hypothesis can be investigated by looking at the behaviour of the mean velocity profile. Figs. 10, 11, and 12 show velocity-dispersion and mean-velocity profiles, and regions likely not contaminated by other clumps and thus reliable for kinematical analysis. Detailed results of this analysis are included in Table 1 where the clumps are named as SW, E, and NE. The analysis of the South-West central region has indicated the presence of a low-velocity clump with a low velocitydispersion (of 300-400 km s −1 according to DS-SW and KMM1a results). Fig. 10 shows how the velocity-dispersion increases with the distance from the South-West X-ray peak. The mean-velocity shows a sharp change very close to the Xray peak, at ∼ 0.2 Mpc. This suggests a strong contamination of galaxies from other structures. We consider two possible uncontaminated regions: one within 0.2 Mpc, where we find σ V ≃ 500 km s −1 , and one within 0.18 Mpc, where we find σ V ≃ 300 km s −1 . Such a sharp change of σ V is induced just by the rejection of two galaxies, one of which has an anomalously high velocity. The value of σ V for the SW-clump is further analyzed in Sect. 3.5 and discussed in Sect. 4.1. The Dressler-Schectman analysis of the East region has indicated the presence of a high-velocity clump with a velocity dispersion of about 600-800 km s −1 . By choosing the X-ray peak as center (Fig. 11), the mean velocity changes at ∼ 0.4 − 0.5 Mpc from the X-ray peak. Inside this region, we obtain σ V ∼ 700 km s −1 for the E-clump Fig. 12 refers to the region around the North-East X-ray peak. The main mass clump is located in this same position, according to the gravitational lensing analysis (Jee et al. 2005). We have shown that this region is mostly populated by galaxies having velocities intermediate between those of the above clumps, and likely forms a high velocity dispersion structure (i.e., KMM1b clump in Sect. 3.1, and DS-M system in Sect. 3.3). Fig. 12 shows an increase of the integral velocity- dispersion profile at about 0.4 Mpc from the X-ray peak, and a corresponding sharp change in the mean velocity. Moreover, galaxies of both DS-SW and DS-E substructures lie beyond 0.4 Mpc from the North-East X-ray peak. Thus, the value σ V ≃ 900 km s −1 , computed within 0.4 Mpc , should be reliable. The three X-ray clumps differ from each other in mean velocities at a c.l. > 99%, according to the means-test (e.g., Press et al. 1992). Assuming that each of the three galaxy clumps is a system in dynamical equilibrium, for each clump we compute the virial radius and the mass contained inside with the same procedure adopted in Sect. 2 (see Table 2). The large uncertainties associated to the mass values are due to poor number statistics. Spectral-type segregation We check for possible spectral-type segregation of galaxies, both in position and in velocity space, by using the classification of Demarco et al. (2005, see their Table 4), i.e. passive galaxies (k), galaxies with significant Balmer lines -likely post-starbursts (k+a/a+k) and galaxies with relevant emission lines (e/k+a+ [OII]). The sample of cluster members contains 56, 7, and 32 passive, post-starburst, and emission-line galaxies, respectively. Figure 13 shows the spatial distribution of galaxies of different types. As already noted by Demarco et al., emission-line galaxies avoid the regions of the subclumps (see also Homeier The dashed vertical line indicates the radius of the extended Xray emission, as defined by Demarco et al. (2005). Top panel: rest-frame velocity vs. projected distance from the clump center: squares and crosses indicate the DS-SW and DS-E as in Fig. 9. Differential (big circles) and integral (small points) mean velocity and LOS velocity-dispersion profiles are shown in middle and bottom panels, respectively. For the differential profiles are plotted: a) the values for eight annuli from the center of the clumps, each of 0.2 Mpc (heavy symbols); the current values of each ten galaxies (faint symbols). For the integral profiles, the mean and dispersion at a given (projected) radius from the clump-center is estimated by considering all galaxies within that radius. The error bands at the 68% c.l. are also shown. et al. 2005). The same behaviour is shown by post-starburst galaxies. When comparing spatial distributions of passive (k) and "active" (k+a/a+k/e/k+a+[OII]) galaxies we find a very strong difference: > 99.99%, according to the 2DKS-test. As for the velocity distributions, no difference is found between passive and "active" galaxies, according to the KS-tests. Moreover, mean velocities and velocity dispersions of the two populations (see Table 1) do not significantly differ according to the means-and F-test (e.g., Press et al. 1992). This suggests that our sample of member galaxies is not significantly contaminated by interlopers. In fact, possible field galaxies would preferably contaminate the sample of "active" galaxies causing a difference in the kinematical properties with respect to the sample of passive galaxies, e.g., enhancing the velocity dispersion or changing the mean velocity. Finally, we perform again the analysis of mean velocity and velocity-dispersion profiles of Sect. 3.4, by considering passive galaxies only. We draw different conclusions only for the SW- clump. Fig. 14 shows that the mean velocity now changes only at ∼ 0.3 Mpc from the South-West X-ray peak and the velocity dispersion does not increase any longer in the central region. Within ∼ 0.3 Mpc, we compute for the SW-clump a value of 321 +132 −59 km s −1 , in good agreement with the lower estimate of σ V obtained in Sect. 3.4 (see Table 1). Table 1). b Using the σ V computed within 0.18 Mpc (see Table 1). Fig. 13. Projected distribution of the 95 member galaxies. Circles, squares, and crosses indicate passive, post-starburst, and emission-lines galaxies, respectively. The three clumps analyzed in Sect. 3.4 are indicated by the three circles and correspond to the regions likely not contaminated by galaxies of other clumps, with radii corresponding to vertical lines in Figs. 10, 11, and 12. Discussion Out of 187 galaxies with available redshift we assign 95 members to RX J0152.7−1357. This galaxy selection is more restrictive than made by Demarco et al. (2005: 102 members giving a velocity dispersion of ∼ 1600 km s −1 ) due to our analysis of position and velocity combined together. In particular, we reject a small group of galaxies at z = 0.864-0.867. In spite of this more restrictive member selection, the value we obtain for the LOS velocity dispersion is still rather high, σ V = 1322 +74 on the L X,bol -σ V plane is consistent with the relation provided by Borgani et al. (1999) for moderately distant clusters and by Wu et al. (1999) for local clusters. As for the σ V -T X relation, assuming the density-energy equipartition between gas and galaxies, i.e. β spec = 1 (e.g., Girardi et al. 1996Girardi et al. , 1998Xue & Wu 2000), where β spec = σ 2 V /(kT/µm p ) with µ = 0.58 the mean molecular weight and m p the proton mass, our value of σ V corresponds to kT = 10.6 +1.2 −1.1 keV. This value is more than 2σ higher than the X-ray temperature determined from BeppoSAX observations (Della Ceca et al. 2000) and more than 3σ higher than those of the North-East and South-West X-ray systems as determined from Chandra data (Maughan et al. 2003;Huo et al. 2004). This suggests a strong departure from dynamical equilibrium and, in fact, we find evidence for non-Gaussianity of the velocity distribution, presence of a velocity gradient and significant substructure. We find no kinematical difference between passive and "active" galaxy populations. This suggests that our sample of member galaxies is not significantly contaminated by interlopers. In fact, possible field galaxies would preferably contaminate the sample of "active" galaxies causing a difference in the kinematical properties with respect to the sample of passive galaxies, e.g., enhancing the velocity dispersion or changing the mean velocity. Instead, our analysis shows that the high value of σ V is due to the complex structure of this system, i.e. to the presence of three galaxy clumps of different mean-velocity. Using optical data only we detect the low-velocity SW-clump in the central regions and the high-velocity E-clump, which lie close to the South-West and East peaks detected by the X-ray analysis. The North-East X-ray peak is then associated to the main galaxy structure. In particular, the high relative velocity between the NE-and SW-clumps, V r = (V NE − V SW )/(1 + z ) = 1531 km s −1 , explains the high value of σ V measured in the central cluster region and the presence of a velocity gradient there (see Figs. 3 and 6), while the global velocity gradient is induced by the presence of the high-velocity E-clump in external cluster regions. The presence of the three galaxy clumps was already suggested by Demarco et al. (2005) from the inspection of the velocity distribution in relation to the spatial location of galaxies. Moreover, the NE-, SW-, and E-clumps correspond to three clumps in the mass distribution as obtained from the weak lensing analysis (Jee et al. 2005: C, F, and A subclumps, respectively). As for the mass of the whole cluster, from the global analysis of Sect. 2 we obtain M(< 2.0 Mpc) = (2.2 ± 0.6) × 10 15 M ⊙ . Since the system is not virialized, but likely bound (see the discussion below), this estimate might overestimate the mass even by a factor two. Adding the mass estimates of each clump within its virial radius (see Table 2, Sect. 3.4), we obtain M = 1.2 +0.5 −0.3 × 10 15 M ⊙ : this estimate should be considered as a lower value within 2.0 Mpc , since it does not consider other small clumps or isolated infalling, bound galaxies, as well the likely possibility that the three clumps extend outside the virial radius. Thus, we conclude that the cluster mass within 2 Mpc lies in the range of (1.2 − 2.2) × 10 15 M ⊙ , which is typical to that of very massive clusters (e.g., Girardi et al. 1998;Girardi & Mezzetti 2001). Our mass estimate is consistent with that of Maughan et al. (2003) of (1.1 ± 0.2) × 10 15 M ⊙ , based on Chandra X-ray analysis and considering only the two central clumps within 1.4 Mpc. To compare with results from weak lensing analyses we also compute the projected mass, by considering the global cluster geometry as formed by the three clumps at the cluster redshift. Each clump is described by the King-like mass distribution (see Sect. 2) or, alternatively, by a NFW profile where the mass-dependent concentration parameter is taken from Navarro et al. (1997) and rescaled by the factor 1+z (Bullock et al. 2001;Dolag et al. 2004). The mass distribution of each clump is truncated at one virial radius or, alternatively, at two virial radii. In the following we indicate the range of our results. We find the projected mass within 1 Mpc from the center of the main clump (NE-clump) to be (9 − 15)×10 14 M ⊙ , higher than that, 5×10 14 M ⊙ , of Jee et al. (2005), but in agreement with the value ∼ > 1 × 10 15 M ⊙ of Huo et al. (2004, see their Figure 10). Indeed, both Huo et al. and Jee et al. compare their weak lensing results with an isothermal sphere with σ V =900-1000 km s −1 , in agreement with the value of σ V that we measure for the main galaxy clump. However, the weaklensing mass lies above or below the isothermal sphere mass for Jee et al. and Huo et al., respectively. Individual clumps Our estimate of σ V for the NE-clump well agrees with that of Demarco et al. (2005) and corresponds to kT = 4.8 +1.8 −0.4 keV, in agreement with the observed gas temperature of ∼ 6 KeV (Maughan et al. 2003;Tozzi et al. 2003;Huo et al. 2004). Similarly, our mass estimate, M(< R vir = 1.35 Mpc) = 7.0 +2.9 −2.1 ×10 14 M ⊙ , well agrees with the X-ray mass by Maughan et al. [2003, M(< 1.4 Mpc) = 7.0 +1.7 −1.5 ×10 14 M ⊙ ]. To compare our results with other studies we rescale M(< R vir ) at their radii by using the King-like profile or, alternatively, the NFW profile (see above). In the following we give the two values obtained from the rescaling, reliable with a 30% lowererror and a 40% upper-error, as derived from the estimate of M(< R vir ). Our estimates well agree with those of other studies: M(< 0.43 Mpc) =(1.9-2.6)×10 14 M ⊙ , cf. with (2.5 ± 0.9)×10 14 M ⊙ by Demarco et al. (2005), based on galaxy dynamics; M(r < 65 ′′ = 0.496 Mpc) =(2.3-2.9)×10 14 M ⊙ , cf. with (3 ± 1)×10 14 M ⊙ by Joy et al. (2001), based on the Sunyaev-Zeldovich effect; M(r < 0.753 Mpc)=(3.8-4.3)×10 14 M ⊙ , cf. with (2.66 ± 0.77)×10 14 M ⊙ by Ettori et al. (2004), based on Chandra X-ray data; M(r < 1 Mpc) =(5.1-5.4)×10 14 M ⊙ , cf. with ∼ 5×10 14 M ⊙ by Huo et al. (2004), based on Chandra Xray data. As for the SW-clump, the results in the literature are not yet clear. In fact, the X-ray temperature suggests that the North-East and the South-West clumps are similar in mass (Maughan et al. 2003;Huo et al. 2004), while both the optical analysis by Demarco et al. (2005) and the weak lensing analysis by Jee et al. (2005) find that the South-West clump is about half massive than the North-East clump. Our analysis of the σ V -profile gives two alternative values for σ V : the larger value is consistent with that found by Demarco et al. (2005, cf . σ V = 503 +439 −96 km s −1 vs. their 737 ± 126 km s −1 ) and with the observed gas temperatures of 5-6 keV (Maughan et al. 2003;Huo et al. 2004), while the lower estimate, σ V = 301 +122 −107 , is significantly different. This uncertainty is due to the fact that the σ V profile increases in central regions (see Fig.10) and thus the σ V estimate strongly depends on the considered region. Demarco et al. considered a region (based on X-ray data) larger than our region (based on kinematical data). Our analysis of passive galaxies also gives a small value, σ V ∼ 300 km s −1 , thus suggesting two alternative hypothesis: 1) high values of σ V are due to galaxy-contamination by other clumps, so that the SW-clump should be considered as a very small group, 2) we are dealing with a very relaxed core hosted in a highσ V , massive cluster. The second hypothesis is consistent with the observations of nearby clusters where σ V of the subsample of bright central elliptical galaxies is lower than σ V of the whole cluster (Biviano & Katgert 2004), a phenomenon possibly due to dynamical friction and galaxy merging (e.g., Menci & Fusco-Femiano 1996). Only a deeper galaxy sample would allow us to better trace and separate the North-East and the South-West systems and thus discriminate between the two hypotheses. However, the SW-clump appears to be so dense of galaxies that we are inclined to believe in the detection of a cluster-core. In this case, we note that: a) our mass estimate would be an underestimate of the global mass of the Southern cluster; b) our results would be reconciled with high values of gas temperature and X-ray luminosity (Maughan et al. 2003;Tozzi et al. 2003;Huo et al. 2004). As for the Eastern clump, the level of X-ray emission in the Chandra image is much lower than those of the North-East or the South-West clumps (see Fig. 1 of Demarco et al. 2005). On the contrary, its gravitational-lensing mass is comparable to that of the South-West clump (see A and F clumps by Jee et al. 2005), and our estimate of velocity dispersion is typical of a massive cluster, σ V ∼ 700 km s −1 . This discrepancy with X-ray luminosity suggests that this galaxy system is far from being virialized, maybe elongated along the LOS (thus giving a high σ V and a high projected lensing mass), with the gas component not very dense. In particular, the Eastern X-ray peak might be associated to a small group embedded in a largescale structure filament connecting to the cluster from the East-South-East region, which is populated by higher velocity -maybe more distant -galaxies (see Fig. 6). In the case of a bound, but non virialized structure, we might have overestimated the mass even by a factor two. We finally compare the projected mass of the three clumps within a radius of 20 ′′ with the results from weak lensing by Jee et al. (2005) Analytic calculations of the dynamical status Here, we investigate the relative dynamics of the NE-and SWclumps in the central cluster region using different analytic approaches which are based on an energy integral formalism in the framework of locally flat spacetime and Newtonian gravity (e.g., Beers et al. 1982). The values of the relevant observable quantities for the two-clump system are: the relative LOS velocity, V r = 1531 km s −1 , the projected linear distance between the two clumps, D = 0.66 Mpc, the mass of the system obtained adding the masses of the two clumps each within its virial radius, M sys = 8.2 +3.6 −2.2 ×10 14 M ⊙ . First, we consider the Newtonian criterion for gravitational binding stated in terms of the observables as V 2 r D ≤ 2GM sys sin 2 αcosα, where α is the projection angle between the plane of the sky and the line connecting the centers of two clumps. The faint curve in Fig. 15 separates the bound and unbound regions according to the Newtonian criterion (above and below the curve, respectively). Considering the value of M sys , the NE+SW system is bound between 30 • and 77 • : the corresponding probability, computed considering the solid angles (i.e., 77 30 cos α dα), is 47%. We also consider the implemented criterion V 2 r D ≤ 2GMsin 2 α V cosα D , which introduces different angles for projection of distance and velocity, not assuming strictly radial motion between the clumps (Hughes et al. 1995). We obtain a binding probability of 44%. Then, we apply the analytical two-body model introduced by Beers et al. (1982) and Thompson (1982; see also Lubin et al. 1998 for a recent application). This model assumes radial orbits for the clumps, with no shear or net rotation of the system. Furthermore, the clumps are assumed to start their evolution at time t 0 = 0 with separation d 0 = 0, and are moving apart or coming together for the first time in their history, i.e. we are assuming that we are seeing the cluster prior to merging. The bimodal model solution gives the total system mass M sys as a function of α (e.g., Gregory & Thompson 1984). Fig. 15 Fig. 15. System mass vs. projection angle for bound and unbound solutions of the two-body model applied to the NE-and SW-clumps (solid and dotted curves, respectively, see text). The thin curve separates the bound and unbound regions according to the Newtonian criterion (above and below the curve, respectively). The horizontal lines give the observational values of the mass system and its 1σ error bands. compares the bimodal-model solutions with the observed mass of the system, which is the most uncertain observational parameter. The present bound outgoing solutions (i.e. expanding), BO, are clearly inconsistent with the observed mass. The possible solutions span these cases: the bound and present incoming solution (i.e. collapsing), BI a and BI b , and the unboundoutgoing solution, UO. For the incoming case there are two solutions because of the ambiguity in the projection angle α. We compute the probabilities associated to each solution assuming that the region of M sys values between 1σ bands is equally probable for individual solutions: P BIa ∼ 65%, P BIb ∼ 35%, P UO < 1 × 10 −4 %. There are several limitations to characterize the dynamics of the central region of RX J0152.7−1357 using these models. For instance, possible underestimates of the masses, e.g., if the clumps extend outside the virial radius or if the SWclump is only the core of the South-West system (see above), lead to binding probabilities larger than those computed above. Moreover, the models do not take into accounts the mass distribution in the clumps when the separation of the clumps is comparable with their size (i.e. at small α) and do not consider the possible effect of the E-clump. Finally, the two-body model breaks down in a regime where the NE-and SW-clumps are already strongly interacting, as suggested by several evidences: the displacement between peaks of gas distribution and of galaxy/dark matter distribution (Maughan et al. 2003;Huo et al. 2004); Jee et al. 2005); the possible presence of a shock front (Maughan et al. 2003); the presence of galaxies showing a very recent star formation episode (Jørgensen et al. 2005); the segregation of star-forming and non star-forming galaxies probably induced by the interaction with the intra-cluster medium (Homeier et al. 2005). Looking at galaxies only we cannot discriminate between a pre-or post-merging phase since the galaxy component is very robust against mergers, e.g., two clusters can pass through one another without destroying the individual optical components (e.g., White & Fabian 1995;Roettiger et al. 1997). Note, for instance, that the properties of the SW-clump resemble those of cluster-cores destined to survive tidal disruption during the merger: size comparable to the cluster core and mass ∼ < 0.05× cluster mass (see González-Casado et al. 1994). These cores will be detectable in the host cluster as a substructure for a long time. Since the gas component shows two well distinct entities in the central cluster regions, we presume that the merging is not too advanced, i.e. well before the coalescence. Under the assumption that the two central clumps are already very close, we apply the above dynamical models to the system made of the [(NE+SW)+E] clumps, too. The values of the relevant observable quantities are: V r = 1401 km s −1 , D = 1.09 Mpc, and M sys = 11.7 +4.7 −2.6 ×10 14 M ⊙ . We obtain that the binding probabilities are 48%, and 45%, according to the Newtonian criterion and its implementation, respectively; while the two-body model gives a probability > 99.9% for the bound incoming solution. Summary & conclusions We present the results of the dynamical analysis of the cluster of galaxies RX J0152.7−1357, one of the most massive structures known at z > 0.8. The X-ray emission is known to have two clumps in the central regions, and a third clump ∼ 1 Mpc to the East. Our analysis is based on velocities and positions of member galaxies taken from the extensive spectroscopic survey performed by Demarco et al. (2005), i.e. 187 galaxies having redshift in the cluster region. We find that RX J0152.7−1357 appears as a well isolated peak in the redshift space at z = 0.836, and select 95 cluster members. We compute a value for the LOS velocity dispersion of galaxies, σ V = 1322 +74 −68 km s −1 , much larger than expected for a relaxed cluster with an observed X-ray temperature of ∼ 5 − 6 keV. We find evidence that this cluster is far from dynamical equilibrium, as shown by: -the non Gaussianity of the velocity distribution according to different tests, at the 90-98% c.l., as well as the presence of significant velocity gaps; -the correlation between velocities and positions of galaxies at the > 99% c.l., and the presence of a velocity gradient; -the presence of significant substructures at the > 99.9% c.l.. To detect and analyze possible subsystems we used different methods. -By applying the KMM method we find that a two-clumps, and likely a three-clumps partition of the velocity distribution is significantly better than a single Gaussian to describe the velocity distribution; in particular, the galaxies of KMM1a group are mainly located in the South-West central region. -By combining positions and velocities in the Dressler & Schectman statistics we detect two substructures, well corresponding in location to the South-West and East X-ray peaks, in addition to the main cluster component identified with the North-East X-ray peak. -Taking advantage of X-ray peak determination, we analyze the three galaxy clumps centered in these peaks through the profiles of mean velocity and velocity dispersion. This analysis allows us to estimate the clump region that is likely not contaminated by galaxies of other clumps and to evaluate the kinematical properties. In summary, our analysis shows that the high value of σ V is due to the complex structure of RX J0152.7−1357, i.e. to the presence of three galaxy clumps of different meanvelocity. Using optical data we detect a low-velocity clump (with σ V =300-500 km s −1 ) in the central South-West region and a high-velocity clump (with σ V ≃700 km s −1 ) in the Eastern region, nicely matching the position of the South-West and East peaks detected in the X-ray emission. The central North-East X-ray peak is associated to the main galaxy structure having intermediate velocity and σ V ∼ 900 km s −1 . The three clumps differ from each other in mean velocities at a c.l. > 99% (relative LOS velocities are > 1000 km s −1 ). The mass of the whole system within 2 Mpc is estimated to be (1.2-2.2)×10 15 M ⊙ , where the upper and lower limits come from the virial analysis of the cluster as a whole and from the sum of virial masses of the three individual clumps, respectively. Analytic calculations, based on the two-body model, indicate that the system is most likely bound, destined to merge. In particular, we suggest that the South-West clump is not a small group, but rather the dense core of a massive cluster, able to survive tidal disruption during the merger. In conclusion, RX J0152.7−1357 reveals a very complex structure, with several clumps likely destined to merge in a very massive cluster. Our results lend further support to the picture that massive clusters at z > 0.8 are dynamically complex and, therefore, likely to be young. This indicates that we are approaching the epoch at which such massive structures take shapes from the evolution of the cosmic web. On-going extensive spectroscopic surveys of such systems at z ∼ 1 and beyond, combined with detailed analyses of their gaseus and dark matter components (now possible with weak lensing analysis of HST-ACS data; Jee et al. 2005;Lombardi et al. 2005), will shed new light on cluster formation processes. Fig. 1 . 1Fig. 1. The redshift galaxy density, as provided by the adaptive-kernel reconstruction method. Unit on the y-axis is normalized to the density of the highest peak Fig. 2 . 2Galaxies in the main peak of Fig. 1. Left panel: restframe velocity vs. projected clustercentric distance; the application of the "shifting gapper" method rejects the galaxies indicated by open squares. Right panel: velocity distribution of all and member galaxies (dotted and solid histograms, respectively). Fig. 3 . 3Integrated mean velocity and LOS velocity-dispersion profiles (upper and lower panel, respectively) Fig. 4 . 4Velocity distribution of radial velocities for the 95 cluster members. Bottom panel: stripe density plot where arrows indicate the position of significant gaps. The first gap lies between 249757 and 249997 km s −1 , the second between 251886 and 252185 km s −1 , and the third between 252305 and 252635 km s −1 . Top panel: velocity histogram with a binning of 500 km s −1 with superimposed the three Gaussians corresponding to KMM1a, KMM1b, and KMM2 in Fig. 5 . 5Spatial distribution on the sky of the 95 member galaxies. Fig. 7 . 7Spatial distribution of cluster members, each marked by a circle: the larger the circle, the larger is the deviation δ i of the local parameters from the global cluster parameters, i.e. there is more evidence for substructure (according to the Dressler & Schectman test, see text). The plot is centered on the cluster center. Fig. 8 . 8The distribution of δ i deviations of the Dressler-Schectman analysis for the 95 member galaxies. The solid line represents the observations, the dashed line the distribution for the galaxies of simulated clusters, normalized to the observed number. Fig. 9 . 9Spatial distribution on the sky of the 95 member galaxies. Squares and crosses indicate galaxies assigned to the South-West and East clumps detected by the Dressler-Schectman analysis, respectively (DS-SW and DS-E): thick symbols indicate DS-SW* and DS-E*. Triangles indicate the remaining galaxies of the main system (DS-M). Fig. 10 . 10Kinematical profiles of the SW clump obtained assuming the X-ray peak as center. The vertical line indicates the region likely not contaminated from other clumps (see Sect. 3.4). Fig. 11 . 11The same as inFig. 10, but referring to the Eclump. Note: here the annuli for the differential profiles are 0.45 Mpc each. Fig. 12 . 12The same as inFig. 10, but referring to the central NE-clump. −1 , and lies in the high-tail of the σ V -distribution of nearby/medium redshift clusters (see, e.g.,Fadda et al. 1996;Mazure et al. 1996;Girardi & Mezzetti 2001). The position Fig. 14 . 14The same as inFig. 10, but considering passive galaxies only. Here the annuli for the differential profiles are 0.3 Mpceach. . The resulting values for projected masses of the NE-, SW-, and E-clumps lie in the ranges (1.6 − 2.3)×10 14 M ⊙ , (0.5 − 0.7)×10 14 M ⊙ , and (1.0 − 1.5)×10 14 M ⊙ , all values somewhat higher than those reported in Table2 of Jee et al. for clumps C, F, and A, respectively. Table 1 . 1Results of Kinematical analysis a We use the biweigth and the gapper estimators bySample N g < V > σ a V km s −1 km s −1 Whole system 95 250530 ± 135 1322 +74 −68 KMM partitions KMM1 76 249730 ± 124 1080 +113 −53 KMM2 19 253634 ± 50 210 +31 −22 KMM1a 19 247290 ± 97 408 +67 −50 KMM1b 57 250498 ± 102 768 +97 −50 Dressler − Schectman structures DS − SW * 6 248338 ± 146 318 +96 −36 DS − E * 8 253488 ± 241 645 +263 −110 DS − SW 10 248613 ± 107 317 +81 −47 DS − E 9 253429 ± 298 848 +330 −183 DS − M 76 250510 ± 149 1293 +94 −69 Properties of X − ray − centered clumps SW − clump(< 0.2Mpc) 10 248535 ± 172 503 +439 −96 SW − clump(< 0.18Mpc) 8 248713 ± 121 301 +122 −107 E − clump(< 0.4Mpc) 7 253506 ± 304 710 +287 −117 NE − clump(< 0.4Mpc) 15 251346 ± 241 888 +152 −75 Segregation analysis passive galaxies 56 250313 ± 171 1268 +114 −81 "active" galaxies 39 250803 ± 226 1410 +139 −125 SW − clump(passive gals)(< 0.3Mpc) 10 248488 ± 110 321 +132 −59 Table 2 . 2Virial mass estimatesSample R vir M(< R vir ) Mpc 10 14 × M ⊙ Whole system 2.0 22 ± 6 SW − clump a 0.8 1.2 +2.1 −0.5 SW − clump b 0.5 0.27 +0.23 −0.22 E − clump 1.1 3.5 +3.0 −1.5 NE − clump 1.3 7.0 +2.9 −2.1 a Using the σ V computed within 0.2 Mpc (see Acknowledgements. 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[]
[ "Don't Forget Your ABC's: Evaluating the State-of-the-Art in Chat-Oriented Dialogue Systems", "Don't Forget Your ABC's: Evaluating the State-of-the-Art in Chat-Oriented Dialogue Systems" ]
[ "Sarah E Finch \nDepartment of Computer Science\nEmory University Atlanta\nGAUSA\n", "James D Finch [email protected] \nDepartment of Computer Science\nEmory University Atlanta\nGAUSA\n", "Jinho D Choi [email protected] \nDepartment of Computer Science\nEmory University Atlanta\nGAUSA\n" ]
[ "Department of Computer Science\nEmory University Atlanta\nGAUSA", "Department of Computer Science\nEmory University Atlanta\nGAUSA", "Department of Computer Science\nEmory University Atlanta\nGAUSA" ]
[]
Despite tremendous advancements in dialogue systems, stable evaluation still requires human judgments producing notoriously high-variance metrics due to their inherent subjectivity. Moreover, methods and labels in dialogue evaluation are not fully standardized, especially for opendomain chats, with a lack of work to compare and assess the validity of those approaches. The use of inconsistent evaluation can misinform the performance of a dialogue system, which becomes a major hurdle to enhance it. Thus, a dimensional evaluation of chat-oriented opendomain dialogue systems that reliably measures several aspects of dialogue capabilities is desired. This paper presents a novel human evaluation method to estimate the rates of many dialogue system behaviors. Our method is used to evaluate four state-of-the-art open-domain dialogue systems and compared with existing approaches. The analysis demonstrates that our behavior method is more suitable than alternative Likert-style or comparative approaches for dimensional evaluation of these systems.A detailed validation of human evaluation meth-ods, including likert scales and pairwise comparisons (Section 7).A comprehensive evaluation of four MTOD chatbots using validated metrics (Section 8).By presenting a detailed picture of MTOD chatbot performance and standard methods to evaluate them, we aid future work's efforts to further understand and improve human-computer interaction. Our evaluation platform, analyses, and data are available at https://github.com/emorynlp/ ChatEvaluationPlatform.ChatbotsTo evaluate the strengths and weaknesses of MTOD models, we select the chatbots for our study using a arXiv:2212.09180v2 [cs.CL] 26 May 2023 two-stage process: (1) a literature review to identify chatbot candidates, and (2) a pilot evaluation to select the final set of bots for our full study.Literature Review To promote diversity among the selected chatbots, we focus our review on four popular themes of the human-computer chat: (1) Knowledge-grounded chat, (2) Empathetic chat, (3) Self-consistent chat, and(4)General open-domain chat with large pre-training resources like Reddit. Candidate chatbots are selected from each theme using the following criteria:1. The bot must demonstrate state-of-the-art performance in a task related to the theme. 1 2. The implementation must be provided. 2 3. The response latency of the bot must be <10 seconds using modern GPU hardware.
10.48550/arxiv.2212.09180
[ "https://export.arxiv.org/pdf/2212.09180v2.pdf" ]
254,854,340
2212.09180
a0808c59391481c5718139583ec6c29ed59983f7
Don't Forget Your ABC's: Evaluating the State-of-the-Art in Chat-Oriented Dialogue Systems Sarah E Finch Department of Computer Science Emory University Atlanta GAUSA James D Finch [email protected] Department of Computer Science Emory University Atlanta GAUSA Jinho D Choi [email protected] Department of Computer Science Emory University Atlanta GAUSA Don't Forget Your ABC's: Evaluating the State-of-the-Art in Chat-Oriented Dialogue Systems Despite tremendous advancements in dialogue systems, stable evaluation still requires human judgments producing notoriously high-variance metrics due to their inherent subjectivity. Moreover, methods and labels in dialogue evaluation are not fully standardized, especially for opendomain chats, with a lack of work to compare and assess the validity of those approaches. The use of inconsistent evaluation can misinform the performance of a dialogue system, which becomes a major hurdle to enhance it. Thus, a dimensional evaluation of chat-oriented opendomain dialogue systems that reliably measures several aspects of dialogue capabilities is desired. This paper presents a novel human evaluation method to estimate the rates of many dialogue system behaviors. Our method is used to evaluate four state-of-the-art open-domain dialogue systems and compared with existing approaches. The analysis demonstrates that our behavior method is more suitable than alternative Likert-style or comparative approaches for dimensional evaluation of these systems.A detailed validation of human evaluation meth-ods, including likert scales and pairwise comparisons (Section 7).A comprehensive evaluation of four MTOD chatbots using validated metrics (Section 8).By presenting a detailed picture of MTOD chatbot performance and standard methods to evaluate them, we aid future work's efforts to further understand and improve human-computer interaction. Our evaluation platform, analyses, and data are available at https://github.com/emorynlp/ ChatEvaluationPlatform.ChatbotsTo evaluate the strengths and weaknesses of MTOD models, we select the chatbots for our study using a arXiv:2212.09180v2 [cs.CL] 26 May 2023 two-stage process: (1) a literature review to identify chatbot candidates, and (2) a pilot evaluation to select the final set of bots for our full study.Literature Review To promote diversity among the selected chatbots, we focus our review on four popular themes of the human-computer chat: (1) Knowledge-grounded chat, (2) Empathetic chat, (3) Self-consistent chat, and(4)General open-domain chat with large pre-training resources like Reddit. Candidate chatbots are selected from each theme using the following criteria:1. The bot must demonstrate state-of-the-art performance in a task related to the theme. 1 2. The implementation must be provided. 2 3. The response latency of the bot must be <10 seconds using modern GPU hardware. Introduction Recent work in human-computer chat has made remarkable progress. Multi-turn open-domain (MTOD) models are capable of holding engaging conversations with humans (Roller et al., 2021;Adiwardana et al., 2020). However, there remain a number of challenges facing MTOD chatbots such as hallucinations , commonsense violations , and consistency issues (Nie et al., 2021). A significant obstacle for research that addresses these challenges is the difficulty in formulating an appropriate evaluation methodology due to the inherent subjectivity in determining chat quality (van Miltenburg et al., 2021). Since existing automatic evaluation metrics have been shown to be biased measures of chat quality (Liu et al., 2016;Sai et al., 2019; Contributed equally to this work as first authors. et al., 2022), evaluation using human judgments is standard, although the type of human judgments varies widely across works . Overall, there are few works comparing and assessing the validity of various human evaluation methods. The result of this gap in the literature is that the relative sensitivity, interpretability, and importance of the metrics used to evaluate chat models are not well understood. A dimensional approach for evaluating chat models that measures different aspects of chat quality would surely aid progress (van Miltenburg et al., 2021). However, to our knowledge, no work has investigated the coverage of a comprehensive set of evaluation metrics. Consequently, existing chat model evaluation results provide an incomplete picture of the strengths and weaknesses of MTOD chatbots. This paper addresses these limitations of previous work through the following three contributions: 1. A novel, dimensional human evaluation method that measures the rate of chatbot behaviors impacting chat quality (Section 5). This review yields the 6 chatbot candidates in Table 1: Blender-Decode (Nie et al., 2021), Blender2 , BART-FiD-RAG , Emora , DukeNet , and CEM (Sabour et al., 2022). Appendix A presents details of our literature review and selection process. Model Theme N Q Pass Blender-Decode Consistency 10 4.1 ✓ Blender2 General 10 3.8 ✓ BART-FiD-RAG Knowledge 10 3.5 ✓ Emora General 10 3.3 ✓ DukeNet Knowledge 9 1.9 ✗ CEM Empathy 12 1.1 ✗ Table 1: The pilot results for 6 bots, showing the theme of the approach (Theme), the number of collected conversations (N), and the avg. dialogue-level Likert quality rating (Q). Pass denotes which models passed the verification criteria and were included in the full study. Chatbot Selection A pilot evaluation using the 6 chatbot candidates is conducted in order to verify the multi-turn dialogue capability of the chatbot candidates. Appendix B provides details on the implementations of each chatbot candidate. 10 students majoring in Computer Science or Linguistics are invited to interact with randomly assigned chatbots in 3-5 text-based conversations, 3 each of which consisted of 30 turns. 4 At the end of each conversation, students are asked to rate the quality from 1 (least) to 5 (most). Based on the pilot results (Table 1), DukeNet and CEM are excluded from our full study because they are unable to hold satisfying multi-turn conversations, despite their reasonable single-turn response generation capabilities. Appendix C shows example dialogues from these systems. Conversation Collection The conversation dataset used for the full study is collected using human interactors in a text-based conversation setting. 46 undergraduates are recruited as interactors. Each interactor is compensated with a $5 gift card for every 6 conversations, and allowed to complete up to 18 conversations. Conversations are collected remotely using Par-lAI's interactive web interface, and links to the web interface are sent to each interactor with instructions to be completed within 2 weeks. For each link, the interactor completes two conversations with a random pair of chatbots, for a minimum of 30 turns per conversation. We impose a similar open-ended, topic-free chatting environment to Adiwardana et al. (2020). Interactors are asked to rate 8 dimensions (Table 3) of each conversation after its completion on a 1-5 Likert scale, and to select the higher-quality conversation along the same 8 dimensions after each conversation-pair (ties allowed). Our final conversation dataset includes 400 human-bot dialogues (100 dialogues per chatbot), averaging 30.3 turns per dialogue (11.3 tokens per user turn). Evaluation Methods For a comprehensive evaluation of MOTD chatbots, a robust dimensional evaluation of their chat capabilities is crucial (van Miltenburg et al., 2021). To have confidence that any evaluation metric yields useful information, its interpretability and sensitivity require validation. In addition, it is important to verify that each evaluation metric provides distinct information relative to the others. Several previous works propose sets of evaluation metrics that could be used for a dimensional evaluation but with insufficient analyses to validate them. present an exhaustive set of metrics based on a literature survey of human evaluation methods, but do not quantitatively validate its interpretability, sensitivity, or per-metric distinctness. Mehri and Eskenazi (2020a) present a set of Likert metrics and analyze their relationship to overall dialogue quality, but do not validate the sensitivity or distinctness of the individual metrics. Mehri and Eskenazi (2020b) present 5 Likert metrics and evaluate their coverage with respect to explaining single response quality, but do not validate their sensitivity or distinctness. Similarly, some works look to identify common chatbot errors. Sanguinetti et al. (2020) and Higashinaka et al. (2021) present error taxonomies empirically grounded by error analyses, but do not present distinctness or sensitivity results for their error categories. See and Manning (2021) identify errors for one dialogue model and analyze the impact of each error on overall quality but do not attempt to verify the generalizability of their results. Furthermore, various works propose novel evaluation methods with varying degrees of validation of the reliability and effectiveness of such methods. Deriu et al. (2020) present Spot the Bot, a pairwise evaluation approach that uses survival analysis to rank bots based on self-chats, but do not directly compare to alternative methodologies other than for cost. Sedoc and Ungar (2020) apply Item-Response Theory (IRT) (Lord and Novick, 2008) to pairwise comparison dialogue evaluation, by using a latent variable Bayesian model to estimate both the ability of the evaluated systems and the informativeness of inputs in the static evaluation set. Their analysis of the utility of IRT for dialogue evaluation does not include comparisons to existing approaches or a dimensional focus since they exclusively consider overall response quality. Ji et al. (2022) propose a continuous-scale method for evaluating multi-turn dialogue systems with quality control measures for mitigating artifacts from human annotators. They validate their proposed method on various dialogue dimensions using replication studies, a sensitivity analysis, and a correlation analysis between dimensions, although they explicitly acknowledge that their set of dimensions is not intended to be comprehensive. Phy et al. (2020) assert 3 dimensions (understandability, sensibleness, and likability) are sufficient for capturing the quality of a dialogue and validate their claims using agreement, correlation analysis, and distinctness analysis on human annotations of their dimensions, although they are not applied to multi-turn dialogues. Two studies, Li et al. (2019a) and Smith et al. (2022), compare pairwise comparison and Likert evaluation methods via a sensitivity analysis. However, neither of them target a high-coverage set of dimensional metrics, as their studies were limited to 4 and 3 metrics respectively. Lee et al. (2020) also investigates pairwise evaluation using the Chat-Eval platform. However, this is not a multi-turn evaluation setup and it does not target a dimensional analysis since the comparisons are based exclusively on the overall quality of the responses. inter-annotator agreement, Sensitivity: validates metric sensitivity via statistical testing, Importance: relates evaluation metrics to overall dialogue quality, Distinctness: analyzes whether metrics provide distinct information about quality. M C P A S I D Overall, the relative validity of human evaluation metrics requires further investigation before a comprehensive and reliable dimensional evaluation of human-computer chat is achieved. Table 2 summarizes the goals and contributions of the previous evaluation works. Our study addresses all existing gaps by conducting a detailed validation study of 4 different human evaluation methods and a wide range of fine-grained metrics. Selected Methods Four human evaluation methods are chosen for our study. Since MTOD chat model evaluation is our goal, any domain-or approach-specific methods or single-response evaluation methods providing chatbots with a specific context are excluded. 5 We also focus on external human evaluation methods, where human evaluators judge conversations they do not participate in. Three of the selected methods represent popular approaches: Dialogue Likert, Turn Likert, and Comparative. The fourth method, ABC-Eval, is our novel evaluation approach. Label Dialogue Turn Comparative Likert Likert (2020)), henceforth referred to using their abbreviations and colors. Consistency Con d Con t Con c Emotion Emo d Emo t Emo c Understanding Engagingness Eng d Eng t Eng c Grammaticality Gra d Gra t Gra c Informativeness Inf d Inf t Inf c Quality Qua d Qua t Qua c Proactivity Pro d Pro t Pro c Relevance Rel d Rel t Rel c Dialogue Likert Annotators provide dialoguelevel ratings from 1 (least) to 5 (most) for the 8 labels shown in Table 3. We use the dimension set proposed in which results from a detailed survey of characteristics used in chat evaluation and has better coverage than alternatives like the set used in ACUTE-Eval (Li et al., 2019a). Bot-level metrics are calculated as the mean rating across all bot dialogues. Turn Likert Annotators provide turn-level ratings on the same scale and labels as those used for Dialogue Likert. The dialogue-level metric is measured as the mean rating of a single dialogue's turns. The bot-level metric is calculated as the mean rating of all turns in all bot dialogues. Comparative Annotators select the dialogue in which chatbot responses better fit a label definition from a side-by-side pair of dialogues, also using the labels in Table 3. A "neither" option is allowed, only for cases where the evaluator cannot distinguish which dialogue was a better fit. Bot-level metrics are calculated as bot pair win/tie/loss proportions between pairing of their dialogues. Behavior Classification: ABC-Eval Annotators provide binary labels on the turn-level indicating the presence or absence of a particular chat characteristic. The included chat characteristics are defined in Table 4. Dialogue-level metrics are calculated as the proportion of turns that display the characteristic of the dialogue. Bot-level metrics are calculated as the proportions of turns that display the characteristic over all bot dialogues. ABC-Eval is described in detail next in Section 5. ABC-Eval Design We hypothesize that binary turn-level behavior labels provide more reliable and informative metrics for quantifying fine-grained aspects of chat quality than alternative approaches such as Likert or Comparative scoring. Our novel method, the Annotation of Behaviors in Chat Evaluation (ABC-Eval), is developed in three stages: (1) collecting a set of behavior label candidates, (2) developing and piloting our annotation instructions and procedure, and (3) selecting a subset of behavior labels based on the validation study results in Section 7. Collecting Behavior Label Candidates Based on a review of recent work in chat-oriented dialogue modeling and evaluation, we identify characteristics of chatbot responses relevant to conversation quality. These characteristics include those presented as error cases, evaluation metrics, or desirable response features. We then curate binarized definitions of these characteristics to create an initial set of behavior label candidates, which are revised through an iterative piloting and development process. Due to its high coverage of error categories, Higashinaka et al. (2021) is the primary source of inspiration for many of our behavior labels. However, we improve upon their presented taxonomies by considering additional labels based on characteristics of chat presented by other work, and by further refining their error categories to improve average Inter-Annotator Agreement (Section 7.1). Table 4 presents the final set and definitions of the 16 candidate behavior labels used in our full study, along with selected works from our review that inspired their inclusion. Appendix D details in full our development process. Annotation Procedure The ABC-Eval procedure includes 16 binary behavior labels divided between 8 independent annotation tasks ( Preference Info Pre b The response expresses the bot's preferences, wishes, or values. 10, 11 Life Info Lif b The response shares information about the bot's life or experiences. Empathetic Emp b The response shows an understanding and reacts appropriately to someone's emotions. 11, 12, 13 Lack of Empathy !Emp b The bot misunderstands or reacts inappropriately to someone's emotions. Commonsense !Com b The response misunderstands or contradicts common knowledge. 2, 14, 15, 16 Contradiction Fact Usage Fac b The response accurately incorporates encyclopedic or expert knowledge. 1, 2, 11, 17, 18, Fact Contradiction !Fac b The response hallucinates or inaccurately presents encyclopedic or expert knowledge. 19, 20 Self Contradiction !Sel b The bot contradicts something it said earlier in the dialogue. 2, 3, 6, 20, 21, Partner Contradiction !Par b The bot contradicts or misremembers something the user said earlier in the dialogue. Redundant Red b The response inappropriately repeats information presented earlier in the dialogue. 22, 23 Ignore Ign b The response ignores what the user just said. 1, 2, 3, 6, 24 Irrelevant !Rel b The response interrupts the current topic of discussion by presenting unrelated information. Follow-up Fol b The response explores, elaborates on, or asks about the ideas shared in the previous turn. Topic Switch Top b The response introduces a new topic of conversation. chatbot turn in a given human-chatbot conversation with each behavior label. Evaluators complete these tasks using a custom web application based on the ParlAI evaluation interface (Appendix G). Training and Screening To improve annotation consistency and detect poorly performing evaluators, we develop automated training sessions each annotation task inspired by van Miltenburg et al. (2021). Each session consists of 3 conversations that evaluators annotate using an identical procedure and web interface to the corresponding task. The 3 conversations used for each session are handcrafted by the authors to represent a variety of positive and negative examples of the behavior labels for the corresponding task (Appendix D). The gold annotations for each training conversation are hidden from evaluators during the annotation; however, after completing each training conversation, any disagreements between the evaluator's annotations and gold labels are displayed along with an explanation to help the evaluator improve. We use the evaluator's performance on the third conversation of each training session to screen evaluators, where performance is measured by the number of turns where their annotations disagree with gold labels. Evaluators are eligible to complete the work on a task if they make mistakes on fewer than 2 turns for the antisociality and uninterpretability tasks, or on fewer than 3 turns for the other 6 tasks. Evaluation Study Our full study consists of the collection of 40 labels per conversation. This collection was split into 18 independent evaluation tasks as follows: • 8 ABC-Eval tasks, each composed of 1 to 4 labels as denoted by groupings in Table 4 • 1 Dialogue Likert task, composed of all 8 labels from Table 3 completed in random order • 8 Turn Likert tasks, each composed of 1 label from Table 3 • 1 Comparative task, composed of all 8 labels from Table 3 completed in random order The 18 evaluation tasks are posted on SurgeHQ's annotation platform 6 to be completed by dedicated remote workers (Surgers) with experience in NLP annotation. Each time an evaluator connects to one of our tasks, they are assigned a randomly selected conversation to annotate. We are allocated a group of 125 Surgers, chosen by a SurgeHQ employee based on high annotation performance on past projects. Evaluators are compensated per annotated conversation per task, at an estimated rate of $20/hr 7 . We allow evaluators to annotate up to 60 conversations per task. We measure the reliability of interpreting each metric's annotation instructions by calculating IAA using our set of 100 double-annotated conversations (Figure 1). High agreement between annotators demonstrates that different people can reliably come to the same conclusions about how a metric's definition applies to each chatbot response. Our results suggest that the definitions of most ABC-Eval metrics can be interpreted more reliably than the definitions of most Dialogue Likert, Turn Likert, and Dialogue Comparison metrics. Likertstyle and comparison-style annotations appear to have similar interpretability, although Qua c was a notable exception that produced higher agreement than Qua d . Importance The importance of each metric is estimated by a predictive validity analysis that measures the extent, to which the metric can predict conversation quality ( Figure 2). We use Qua d and Qua c from interactors that participated in the conversations (Section 3) to avoid cases where the same evaluator produced the quality label and explanatory metric. 8 Only 192 of our 200 dialogue pairs were evaluated with Comparative labels due to a collection mistake 9 Bias-corrected and accelerated confidence intervals with k =10,000 Monte Carlo case resamples. 10 !Soc b and !Int b 's confidence intervals are largely due to a low rate of positive examples (see Figure 4). The predictive validity of each metric was measured by fitting univariate linear or logistic regression models to predict Qua d or Qua c , respectively. Qua c was represented as a binary encoding, where 0 and 1 represent choosing the first and second conversation, respectively. We excluded any conversation pairs in which the interactor could not distinguish a difference in quality between conversations, and fitted models on the remaining set of 184 conversations. To use non-comparative predictors for predicting Qua c , the difference in metric value between each pair of conversations was used. Our results suggest that dialogue quality is substantially related to emotional understanding metrics (Emo, Emp b , !Emp b ), relevance-related metrics (Rel, !Rel b , Ign b ), and consistency metrics (Con, !Sel b , Red b , !P ar b ). Within these metric groupings, ABC-Eval metrics were overall more predictive of quality than their Likert or comparative analogs, while comparative metrics were least predictive of quality. Chatbots' ability to express knowledge (Inf , F ac b , !F ac b Lif b , P ref b ) was an overall poor predictor of quality; however, commonsense knowledge errors (!Com b ) was one of the strongest predictors. Sensitivity We investigate the sensitivity of each metric using two analyses. First, we use the fitness of the univariate regression models described in the previous section as one source of evidence for metric sensitivity, since a metric must be sufficiently sensitive in order to distinguish conversations of low and high quality. Second, we follow Li et al. (2019a) and run hypothesis tests to count the number of statistically significant differences each metric is able to detect between the 6 pairings of our 4 chatbots ( downsample the conversations used for hypothesis testing to 32 conversations per bot for our Dialogue Likert, Turn Likert, and ABC-Eval metrics to match the 32 conversation-pairs per bot-pair produced by our Comparative evaluation. α !Soc b !Com b Fac b Emp b Fol b Ign b !Fac b !Rel b !Emp b Lif b !Par b Pre b Red b !Sel b Top b !Int b Con t Emo t Our results show that the Likert evaluations were more sensitive than the Comparative evaluation for most labels. ABC-Eval metrics have a wide range of sensitivity, with some ABC-Eval metrics appearing to be more sensitive analogs of similar likert metrics. For example, the results suggest that !Sel b and Red b are more sensitive than Con, that F ac b and !F ac b are more sensitive than Inf , and that Emp b and !Emp b are more sensitive than Emo. On the other hand, the likert-style Rel metric shows similar or slightly superior sensitivity compared to the analogous Ign and !Rel behavior metrics. Coverage & Distinctness We investigate the coverage and distinctness of our metrics via incremental validity analysis. For this analysis, we perform backwards stepwise regression that determines (1) the ability of an evaluation method as a whole to explain conversation quality, and (2) whether each metric contributes distinct information about quality above and beyond other metrics (Figure 3). Specifically, we fit a multivariate regression model for each of our 4 evaluation methods. These models are fit similarly to those presented in Section 7.2, but include all non-quality metrics within an evaluation method as predictors. Then, we remove predictors from each model one at a time based on a beam search (k=100) of which removed predictor results in the smallest decrease in model fitness (adjusted R 2 or adjusted pseudo-R 2 ). We perform this stepwise regression analysis twice to predict both Qua d and Qua c given by interactors, similar to our analysis in Section 7.2. Our results suggest that ABC-Eval has overall better coverage than other evaluation methods for explaining conversation quality. Furthermore, most ABC-Eval metrics that have a strong relationship with conversation quality appear to be appropriately distinct in the information they provide, espe- cially !Emp b , !Sel b , Red b , !Rel b , Emp b , !Com b , and Ign b . Similar distinctness can also be seen in Turn Likert metrics, whereas dialogue-level metrics show relatively low distinctness. Final ABC-Eval Metrics Given the results of our metric analysis, we select the final set of ABC-Eval metrics bolded in Table 4. In our analyses, this final set had better interpretability (Section 7.1), a wider coverage of distinct characteristics of chat that impact quality (Section 7.2 and Section 7.4), and overall higher measurement sensitivity (Section 7.3) than alternative evaluation methods. Furthermore, the fi- ABC-Eval Chatbot Evaluation To evaluate the strengths and weaknesses of our 4 selected chatbots, we present results for the 400 collected conversations across all ABC-Eval metrics ( Figure 4 and Figure 5), Likert Dialogue metrics ( Figure 6), Likert Turn metrics (Figure 8), and Comparative metrics (Figure 7). We focus our discussion on the final set of ABC-Eval metrics since they performed best in our metric analysis. The results highlight the notable recent progress in human-computer chat, as the vast majority of chatbot turns are interpretable, relevant responses to the dialogue context. Less than 1% of re-11 See Appendix F for detailed cost results. sponses have interpretability issues, and Blender2 and BART-FiD-RAG each achieve a relevant response rate of nearly 90%. Blender2 specifically is also able to incorporate factual knowledge into about 20% of its responses while hallucinating factual information at a remarkably low rate, less than 1%. Furthermore, the chatbots almost never produce responses with offensive language. 12 The chatbots also show a high rate of emotional understanding, with 40% of their responses containing emotionally-appropriate reactions to the user. Despite these strengths, our results also show several clear directions for improvement. Commonsense violations are present in about 15-20% of the bots' responses. Consistency issues are prevalent across all bots: self-contradictions, partner contradictions, and redundancies appear in about 5% of the bots' responses overall. Also, all chatbots have a substantial rate of violating natural dialogue structure: about 10% of responses are judged as ignoring the user, and depending on the chatbot, 12 Note that our experiments are conducted with cooperative human interactors. Chatbots similar to those we test have been shown to reliably produce offensive language when responding to provocative inputs (Dinan et al., 2022). !Soc b !Com b Ign b !Fac b !Rel b !Emp b !Par b Red b !Sel b Top b !Int b 0.0 0.1 0.2 0.3 BART-FiD-RAG Emora Blender2 Blender-Decode Figure 4: Proportions of turns expressing undesirable behaviors, with 95% Wilson score confidence intervals. around 10-20% of responses include irrelevant contributions to the dialogue. Additionally, 5-15% of the chatbots' responses show a lack of empathy or other emotional misunderstandings. The reality of these observed rates of problematic behaviors is that, in most 30-turn conversations with these chatbots, a human interactor is likely to experience several issues that impact conversation quality. Fac b Emp b Fol b Lif b Pre b 0.0 0.2 0.4 0.6 BART-FiD-RAG Emora Blender2 Blender-Decode Conclusion As illustrated here, dialogue quality is a complex construct with many dimensions. Depending on the approach, dialogue systems can have markedly different weaknesses among these quality dimensions. Our research highlights several outstanding challenges, especially regarding the relevance, consistency, common-sensibility, and emotional understanding of chat model responses. Our analyses not only demonstrate that these four dimensions have a high impact on conversation quality, but also that current chatbots have substantial response error rates in these areas. To efficiently address the challenges facing opendomain dialogue models, we need a reliable, dimensional evaluation method; however, our results show that popular evaluations such as dialoguelevel Likert and comparative methods may not be suitable. The presented ABC-Eval serves as a promising alternative in this direction. Although the popular dialogue-level likert evaluation method may be the most cost-effective and robust method for measuring overall dialogue quality, we recommend that researchers additionally use the final set of ABC-Eval metrics, or a subset relevant to their scientific goals, to evaluate specific strengths and weaknesses of new chat models. Overall, we hope future work can use insights from our study to make better-informed decisions about which evaluation method to use, and to tackle the challenges facing current chatbots. Limitations There are several characteristics of the presented analyses that limit the scope of conclusions that can be drawn. We discuss how each of these limitations affect the takeaways of our results below. Number of Chatbots The generalizability of our metric analysis results (Section 7) is constrained by the fact that we were only able to include conversations from 4 chatbots in our analyses. We did our best to choose chatbots representative of the field and seem to have selected a fairly diverse group of models (Section 8). However, it is possible that not all results we found in our metric analyses will generalize when evaluating other chat models. One possible example is the number of partner contradictions we observed among our 4 chatbots ( Figure 4), which may be similar by coincidence. If other chatbot models indeed differ more substantially in partner contradiction rates, our sensitivity metric analysis may have underestimated the sensitivity of our partner contradiction metric (Section 7.3). In general, including a larger number of chatbots in a metric analysis will improve the chance that its results will apply to new chatbot models. Future work that performs metric analyses like those we presented, but with different chatbots than the 4 selected in this work, would aid further analysis of our results' generalizability. Use of Surgers as Evaluators We perform our analyses using only a single evaluator group (Surgers). This choice of evaluator group does not harm the replicability of our methods, as other researchers have access to use of SurgeHQ or similar third-party annotation companies. However, several other evaluator groups are more popularly used for chat model evaluation, such as university students and Amazon Mechanical Turkers (MTurkers). We attempted to carry out our study with three evaluator groups (see Appendix E for details), but were unable to proceed with student and MTurker evaluator groups due to time constraints. Consequently, it is unclear to what extent our metric analysis results will generalize to other choices of evaluator. Number of Collected Conversations As with any study involving a sampling procedure, resource constraints limit the number of collected samples, which in turn limits the statistical power of the study's analyses. Our study included 400 conversations, which provided more than adequate statistical power for most of our analyses. For example, our investigation of each metric's predictive validity (Section 7.2) relied on a simple linear regression analyses. At a significance level of α=0.05, our 400 conversation samples would yield a statistical power of 1-β=0.80 to detect effect sizes of f 2 =0.14 2 by F-test for each metric's regression. However, our analyses with the weakest statistical power are our dialogue-level analyses that compare bots with only 100 samples per bot. At 100 samples per bot, and assuming a standard deviation of 1.0 Likert points, 13 a two-tailed t-test of mean Dialogue Likert rating would have a statistical power of 1-β=0.80 to detect differences of an effect size of Cohen's d=0.40. This is still a reasonable amount of statistical power, but leaves room for our study to produce inconclusive results when the true differences between chatbots are small. Ethics Statement The presented work aims towards improving the scientific methodology of chat model evaluation. To this end, we present a battery of analyses comparing several aspects of metric validity for four different evaluation methods (Section 7). Our results allow other researchers in the field to make betterinformed decisions regarding appropriate evaluation methodology in human-computer chat. To ensure replicability of our methods we publicly release the annotation software and chatbot implementations used to collect our conversation and evaluation data. Additionally, we provide full transparency in our analyses by releasing the code for all our presented analyses. Finally, to aid future research efforts in human-computer chat modelling and evaluation, we release an anonymized version of our conversation and evaluation data. One ethical consideration involved in our work involved managing human workers in our data collection processes. All worker participation in our study was voluntary and involved zero subjective screening processes, with a complete description of worker tasks, workload, and timeframe provided before work was assigned. Workers could opt out of our study at any time for any reason. As compensation for work completed, we targeted a compensation rate of $10/hour for student 14 and Amazon Mechanical Turk workers, and a rate of $20/hour for Surgers. We compensated on a pertask-completed basis to ensure timely completion of work, but verified that target hourly rates were reasonably approximated throughout the course of the study by measuring workers' median task completion times (see Appendix F for details). These measures ensured that all human work in our study was fair, transparent, and mutually-beneficial. Other ethical considerations arise in our study's conversation collection. Unlike the collection of evaluation or annotation data, collecting interactive conversation data from human-computer interaction poses a small but meaningful risk that sensitive, damaging, or personally identifying information could get collected. We mitigated this risk in three ways. First, students were notified in multiple email communications and before each conversation that their conversations with our chatbots would be publicly released. Included in these notices was the instruction to refrain from releasing any personally identifiable or damaging information. Our instructions suggest that students fabricate personal information at any time during the conversations if it would make them feel more comfortable. Second, we hand-checked all 400 conversations to ensure the non-presence of any sensitive information. Third, we anonymize all data before public release. Our study's collection and analysis of conversation data did not investigate interactors as human subjects, and we did not seek institutional review board approval. Finally, there is a concern in our study about the potential of the chatbots to respond to student interactors with toxic, insensitive, or vulgar language. The data-driven nature of some of our evaluated chat models means the chatbots are prone to reflecting any biases, toxicity, and vulgarity present in the training data (see Dinan et al. (2022) for a quantitative analysis). A high rate of antisocial behaviors among our evaluated models could potentially make human interactors' experience talking with the bots quite uncomfortable, and would poorly reflect on the research field's potential for social good. To mitigate this risk, the authors extensively hand-tested all evaluated chat models, as well as conducting a pilot evaluation among the authors' lab group. As confirmed further in our results (Section 8), our chatbots exhibited negligible rates of antisocial behavior. Acknowledgements Thank you to Bradley Webb, Scott Heiner, and the rest of the SurgeHQ team for their guidance in running our annotation projects on their platform. We are also grateful to our colleagues at Emory for their participation in piloting the bots and refining the annotation interfaces. Lastly, a thank you to our reviewers for their helpful feedback. C Bot Pilot Examples Tables 6 and 7 show two of our chatbot pilot (Section A) conversations, one from CEM (Sabour et al., 2022) and one from Blender2 , that exemplify the difference between single-turn and multi-turn dialogue response generation models. The CEM model is trained to give an empathetic response to a dialogue context, and achieves good performance towards this goal. However, as shown in the example, this response policy does not translate well for multi-turn interaction with a human. By contrast, Blender2 is trained and evaluated specifically to achieve multi-turn dialogue. D Pilots and Development The final ABC-Eval label set, annotation procedure, and software application are created using an iterative process of development and piloting. 14 students are invited to serve as evaluators for piloting the evaluation. To avoid overfitting the evaluation design, our pilots evaluated conversations collected between Blender (Roller et al., 2021) and one of the authors, and a new set of conversations was used for each pilot round. We ran 4 pilot rounds, making revisions after manually reviewing each round's annotation. Table 8 presents a summary of the major changes made in each pilot round and IAA metrics. It is important to note that each annotator performed all of the annotation tasks in one sitting in sequence for each pilot. These piloting rounds are not necessarily directly comparable to one another when taken as a whole, since the annotator groups and dialogues to be annotated varied between each round. Instead, we will discuss below the major takeaways afforded by different splits of the pilots that informed the final design of ABC-Eval. Subtask Formulation The decision to format ABC-Eval into several small annotation subtasks, each with a tailored subset of the behavior labels, was made from the results of Pilot 1. In Pilot 1, we divided the initial set of annotation labels into 3 annotation subtasks each with 4-9 labels: errors, information usages (commonsense, world knowledge, etc.), and utterance types (request, presentation, etc.). Each annotator performed the annotation tasks in one sitting in sequence. The overall interannotator agreement was quite low (α = 0.18), which was concerning. Based on ad-hoc feedback from the pilot annotators, the consensus was that each subtask demanded an unreasonable cognitive load on annotators due to the large number of labels to keep track of. For Pilot 2 we increased the number of annotation tasks such that each covered a small and related scope of behavior labels, with 1-4 labels per task. Table 8 shows the boost to interannotator agreement between Pilots 1 and 2. However, this agreement increase could have resulted from an increase in the quality of the annotators (as Pilot 2 was composed primarily of annotators with a graduate-level education whereas Pilot 1 was more evenly split between annotators with an undergraduate-level education and graduate-level education). To remove this confound, we calculated the agreement in Pilots 1 and 2 when only considering graduate-level annotators. Although it was less dramatic, there remained an increase in agreement from 0.39 to 0.50, which encouraged the decision to maintain the smaller annotation subtasks. Dividing the annotation into tailored subtasks seemed to reduce the cognitive load on annotators, thus allowing them to perform more accurate annotations per task. Training and Screening Manual analysis of the pilot annotations from Pilots 1 and 2 revealed some recurring annotation mistakes, arising from misunderstandings of the guidelines for the tasks. In an attempt to correct such misunderstandings, a training procedure was introduced for each task. Each round of training consists of 1 curated conversation with ground-truth labels and explanations that are shown as feedback to the annotator after they complete the training round. We used the results of Pilots 1 and 2 in order to develop these curated conversations as follows: 1. Label Specifications: We constructed a label specification that consisted of a comprehensive enumeration of positive and negative cases of the label with the goal of defining a decision boundary that the annotators should strive towards. We especially focused on the utterances for which several of the annotators failed to produce labels that matched the ground truth annotations we had defined for each of the Pilots. 2. Training Conversation Selection: We selected 3 conversations between Blenderbot and a human (from a collection within our lab) for each label to be used as training conversations for it. This selection was manually done by ranking the conversations on their coverage of the label specification. Training Conversation Modification: We heavily revised the selected conversations by hand to ensure that all of the cases identified in the specification were adequately represented, most often by inserting new utterances that corresponded to any underrepresented cases. To evaluate the utility of this training process, a third pilot was conducted using 4 undergraduates. We observed a general upwards trend in annotation performance between the training rounds for the annotators, suggesting that the training was aiding in the annotation accuracy for the annotators. The final agreements were 0.43 and 0.45 between all annotators and annotators who passed the training, respectively, on the annotated conversations. Due to the small nature of this pilot, we are unable to conclude whether this difference is meaningful. However, ad-hoc feedback from the annotators suggested that the training rounds were useful towards their understanding of the tasks, although the amount of training did increase the overall workload of participation. Accordingly, the decision was made to treat each subtask independently, rather than require all subtasks to be completed for one dialogue in a single sitting for each annotator. General Revisions Throughout each of these pilot rounds the annotation instructions, examples, and training rounds were updated based on manual review of the annotations in an attempt to correct any unclear or misleading information. E Evaluator Training and Screening We attempted to use three different groups of evaluators for our full evaluation study: Students Undergraduate students were recruited from the authors' university via word-of-mouth and email advertisements sent to computer science, psychology, and quantitative methods departmental mailing lists. 19 MTurkers Our 20 evaluation tasks were posted to the Amazon Mechanical Turk crowdsourcing platform. 20 Surgers Our 20 evaluation tasks were posted on SurgeHQ's annotation platform 21 to be completed by dedicated workers with experience in NLP annotation. A group of 125 Surgers were qualified to participate in our tasks, chosen by a SurgeHQ employee on the basis of high annotation performance on past projects. All three groups were compensated per task per annotated conversation, at an estimated rate of $10/hr for Students and MTurkers, and $20/hr for Surgers. To check the viability of each worker group to produce evaluation data for our full study, we released a random 5 conversations out of our set of 400, to be fully evaluated by each worker group in each of our 8 ABC-Eval tasks. After a two week period, Surgers were the only worker group that were able to fully evaluate the 5 conversations in all 8 ABC-Eval tasks. This was due to an overall lack of participation from the Student group, and due to low training pass rates from the MTurk group (see Figure 9 for quantitative outcomes). Although worker group differences in work rate and training performance might be explained by the difference in compensation structure, we decided to proceed with the Surgers group only for our full study to collect our evaluation data in a timely manner. F Collection Cost Compensation rates are based on per-task completion times from an internal pilot run. The rates per task paid to Surgers are shown in Table 9. We also present the real and theoretical costs for collecting each method included in our evaluation data (Table 10). As expected, turn-level annotation tasks are an order of magnitude more expensive to collect than dialogue-level tasks. Notably, the final set of ABC-Eval labels (Table 4) are, on average, less expensive to collect than turn-level Likert labels. G Evaluation Interfaces Examples of the annotation interfaces for each annotation task of ABC-Eval are provided in Figures 10 -17, and an example for the conversation collection interface is provided in Figure 21. Examples of the annotation interfaces for Dialogue Likert, Turn Likert, and Comparative evaluations are provided in Figures 18, 19, and 20, respectively. The definitions that were shown to the annotators in the interface for each of the 8 dimensions of Dialogue Likert, Turn Likert, and Comparative are taken verbatim from Finch and Choi (2020). Figure 1 : 1IAA (Krippendorff's alpha) for all metrics. Error bars denote 95% bootstrap 9 confidence intervals. 10Our final evaluation dataset consists of 400 conversations, each with results for all 40 labels. 8 Additionally, a randomly-selected subset of 100 conversations (and 50 of the conversation pairs) is evaluated a second time by a different Surger in order to measure IAA. Figure 5 :Figure 6 : 56Proportions of turns expressing desirable behaviors, with 95% Wilson score confidence intervals. Con d Emo d Eng d Gra d Inf d Pro d Qua d Rel d Average Dialogue Likert ratings of the conversations, with 95% Student's t confidence intervals. Figure 7 : 7Win/tie rates of each bot vs all other bots, with 95% Wilson score confidence intervals for win proportion. Transparent segments denote tie rates. Figure 8 : 8Con t Emo t Eng t Gra t Inf t Pro t Qua t Rel t Average Turn Likert ratings of the conversations, with 95% Student's t confidence intervals. Figure 9 : 9The training pass rate of each annotation workforce for the behavior evaluation tasks. Figure 10 :Figure 11 : 1011Interface for uninterpretable Interface for antisocial Figure 12 :Figure 13 : 1213Interface for preference info and life info Interface for empathetic and lack of empathy Figure 14 :Figure 15 : 1415Interface for commonsense contradiction Interface for fact usage and fact contradiction Figure 16 :Figure 17 : 1617Interface for self contradiction, partner contradiction, and redundant Interface for ignore, irrelevant, follow-up, and topic switch Figure 18 :Figure 19 :Figure 20 :Figure 21 : 18192021Interface for one dimension of Dialogue Likert Interface for one dimension of Turn Likert Interface for one dimension of Comparative Interface for conversation collection Table 2 : 2Recent studies of human evaluation metrics, in order of mention in Section 4. Multi-turn: investigates multi-turn response generation, Comprehensive: a set of metrics intended to explain dialogue quality, ComPared: compares alternative evaluation methods, Agreement: Table 3 : 3The 8 labels for Likert and Comparative evalu- ations (taken from Finch and Choi Table 4 ) 4. Table 4 : 4The 16 behavior labels within ABC-Eval. Row separators denote evaluation task groupings.Bold indicates Table 5 ) 5. To make results comparable, weFigure 2: Extent to which each evaluation metric can explain variance in conversation quality by fitting a univariate regression model (R 2 for predicting Qua d using linear regression, McFadden's Pseudo-R 2 for predicting Qua c using logistic regression). Comparative metrics cannot predict Qua d so only results for variance of Qua c are shown.!Socb !Comb Facb Empb Folb Ignb !Facb !Relb !Empb Lifb !Parb Preb Redb !Selb Topb !Intb Cont Emot Engt Grat Inft Prot Relt Cond Emod Engd Grad Infd Prod Reld Conc Emoc Engc Grac Infc Proc Relc 0.00 0.02 0.04 0.06 0.08 0.10 Interactive Quac Interactive Quad Table 5 : 5The number of statistically significant differences detected by each metric when comparing bot-pairs using z-tests of proportions (ABC-Eval), t-tests (Turn Likert and Dialogue Likert), and sign tests (Comparative) at three significance thresholds. Figure 3: Incremental validity of metrics within 4 evaluation methods, obtained using backwards stepwise regression. Points represent the extent to which a model can explain variance in quality (R 2 for predicting Qua d with a linear model, McFadden's pseudo-R 2 for predicting Qua c with a logistic model) using all metrics on the same line and to the left as predictors. Filled marker symbols denote steps where the model's predictors all contributed positively to adjusted R 2 or adjusted pseudo-R 2 values; otherwise, marker symbol is unfilled. Comparative metrics cannot be used to predict Qua d so only results for explaining variance of Qua c are shown.Turn Likert Dialogue Likert Comparative 0.05 0.10 0.15 0.20 0.25 !Emp !Sel !Fac Red Fol !Int Lif !Com Emp Fac Pre !Rel Ign Top !Soc !Par Pro Con Inf Rel GraEmoEng Eng Emo Pro Con Inf Gra Rel Emo Inf Eng Gra Rel Con Pro !Emp Red !Sel !Rel Emp !Com Ign Fac Fol !Par !Int Pre !Fac Top Lif !Soc Eng Pro Rel Con Emo Gra Inf Rel Emo Pro Eng Gra Con Inf Interactive Qua c Interactive Qua d nal ABC-Eval metrics are less costly 11 (a median of 15.2 min/dialogue) to collect than Turn Likert metrics (19.9 min/dialogue). Although dialogue- level evaluations are least costly (2.8 min/dialogue for Dialogue Likert, 4.4 min/dialogue for Com- parative), our results suggest that dialogue-level judgements may be ill-suited for dimensional eval- uation, since the dialogue-level metrics we tested had worse coverage and distinctness (Section 7.4). Table 7 : 7Chatbot pilot conversation: Blender2 Table 8 : 8The distribution of dialogues and annotators for each annotation pilot. Pilot 1 included full label annota- tion of each dialogue where dialogues were distributed among groups of annotators. Pilots 2 and 3 had a subset of labels annotated for each dialogue, but all annotators annotated each dialogue. α: Krippendorf's alpha, U: Undergraduate annotators, G: Graduate level annotators, ALL: Undergraduate and graduate annotators. Table 9 : 9Payment per annotation task in USD. The payment for Turn Likert is per label whereas the indicated payment for Dialogue Likert and Comparative covers all labels, due to how the annotation tasks were constructed (Section 6).Metric TI TP EC OC Dialogue Collection 8.08 7.43 1077.14 333.33 Dialogue Likert 2.81 21.37 374.36 240.00 Comparative 4.35 13.81 289.68 286.67 Turn Likert 19.94 3.01 2658.40 2240.00 ABC-Eval all 25.60 2.34 3413.58 3422.67 ABC-Eval f inal 15.17 3.95 2022.98 - Table 10 : 10The data collection costs for each task in United States Dollars. TIme is the median completion time in minutes for one dialogue. ThroughPut represents the number of completed dialogues per hour. Estimated Cost is calculated using median completion time, 400 dialogues, and $20/hr rate. Our Cost is the total amount paid in this work to collect a dataset of 400 conversations (single-annotated). Note that selection occurred in October 2021.2 We accepted either a trained English model or codebase with a fully-specified procedure to replicate the model.3 We use the web interface provided by ParlAI(Miller et al., 2017) hosted on our local webserver. A "turn" is defined as ONE message from a single interactor. We do not include a turn-level comparative evaluation because controlled comparisons require comparing turns with identical historical contexts which is not viable for real human-bot dialogues like those used in this work. https://www.surgehq.ai; Appx. E details annotator selection. 7 Per-task payment rates provided in Appendix F. Smith et al. (2022) reports standard deviations of Likert metrics between 0.8 and 1.3 14 Students' compensation is given as an Amazon Gift Card for convenience; students are informed of this prior to any work being completed Students were compensated with an Amazon gift card at the completion of the data collection. 20 https://www.mturk.com/ https://www.surgehq.ai/ A Chatbot Selection DetailsThis appendix discusses the details of our literature review for each of the four chosen research themes.General Our work focuses primarily on opendomain chat. Large-scale language modeling using dialogue-like pretraining data can produce surprisingly human-like conversation on virtually any popular conversation topic(Roller et al., 2021). Of these approaches, we chose Blender2 , which reportedly outperformed its previous iteration Blender(Roller et al., 2021)who had surpassed DialoGPT(Zhang et al., 2020)and Meena(Adiwardana et al., 2020).There had been also several chatbots produced by the Amazon Alexa Prize Socialbot Grand Challenge(Ram et al., 2018)focusing on general, opendomain chat, most of which incorporate rule-based methods to ensure interesting and consistent responses . Since these chatbots performed well in practice but lack comparison to SOTA data-driven models, we selected the bot with the all-time highest final score, Emora 15 , as one of our candidates.Knowledge Grounding chat with supplementary knowledge resources is a common way to improve engagingness and control the topic of conversation(Li et al., 2019b;Ye et al., 2020).ColV (Zhan et al., 2021)achieved SOTA performance in knowledgegrounded dialogue response generation on the popular WoW dataset(Dinan et al., 2019); however, no implementation was publicly available. DukeNet and PIPM report next-best performance in this task. DukeNet's implementation was available while PIPM's was not, therefore we selected DukeNet as a candidate.BART-FiD-RAG also reported compelling performance for knowledge-grounded chat , but did not compare to other SOTA models we identified. Since BART-FiD-RAG's inclusion in ParlAI provided easy replication, we included it in our bot pilot.Consistency Improving consistency of chatbot responses is noted as a challenge and addressed in several works(Welleck et al., 2019;Nie et al., 2021;. DECODE(Nie et al., 2021)reported SOTA performance for general inconsistency avoidance, improving upon an approach that used unlikelihood training with dialogue natural 15 https://github.com/emora-chat/emora_ap3_parlai language inference data . Note that there were several works focusing specifically on persona consistency(Song et al., 2020;Kim et al., 2020;Song et al., 2021), which we did not consider due to their narrower contradiction scope.Empathy Several works demonstrated the importance of emotional understanding in chat(Partala and Surakka, 2004;Prendinger and Ishizuka, 2005;Sabour et al., 2022). To provide contrast with our knowledge-grounded candidates, we selected CEM(Sabour et al., 2022), which reported SOTA results in empathetic response generation. Many related works investigated controllable emotional response generation(Song et al., 2019;Zhong et al., 2021), but we did not consider models requiring an emotion label as input.B Chatbot Implementation DetailsFor each selected candidate model, a brief overview of the implementation details required to use them as interactive models in this work is below:Emora We implement a ParlAI agent using the interactive chatting mode provided for the Emora system .BART-FiD-RAG An interactive chatting mode for BART-FiDRAG is provided through ParlAI. We use the default arguments, except for compressed-indexer-nprobe and beam-context-block-ngram which we set to 128 and 3, respectively, following internal testing by the authors before the pilot.Blender2 An interactive chatting mode for Blender2 16 is provided through ParlAI; however, the internet-search augmented functionality as described inKomeili et al. (2022)is not provided. Although there is an available alternative internet search server 17 that is compatible with Blender2, it increased the response latency of Blender2 such that it no longer satisfied our response latency requirement. As a result, we use the 3B parameter version of Blender2 that does not include the internet search functionality.Blender-DecodeWe implement a custom agent in the Parlai framework following the bestperforming setup ofNie et al. (2021)of finetuning a consistency classification model and using it to 16 https://parl.ai/projects/blenderbot2/ 17 https://github.com/JulesGM/ParlAI_SearchEngine rerank generated responses from Blender(Roller et al., 2021)such that the least contradictory response is selected.DukeNet The published code for DukeNet is missing an interactive chatting mode, although a trained model on the multi-topic Wizard of Wikipedia (WoW)(Dinan et al., 2019)is provided. We modified the published code to support an interactive chatting mode. However, one major difference is that the relevant knowledge documents need to be retrieved in interactive chatting mode, whereas they are predetermined in the training and evaluation environments. To remedy this, we use a trained Dense Passage Retrieval model(Karpukhin et al., 2020)provided through Sentence-Transformers 18 to select knowledge at each dialogue turn. We implement a ParlAI agent using the available DukeNet model and our custom interactive chatting mode.CEM The published code for CEM is missing both an interactive chatting mode and a trained model. We trained a CEM model using the default arguments from the repository and modified the published code to support an interactive chatting mode. We implement a ParlAI agent using this trained model and our custom interactive chatting mode.User Daniel Adiwardana, Minh-Thang Luong, David R So, Jamie Hall, Noah Fiedel, Romal Thoppilan, Zi Yang, Apoorv Kulshreshtha, Gaurav Nemade, Yifeng Lu, V Quoc, Le, 10.48550/arXiv.2001.09977ArXiv:2001.09977Towards a Human-like Open-Domain Chatbot. cs, statDaniel Adiwardana, Minh-Thang Luong, David R. So, Jamie Hall, Noah Fiedel, Romal Thoppilan, Zi Yang, Apoorv Kulshreshtha, Gaurav Nemade, Yifeng Lu, and Quoc V. Le. 2020. Towards a Human-like Open- Domain Chatbot. ArXiv:2001.09977 [cs, stat]. Aviel Rubin, and Shari Watkins. 2022. 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[ "https://github.com/emorynlp/", "https://github.com/emora-chat/emora_ap3_parlai", "https://github.com/JulesGM/ParlAI_SearchEngine" ]
[ "The Malyuzhinets-Popov diffraction problem revisited", "The Malyuzhinets-Popov diffraction problem revisited" ]
[ "Ekaterina A Zlobina \nDepartment of V. A. Steklov Institute of Mathematics\nRussian Academy of Sciences\nFontanka Emb. 27191023St, Petersburg, St. PetersburgRussia\n\nSt. Petersburg State University\nUniversitetskaya Emb. 7-9199034St. PetersburgRussia\n", "Aleksei P Kiselev \nDepartment of V. A. Steklov Institute of Mathematics\nRussian Academy of Sciences\nFontanka Emb. 27191023St, Petersburg, St. PetersburgRussia\n\nSt. Petersburg State University\nUniversitetskaya Emb. 7-9199034St. PetersburgRussia\n\nInstitute for Problems of Mechanical Engineering of Russian Academy of Sciences\nVasilievsky Ostrov Bolshoy Prospect 61199178St. PetersburgRussia\n" ]
[ "Department of V. A. Steklov Institute of Mathematics\nRussian Academy of Sciences\nFontanka Emb. 27191023St, Petersburg, St. PetersburgRussia", "St. Petersburg State University\nUniversitetskaya Emb. 7-9199034St. PetersburgRussia", "Department of V. A. Steklov Institute of Mathematics\nRussian Academy of Sciences\nFontanka Emb. 27191023St, Petersburg, St. PetersburgRussia", "St. Petersburg State University\nUniversitetskaya Emb. 7-9199034St. PetersburgRussia", "Institute for Problems of Mechanical Engineering of Russian Academy of Sciences\nVasilievsky Ostrov Bolshoy Prospect 61199178St. PetersburgRussia" ]
[]
In this paper, the high-frequency diffraction of a plane wave incident along a planar boundary turning into a smooth convex contour, so that the curvature undergoes a jump, is asymptotically analysed. An approach modifying the Fock parabolic-equation method is developed. Asymptotic formulas for the wavefield in the illuminated area, shadow, and the penumbra are derived. The penumbral field is characterized by novel and previously unseen special functions that resemble Fock's integrals.
10.1016/j.wavemoti.2023.103172
[ "https://export.arxiv.org/pdf/2206.05444v2.pdf" ]
254,854,388
2206.05444
b728a62b2dba47d7d38d7f05e97d690f6f1ef2c5
The Malyuzhinets-Popov diffraction problem revisited Ekaterina A Zlobina Department of V. A. Steklov Institute of Mathematics Russian Academy of Sciences Fontanka Emb. 27191023St, Petersburg, St. PetersburgRussia St. Petersburg State University Universitetskaya Emb. 7-9199034St. PetersburgRussia Aleksei P Kiselev Department of V. A. Steklov Institute of Mathematics Russian Academy of Sciences Fontanka Emb. 27191023St, Petersburg, St. PetersburgRussia St. Petersburg State University Universitetskaya Emb. 7-9199034St. PetersburgRussia Institute for Problems of Mechanical Engineering of Russian Academy of Sciences Vasilievsky Ostrov Bolshoy Prospect 61199178St. PetersburgRussia The Malyuzhinets-Popov diffraction problem revisited High-frequency asymptoticsnon-smooth obstaclesHelmholtz equationboundary-layer method 2010 MSC: 35J2535L0535Q60 In this paper, the high-frequency diffraction of a plane wave incident along a planar boundary turning into a smooth convex contour, so that the curvature undergoes a jump, is asymptotically analysed. An approach modifying the Fock parabolic-equation method is developed. Asymptotic formulas for the wavefield in the illuminated area, shadow, and the penumbra are derived. The penumbral field is characterized by novel and previously unseen special functions that resemble Fock's integrals. Introduction In a remarkable paper [1], Alexey V. Popov addressed the high-frequency diffraction of a plane wave incident along a planar boundary (with the Neumann condition), passing into a parabola at its apex (Fig 1). He aimed at describing a diffracted cylindrical wave that emerges from the point of non-smoothness [2,3,4] and, by means of virtuoso calculations, obtained an explicit formula for it. Popov's research was inspired by Malyuzhinets' paper [5], in which this problem was first clearly formulated and qualitatively investigated. The Malyuzhinets-Popov problem has much in common with the so-called Fock problem (see [6,7,8,9]), which consists of investigation of a high-frequency field in the vicinity of a tangency point in the diffraction of a plane wave by a smooth, convex obstacle. A significant qualitative difference is the presence of a diffracted cylindrical wave in the illuminated region instead of a reflected one. Fock's approach (inspired by a physical idea due to Leontovich) is based on introduction of an approximate equation called the parabolic equation. Fock dealt with a simplified problem that admits an explicit but rather complicated solution that he explored asymptotically. He was attracted to the description of the field at a short distance from the obstacle, primarily in the vicinity of the limit ray, for which he developed a spectacular analytical technique. A. Popov ingeniously used a complex combination of the parabolic-equation method, Malyuzhinets' technique and Kirchoff's approach to derive an expression for the diffracted wave. However, he did not address the wavefield in the shadow and the penumbra surrounding the limit ray that separates illuminated and shadowed areas (Fig. 1). We observed a certain inaccuracy in the paper by A. Popov [1], which apparently did not affect the expression for the diffracted wave, but would impact upon a detailed description of the field in the penumbra. There are two reasons for revisiting this problem. First, for half a century, steady interest has been seen in the description of the effects of high-frequency diffraction by contours in which the curvature is neither strictly positive nor strictly negative (see, e.g., [1,10,11,12,13,14,15,16,17,18,19,20,21,22]). A. Popov's work is historically the first in which such phenomena were quantitatively approached and one of the few (see, e.g., [22]) where the sign of curvature changes not in a smooth manner, but in a jump. Second, the effects of a jump in the curvature (as well as that of weaker singularities; see [23]) do not allow a description from simple model problems because they do not exist. These effects were studied for non-tangential incidence by heuristic approaches, such as the Kirchhoff method (see, e.g., [4,24,25,26,27]), and by the boundary-layer theory [28,29]. For problems with tangential incidence, no close inspection based on the boundary-layer method is available. We address the Malyuzhinets-Popov problem as a relatively simple problem with a tangential incidence on the boundary with a jump in the curvature. We use a systematic boundary-layer technique, which goes back to the research of Fock (see [6], [7, ch. 7]) and was further developed by Brown [8] and Babich and Kirpichnikova [9, ch. 6]. Similar to these studies, we examined wavefields in a small neighborhood of the singular point of the boundary. We had to overcome significant analytical difficulties in deriving an expression for the diffracted wave (which agrees with the findings of A. Popov [1]) and in investigating the Fresnel field and an analog of Fock's background field in the penumbra. In addition, we found an expression for the field in deep shadow that matches with the creeping waves. At all stages, we carefully estimated the remainder terms, which allowed us to reliably indicate the domains of validity of the resulting expressions. Formulation of the problem We consider the wavefield u governed above the contour C by the Helmholtz equation ∂ 2 x + ∂ 2 y + k 2 u = 0(1) with large wavenumber k, k 1,(2) and satisfying the Neumann boundary condition ∂ n u| C = 0.(3) Here, ∂ n is the derivative along the inner normal to C. The contour consists of the flat part C − touching the smooth curved part C + at point O (Fig. 1). Near O the curvature ae of contour C has the following form: ae(x) = hθ(x),(4) where θ(x) =      1, x > 0, 0, x ≤ 0,(5) is the Heaviside step function, and h = 0 is the magnitude of the jump of curvature. The total wavefield u is the sum of the incident plane wave u inc = e ikx that travels left to right along C − towards O and the outgoing wave u out : u = u inc + u out .(6) The problem is to describe the outgoing wave. The problem under consideration is similar to that of Fock in that the limit ray is surrounded by several boundary layers, where the wavefield behavior differs. The principal distinction is that the diffracted wave u dif emerges at the non-smoothness point O, for which the GTD [3,4] predicts the following approximation far above the limit ray: u dif ≈ A(ϕ; k) e ikr √ kr , kr 1.(7) Here, A(ϕ; k) is a diffraction coefficient and (r, ϕ) denotes the classical polar coordinates centred at O: x = r cos ϕ, y = r sin ϕ, 0 ≤ r < ∞, −π ≤ ϕ < π. Unlike A. Popov, who was interested exclusively in the diffracted wave, we provide asymptotic descriptions of wavefields in boundary layers separating the illuminated region from the deep shadow, as shown in Fig. 3. These descriptions employ functions resembling Fock's integrals [7,9]. however, in contrast to the classical case [7,9], the mapping (x, y) → (s, n) is not smooth. Furthermore, s and n and, accordingly, x and y will be small in comparison with the curvature radius h −1 . We use the following elementary relation which results from (4) by a simple calculation:          x = s + θ(s) hns − h 2 s 3 6 + O h 3 s 3 (n + hs 2 ) , y = n − θ(s) hs 2 2 + O h 2 s 2 (n + hs 2 ) . (9a) (9b) We also use the expression for the Laplacian [9]: ∂ 2 x + ∂ 2 y = 1 (1 + κ(s)n) 2 ∂ 2 s − nκ (s) (1 + κ(s)n) 3 ∂ s + ∂ 2 n + κ(s) 1 + κ(s)n ∂ n .(10) Reduction to the problem for parabolic equation We start with the observation (traceable to Leontovich and Fock) that, as in the case of the tangential incidence of a plane wave at a smooth obstacle, in the vicinity of point O u inc = e ikx = e iks V,(11) where V oscillates slower than the exponentials. This inspired us to follow the classical research strategy [7,8,9] based on separating out a rapidly oscillating We introduce dimensionless stretched coordinates by σ = (h 2 k/2) 1 3 s, ν = (2hk 2 ) 1 3 n,(12) with the same powers of k as in the original definition of Leontovich and Fock [7]. The dimensionless large parameter k/h 1(13) of our problem is in agreement with that of Fock. We seek the outgoing wavefield in a form similar to that of the Leontovich-Fock ansatz [7]: u out = e iks U (σ, ν) = e iks (U 0 (σ, ν) + . . .) ,(14) where the attenuation factor U oscillates slower than e iks and the dots stand for smaller-order terms with respect to the large parameter k/h. Formulas connecting coordinates (σ, ν) with the Cartesian coordinates follow from (9):              k(x − s) = θ(σ) νσ − σ 3 3 + O h k 2 3 σ 3 (ν + σ 2 ) ,(2h)1 3 k 2 3 y = ν − θ(σ) σ 2 + O h k 2 3 σ 2 (ν + σ 2 ) . (15a) (15b) These formulas entail useful relations involving polar coordinates r = x 2 + y 2 and ϕ = arctan y/x k(r − s) = 1 2 ν 2 2σ + θ(σ) νσ − σ 3 6 + O h k 2 3 σ 3 (ν + σ 2 ) + σ ν − σ 2 3 2 ,(16)ϕ = 2h k 1 3 ν − θ(σ)σ 2 2σ + O h k σ(ν + σ 2 ) + (ν − σ 2 ) 2 σ .(17) With the help of (15a), we rewrite the factor V in (11) as V (σ, ν) = 1 − θ(σ) + θ(σ)e i σν− σ 3 3 + O h k 2 3 σ 3 (ν + σ 2 )(18) and, following Fock, require the smallness of the remainder terms. This implies the restrictions on distance from the singular point O which can be summarized as σ (k/h) 2 15 , ν (k/h) 4 15 .(19) Further research refers to the area characterized by the conditions (19). 1 Rewriting (10) in stretched coordinates (12), expanding the coefficients in powers of (h/k) 2 3 and substituting (18) and (14) into the Helmholtz equation (1) and boundary condition (3), we immediately obtain the problem for the main term U 0 of the outgoing wave:    ∂ 2 ν U 0 + i∂ σ U 0 + νθ(σ)U 0 = 0, ∂ ν U 0 | ν=0 = −iθ(σ)σe −i σ 3 3 . (20a) (20b) We note that in a remarkable paper by A. Popov [1], which we followed up until now, the exponential on the right-hand side of (20b) was missing, and thus, his further results require refinement. 1 Limitations for the corresponding area in the Fock problem, presented in [9], were different because the speed of propagation was assumed nonconstant. Formal solution of (20) We seek the main term of the attenuation factor U 0 in the form of a Fouriertype integral as follows: U 0 (σ, ν) = 1 2π ∞ −∞ U 0 (ξ, ν)e iσξ dξ.(21) Using the representation for the Heaviside function θ(σ) = − i 2π ∞ −∞ e iσξ ξ − i0 dξ,(22) where (ξ − i0) −1 := lim ε→+0 (ξ − iε) −1 , and the convolution theorem [30], we arrive at            ∂ 2 ν U 0 (ξ, ν) − ξ U 0 (ξ, ν) + iν 2π ∞ −∞ U 0 (t, ν) t − (ξ − i0) dt = 0, ∂ ν U 0 ν=0 = I (ξ). (23a) (23b) Here, I(ξ) = ∞ 0 e −i σξ+ σ 3 3 dσ(24) is an inhomogeneous Airy function (Appendix A), and denotes the differentiation with respect to ξ. The function I(ξ) vanishes in the lower half of the complex plain C − as |ξ| → ∞. Assume that U 0 is analytic with respect to ξ and decreases to zero in C − as |ξ| → ∞. 2 The integral in (23a) can be evaluated with the help of Jordan's lemma and residue theorem, and problem (23) takes the form      ∂ 2 ν U 0 (ξ, ν) − (ξ − ν) U 0 (ξ, ν) = 0, ∂ ν U 0 ν=0 = I (ξ). (25a) (25b) The desired solution of (25) is U 0 (ξ, ν) = − I (ξ) w 1 (ξ) w 1 (ξ − ν),(26) where w 1 is the Airy function in Fock's definition (Appendix A). We choose this particular solution of the Airy equation (25a), guided by the same argument as in Fock [7] and Babich and Kirpichnikova [9], as it corresponds to the outgoing wave for the harmonic time-dependence e −ikt , which we omit. Therefore, U 0 (σ, ν) = − 1 2π ∞ −∞ I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ dξ.(27) The zeroes of w 1 are located at the upper half-plane; thus, the integrand is analytic with respect to ξ in C − and decreases there as |ξ| 1 (Appendix A). Hence, U 0 ≡ 0 as σ ≤ 0, and hereinafter we assume σ > 0. The main term of the attenuation factor for the total wavefield is, according to (6), the sum of the main terms of (14) and (18): W 0 = θ(σ)e i(σν− σ 3 3 ) − 1 2π ∞ −∞ I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ dξ.(28) This can be equivalently rewritten in the form W 0 = 1 2π ∞ −∞ I(ξ − ν) − I (ξ) w 1 (ξ) w 1 (ξ − ν) e iσξ dξ,(29) convenient for studying the wavefield in the shadow zone. The expression (27) is obtained under the assumption that the inequalities (19) are satisfied. When neither σ nor ν is large (the area indicated in Fig. 3 by D 1 ), the integral does not allow for a high-frequency asymptotic simplification. Henceforth, we investigate (27) in areas D 2 -D 6 where at least one of σ and ν is large, ν + σ 1.(30) The condition (30) guarantees that all integrals that will be encountered below will contain a large parameter and can be subjected to an asymptotic analysis. The foregoing analysis relies on a well-developed asymptotic evaluation technique of the integrals of rapidly oscillating functions [31]. For different parts of the integration interval, all or some special functions in (27) can be replaced by their approximations. We will transform expression (27) of Fock [7], Brown [8] and Babich and Kirpichnikova [9], implying the simplification of integrands in small vicinities of critical points of respective phase functions. We matched the asymptotics of (27) derived in the well-illuminated area D 2 with the cylindrical wave (7), which yields an expression for the diffraction coefficient. The approximation of (29), which we constructed inside the deep shadow area D 6 , allows matching with the Friedlander-Keller formulas. Neighboring areas defined by the corresponding inequalities intersect in the vicinity of their common boundaries, and the respective asymptotic formulas that we construct match in these overlapping zones. Illuminated area D 2 Let the observation point be positioned far above the limit ray in the illuminated zone D 2 (Fig. 3), where ν 1, ν − σ 2 σ.(31) In polar coordinates, the second inequality means that (17) ϕ (h/k) 1 3 .(32) We separately considered segments of the real axis where the integrand of (27) behaves differently. 1 On the half-line −ξ 1, all special functions in (27) can be replaced by their approximations (Appendix A). The integrand takes the form I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ = √ πe iΨ1(ξ)−i π 4 (ν − ξ) 1 4 − e iΨ2(ξ) (ν − ξ) 1 4 (−ξ) 9 4 × 1 + O (−ξ) − 3 2 + O (ν − ξ) − 3 2 ,(33) with the phases Ψ 1 (ξ) = 2 3 (ν − ξ) 3 2 + σξ,(34)Ψ 2 (ξ) = 2 3 (ν − ξ) 3 2 − 2 3 (−ξ) 3 2 + σξ.(35) The phase Ψ 1 has a unique critical point ξ 1 where Ψ 1 (ξ 1 ) = 0: ξ 1 = ν − σ 2 .(36) Under the condition (31), ξ 1 1, whence the first term on the right-hand side of (33) has no critical point on the half-line under consideration and its contribution is negligible. Now, we address the second term. The equation for the critical point ξ 2 of Ψ 2 is 0 = Ψ 2 (ξ 2 ) ≡ σ − (ν − ξ 2 ) 1 2 + (−ξ 2 ) 1 2 ,(37) whence (−ξ) 1 2 = (ν − σ 2 )/(2σ). Because the inequality (31) guarantees that the right-hand side is positive and large, the phase has a critical point ξ 2 = − ν − σ 2 2σ 2 ,(38) on the half-line under consideration. Segments |ξ| < const On the interval |ξ| < const, only the function w 1 (ξ − ν) in (27) can be replaced by its approximation (A.4), which gives I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ = I (ξ)e iΨ1(ξ)+i π 4 w 1 (ξ)(ν − ξ) 1 4 1 + O (ν − ξ) − 3 2 ,(39) with Ψ 1 introduced in (34). Under condition (31) its critical point is far outside the interval and the respective contribution can thus be ignored. We split the half-line ξ 1 into three pieces: 1 ξ ν, |ξ − ν| < const and ξ ν. Where 1 ξ ν, we can substitute into (27) I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ = − e Ψ3(ξ)−i π 4 ξ 9 4 (ν − ξ) 1 4 1 + O ξ − 3 2 + O (ν − ξ) − 3 2 (40) with Ψ 3 (ξ) = i σξ + 2 3 (ν − ξ) 3 2 − 2 3 ξ 3 2 .(41) On the interval |ξ − ν| < const, functions I (ξ) and w 1 (ξ) can be replaced by their approximations ((A.4) and (A.10)), but w 1 (ξ − ν) cannot. Hence, we have I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ = i ξ 9 4 w 1 (ξ − ν)e Ψ4(ξ) 1 + O ξ − 3 2 ,(42) where Ψ 4 (ξ) = iσξ − 2 3 ξ 3 2 .(43) Finally, as ξ ν, all special functions can be asymptotically approximated ((A.4) and (A.10)), and the integrand becomes I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ = ie Ψ5(ξ) ξ 9 4 (ξ − ν) 1 4 1 + O ξ − 3 2 + O (ξ − ν) − 3 2 ,(44) with Ψ 5 (ξ) = iσξ − 2 3 ξ 3 2 − (ξ − ν) 3 2 .(45) From formulas (40)-(45) we conclude that the contribution of semi-axis ξ 1 to (27) is exponentially small. Matching with cylindrical wave The analysis above shows that only the critical point (38) contributes to (27). Then, the standard stationary phase method [31] yields U 0 = 2e −i π 4 √ πσ (2σ) 4 (ν − σ 2 ) 4 e iΨ2(ξ2) 1 + O σ 3 (ν − σ 2 ) 3 ,(46) where Ψ 2 (ξ 2 ) = 1 2 σν + ν 2 2σ − σ 3 6 .(47) Relations (16) and (17) allow us to match the expression (46) with a cylindrical wave (7) in the area where σ 1, but the remainder terms in (16) are small, which can be written as follows: k(x − s) 2 s.(48) The matching easily gives the following formula for the diffraction coefficient A(ϕ; k): A(ϕ; k) = 2 π h k 2 ϕ 4 e −i π 4 .(49) This formula is in agreement with the result of A. Popov [1]. Similar to the case of non-tangential incidence (see, e.g., [26,28,29]), the diffraction coefficient is linear in the jump of the curvature h, but the singularity when approaching to the limit ray is different. We establish that in the area described by kr 1 and (32) the total wavefield is the sum of incident and diffracted waves. Preliminary analysis of Fock-type integrals in the penumbra Investigations on wavefields in the penumbra for a smooth contour [7,8,9] suggest a transformation of the expression (27). First, the classical Airy function w 1 (Appendix A) is connected with inhomogeneous Airy functions I (24) and H, H(z) = ∞ 0 e zt− t 3 3 dt,(50) by the simple relation √ πw 1 (z) = iI(z) + H(z).(51) This identity allows us to rewrite (27) as follows: U 0 = F + D + G.(52)F = i 2 √ π 0 −∞ w 1 (ξ − ν)e iσξ dξ,(53)D = − i 2π 0 −∞ H (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ dξ(54) and G = − 1 2π ∞ 0 I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ dξ.(55) The decomposition (52), as we will see further, is essential in penumbral areas D 4 and D 6 and will be helpful in the illuminated area D 3 and shadow zone D 7 (Fig. 3). We study functions F , D and G, which we refer to as Fock-type integrals, separately by applying a specific analytical technique developed by Babich and Kirpichnikova [9]. Function F corresponds, by Fock's terminology, to the Fresnel part of the wavefield. Functions D and G are analogous to Fock's background part of the wavefield. Function F As ν 1, replacing the function w 1 by its asymptotic expansion (A.4) gives F = − e −i π 4 2 √ π 0 −∞ e iΨ1(ξ) (ν − ξ) 1 4 1 + O (ν − ξ) − 3 2 dξ,(56) where the phase Ψ 1 introduced in (34) has the critical point ξ 1 (36). When ξ 1 is negative and not small, it contributes to the integral; however, when ξ 1 is positive and not small, it does not. These cases correspond to the location of the observation point not too close to the limit ray in the shadow zone and illuminated region, respectively. Accordingly, these areas are characterized by the inequality ν 1 4 | √ ν − σ| 1,(57) which is ( (16), (17)) equivalent to krϕ 2 1.(58) If the critical point of phase ξ 1 and the endpoint 0 merge, then ξ 1 contributes to the integral whenever it is negative or positive. The standard stationary phase method is not applicable; however, the expression (56) can be rewritten in terms of the Fresnel integral Φ(z) = e −i π 4 √ π z −∞ e it 2 dt,(59) as follows. Consider a very narrow neighborhood of the limit ray where the inequalities ν 1, | √ ν − σ| 1(60) hold true. In polar coordinates the second one has the form |ϕ| (h/k) 1 3 .(61) Under the condition (60), |ξ 1 | √ ν, and the main contribution to the integral (56) is given by the interval 0 ≤ −ξ √ ν. Expanding the integrand of (56) in powers of 1/ν yields Ψ 1 (ξ) = 2 3 ν 3 2 + ξ(σ − √ ν) + ξ 2 4 √ ν + O ξ 3 ν 3 2 .(62) On the interval under consideration, the cubic term in ξ and higher-order terms are small. Taking a quadratic approximation of the phase and main-term approximation of amplitude and extending integration to the half-line (−∞−iε, 0], ε > 0, we derive the following expression for F : F = − e −i π 4 2 √ π e i 2 3 ν 3 2 ν 1 4 0 −∞−iε e i (σ− √ ν)ξ+ ξ 2 4 √ ν 1 + O ξ 3 ν 3 2 dξ.(63) This can be immediately rewritten in terms of the Fresnel integral (59) F = −e iΘ Φ (−Z) 1 + O ( √ ν − σ) 3 (64) with Θ = 2 3 ν 3 2 − √ ν(σ − √ ν) 2 , Z = ν 1 4 ( √ ν − σ).(65) Under the condition (60), the remainder terms in (64) are small. In the area described by (60), the phase of the exponential and the argument of the Fresnel integral allow for the following geometrical interpretation (see (15)- (17)): Θ ≈ k(x − s), Z ≈ kr 2 ϕ.(66) Variable Z is typical of wavefields description in areas where waves of different natures merge [3]. In a part of the area characterized by (60) (namely, where (57) holds), Z can be large, and the Fresnel integral allows an asymptotic expansion. Functions D and G We start with the analysis of the function D (54). In the part of the integration interval where const < ξ ≤ 0, the function H (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ = ie iΨ2(ξ) (ν − ξ) 1 4 (−ξ) 9 4 1 + O (−ξ) − 3 2 + O (ν − ξ) − 3 2 .(67) Here, Ψ 2 is introduced in (35). In Section 4, we show that under the condition (31), phase Ψ 2 has one critical point ξ 2 (38), of which the contribution to U 0 has the form (46). Now, we consider the vicinity of the limit ray, where the inequalities ν 1, √ ν − σ ν 1 8(68) hold true. In polar coordinates, they read kh 2 r 3 1, krϕ 4 (hr) 2(69) The area characterized by (68) is wider than that described by (60). The inequality (68) implies that −ξ 2 ν 1 4 , whence the main contribution to D is given by the segment 0 ≤ −ξ ν H (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ = H (ξ) w 1 (ξ) e iΨ1(ξ) (ν − ξ) 1 4 1 + O (ν − ξ) − 3 2 .(70) Here, Ψ 1 is introduced in (34). Then, we expand the integrand in powers of 1/ν. Phase Ψ 1 is rewritten as (62). As |ξ| ν 1 4 , the quadratic term in ξ and higher-order terms in (62) are small and can be discarded. Therefore, we rewrite the integrand as follows: e i 2 3 ν 3 2 ν 1 4 H (ξ) w 1 (ξ) e i(σ− √ ν)ξ 1 + O ξ 2 √ ν + O ξ ν + O (ν − ξ) − 3 2 .(71)ξ 2 = −( √ ν − σ) 2 ,(72) provided that √ ν − σ 1, cf. (31). The condition (68) implies that − ξ 2 ν 1 4 , whence the integral of (71) over the half-line −ξ ≥ const ν 1 4 is negligible. Therefore, under the conditions (68), function D has the following asymptotic representation: D = e −i π 4 2π e i 2 3 ν 3 2 ν 1 4 0 −∞ H (ξ) w 1 (ξ) e iξ(σ− √ ν) dξ (1 + o (1)) .(73) A similar manipulation with G yields G = − e i π 4 2π e i 2 3 ν 3 2 ν 1 4 ∞ 0 I (ξ) w 1 (ξ) e iξ(σ− √ ν) dξ (1 + o(1)) .(74) Penumbral area D 4 Let the observation point be positioned inside the penumbral region D 4 characterized by inequalities (60). The results in Section 5 imply the following representation of (28): W 0 = e iΘ Φ (Z) 1 + O ( √ ν − σ) 3 + e −i π 4 2π e i 2 3 ν 3 2 ν 1 4   0 −∞ H (ξ) w 1 (ξ) e iξ(σ− √ ν) dξ − i ∞ 0 I (ξ) w 1 (ξ) e iξ(σ− √ ν) dξ   (1 + o(1)) ,(75) where Θ and Z were introduced in (65). The first term on the right-hand side of (75) is the Fresnel part of the wavefield, which is exactly the same as that in the Fock case [7,8,9], and, like The inequality (60) characterizes the transition zone where u inc cannot be considered as an individual wave. Illuminated area D 3 We resume consideration of the illuminated area, addressing its part D 3 (Fig. 3), where conditions (57) and (68) hold and ϕ > 0.(76) Here, we employ the representation (52) for the outgoing wavefield. Particularly, (68) describes the area wider than that characterized by (60). Thus, the representation (64) for the function F (53) is not applicable in the entire area D 3 . We consider function F (53), starting with its approximation (56). Although the phase Ψ 1 (34) of the integrand in (56) has a critical point ξ 1 (36), it is positive and large when (76) and (57) hold and does not contribute to F . Thus, it is sufficient to consider only a small neighborhood of endpoint 0. As in Section 5.1, we expand the integrand of (56) in powers of 1/ν, and the phase takes the form (62). Now, because |ξ| is small, we discard not only the cubic term in ξ and higher-order terms but also the quadratic term. We retain the linear term in the phase and then extend the integration to the half-line (−∞ − iε, 0], ε > 0. This manipulations result in the following approximation for F : F = e −i π 4 2 √ π e i 2 3 ν 3 2 ν 1 4 0 −∞−iε e i(σ− √ ν)ξ dξ (1 + o(1)) .(77) Although the integral (77) can be explicitly evaluated, it is convenient to leave it as it is. We now sum the functions F , D and G, utilizing their representations (77), ∞e −iπ/3 0 √ π − i I (ξ) w 1 (ξ) e i(σ− √ ν)ξ dξ (1 + o(1)) = e −i π 4 2π e i 2 3 ν 3 2 ν 1 4 ∞e −iπ/3 0 H (ξ) w 1 (ξ) e i(σ− √ ν)ξ dξ (1 + o(1)) . (78) Finally, using the expression (73) for D, we come up with U 0 = F + D + G = e −i π 4 2π e i 2 3 ν 3 2 ν 1 4    0 −∞ + ∞e −iπ/3 0    H (ξ) w 1 (ξ) e iξ(σ− √ ν) dξ (1 + o (1)) .(79) We consider the expression (79) well apart from the limit ray in the illuminated area, where the condition (31) is satisfied. On the segments of integration contours where |ξ| 1, the special functions on the right-hand side of (79) can be replaced by their approximations (Appendix A). Now, for the integral over [0, ∞e −iπ/3 ), it can be easily shown that its asymptotics is given by the contribution of the endpoint 0. We consider the integral over (−∞, 0]. As follows from the results in Section 5.2, its asymptotics is given by the contributions of the critical point (72) and the endpoint 0, and the latter fully cancels with the aforementioned approximation of the integral over [0, ∞e −iπ/3 ). Consequently, the asymptotics of (79) is given by the critical point of the phase: U 0 = 2e −i π 4 √ πν 1 4 ( √ ν − σ) 4 e i 2 3 ν 3 2 + ( √ ν−σ) 3 3 (1 + o(1)).(80) When the conditions (31) and (68) are satisfied, this formula agrees with the expression (46), which matches the diffracted wave. Shadow area D 5 Now, we consider the wavefield in the shadow area D 5 , ϕ < 0,(81) described by the inequalities (57) and (68) (Fig. 3). Preliminary transformation of the integral (29) Using the relation (51) connecting functions w 1 , I and H, we rewrite the expression (29) for the attenuation factor of the total wavefield as follows: W 0 =   i 2π 0 −∞ H(ξ − ν)e iσξ dξ + 1 2π ∞ 0 I(ξ − ν)e iσξ dξ   − 1 2π ∞ 0 I (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ dξ − i 2π 0 −∞ H (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ dξ.(82) We observe that i 2π 0 −∞ H(ξ − ν)e iσξ dξ + 1 2π ∞ 0 I(ξ − ν)e iσξ dξ = − i 2 √ π ∞e iπ/3 0 w 1 (ξ − ν)e iσξ dξ.(83) Let us transform the integral on the right-hand side of (83). Using the asymptotic formula (A.4) for w 1 (ξ − ν), we observe that the integrand decays exponentially on segments |ξ − ν| < const and |ξ| ν as |ξ| increases. Hence, up to an exponentially small error, it is sufficient to deal with the segment 0 ≤ |ξ| ν, whence − i 2 √ π ∞e i π 3 0 w 1 (ξ − ν)e iσξ dξ = e −i π 4 2 √ π 0≤|ξ| ν arg ξ=π/3 e iΨ1(ξ) (ν − ξ) 1 4 1 + O (ν − ξ) − 3 2 dξ(84) with Ψ 1 introduced in (34). Next, we expanded the integrand in (84) Expressions similar to (85)-(86) can be found in [9]. Substituting the expansion (85) into (84) and then extending the integration to the half-line arg ξ = π/3, we obtain, with superpower accuracy, Similarly, with superpower accuracy, we derive the approximation for the second term on the right-hand side of (82): − i 2 √ π ∞e iπ/3 0 w 1 (ξ − ν)e iσξ dξ = e −i π− 1 2π ∞ 0 I (ξ) w 1 (ξ) w 1 (ξ−ν)e iσξ dξ = − e iπ/4 2π e i 2 3 ν 3 2 ν 1 4 ∞e iπ/4 0 I (ξ) w 1 (ξ) e i(σ− √ ν)ξ (1 + Q) dξ,(88) and for the third term: − i 2π 0 −∞ H (ξ) w 1 (ξ) w 1 (ξ − ν)e iσξ dξ = e −i π 4 2π e i 2 3 ν 3 2 ν 1 4 0 −∞ H (ξ) w 1 (ξ) e iξ(σ− √ ν) (1 + Q) dξ.(89) We now return to (82). Summing the right-hand sides of (87) and (88) with the help of (51), we derive e −i π 4 2π e i 2 3 ν 3 2 ν 1 4 ∞e iπ/4 0 √ π − i I (ξ) w 1 (ξ) e i(σ− √ ν)ξ (1 + Q) dξ = e −i πW 0 = e −i π 4 2π e i 2 3 ν 3 2 ν 1 4    0 −∞ + ∞e iπ/4 0    H (ξ) w 1 (ξ) e i(σ− √ ν)ξ (1 + Q) dξ(91) describing W 0 with superpower accuracy. The integrand in (91) has an infinite number of poles at zeroes {ζ j } ∞ j=1 of function w 1 (Appendix A) located at C + . When the observation point goes from the shadowed portion of penumbra deeper into the shadow, and −ϕ (h/k) 1 3 (i.e. σ − √ ν 1; see (17)), the evaluation of the integral by residues becomes feasible. We delayed the discussion of such a representation to Section 9. Deep shadow area D 6 We address the wavefield in the area D 6 in the vicinity of contour C (Fig. 3), as described by σ 1, σ 2 − ν σ.(92) In polar coordinates, this reads kh 2 r 3 1, −ϕ (h/k) 1 3 ,(93) cf. (31). Further calculation is much in the same way as that presented by Babich and Kirpichnikova [9]. The integrand in (29) has an infinite number of poles at zeroes {ζ j } ∞ j=1 of w 1 (ξ), which are located at the half-line arg ξ = π/3 (Appendix A). The asymptotic formulas for functions I and w 1 and their derivatives ((A.4) and (A.10)) shows that integrand decays as |ξ| −1 when | ξ| → ∞ and ξ > 0. Raising the integration contour allows for the representation W 0 = N j=1 A j w 1 (ζ j − ν)e iσζj + E ,(94) with E = e −εσ 2π ∞ −∞ I(ξ + iε − ν) − I (ξ + iε) w 1 (ξ + iε) w 1 (ξ + iε − ν) e iσξ dξ, where ζ N < ε < ζ N +1 , N is a positive integer, and A j = − iI (ζ j ) ζ j w 1 (ζ j ) .(96) Similarly to the Fock case, the remainder E is of a smaller order than each of the residues: E decays as e −εσ , whereas the residues decay as e −σ ζj , and 0 < ζ j < ε [9]. Constants A j , j = 1, . . . , N , can be interpreted as the excitation coefficients of creeping modes, and they differ from those found for the Fock problem by Babich and Kirpichnikova [9]. At a distance from the contour C, such that ν 1, functions w 1 (ζ j − ν) can be replaced by their approximations (A.4), which yields W 0 = e i π 4 ν 1 4 N j=1 A j e i 2 3 ν 3 2 +ζj (σ− √ ν) 1 + O 1 √ ν + E .(97) Accounting the equations (51) and w 1 (ζ 1 ) = 0, one can easily observe that I (ζ j ) = iH (ζ j ) and rewrite formula (96) as A j = − iH (ζ j ) ζ j w 1 (ζ j ) ,(98) which makes it possible to indicate that (97) agrees with the representation for W 0 resulting from the evaluation of the integral in (82) by residues. Moreover, the expression (97) matches with the Friedlander-Keller asymptotic formulas (see, e.g., [9]). Conclusions We have explored the simplest problem of diffraction on non-smooth contours, where the incidence is tangential and the contour curvature changes with jump. We hope that the developed technique will be useful in the study of similar problems with concave-convex transitions, which attract significant attention in the smooth case (see, e.g., [12,13,14,15,16,17,18,19,20,21]). Our consideration, similarly to the pioneering research by Fock [6,7], was limited to the small area characterized by inequalities (19) resulted from the analysis of approximations which led to the parabolic equation. In subsequent papers [8,9,32,33,34,35,36], steps were taken towards describing the field at a larger distance. Perhaps, these techniques will be helpful in the further study of the Malyuzhinets-Popov problem. as follows: I(z) = πe −i π 6 Hi ze −i 2π OFigure 1 : 1of the contour C in accordance with the Geometrical Theory of Diffraction Geometry of the problem (GTD) It is natural to characterize the position of an observation point M near point O by coordinates (s, n), where n ≥ 0 is the length of the perpendicular from M to the contour C and s is the length of the arc between O and the foot of the perpendicular, as shown in Fig. 2. These coordinates are orthogonal; Figure 2 : 2Coordinates s and n factor and introducing stretched coordinates, leading to a problem for the socalled parabolic equation. Figure 3 : 3Schematic sketch of areas under consideration asymptotic expressions for w 1 (ξ − ν), I (ξ) and w 1 (ξ) ((A.4) and (A.10)), and the integrand takes the form w 1 1(ξ − ν) can be replaced solely by its asymptotic expansion (A.4). As established in Section 4, the derivative of the phase of the integrand vanishes at the point ξ 1 (36). Even if the critical point ξ 1 lies inside the interval under consideration, it gives no stationary-phase contribution because the integrand does not rapidly oscillate there. On the half-line −ξ 1, all special functions can be replaced by their approximations ((A.4) and (A.9)), which gives 1 4 . 4We replace w 1 (ξ − ν) by its approximation (A.4): On the half-line −ξ 1, the functions w 1 and H can be replaced by their approximations (A.4) and (A.9), respectively, and we arrive at an integrand with the phase Ψ 2 = to that concerning the phase Ψ 2 ; see(35)-(38)) shows that Ψ 2 has exactly one critical point theirs, does not depend on the curvature of the contour (66). The second term in (75) is analogous to Fock's background part of the wavefield and transforms into it after replacing functions I and H with the classical Airy functions v and w 2 (Appendix A), respectively. Similarly to Fock's case, the Fresnel part is superimposed on the background. ( 73 )F 73and (74), and obtain the expression for U 0 . We deform the integration contours in (77) and (74) to the half-line arg ξ = −iπ/3. Accounting (51) allows + in powers of 1/ν, where Ψ 1 takes the form of (62). We observed an Ψ 1 > 0 on the integration interval. Indeed, in accordance with (81), ξ(σ − √ ν) > 0. Evidently, the imaginary part of the quadratic term in ξ is also positive. Thus, the integrand exponentially decays as |ξ| increases, and the asymptotics of the integral is given by the contribution of the endpoint 0. We retain only the term linear in ξ in the phase and finally obtain w 1 (ξ − ν)e iσξ = Q is the expansion in the positive integer powers of ξ ξ (1 + Q) dξ. (90) Employing equation (89), we finally come up with the formula Figure A. 4 : 4Behavior of functions w 1 (z), I(z) and H(z) in the complex z-plane solving the inhomogeneous Airy equation w (z) − zw(z) = 1 π (A.7) This assumption is a form of causality condition and is equivalent to the condition U 0 (σ) = 0 for σ < 0. , H(z) = πHi(z).(A.8) Appendix A. Airy and inhomogeneous Airy functionsHere, some properties of Airy and inhomogeneous Airy functions[37,38]are briefly summarized.The notation w 1 (z) was introduced by Fock[7]for a particular solution to the homogeneous Airy equation:defined by the integral representationwhere contour γ goes from infinity to zero along the line arg t = −2π/3 and then from zero to infinity along the line arg t = 0. Function w 1 is related to another common notation[38]as follows:Both function w 1 and its derivative w 1 have an infinite number of simple zeroes located on the line arg z = π/3. 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[ "A Coarse-Grained Simulation Model for Colloidal Self- Assembly via Explicit Mobile Binders", "A Coarse-Grained Simulation Model for Colloidal Self- Assembly via Explicit Mobile Binders", "A Coarse-Grained Simulation Model for Colloidal Self- Assembly via Explicit Mobile Binders", "A Coarse-Grained Simulation Model for Colloidal Self- Assembly via Explicit Mobile Binders" ]
[ "Gaurav Mitra ", "Chuan Chang ", "Angus Mcmullen ", "Daniela Puchall ", "Jasna Brujic ", "Glen M Hocky ", "Gaurav Mitra ", "Chuan Chang ", "Angus Mcmullen ", "Daniela Puchall ", "Jasna Brujic ", "Glen M Hocky " ]
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Colloidal particles with mobile binding molecules constitute a powerful platform for probing the physics of self-assembly. Binding molecules are free to diffuse and rearrange on the surface, giving rise to spontaneous control over the number of droplet-droplet bonds, i.e., valence, as a function of the concentration of binders. This type of valence control has been realized experimentally by tuning the interaction strength between DNA-coated emulsion droplets. Optimizing for valence two yields droplet polymer chains, termed 'colloidomers', which have recently been used to probe the physics of folding. To understand the underlying self-assembly mechanisms, here we present a coarse-grained molecular dynamics (CGMD) model to study the self-assembly of this class of systems using explicit representations of mobile binding sites. We explore how valence of assembled structures can be tuned through kinetic control in the strong binding limit. More specifically, we optimize experimental control parameters to obtain the highest yield of long linear colloidomer chains. Subsequently tuning the dynamics of binding and unbinding via a temperature-dependent model allows us to observe the heptamer chain collapse into all possible rigid structures, in good agreement with recent folding experiments. Our CGMD platform and dynamic bonding model (implemented as an open-source custom plugin to HOOMD-Blue) reveal the molecular features governing the binding patch size and valence control, and opens the study of pathways in colloidomer folding. This model can therefore guide programmable design in experiments.
10.1039/d3sm00196b
[ "https://export.arxiv.org/pdf/2212.11946v2.pdf" ]
254,974,479
2212.11946
5b0867c9f80c7f6f0d3a43dbae6bf3ff5f3f5bf9
A Coarse-Grained Simulation Model for Colloidal Self- Assembly via Explicit Mobile Binders Gaurav Mitra Chuan Chang Angus Mcmullen Daniela Puchall Jasna Brujic Glen M Hocky A Coarse-Grained Simulation Model for Colloidal Self- Assembly via Explicit Mobile Binders Colloidal particles with mobile binding molecules constitute a powerful platform for probing the physics of self-assembly. Binding molecules are free to diffuse and rearrange on the surface, giving rise to spontaneous control over the number of droplet-droplet bonds, i.e., valence, as a function of the concentration of binders. This type of valence control has been realized experimentally by tuning the interaction strength between DNA-coated emulsion droplets. Optimizing for valence two yields droplet polymer chains, termed 'colloidomers', which have recently been used to probe the physics of folding. To understand the underlying self-assembly mechanisms, here we present a coarse-grained molecular dynamics (CGMD) model to study the self-assembly of this class of systems using explicit representations of mobile binding sites. We explore how valence of assembled structures can be tuned through kinetic control in the strong binding limit. More specifically, we optimize experimental control parameters to obtain the highest yield of long linear colloidomer chains. Subsequently tuning the dynamics of binding and unbinding via a temperature-dependent model allows us to observe the heptamer chain collapse into all possible rigid structures, in good agreement with recent folding experiments. Our CGMD platform and dynamic bonding model (implemented as an open-source custom plugin to HOOMD-Blue) reveal the molecular features governing the binding patch size and valence control, and opens the study of pathways in colloidomer folding. This model can therefore guide programmable design in experiments. Introduction Self-assembly of colloidal materials can create non-trivial and programmable structures with wide-ranging and tunable material properties. The spatio-temporal visualization of colloids renders them as useful model systems for probing the underlying physics behind assembly processes of molecular systems [1][2][3][4] . The synthesis of colloidal particles with chemically or physically patterned solid surfaces-"patches"-has been an active area of research due to the desire to control the bond valence and orientation, going beyond what can be achieved through isotropic interactions [4][5][6][7] . For example, a long-desired target is a diamond lattice, with an open structure that is difficult to achieve without an imposed tetrahedral symmetry 8 . The most common approach to engineering specific interactions between patches is to use complementary neighbors with complementary binding molecules, minimizing the total free energy of the system. Experimentally, using oil droplets in water provides a mobile interface on which the DNA linkers segregate into patches by diffusion to give rise to e.g., dimers when all the DNA is recruited into a single patch or droplet chains at higher concentrations of DNA where two patches per droplet are favored 23 . Alternatively, solid colloids can be coated with fluid lipid bilayers 24,25 . The mobility of the linkers broadens the melting temperature window of the DNA, facilitating equilibrium self-assembly 2,26 . Particles with mobile binders can also serve as a physical mimic for biological adhesion, where cells use a variety of dynamic binding molecules to stick to each other and to surfaces 27 . To this end, biomimetic emulsion droplets have been successfully functionalized with adhesive mobile proteins, such as biotinstreptavidin complexes 28,29 , cadherin ectodomains 30 , or other ligand-binder pairs relevant for immunotherapy 31 . Early theoretical work expanding on the model of Ref. 20 predicts that such particles could have an equilibrium valence that depends on the number of available neighboring particles in the system 32 . Our previous work shows that monodisperse PDMS oil droplets functionalized with different flavors of singlestranded DNA on the surface can self-assemble into structures of tunable valence 23,25,[33][34][35][36] . Under conditions where DNA bonds are reversible at room temperature, these systems achieve their equilibrium valence configuration. These results are predicted by a free-energy functional that takes into account the molecular properties of the system, including DNA binding strength, flexibility, steric repulsion, and concentration 37 . Optimizing for valence two, the self-assembly of complementary DNA-coated droplets yields linear colloidomer chains 23 . Further programming the secondary interactions along the chains offers a physical model system to probe the energy landscape of biopolymer folding, and for building small 'foldamer' structures that can serve as the basis for larger scale assemblies [38][39][40] . A complementary work demonstrated the formation of reconfigurable colloidal molecules using polydisperse droplets surrounded by ligands 41 . Motivated by these experimental studies, our work develops a coarse-grained molecular dynamics (CGMD) simulation model and framework to study the self-assembly of these colloidal chains with mobile binders, and their subsequent folding. The crucial feature of our model is the use of explicit mobile linkers with bonds between complementary binding partners. Prior work on these types of systems used implicit models of binding between neighboring droplets via the formation of adhesion patches, and some models included approximations to account for the dynamics of adhesion 32,[42][43][44][45][46] . The use of explicit mobile binders allows us to test the underlying assumptions in a more realistic model, albeit at the cost of additional complexity. For example, our model explicitly shows how steric repulsion between binders (designed to mimic electrostatic repulsion between DNA strands) affects the adhesion patch size and the concentration of binders therein. These results in turn explain the overall valence distribution that results from the assembly process. To demonstrate the capabilities of our model, we describe the parameters that optimize the formation of colloidal chains under kinetic control, and show that it exhibits the folding behavior of twodimensional colloidal chains commensurate with what has been recently demonstrated experimentally 40 . 2 Description of the model Coarse-Grained model for colloidal particles with explicit mobile binders The central unit of our simulation model is a droplet, as shown in Fig. 1a. Each droplet consists of a central spherical particle of radius R (type A) with N b binders distributed on the surface. The positions of the binders are initialized in a "Fibonacci" arrangement to prevent overlap between particles in the initial configurations 47 . Each binder is composed of two particles (types B and C) -the outer particle is responsible for binding complementary partners, while the inner one is used to modulate excluded volume between binders. This pair of particles mimics the combination of the double-stranded tether and the single-stranded sticky-end DNA used in experiments 23 (Fig. 1a). This configuration also allows us to apply an angular term to tune the propensity of the binder to stand vertically from the surface. The binders diffuse on the surface 25 due to a harmonic bond between the center of the droplet and that of the inner binder particle with a spring constant k AB and rest length l 0 AB . The inner and outer particles in the binder are similarly held together by a harmonic bond with spring constant k BC and l BC = r B + r C , the sum of their radii. To have the binder stick outward from the droplet, the two binder particles are forced to align along the radial vector from the center of the droplet, by introducing a harmonic angular potential between the triplet of particles with parameters k ABC and rest angle θ 0 ABC = 180 • . In each case, the spring constants are chosen such that the thermal standard deviation σ = k B T /k (where k B is the Boltzmann's constant) is a small fraction of the rest length or angle (see Tab. S3). As a trade-off between enforcing rigid bonds and using an infinitesimal MD time step, we chose to use spring constants where the standard deviation is 1 − 2%. Fig. 1 (a) The initial configuration of a droplet with binders adhered to its surface, arrayed in their initial "Fibonacci" structure. As shown in the inset, each binder consists of two constituent particles, which in the case of DNA corresponds to a spacer double-stranded sequence and a single stranded sequence which is available to bind to a complementary strand. (b) A schematic showing dynamic binding between the outer binder particles of two droplets. Different particle types used in our python framework are schematically labeled A-D, with C and D representing a pair of complementary binders. Non-bonded interactions To prevent overlap between particles, we use a soft repulsion given by Eq. A1, which is applied between all particle types except between pairs of outer binder particles of complementary types (otherwise repulsion can prevent binding). By tuning the effective diameter of B particles, we tune the steric repulsion between adjacent binders, which corresponds to the case of altering the screening of the electrostatic interactions between DNA strands on the surface 37 . Our CGMD model can be used to study droplet interactions in three dimensions, but we implement 2D confinement for comparison with recent experiments. To replicate this quasi two-dimensional arrangement in our system, we confine droplets above and below using a forceshifted Lennard-Jones wall potential 48 on each droplet, between a fixed z-position and the center of the droplet A particle. The origins of the walls are given by (0, 0, 2.5R) and (0, 0, −2.5R). Dynamic bonding model We model interactions between binders through covalent bonds. To do so, we develop a plugin to HOOMD-Blue 49 50 that builds upon a model for epoxy binding developed in Ref. 51. Adhesive bonds form only between complementary outer binder particles of respective droplets, as shown in Fig. 1b. In the simplest case, we have a mixture of droplets containing outer binder particles that are 100% of types C and D, respectively. The model allows for individual droplets to contain mixtures of binder types, and there may be many more than two types, as designated by the user. In this study, harmonic bonds are added with spring constant k dyn and length l dyn = r C + r D , the sum of the radii of particles forming a bond. Here, we choose harmonic bonds, but any form of bond implemented in HOOMD-Blue could be used, since our algorithm only changes the bond table within the MD simulation, and does not compute or apply forces. * The effective free energy difference between a bound and an unbound state in a two-state binding reaction is given by ∆G = k B T ln k on k off ≡ ε(1) Here, k on and k off are the rate constants for binding and unbinding, respectively. Below, we tune ε to modulate the affinity between individual binders. In Sec. A3 we describe details of the algorithm for binding and unbinding, satisfying detailed balance. In addition, the temperature dependence is also described (Sec. A4), in which the binding and unbinding rates are tuned as a function of temperature in such a way that the fraction of bound pairs at the equilibrium distance tends smoothly to zero at high T , with 50% bound pairs at a specified melting temperature T melt . Comparison of model and experiment geometry This CGMD model is generic, while Fig. A1 shows how two particles describe the experimental setup in Ref. 23. The radius of one binder sphere corresponds to the sticky end of length ∼ 5.1nm, which is our reduced unit of length. Therefore, a droplet radius of R = 300 corresponds to the droplet size of ∼ 1530nm used in Ref. 23. Smaller particle sizes studied in Sec. 4 are also used in experiments. Practically, our simulations are computationally limited to hundreds of binders per droplet. Therefore, we equate the scale of the simulations to experiments by matching Fig. 2 Illustration of adhesion patch formation for a dimer of droplets. Patch formation under conditions with high binding affinity occurs via a process with two time scales (τ 1 and τ 2 ), one for recruitment of most linkers into a patch, and a second proceeding to saturation. The conditions for this particular simulation are R = 50, N b = 100 and ε = 20.7. See Sec. S1.1 and Tab. S1 for more details. the excluded volume of all binders to that of DNA surface coverage in the experiment. More specifically, the surface coverage, p is defined as p = 4πr B 2 N b 4πR 2 = N b r B R 2 .(2) In the experiment, for droplets of radius R = 1530 nm, an effective repulsive radius of DNA of 1.5nm 37,52,53 , and an estimated 1 × 10 3 − 2 × 10 4 DNA strands 23,37 , p ranges from ∼ 0.001 − 0.02 or 0.1 − 2% coverage. For many simulations below, we use r B = 1, R = 50, and N b = 100, in which case p = 0.04. From this perspective, each binder plays the collective role of hundreds of DNA. Simulation methods MD simulations of droplets coated with mobile binders were performed using HOOMD-blue version 2.9.6 49,50,54 . The Langevin integrator 55 was used to integrate all particles forward in time. Two different values of the drag coefficient γ were used: one for the droplets (γ A ) and the other for the binders (γ binder ). The equation of motion for each particle i in Langevin dynamics 56 is given by: m i¨ r i t = F i − γ i˙ r i t + 2γ i k B T η t(3) where, m i is the mass of the particle, k B is the Boltzmann constant, γ i is the drag coefficient,˙ r i (t) is the velocity of the particle, F i = −∇U i is the force on particle i derived from the total potential energy function of the system, and η(t) is the delta-correlated random white noise, with zero mean and unit variance. We use the tree neighbor list 57,58 to accelerate non-bonded calculations, and in the construction of our list of possible pairs to bond as described above. Results and Discussion Main objectives In this section, we optimize simulation conditions to robustly self-assemble long colloidomers, without branched structures. We explore both the molecular properties of the system (droplet radius, binder concentration, and binder interaction strength), and the experimental conditions for assembly (particle concentration and solution viscosity). The detailed parameters used for our MD simulations are listed in tables S3 to S6. Our results indicate that kinetic factors can be rationally employed to target the desired outcome with high yield and fidelity for fixed-time experiments. We subsequently show that our model allows us to study the folding process for self-assembled colloidomer chains. Adhesion patch formation is a two-stage process for high bond strengths The formation of chains requires that each droplet has two contacting neighbors. Monomers first form dimers, after which they either combine with monomers to make trimers, or with other dimers to make tetramers. We therefore first probe the physical processes involved in forming a patch in a dimer or trimer configuration, and then consider de novo assembly in Sec. 4.5 . Simulations of patch formation begin with dimers and trimers in an initial configuration with a single bond already formed between the droplets. Subsequently, the patches progressively grow until they reach steady-state. We find that patch formation (at intermediate and high binding affinities, ε ≥ 10) happens in two stages, as illustrated in Fig. 2 (see Fig. S1). Fitting the fraction of unbound binders versus time with a double exponential function reveals two time scales, as shown in Sec. S1.1. Initially, the fast time scale of recruitment of binders describes the formation of a stable adhesion patch (τ 1 ), while the satura- tion of the patch is captured by a 1-2 orders of magnitude slower timescale (τ 2 ). Tab. S1 reports values of τ 1 and τ 2 for some of these conditions. Optimizing bond strength, droplet size, and binder concentration for colloidomerization Optimal conditions for colloidomerization require considering both the dimer and the trimer assembly, since we must find a condition where not all binders are exhausted in the dimer, where ∼ 50% of binders are available on a terminal droplet of a trimer, and where very few are left on the trimer middle droplet, preventing branching. Fig. 3a shows a systematic study of the self-assembly of droplets of varying R and ε, for a fixed total number of binders N b = 100. The heat maps show the fraction of unbound binders in three configurations: a droplet-droplet dimer, a terminal droplet in a trimer, and a central droplet in a trimer. Conditions for colloidomerization are satisfied in the region where the dashed ovals overlap. Later, we choose R = 50, ε = 20.7 and N b = 100 to demonstrate that this condition results in the robust assembly of colloidomers. The number of binders in a patch at equilibrium is dictated by the free energy of patch formation, which can be considered as the difference in the chemical potential inside and outside the patch 37 . The driving force for a binder to enter the adhesion patch is determined by the energy of forming individual bonds, ε, and opposed by steric repulsion between binders, the stretching of binders at the interface, as well as the loss of entropy as the binder motion is constrained in a patch. As ε increases, the fraction of free binders decreases monotonically in the case of both dimers and trimers until it reaches an asymptotic value that is limited by the steric repulsion between binders. Similarly, increasing droplet size in the strong binding limit recruits progressively more binders into the patch, since there is more space at the interface between larger droplets (Fig. 3a). For large R in both dimers and trimers, we observe that the transition from no binding to all binders in the patch is very sharp as energy gain overtakes entropic losses without a penalty from steric repulsion. In contrast, for small radii, the recruitment of binders into the patch is more gradual with ε due to crowding. In addition to binding enthalpy gain and crowding, there is also the entropic cost of patch formation, which increases with droplet size. Therefore, at intermediate values of ε we observe a non-monotonic recruitment of binders into the patch as a function of droplet size. While crowding dominates in small droplets, entropic costs dominate in large droplets, giving rise to an optimal size for binder recruitment, e.g. R = 120 for ε = 15 in the dimer configuration. We can now consider the transition from dimer to trimer, where two adhesion patches are formed. Competition between the two adhesion patches results in an even split of binders on the middle droplet; therefore, even in cases where more than half of binders can pack into a patch, only half of the binders on each terminal droplet are exhausted. This situation occurs for higher ε, whereas for weaker binding, entropy dominates and many binders remain outside of the two adhesion patches on both the central and termi-nal droplets. In Fig. 3b we fix ε = 20.7 at the goldilocks value from Fig. 3a, and explore the effect of varying binder surface coverage. In this strong binding regime, we eventually saturate the geometric limit set by the droplet size and binder repulsion (see also Fig. S2). If N b is increased above this limit, it simply results in additional free binders. For small droplets, we quickly reach the situation where not all binders on the middle droplet of a trimer can fit into two patches, but we see that for larger droplets there is a very wide tolerance for binder concentration which might be useful for colloidomerization. In summary, for certain parameters (including ε = 20.7, R = 50, N b = 100) we find that < 100% of the binders are recruited for dimers, while in trimers patches contain exactly half of the binders due to the competition between neighbors. This is an optimal situation for the self-assembly of colloidomers, and we proceed to study assembly of these droplets in Sec. 4.5. Every system is the same size, but each snapshot has identically sized field of view by area, meaning droplets are cropped at φ < 0.4. Droplets are colored according to structure as shown in the key. Each condition is accompanied with bar charts representing quantities computed over 10 independent runs that can quantitatively help in collectively deciding the 'winning condition' for colloidomer formation. These quantities are labeled in the second key, and described in detail in the main text, and for each quantity, optimal would be a larger bar. The few droplets that have valence 3 or higher are also marked with a 'cross' in the representative configurations to reinforce that even the branched structures which are considered part of 'errors' have long segments of droplets with valence=2. The condition we consider optimal, φ = 0. Fig. 4 illustrates scenarios that are predicted to be good or bad for colloidomer assembly, as described above; full trajectories for these conditions are also shown in Movies M1 and M2. Considering the dimer, we see that large droplets and high ε allow almost all the binders to fit into a patch, which is detrimental for colloidomer assembly because it terminates the polymerization reaction. Decreasing droplet size to R = 50 limits binders due to their steric repulsion in the patch, leaving just enough binders to seed a trimer with no remaining binders on the middle droplet, thus imposing the self-assembly of linear chains. Lowering the droplet size further or decreasing the binding strength leaves too many binders free on the trimer middle droplet at equilibrium, which would eventually lead to the self-assembly of branched colloidomers. Fig. 4b illustrates that for a fixed droplet size and bond strength, the number of binders must be chosen such that dimers have free binders to grow the chain, while trimers have no free binders for branching via the middle droplet, as is the case for N b = 100. On the other hand, N b = 50 has almost all the binders (97%) recruited in the case of the dimer, terminating colloidomer assembly. For N b = 300, the central droplet of a trimer has 21% binders remaining on the surface, which would result in the branching of colloidomers. Illustrating a molecular recipe for colloidomers Kinetic optimization of colloidomerization Beyond dimers and trimers, the self-assembly of chains requires further optimization of competing experimental timescales. Combining a 1:1 mixture of droplets containing complementary binders of particles type 'C' and 'D' mimics the experiments in Ref. 23. Tuning the density φ (area fraction) of droplets for a finite assembly time effectively controls the collision time between droplets and allows us to optimize the formation of long colloidomers out of equilibrium. Given that the droplets are undergoing simple diffusion, the collision time τ collision ∼ l 2 /D, where l is the distance between droplets and D is the diffusion constant. In two dimensions, l 2 ∼ φ −1 , such that the collision time is inversely proportional to the droplet density. The Einstein relation gives D = k B T /γ, therefore τ collision ∼ γ, the drag on a particle. As described in the simulation methods Sec. 3, γ for binder particles is kept small and constant, while γ on the droplet particle ' A' is varied. In Fig. 5, we show that by fixing the previously optimized R, N b , ε parameters and varying φ and γ, we can maximize the yield of colloidomers. Although we would predict that we would not observe any branching in equilibrium, since droplet collisions are fast in dense suspensions, we do observe many droplets with valence three at the highest initial density. Increasing γ slows down the collision rates and therefore improves the yield of linear chains and avoids loop formation (top row); full trajectories for the upper row are also provided in Movie M3. More quantitatively, we compute the distribution of structures produced at the end of 10 8 steps, averaging over 10 independent simulations. We partition every interconnected assembly using an algorithm described in Sec. S2.1, and then classify these structures as monomers, dimers, linear chains (N ≥ 3), loops, and 'other' (at least one droplet has valence ≥ 3). To define the best conditions for chain assembly, we sought to maximize three quantities that are shown in bar charts next to each condition in While conditions at φ = 0.4 have the longest chains and a high fraction of particles with valence 2, there are also many 'errors' due to chain branching, and so we eliminate high density. Conditions at lower φ have fewer errors but also shorter chains. Based on maximizing these three metrics, we choose the condition (φ = 0.3, γ = 1.0) as our best, and we provide more details about the structures observed at this condition in Fig. 6. In Fig. 6a, we show the evolution of bond valence versus time, which reflects on average, colloidomers are formed by conversion of monomers to dimers, followed by the conversion of dimers to trimers. Fig. 6b shows the distribution of valence, and for this condition, we observe that only ≈ 5% of droplets had valence higher than 2, which is indicative of a small number of branching points in chains. This result actually has a higher yield of linear chains than the best condition found experimentally in Ref. 23. However, even a small fraction of droplets with valence 3 can prevent extremely high quality assembly of only chains, as shown in Fig. 6c, where we observe that ≈ 30% of droplets are present in structures that contain at least one particle of valence 3 (cyan bar), and hence are considered as errors. Finally, Fig. 6d shows that the distribution of chain lengths seems to follow an exponential distribution, with an average length longer than in the optimal conditions in Ref. 23. An exponential distribution implies that it will be challenging to obtain a large median chain length via a simple self-assembly strategy. This comports with experimental findings, where for efficiency, a new methodology was developed to engineer longer chains using magnetic fields applied to a dispersion in a ferrofluid 40 . Application of the model to colloidomer folding Folding of colloidomer chains and the study of their pathways towards stable structures is an area of active research 40 . Here, we demonstrate that our CG model can capture experimentally relevant folding behavior of colloidomer chains and use this to highlight additional features of our CG framework. Our CG framework allows the user to generate an initial linear colloidomer chain of any length and an arbitrary sequence of binder types. Moreover, there can be multiple types of binders on the same droplet. To study colloidomer folding, we mimic the experimental setup of having two types of binders on the same droplet, C and D, each of which is self-complementary. Here, C-C bonds are intended to make up the backbone of the chain, while D-D bonds are the ones driving folding. As described earlier, our dynamic bonding model can also have temperature dependent binding/unbinding (see Sec. A4 for full details). We therefore choose different melting temperatures such that D-D bonds melt at a lower temperature, while C-C bonds stay in place. Here, we generate folded structures using a square wave heating and cooling cycle. During the first segment at higher temperature, the backbone adhesion patches form and the chain explores unfolded configurations. After that, repeated cooling and heating are used to generate low energy structures. In contrast to our work in Sec. 4.3 on optimizing colloidomer assembly, here we do want to produce higher valences. This is achieved through the use of smaller R = 20 and lower ε DD = 4.6, where reversible bonding allows for rearrangements such that equilibrated structures form. Low R has the additional benefit of faster folding times, allowing us to generate many structures in relatively little computational time. As a benchmark case to confirm that our model would be applicable in future studies of colloidomer chain folding, we investigate the heptamer case consisting of N = 7 droplets that has been experimentally realized in Ref. 40. There are four possible rigid structures which are the low energy states of the heptamer. Scanning a small range of R and ε DD over 15 heating-cooling cycles we uncover the aforementioned condition (R = 20, ε DD = 4.6) where all four possible structures are observed in a single simulation, as shown in Fig. 7a, and Movie M4. Having obtained all structures in a single long simulation, we wished to quantify the population of each stable folded state. We ran 300 independent simulations each consisting of 5 folding/unfolding cycles and show in Fig. 7b the yields of each structure, which are in good agreement with those reported in Ref 40. The ladder and the chevron structures are kinetically accessible and have a higher yield than the rocket and flower geometries. From these preliminary studies of folding within our model, we have learned several key principles. Firstly, we only observe the flower structure-which has the highest bond count-when ε DD is very low;this is because folding to this structure is not fully downhill, and requires the breaking of a droplet-droplet bond (dissolving an entire adhesion patch) to reach the final state. Second, every droplet contains both C and D binders, and so it could be a concern that the D's become exhausted in forming backbone bonds, which would tend to form faster than bonds farther away in sequence space. In our simulation, this is prevented by choosing R and N b where the patch is fully saturated by C's (see Fig. S2 and S9), meaning that there is no opportunity for D's to enter the backbone. Moreover, the exclusion of D's from the adhesion patches means there is a smaller area for them to occupy, which results in faster formation of bonds during the folding process. The factors which contribute to the speed of folding are also important to our results, since both in simulations and in experiments we do not want to wait arbitrarily long times in the cooling phase when generating low energy structures. In this work, we report a CG model and simulation framework for colloidal liquid droplets with explicit mobile binders. The core of this model is a dynamic bonding protocol that satisfies detailed balance, that is very flexible in allowing one to control separately binding and unbinding rate constants, as well as implementing a tunable temperature dependence. Both the dynamic bonding code and the pyColloidomer framework are easy to use and freely available with examples from https://github.com/ hocky-research-group/pyColloidomer_2023. Previous modeling works have studied colloidal liquid droplets with implicit mobile linkers, such that bonds are formed or removed based on a statistical mechanical model that predicts the strength of a patch and which can include timescales for bonding. Our model with explicit binders complements these studies-while having many explicit binders makes the model higher resolution, and hence slower, it also allows us to build insight into the adhesion patch formation process. For example, the use of explicit binders allowed us to see the effects of excluding particles from patches once they are formed, which had major consequences for ensuring valence=2 structures in optimizing colloidomer assembly, and in preventing binders from being used up in colloidomer backbones in our folding studies. We also observe that above a certain binding strength (ε ≈ 10), the growth of a patch can follow a process characterized by two timescales, where saturation can take much longer than initial formation and recruitment. This could have an important effect at higher densities and lower viscosities, where collisions can take place before patches are fully recruited; this effect could be incorporated into simulation models that use a parameterized equation for the recruitment process, such as recently done in Ref. 46. In ongoing work, we are now using these explicit binders to test the contributions to the free energy of patch formation and patch shape predicted experimentally in Ref. 37. We are also expanding our dynamic bonding model to include the effect of force on unbinding rates 59 to probe its effect on adhesion patch formation, a dependence which is known to play an important role in the behavior of biomimetic assemblies of cellular adhesion proteins 29 . Preliminary data shows that our model captures observed behavior for folding of two dimensional colloidomer homopolymer chains; in future work we can use our model to compare the structures and pathways generated through the use of explicit binders with those using very reduced models 40,60 . We can also trivially expand our folding studies to three dimensions by removing confinement, which will allow us to detail folding pathways in ways that are difficult to quantify in experiment. Although our model captures what we believe to be the most crucial features, there are simplifications whose effects we would like to investigate in the future. For example, the presence of a spring between the center of the droplet and binders allows the binders' vertical position to vary, and by tuning this parameter we can explore the tendency to form a planar adhesion patch -however, we are missing the lateral coupling between binders that could be important in the case of deformable droplets. Our work also currently employs harmonic springs, and it would be fascinating to investigate the differences where more complex stretching behavior is taken into account. Our powerful and flexible framework is freely available and simple to use, and so we hope others will build upon our work and take these studies in new directions. A2 Non-bonded interaction details U soft r =      ε soft 1 − r r cut 4 if r < r cut 0 if r ≥ r cut (A1) Here, ε soft is the strength of the interaction potential (in units of k B T ) and r cut is the cut-off distance, as shown by the dotted green curve in Fig. A2. A smoothing function was applied to this potential U soft (r) that results in both the potential energy and the force going smoothly to 0 at r = r cut , in this case the XLPOR smoothing 61 function S(r), given by S r =        1 if r < r on (r 2 cut −r 2 ) 2 (r 2 cut +2r 2 −3r 2 on ) (r 2 cut −r 2 on ) 3 if r on ≤ r ≤ r cut 0 if r > r cut (A2) Here, r on is chosen as the point at which the smoothing starts. We set r on = 0.1r cut for our simulations. The modified potential is shown in Fig. A2 and is given by V so f t r = S(r)U soft (r) if r on < r cut U soft (r) −U soft (r cut ) if r on ≥ r cut (A3) The soft potential was implemented by using HOOMD-Blue's tabulated potential option (with 1000 interpolation points between r min = 0 and r max = 1.05r cut ). The soft repulsive pair potential V (r) as a function of the distance r between two binder particles of radius r C = 1. In this figure, r cut = 2 is indicated by the vertical dotted line. The green dotted curve shows the potential U(r) without any smoothing function applied to it and the red solid curve shows the potential V (r) after it is multiplied by a suitable smoothing function S(r). The wall potential is given by a shifted LJ potential with r cut = 2 1/6 (2R). V wall r = V FLJ r −V FLJ r cut (A4) where, V FLJ (r), the force-shifted Lennard-Jones pair potential is given by, V FLJ r =              4ε wall σ r 12 − α σ r 6 if r < r cut +∆V r 0 if r ≥ r cut (A5) ∆V r = −(r − r cut ) ∂V LJ ∂ r (r cut ) (A6) where α = 1. A3 Algorithm for binding and unbinding Every n steps of the MD simulation (run using HOOMD-Blue), a Dynamic Bond Updater is called. The Dynamic Bond Updater is an open source C++ plugin (based on previous work on epoxy curing 51 ) that stochastically adds or removes dynamic bonds during the course of the MD simulation. If there is more than one dynamic bond type present in the system, the Bond Updater has to be configured for each independently. Each time the Bond Updater is called, it first iterates over all the existing dynamic bonds to attempt unbinding. The probability of unbinding is calculated, given by P 0 off = n k off dt (A7) where, dt is the timestep of the MD simulation. For each bond in sequence, a uniform random number r ∈ [0, 1) is generated, and if r < P off the bond is added to a list of bonds to be removed from the bond table. Once all the possible unbinding events are taken into consideration, we iterate over the list of bonds to be removed and perform unbinding. After performing unbinding, we create a list of proposed bonds to add. We iterate over particles which were unbound at the start of the binding update (but not those freed by unbinding), and find the closest available complementary particle which is also unbound and whose distance from the particle under consideration is between l min and l max (by iterating over HOOMD-Blue's neighbor list). If an eligible neighbor exists, , and neither particle is already in the proposed bonds list, then this pair is appended. We then iterate over the proposed bond list, generating new uniform random numbers and creating a bond if r < P on . By default, P 0 on = n k on dt. To ensure detailed balance for individual binding reactions, the probability of binding is modified such that P on = P 0 on e −∆U(d)/k B T , where ∆U is the additional energy added by creating a bond of length d, possibly away from its rest length. As in Ref. 62, we are putting all of the energetic dependence into the binding step and none in the unbinding step. Since we are only using harmonic bonds in this work, P on (d) = P 0 on e −k dyn (d−l dyn ) 2 /(2k B T ) ,(A8) where k B T is the instantaneous temperature of the system. Detailed balance for the overall binding/unbinding reaction is satisfied to the best of our ability when performing a large set of binding/unbinding reactions at once (as compared to only doing one single binding/unbinding per trial) since every event is independent, and each individually satisfies a Metropolis criterion. The only possibly weak breaking of detailed balance comes in the rare situation where upon unbinding, one or both of the particles was assigned a different binding partner since it was not bound to the neighbor from whom its distance is most close to the equilibrium bond length. In practice, because the configuration evolves n steps between binding/unbinding trials, we do not expect this to cause any substantial non-equilibrium effects. A4 Temperature dependence of binding/unbinding Our dynamic bonding model allows us to use non-constant values of k off , k on . As one example, for this work we have incorporated an optional dependence of the rate constants on temperature. Since our binders here might represent double stranded DNA, which dissociates in a cooperative manner, we implemented an optional tunable sigmoidal dependence on temperature for the rate constants. Without trying to match the behavior of any specific module, we implemented a two parameter sigmoidal dependence on temperature to represent cooperative melting, g T = 1 2 tanh α T − T melt + 1 ,(A9)k on/off T = k on/off init 1 − g T + k on/off melt g T (A10) where, k on/off melt is the value of the binding (or unbinding) rate constant after the melting of bonds. Here, we can set the steepness of the transition with the parameter α (which in experiment could be tuned by changing the DNA sequence and sequence length). T melt should be the temperature where the fraction of bonds formed is 0.5. For a two-state model, the bound fraction is given by f bound T = K eq (T ) 1 + K eq (T ) (A11) where, K eq (T ) = k on (T )/k off (T ). Ensuring f bound (T melt ) = 0.5 requires k on (T melt ) = k off (T melt ). Therefore k on init + k on melt = k off init + k off melt (A12) since, g(T melt ) = 0.5. We choose k on melt = 0, so that there is no binding at T T melt , at which point k off melt can be determined from Eq. A12. Fig. A3(b) shows how f bound depends on T melt using this model. The fraction of the total number of binders not present in the adhesion patch is monitored over time for the case of a dimer, the terminal droplet of a trimer and also its central droplet. For low and high ε, we consider different ways of fitting the curves in order to obtain the converged value of the fraction of free binders. Supporting Information S1 Additional results for Dimers and Trimers S1.1 Convergence of the fraction of binders not recruited in an adhesion patch with simulation time For binding affinity ε ≥ 10, as shown in Fig. S1b,d,f for ε = 20.7, we fit the fraction of binders not in a patch f (t) to a double exponential function of analytical form f (t) = ( f (0) − a) exp(−k 1 t) + (a − b) exp(−k 2 t) + b, where we interpret k 1 and k 2 to be related to the two different time scales-the recruitment time of the binders and the time taken for the adhesion patch to saturate, explained in Sec. 4.2. The fraction of unrecruited binders at saturation of the patch f (t)| t=∞ = b according to this expression. The fitted values for the parameters a, b, k 1 , k 2 are obtained using the curve fitting feature of SciPy 63 . From the fitted values of k 1 and k 2 , we can estimate the recruitment time τ 1 and the patch saturation time τ 2 , as shown in Tab. S1. At larger values of R > 100, we find that the curves for trimers are well fit by a single exponential, and this emerges naturally in our double exponential fit with k 1 and k 2 being identical. For binding affinity ε < 10, as shown in Fig. S1a,c,e for ε = 9.2, we find that the fraction remains fairly constant over time (for all the droplet radii) with fluctuations characteristic of low ε. For each of these situations, we obtain a mean of this fraction over the last 50% of the simulation time (in this case, between 10 8 steps and 2 × 10 8 steps) and fit the curve to this constant mean value. For all cases the final value of fraction at the final time (at 2 × 10 8 steps) has been found to lie within 5% of the fit value. Fig. S2 shows that the number of binders in an adhesion patch saturates due to steric repulsion with N bonded < N b . S2 Additional data and procedures for assembly of 50/50 mixtures of C and D droplets S1.2 Saturation of adhesion patch S2.1 Clustering via a segmentation algorithm to identify unconnected assemblies and classification of the structures For every frame in our simulation, we identify the unconnected assemblies of droplets bonded to each other via a simple segmentation algorithm, using the bond table of droplet pairs. The bond table is a list of unique pairs of droplets that have at least one dynamic bond between them. From this list of bonded droplet pairs, we want to find the components which have common elements between them. We follow an algorithm to continuously merge sets of pairs that have common elements † to perform the clustering: (i) We take the first set, say ' A' from the list and separate the rest of the list from it. (ii) Next, for every other set B in the list, if B has common element(s) with A then we merge sets A and B and remove B from the list. (iii) This step is repeated until none of the other sets have any overlap with A. (iv) The set A is then added to the output of the clustering. (v) Step (i) is repeated, but now with the rest of the list (excluding ' A'). In this way, we ultimately end up with a list of lists where each sub-list consists of droplets which are 'bonded' to each other. Once the set of unconnected assemblies (or 'clusters') is obtained, the next step is to classify them into structuresmonomers, dimers, linear chains (N ≥ 3), loops and 'other' (which includes any branched colloidomer chains or gels/aggregates with higher droplet valences). Monomers are identified as those that do not appear in the bond table. Next, in order to classify the remaining droplets into the other structures, for every cluster we obtained, we first calculate the valence of each droplet in that cluster from the bond table of droplets, and then count the number of droplets with valence 1 and 2 respectively. If the size of the cluster is 2, then both of the droplets belong to a 'dimer'. However, if the size of the cluster list is greater than 2, then the valence information will help us to further identify if it is a linear chain, a loop, or 'other'. If the number of droplets in the given cluster with valence 1 is 0 and the number of droplets with valence 2 is equal to the total size of the cluster, then the structure is a 'loop'. If the number of droplets with valence 1 is 2 and the number of droplets with valence 2 is equal to the (size of the cluster)-2, then the structure is a 'linear colloidomer chain' (for which only the 2 terminal droplets have valence 1 and the rest have valence 2). Else, the cluster is classified as 'other'. S2.3 Effect of kinetic factors on self-assembly in reversible regimes (low and intermediate ε) In Sec.4.5 we described how kinetic factors such as φ and γ A can dictate the structures formed in self-assembly for a high ε. In this regime, once an adhesion patch forms, binders are very unlikely to redistribute into bonds with other droplets. However, for reversible binding in case of lower ε, we find that this kinetic trapping effect is not observed because redistribution of binders between droplets is allowed here, allowing droplets to potentially achieve their equilibrium valence. We show a case of lower (ε = 9.2) and intermediate (ε = 13.8) binding affinity here to demonstrate the effect of φ and γ A for reversible binding scenarios (final configurations shown in Figures S6 and S7 respectively.). The distribution of bond valences over time for φ = 0.3 and γ A = 1.0 is shown in Fig.S8 for lower and intermediate ε. The difference between the two cases is the most profound for lower densities such as φ = 0.2, where for ε = 9.2, we already obtain aggregates whereas for ε = 13.8, we end up mostly in chains, with and without branching. For ε = 9.2 and high densities such as φ = 0.4, it is almost impossible to differentiate the effect that low and high γ A have, because of system-spanning gels forming in both situations. S2.4 Bootstrapping procedure From the original data set of chain lengths for 10 final configurations, we create 100 new resampled sets of data with replacements (each resampled set is of the same size as the original data set). To create a bootstrap sample, the resampling is done randomly, by choosing a random integer between 0 and (the size of the original data set)-1. The element from the original data set with this random index is then added to the bootstrap sample and this process is repeated for as many times as the size of the original data set. Ultimately from the 100 bootstrap samples obtained, we can calculate the mean fraction of all chain sizes and also the standard deviation. By bootstrapping, we obtain an estimate of the error resulting from having a small number of samples 64 . S3 Folding/unfolding simulations S3.1 Identification of the folded structures for the heptamer The folded structures obtained in our 300 independent simulations for the heptamer can be classified as either a ladder, a chevron, a rocket or a flower (see Sec. 4.6, and Ref. 40). These structures can be differentiated from one another on the basis of the valence of the droplets present in the structure. A valence list of all the 7 droplets is obtained for every folded structure. The number of droplets with valence 2,3,4,5 and 6 is calculated for each of these structures from the list of valences. We assign the structure based on the counts for each of these valences (see Tab. S2). We found that only 5 of the 1500 folded structures obtained could not be assigned to any of the four folded states mentioned above. Table S3 A table containing 1.368 (l dyn − 2σ ) Dynamic bonding maximum distance (l max ) 2.632 (l dyn + 2σ ) Dynamic bond checksteps (n) 10 Table S4 A table containing important parameters specific to the simulations S4 Detailed simulation parameters Fig. 3 3Heat maps showing the percentage of binders remaining on the surface of a droplet in a dimer geometry, and on the terminal or central droplets in a trimer geometry. Conditions predicted to be "good" for colloidomer assembly are indicated by dashed ovals, and selected conditions marked by red open circles are illustrated in Fig. 4. (a) Fraction of remaining binders at fixed number of binders, varying droplet radius and bond strength. (b) Fraction of remaining binders at fixed high bond strength, varying number of binders and droplet radius. Fig. 4 4Illustrations showing the dimer and trimer geometries simulated inFig. 3, and the "molecular" features of our droplet model that can be tuned to optimize for linear chains. Both results for dimers and trimers must be considered to predict the resulting polymerization reaction. (a) Varying R and ε at fixed number of binders N b = 100, with dimer on left and trimer on right. (b) Varying the number of binders N b at fixed R = 50 and ε = 20.7. In (a) and (b), the % of free binders available (averaged over 10 independent runs) is indicated in parentheses beside each of these conditions for a droplet in a dimer as well as the terminal and central droplet(s) of a trimer. Based on the values of the % of free binders available, the conditions which are not suitable for colloidomer assembly (almost all used up in dimer, too many remaining on central droplet in a trimer) are indicated by a red X. Fig. 5 5Effect of area fraction φ and the droplet drag coefficient γ A on self-assembly in a 1:1 mixture of 81 R = 50 droplets containing N b = 100 complementary binders of type C and D (with one surplus droplet of C/D). ε = 20.7. Here, φ = 0.1, 0.2, 0.3, 0.4 (increasing from left to right) and γ A = 0.01, 1.0 (bottom and top). 3, γ A = 1.0 is highlighted with a red box. Snapshots at φ = 0.4 show periodic images to emphasize that structures are extended across the periodic boundaries. The distributions of structures obtained and of the colloidomer chain lengths for each of these 8 conditions are illustrated in Fig. S3 and Fig. S5 respectively. Fig. 6 6Best choice of parameters for obtaining maximum quality of colloidomer chains: N b = 100, R = 50, φ = 0.3, ε = 20.7, γ A = 1.0. (a) Fraction of droplets in each possible valence versus time. (b) Final valence distribution obtained across the 10 simulations. Open squares show optimal valence distribution from Fig. 2g in Ref. 23. (c) A combined histogram showing the fraction of the total number of droplets present as a particular kind of structure from 10 different final configurations for this condition. (d) Distribution of linear chain lengths P(N) obtained for this condition, with the errorbars computed using a bootstrapping procedure described in Sec. S2.4. Open squares show experimental length distribution from Fig. 3a in Ref. 23. Fig average maximum chain length (blue bar, which we scale by the highest value N max highest = 20 observed for the condition φ = 0.4, γ A = 1.0). Fig. 7 7(a) Folding and unfolding cycles shown for a heptamer of droplets with 100 'C' binders and 100 'D' binders on each droplet, R = 20, ε DD = 4.6, ε CC = ∞, and γ A = 0.1. T melt = 1.2 for D-D bonds and temperature is cycled between 1.0 and 1.3. The variation of the average bond valence B n (red) with simulation time is shown, as the temperature (navy blue) is alternately raised and lowered. An expected B n = 1.7 is obtained when the structures unfold back to chains. (b) Histogram showing the yield of each of the four possible folded rigid structures (ladder, chevron, rocket and flower) 40 from a total of 1500 folded structures obtained from 300 independent simulations each consisting of 5 folding and unfolding cycles. (only 5/1500 did not reach one of these structures, and are not shown in this histogram). Fig. A1 A1Complementary binders in our CG model mapped onto the dsDNA and ssDNA configuration of Ref. 23 for scale. Complementary outer binder particles (red and blue) form a dynamic bond representing the interaction between complementary DNA strands (Figure created with https://biorender.com/). Fig. A2 A2Fig. A2 The soft repulsive pair potential V (r) as a function of the distance r between two binder particles of radius r C = 1. In this figure, r cut = 2 is indicated by the vertical dotted line. The green dotted curve shows the potential U(r) without any smoothing function applied to it and the red solid curve shows the potential V (r) after it is multiplied by a suitable smoothing function S(r). Fig. A3 (a) g(T ) vs. T (b) Fraction f bound T of DNA that is bound for a single DNA pair vs. T for two different binding strengths ε = 3.0 and 16.0. (The temperature is in units of T * ). Fig. S1 S1Plots showing the convergence of the fraction of the total binders N b = 100 not in any adhesion patch with the simulation time, for two different binding affinities ε = 9.2 (low)-shown in (a),(c),(e) and ε = 20.7 (high)-shown in (b),(d),(f). (a) and (b) show these cases for the dimer, (c) and (d) for the terminal droplet of a trimer, (e) and (f) for the central droplet of the trimer. Each plot shows the convergence for 8 different droplet radii R ranging from 20 to 200. These simulations are all run for 2 × 10 8 HOOMD steps which is sufficient for all the individual curves to attain convergence. S1 A table showing the values of the recruitment time (τ 1 ) and the adhesion patch saturation time (τ 2 ) obtained from the double exponential fit for ε = 20.7 and N b Fig. S2 A S2heat map of the number of binders N b and the droplet radius R showing the number of binders in a saturated adhesion patch (N bonded t=∞ ) for high binding affinity (ε = 20.7) for a droplet in a dimer. S2. 2 2Distributions of droplet valences, structures and colloidomer chain lengths in the final configurations obtained from 10 independent runs for all choices of (φ , γ A ) Fig. S3 Distributions of structures obtained from all final configurations for each of the (φ , γ A ) pairs, illustrated in Fig. 5. φ = 0.1, 0.2, 0.3, 0.4 (increasing from left to right) and γ A = 0.01, 1.0 (bottom and top). Fig. S4 S4Distributions of droplet valences obtained from all final configurations for each of the (φ , γ A ) pairs, illustrated in Fig. 5. φ = 0.1, 0.2, 0.3, 0.4 (increasing from left to right) and γ A = 0.01, 1.0 (bottom and top). Fig. S5 S5Distributions of colloidomer chain lengths from all final configurations for each of the (φ , γ A ) pairs, illustrated in Fig. 5. φ = 0.1, 0.2, 0.3, 0.4 (increasing from left to right) and γ A = 0.01, 1.0 (bottom and top). Fig. S6 S6Effect of φ and γ A on the kind of self-assembled structures for our system of 81 droplets with N b = 100, R = 50 for intermediate binding affinity ε = 13.8. The droplets are colored according to the type of structure to which they belong, as explained inFig. 5 Fig. S7 S7Effect of φ and γ A on the kind of self-assembled structures for our system of 81 droplets with N b = 100, R = 50 for lower binding affinity ε = 9.2. The droplets are colored according to the type of structure to which they belong, as explained inFig. 5 Fig. S8 S8Fraction of droplets with a given bond valence (B n ) as a function of the simulation time (in HOOMD units) for the optimized condition which gives maximum quality of colloidomer chains at high binding energy: N b = 100, R = 50, φ = 0.3, γ A = 1.0 but for (a) lower binding affinity ε = 9.2 and (b) intermediate binding affinity ε = 13.8. The corresponding plot for ε = 20.7 is shown inFig. 6a. S3. 2 2Number of bonds of C-C and D-D types in an adhesion patch with an adjacent droplet during folding/unfolding simulationsFig. S9 Plot showing the average number of bonds of C-C (backbone) and D-D (secondary) types per adhesion patch between adjacent droplets, as a function of the simulation time for the square wave heating and cooling cycle described in Sec. 4.6. The conditions for this simulation are: N b = 200 (100 binders of types C and D respectively on every droplet), R = 20, γ A = 0.1. The choice of R and N b used here ensures that on an average, ∼ 44% of the C binders get recruited whereas ∼ 0% D binders are able to go into an adhesion patch between any two adjacent droplets. As a result, the adhesion patches between adjacent droplets are always fully saturated by C's (See Sec.4.6) for soft repulsive potential (ε soft ) 200.0-5000.0 Cut-off distance for soft potential: (i) r cut,AA 110.0 (R = 50.0) (ii) r cut,AC/AD 53.0 (R = 50.0, r C =1.0) (iii) r cut,BB 2.0 (r B =1.0) (iv) r cut,CC/DD 2.0 (r C =1.0) Epsilon for Wall Potential (ε wall ) 10.0 (d = 2) Cut-off distance for Wall Potential (r cut,wall ) 112.25 (R = 50.0) z-coordinate of upper wall plane origin 125.0 (R = 50.0) z-coordinate of lower wall plane origin -125.0 (R = 50.0) Initial rate constant for binding (k init on ) 100.0-200.0 Initial rate constant for unbinding (k init off ) 10 −9 -5.0 Rate constant for binding after melting (spring constant (k dyn ) 10.0 Dynamic bonding minimum distance (l min ) 1 1constant of binding for CC (k on init,CC ) 100.0 Initial rate constant of unbinding for CC (k off init,CC ) 10 −9 -5.0Table S5A table containing important parameters specific to the self-assembly simulations for a 9 × 9 lattice) Number of binders on each droplet (N b ) 100 Drag coefficient of droplet (γ A ) 0.01,1.0 Initial rate constant of binding for CD (k on init,CD ) 100.0 Initial rate constant of unbinding for CD (k off init,CD ) 10 −9 -5.0 rate constant of binding for CC/DD (k on init,CC/DD ) 200.0 Initial rate constant of unbinding for CC (k off init,CC ) 0 Initial rate constant of unbinding for DD (k off init,DD ) 2.0 Melting Temperature for DD (T melt,DD ) 1.2 Table Table S2 A S2table showing the counts of the droplet valences for the different folded geometries for N=7 in two dimensions Valence Ladder Chevron Rocket Flower Table S6 A S6table containing important parameters specific to the folding/unfolding simulations for 300 independent simulations with 5 folding/unfolding cycles in each AcknowledgementsWe thank Joshua A. 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J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey,İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Har- ris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt and SciPy 1.0 Contributors, Nature Methods, 2020, 17, 261-272. An introduction to the bootstrap. B Efron, R J Tibshirani, CRC pressB. Efron and R. J. Tibshirani, An introduction to the bootstrap, CRC press, 1994. S5 Additional movies M1: Movies showing the growth and saturation of the adhesion patch in a dimer for the exact same conditions as illustrated in Fig. 4S5 Additional movies M1: Movies showing the growth and saturation of the adhesion patch in a dimer for the exact same conditions as illustrated in Fig. 4). M2: Movies showing the growth and saturation of the two adhesion patches in a trimer for the exact same conditions as illustrated in Fig. 4M2: Movies showing the growth and saturation of the two adhesion patches in a trimer for the exact same conditions as illustrated in Fig. 4). M3: Movies showing self-assembly of 50/50 mixtures of C and D droplets with R = 50. N b = 100, ε = 20.7, γ A = 1.0 for 4M3: Movies showing self-assembly of 50/50 mixtures of C and D droplets with R = 50, N b = 100, ε = 20.7, γ A = 1.0 for 4 M4: A movie showing 15 heating-cooling cycles for a 7-mer of droplets with R = 20. N b = 200, γ A = 0.1, ε DD = 4.6, ε CC = ∞ and T melt,DD = 1.2 (See Fig. 7a)M4: A movie showing 15 heating-cooling cycles for a 7-mer of droplets with R = 20, N b = 200, γ A = 0.1, ε DD = 4.6, ε CC = ∞ and T melt,DD = 1.2 (See Fig. 7a).
[]
[ "Generative AI meets 3D: A Survey on Text-to-3D in AIGC Era", "Generative AI meets 3D: A Survey on Text-to-3D in AIGC Era" ]
[ "Sung-Ho China ", "Kyung Hee Bae ", "South Hee University ", "Choong Korea ", "Seon ", "Kyung Hee Hong ", "South Hee University ", "; / Chenghao Korea ", "KAISTKaist Li ", "South Korea ", "\nLIK-HANG LEE\nHong Kong Polytechnic University\nHong KongChina\n", "\nFrancois Rameau\nFRANCOIS RAMEAU\nState University of New\nUniversity of Electronic Science\nLik-Hang Lee, Hong Kong Polytechnic Uni-versityHong Kong(China\n", "\nState University of New York\nKorea\n", "\nUniversity of Electronic Science and technology\nChina\n" ]
[ "LIK-HANG LEE\nHong Kong Polytechnic University\nHong KongChina", "Francois Rameau\nFRANCOIS RAMEAU\nState University of New\nUniversity of Electronic Science\nLik-Hang Lee, Hong Kong Polytechnic Uni-versityHong Kong(China", "State University of New York\nKorea", "University of Electronic Science and technology\nChina" ]
[ "South Korea" ]
Generative AI (AIGC, a.k.a. AI generated content) has made remarkable progress in the past few years, among which text-guided content generation is the most practical one since it enables the interaction between human instruction and AIGC. Due to the development in text-to-image as well 3D modeling technologies (like NeRF), text-to-3D has become a newly emerging yet highly active research field. Our work conducts the first yet comprehensive survey on text-to-3D to help readers interested in this direction quickly catch up with its fast development. First, we introduce 3D data representations, including both Euclidean data and non-Euclidean data. On top of that, we introduce various foundation technologies as well as summarize how recent works combine those foundation technologies to realize satisfactory text-to-3D. Moreover, we summarize how text-to-3D technology is used in various applications, including avatar generation, texture generation, shape transformation, and scene generation.
10.48550/arxiv.2305.06131
[ "https://export.arxiv.org/pdf/2305.06131v1.pdf" ]
258,588,157
2305.06131
286940a286f6a702cccc877a4e7cd8630aeb2fd4
Generative AI meets 3D: A Survey on Text-to-3D in AIGC Era Sung-Ho BaeCopyright Sung-Ho Bae2022. May 2022 Sung-Ho China Kyung Hee Bae South Hee University Choong Korea Seon Kyung Hee Hong South Hee University ; / Chenghao Korea KAISTKaist Li South Korea LIK-HANG LEE Hong Kong Polytechnic University Hong KongChina Francois Rameau FRANCOIS RAMEAU State University of New University of Electronic Science Lik-Hang Lee, Hong Kong Polytechnic Uni-versityHong Kong(China State University of New York Korea University of Electronic Science and technology China Generative AI meets 3D: A Survey on Text-to-3D in AIGC Era South Korea Sung-Ho Bae112022. May 2022Manuscript submitted to ACM Manuscript submitted to ACMAuthors' addresses:CCS Concepts: • Computing methodologies → Reconstruction; Shape modeling Additional Key Words and Phrases: text-to-3Dgenerative AIAIGC3D generationmetaverse ACM Reference Format: Generative AI (AIGC, a.k.a. AI generated content) has made remarkable progress in the past few years, among which text-guided content generation is the most practical one since it enables the interaction between human instruction and AIGC. Due to the development in text-to-image as well 3D modeling technologies (like NeRF), text-to-3D has become a newly emerging yet highly active research field. Our work conducts the first yet comprehensive survey on text-to-3D to help readers interested in this direction quickly catch up with its fast development. First, we introduce 3D data representations, including both Euclidean data and non-Euclidean data. On top of that, we introduce various foundation technologies as well as summarize how recent works combine those foundation technologies to realize satisfactory text-to-3D. Moreover, we summarize how text-to-3D technology is used in various applications, including avatar generation, texture generation, shape transformation, and scene generation. INTRODUCTION Generative AI Generative AI (AIGC, a.k.a. AI-generated content) [Zhang et al. 2023d] has attracted significant attention in the past few years. Text-guided content generation, like ChatGPT [Zhang et al. 2023b] and text-to-image [Zhang et al. 2023c], is the most practical one since it enables the interaction between human instruction and AI generation [Zhang et al. 2023d]. The achievements of generative AI in text-to-image [Zhang et al. 2023c] are remarkable, but our world is three-dimensional, thus it is necessary to extend AIGC to the 3D domain. There is a great demand for 3D digital content in many application scenarios, including gaming, movies, virtual reality, architecture, and robotics. However, it takes a lot of artistic and aesthetic training as well as expertise in 3D modeling to bring up a professional 3D modeling artist. Therefore, there is a need to use generative AI to generate high-quality and large-scale 3D models. In addition, content creation guided by natural language can greatly help newbies and professional artists to realize democratized creation of 3D content. Nowadays, there are various types of industrial demands for 3D contents, such as 3D character generation, 3D texture generation, 3D scene generation, etc. Neural Radiance Fields (NeRF) [Mildenhall et al. 2021] is an emergent 3D data representation approach. Initially, NeRF was found to perform well in the 3D reconstruction task, and recently NeRF and other neural 3D representations have been applied to new view synthesis tasks that can use real-world RGB photos. NeRF is trained to reconstruct images from multiple viewpoints. As the learned radiance fields are shared between viewpoints, NeRF can smoothly and consistently interpolate between viewpoints. Due to its neural representation, NeRF can sample with high spatial resolution, unlike voxel representations and point clouds, and is easier to optimize than meshes and other explicit geometric representations, since it is topology-free. The advent of NeRF breaks the stalemate of 3D data scarcity and serves as a soft bridge from 2D to 3D representation, elegantly solving the problem of 3D data scarcity. With the remarkable progress of multimodal AI ] and diffusion models [Ho et al. 2020], text-guided image Manuscript submitted to ACM content AI generation has made significant progress. The key driving factor is the large-scale datasets of billions of text-image pairs obtained from the Internet. Recent works have emerged that guide 3D modeling optimization by leveraging the prior of a pre-trained text-to-image generation model. In other words, they often text-guided 3D model generation with text-to-image priors. Overall, this work work conducts the first yet comprehensive survey on text-to3D. The rest of this work is organized as follows. Sec 2 reviews the different expressions of 3D. Sec 3 first introduces the technology behind text-to-3D, and then summarizes the recent papers. Sec 4 introduces the application of text-to-3D in various fields. 3D DATA REPRESENTATION 3D data can have different representations [Ahmed et al. 2018], divided into Euclidean and non-Euclidean. 3D Euclidean data has a potential grid structure, which allows global parameterization and a common coordinate system. These properties make extending existing 2D deep learning paradigms to 3D data a simple task, where convolution operations remain the same as 2D. On the other hand, 3D non-Euclidean data does not have a grid array structure and is not globally parameterized. Therefore, extending classical deep learning techniques to such representations is a challenging task. In real life, the research of deep learning techniques in non-Euclidean domains is of great significance. This is referred to as geometric deep learning [Cao et al. 2020]. Euclidean data The Euclidean data preserves the attribute of grid structure, with global parameterization and common coordinate system. The major 3D data representations in this category include voxel grids and multi-view images. 2.1.1 Voxel Grids. Voxels can be used to represent individual samples or data points on a regularly spaced threedimensional grid, which is a Euclidean structured data structure similar to pixels [Blinn 2005] in 2D space. The data point can contain a single data, such as opacity, or multiple data, such as color and opacity. Voxels can also store high-dimensional feature vectors in data points, such as geometric occupancy [Mescheder et al. 2019], volumetric density [Minto et al. 2018], or signed distance values [Park et al. 2019]. A voxel only represents a point on this grid, not a volume; the space between voxels is not represented in the voxel-based dataset. Depending on the data type and the expected use of the dataset, this lost information can be reconstructed and/or approximated, for example, by interpolation. The representation of voxels is simple and the spatial structure is clear, which is highly extensible and can be easily applied to convolutional neural networks ]. However, the efficiency is low, as it represents Manuscript submitted to ACM both the occupied parts of the scene and the unoccupied parts, which leads to a large amount of unnecessary storage requirements. This leads to voxels being unsuitable for representing high-resolution data. Voxel grids have many applications in rendering tasks Rematas and Ferrari 2020]. Early methods store high-dimensional feature vectors in voxels to encode the geometry and appearance of the scene, usually referred to as a feature volume, which can be interpreted as a color image using projection and 2D cellular neural networks. This also includes volumetric imaging in medicine, as well as terrain representation in games and simulations. 2.1.2 Multi-view images. With the development of computer vision technology and the remarkable improvement in computing power, coupled with the latest developments in modern digital cameras, it is now possible to easily capture large amounts of high-resolution images. There is an urgent need to extract 3D structure from these images for many applications, such as 3D reconstruction [Jin et al. 2020]. A multi-view image dataset is an aggregation of multiple images, each representing an object or scene from different perspectives, such as the front, side and top, aggregated together to form a multi-view image dataset. Since collecting 3D data from the real world is time-consuming, and deep learning paradigms rely on large amounts of data for training, the availability of multi-view image datasets is its greatest advantage. Its drawback is that multi-view images cannot be strictly defined as 3D model data, but it provides a bridge between 2D and 3D visualizations. Recently, NeRF [Mildenhall et al. 2021] has emerged as a novel approach for 3D reconstruction, which is well-suited for the massive data requirement of the learning-based generalizable NeRF methods with large scale multi-view datasets . It can also be applicable to multi-view stereo [Furukawa et al. 2015] and view-consistent image understanding [Dong et al. 2022] tasks. non-Euclidean data The second type of 3D data representation is non-Euclidean data. This type of data does not have global parametrization or common coordinate systems, which makes it difficult to extend 2D deep learning paradigms. Much effort has been made in learning this data representation and applying DL techniques. Researching deep learning techniques in non-Euclidean domains is of great importance, this is referred to as Geometric Deep Learning [Cao et al. 2020]. The main types of non-Euclidean data are point clouds, 3D meshes and implicit. 2.2.1 Meshes. 3D meshes [Wang and Zhang 2022] are one of the most popular representations of 3D shapes. A 3D mesh structure is composed of a set of polygons, termed faces, which are described based on a set of vertices that describe the presence of coordinates in 3D space. These vertices are associated with a connectivity list that describes how these vertices are interconnected. As meshes only model the surface of the scene, they are more compact. The meshes provide connectivity of surface points for modeling point relationships. Due to these advantages, polygon meshes are widely used in traditional computer graphics [Zhou et al. 2021b] applications such as geometry processing, animation, and rendering. However, on a global level, meshes are non-Euclidean data, and the local geometry of the mesh can be represented as a subset of the Euclidean space where the known properties of the Euclidean space are not well-defined, such as shift-invariance, vector space operations, and global parameterization systems. Thus, deep learning of 3D meshes is a challenging task. However, with the development of graph neural networks , meshes can be seen as graphs. MeshCNN [Hanocka et al. 2019] specifically designs convolutional and pooling layers for mesh edges and extracts edge features for shape analysis. 3D meshes are important in many fields and industries, such as architecture and building, furniture and home living, gaming and entertainment, product design, medical and life sciences, etc. They can be used for designing, visualizing, analyzing architecture and products, creating character objects for games, movies, etc., designing new products, visualizing and analyzing anatomical structures, and helping to increase understanding of diseases and treatment methods. Point Clouds. With the trend of inexpensive and convenient point cloud acquisition equipment, point clouds have been widely used in modeling and rendering, augmented reality, autonomous vehicles, etc [Guo et al. 2020]. Point clouds are a disordered set of discrete samples of three-dimensional shapes on three-dimensional space. Traditionally, point clouds are non-Euclidean data because point cloud data is non-structured globally. However, point clouds can also be realized as a set of globally parametrized small Euclidean subsets. The definition of point cloud structure depends on whether to consider the global or local structure of the object. Since most applications strive to capture the global characteristics of the object to perform complex tasks, traditionally point clouds are non-Euclidean data. Point clouds are the direct output of depth sensors ] and therefore are very popular in 3D scene understanding tasks. Despite being easily obtainable, the irregularity of point clouds makes them hard to process with traditional 2D neural networks. Numerous geometric deep learning [Cao et al. 2020] methods have been proposed to effectively analyze three-dimensional point clouds, such as PointNet [Qi et al. 2017], which is a deep learning network structure based on raw point cloud data and can directly use raw point cloud data as input and use a set of sparse keypoints to summarize the input point cloud, and can effectively process data and have robustness to small perturbations of the input and can achieve good performance in tasks such as shape classification, part segmentation, and scene segmentation. 3D point cloud technology can be applied to multiple fields such as architecture, engineering, civil building design, geological survey, machine vision, agriculture, space information, and automatic driving, and can provide more accurate modeling and analysis, as well as more accurate positioning and tracking. Neural fields. Neural fields are a domain that is either wholly or partially parameterized by neural networks and represented entirely or partially by neural networks for scenes or objects in 3D space.At each point in 3D space, a neural network can map its associated characteristics to attributes. Neural fields are capable of representing 3D scenes or objects in any resolution and unknown or complex topology due to their continuous representation. Additionally, compared to the above representations, only the parameters of the neural network need to be stored, resulting in lower memory consumption than other representations. The earliest work on neural fields was used for 3D shape representation [Peng and Shamsuddin 2004]. NeRF [Mildenhall et al. 2021], a recent emerging representation for 3D reconstruction in neural fields, has the advantages of high-quality and realistic 3D model generation, presenting realistic object surfaces and texture details at any angle and distance. Furthermore, it can generate 3D models from any number of input images without specific processing or labeling of the inputs. Another advantage of neural fields is that the neural network can be operated on low-power devices after it is trained. Polygon ray tracing renders high-resolution and realistic scenes at high frame rates, which requires expensive graphic cards, but high-quality neural fields can be rendered on mobile phones and even web browsers. However, there are also some drawbacks of neural field technology, such as the need for a large amount of computational resources and time for training, as well as difficulty in handling large-scale scenes and complex lighting conditions, and its inability to be structured data, which makes it difficult to be directly applied to 3D assets. Neural fields are a new emerging 3D representation technology with strong application prospects and can be used in 3D fields such as VR/AR and games. We are the first to survey text-to-3D generation. TEX-TO-3D TECHNOLOGIES In the past few years, the success of deep generative models [Ho et al. 2020] in 2D images has been incredible. Training generative models in 2D space cannot meet the needs of some practical applications, as our physical world actually operates in 3D space. 3D data generation is of paramount importance. The success of Neural Radiance Fields [Mildenhall et al. 2021] has transformed the 3D reconstruction race, bringing 3D data to a whole new level. Combining prior knowledge from text-to-image models ], many pioneers have achieved remarkable results in text-to-3D generation. In this section, we will first review the key techniques underlying text-to-3D generation. Secondly, we will survey recent text-to-3D models. Foundation Technologies Fig. 6. An overview of our neural radiance field scene representation and differentiable rendering procedure, the picture is obtained from [Mildenhall et al. 2021]. 3.1.1 NeRF. Neural Rediance Field (NeRF) [Mildenhall et al. 2021] is a neural network-based implicit representation of 3D scenes, which can render projection images from any viewpoint and any position. Specifically, given a 3D point x ∈ R 3 and an observation direction unit vector d ∈ R 3 , NeRF encodes the scene as a continuous volumetric radiance field , yielding a differential density and an RGB color c: (x, d) = ( , c). Rendering of images from desired perspectives can be achieved by integrating color along a suitable ray, r, for each pixel in accordance with the volume rendering equation: C( ) = ∫ ( ) ( )c( ) (1) ( ) = − ∫ ( )(2) The transmission coefficient ( ) is defined as the probability that light is not absorbed from the near-field boundary to . In order to train NeRF network and optimize the predicted colorĈ to fit with the ray R corresponding to the pixel in the training images, gradient descent is used to optimize the network and match the target pixel color by loss: L = ∑︁ r∈R ∥C(r) −Ĉ(r)∥ 2 2 (3) 3.1.2 CLIP. Recent advances in multimodal learning have enabled the development of cross-modal matching models such as CLIP ](Contrastive Language-Image Pre-training) which learn shared representations from image-text pairs. These models are able to produce a scalar score that indicates whether an image and its associated caption match or not. In training, the standard image model is used to jointly train the image feature extractor and linear classifier to predict some labels; CLIP jointly trains the image encoder and text encoder to predict the correct pairings of a batch of (image, text) training samples. The symmetric InfoNCE loss is used to train the image and text encoders, which can then be used for a number of downstream tasks. In testing, the learned text encoder synthesizes a zero-shot linear classifier by embedding the names or descriptions of the target dataset categories. Building on this, a volume has been optimized to produce a high-scoring image rather than reranking. The CLIP structure is shown in Figure 7. Diffusion model. In the past few years, the use of diffusion model [Ho et al. 2020] has seen a dramatic increase. Also known as denoising diffusion probabilistic models (DDPMs) or score-based generative models, these models generate new data that is similar to the data used to train them. Drawing inspiration from non-equilibrium thermodynamics, DDPMs are defined as a parameterized Markov chain of diffusion steps that adds random noise to the training data and learns to reverse the diffusion process to produce the desired data samples from the pure noise. In the forward process, DDPM destroys the training data by gradually adding Gaussian noise. It starts from a data sample x 0 and iteratively generates noisier samples x with (x | x −1 ), using a Gaussian diffusion kernel: ( 1: | 0 ) := =1 ( | −1 ),(4)( | −1 ) := N ( ; √︁ 1 − −1 , )(5) where and are the steps and hyper-parameters, respectively. We can obtain noised image at arbitrary step with Gaussian noise transition kernel as N in Eq. 5, by setting := 1 − and¯:= =0 : ( | 0 ) := N ( ; √¯0 , (1 −¯) )(6) The reverse denoising process of DDPM involves learning to undo the forward diffusion by performing iterative denoising, thereby generating data from random noise. This process is formally defined as a stochastic process, where the optimization objective is to generate ( 0 ) which follows the true data distribution ( 0 ) by starting from ( ): Fig. 8. Overview of DDPM, the picture is obtained from [Ho et al. 2020] 3.1.4 Pretrained text-guided image generation model. Recently, with the emergence of diffusion models, pre-trained models for text-to-image generation based on diffusion models have become good priors for text-to-3D tasks [Zhang et al. 2023c]. The pioneering works into image framework can be roughly categorized according to the diffusion priors in the pixel space or latent space. The first class of methods directly generates images from high-dimensional pixel-level, including GLIDE [Nichol et al. 2021] and Imagen [Saharia et al. 2022]. Another approach suggests compressing the image to a low-dimensional space first, and then training a diffusion model in this latent space. Representative methods of latent space include Stable Diffusion [Rombach et al. 2022] and DALL-E-2 [Ramesh et al. 2022]. ∼U (1, ),x 0 ∼ (x 0 ), ∼N (0,I) ( ) ∥ − (x , )∥ 2(7) The training process of text-to-image generation can be roughly divided into three steps. Firstly, the CLIP model is employed to learn the correlations between text and visual semantics and map the text into the image representation space. Secondly, a prior model is used for inversion to map the image representation space back to the text space, and thus generate text-conditional images. Lastly, diffusion models are utilized to learn the mapping between text encoding and image encoding, providing prior models for text-conditional image generation. The structure of DALLE-2 is shown in figure 9 Successful attempts Recent pioneering studies have demonstrated the utility of pre-trained text-to-image diffusion models to optimize neural radiance fields, achieving significant text-to-3D synthesis results. However, this paradigm also has some issues, and some followers aim to solve these problems. In this section, the first part presents the pioneers of this paradigm, and the second part presents some enhancement works of this paradigm. Poinners of CLIP-Based Text-Guided 3D Shape Generation. In recent years, with the success of text-to-image generation modelling [Zhang et al. 2023c], text-to-3D generation has also attracted the attention of the deep learning community Lin et al. 2022 optimizing the text-to-3D synthesis results of NeRF [Mildenhall et al. 2021]. The framework can create high-quality 3D mesh models within 40 minutes, being twice as fast as DreamFusion while achieving higher resolution and being rated better by 61.7% of evaluators. Furthermore, it also provides users with a new approach for controlling 3D synthesis, opening up new avenues for a variety of creative applications. SJC (Score Jacobian Chaining) [Wang et al. 2022b] proposed a method to promote 2D diffusion models to 3D by applying chain rules. The paper also pointed out the effectiveness of SJC in 3D text-driven generation tasks. Dream3D [Xu et al. 2022b] is the first attempt to incorporate explicit 3D shapes into the CLIP-guided 3D optimization process. ] proposed a novel multi-class diffusion model for addressing the challenges in semantic-driven 3D shape generation. To solve the problems of single-class generation, low-frequency details, and the need of a large amount of paired data, the authors employ a pre-trained CLIP model to establish a bridge between text, 2D images, and 3D shapes, and apply a conditional flow model to generate Followers of CLIP-Based Text-Guided 3D Shape Generation. Pioneers in the text-to-3D field have achieved new heights with emerging technologies, while also bringing new challenges, such as low textual 3D model matching, slow rendering speed, low resolution of generated 3D models, etc. Numerous research endeavors aim at addressing these issues, which are investigated in this section. DITTO-NeRF [Seo et al. 2023b] is a novel pipeline that can generate high-quality 3D NeRF models from textual prompts or single images. It introduces a progressive 3D object reconstruction scheme, including scale, orientation and mask, which can propagate high-quality information from IB to OB. Compared with the previous artworks from image/text to 3D such as DreamFusion and NeuralLift-360 [Xu et al. 2022a], DITTO-NeRF achieves significant advantages in both quality and diversity, with shorter training time as well. 3D-CLFusion [Li and Kitani 2023] presented a novel text-to-3D creation method which utilizes pre-trained latent variable-based NeRFs to rapidly complete 3D content creation within less than a minute. To tackle the challenges faced by NeRFs, 3D-CLFusion adopts a novel approach called view-invariant diffusion, which utilizes contrastive learning [He et al. 2020] to learn the latent variables and thus can generate high-resolution 3D content rapidly during the inference stage. Experimental results demonstrate that 3D-CLFusion is up to 100 times faster than DreamFusion, and can serve as a plug-and-play tool for text-to-3D with pre-trained NeRFs. 3DFuse [Seo et al. 2023a] proposes a novel framework, incorporating 3D awareness into pre-trained 2D dispersion models to improve the robustness and 3D consistency of the 2D dispersion model-based approach. The view inconsistency problem in score-distilling text-to-3D generation, also known as Janus problem [Hong et al. 2023](as shown in Figure 11) is a big challenge that have to overcome. 3DFuse builds a rough 3D structure given the text prompt and then utilizes the projected specific depth map as the condition of the dispersion model. In addition, a training strategy is introduced to enable the 2D dispersion model to learn to handle the errors and sparsity in the rough 3D structure and a method to ensure semantic consistency among all viewpoints in the scene. Experimental results show that 3DFuse effectively solves the problem of 3D consistency and develops a new approach for 3D reconstruction with 2D dispersion models. In another work [Hong et al. 2023] proposes two de-biased methods to address the Janus problem in fractional distillation for 3D generation. These methods reduce artifacts and improve realism, while achieving a good balance between fidelity of 2D diffusion model and 3D consistency with low overhead. Although Text-to-3D can produce impressive results, it is essentially unconstrained and may lack the ability to guide or enforce 3D structure. Latent-NeRF [Metzer et al. 2022] incorporates both textual guidance and shape guidance for image generation and 3D model generation, as well as a latent-dispersal model for direct application of dispersed rendering on 3D meshes. CompoNeRF ] proposes a novel framework that explicitly combines editable 3D scene layouts, providing effective guidance at both the local and global levels for NeRFs, to address the Guidance collapse problem faced in text-to-3D generation. CompoNeRF allows for flexible editing and recombination of trained local NeRFs into a new scene via manipulation of the 3D layout or textual hints, achieving faithful and editable text-to-3D results, as well as opening up potential directions for multi-object composition via editable 3D scene layouts guided by text. DreamFusion generates volumetric representations instead of mesh representations, which makes it impractical in many downstream applications such as graphics, which require standard 3D representations such as meshes. Text2Mesh [Michel et al. 2022] presents a novel framework, which is capable of editing the style of 3D objects such as colors and local geometry details from textual descriptions. The framework uses a fixed mesh and a learned neural field to handle low-quality meshes without UV parameterization. Furthermore, Text2Mesh does not require any pre-trained generative models or specialized 3D mesh datasets. Thus, it can achieve style synthesis for various shapes of 3D meshes. TextMesh [Tsalicoglou et al. 2023] presents a new method for generating highly realistic 3D meshes from textual prompts, which solves the problem of NeRF being infeasible for most practical applications. To this end, the method extends NeRF to adopt a SDF framework, thus improving mesh extraction, and introduces a new method for fine-tuning mesh textures, eliminating over-saturation effects and enhancing the detail of the output 3D mesh. TEXT-TO-3D APPLICATIONS With the emergence of text-to-3D models guided by text-to-image priors, more fine-grained application domains have been developed, including text-to-avatar, text-to-texture, text-to-scene, etc. This section surveys the text-guided 3D model generation models based on text-to-image priors. Manuscript submitted to ACM Fig. 11. Comparison of the model with Janus problem (left) [Wang et al. 2022b] and the upgraded model (right) [Hong et al. 2023], the picture is obtained from [Hong et al. 2023]. Text Guided 3D Avatar Generation In recent years, the creation of 3D graphical human models has drawn considerable attention due to its extensive applications in areas such as movie production, video gaming, AR/VR and human-computer interactions, and the creation of 3D avatars through natural language could save resources and holds great research prospects. DreamAvatar ] proposed a framework based on text and shape guidance for generating high-quality 3D human avatars with controllable poses. It utilizes a trainable NeRF to predict the density and color features of 3D points, as well as a pre-trained text-to-image diffusion model to provide 2D self-supervision. SMPL [Bogo et al. 2016] model is used to provide rough pose and shape guidance for generation, as well as a dual-space design, including a canonical space and an observation space, which are related by a learnable deformation field through NeRF, allowing optimized textures and geometries to be transferred from the canonical space to the target pose avatar with detailed geometry and textures. Experimental results demonstrate that DreamAvatar significantly outperforms the state of the art, setting a new technical level for 3D human generation based on text and shape guidance. AvatarCraft [Jiang et al. 2023] utilizes a diffusion model to guide the learning of neural avatar geometry and texture based on a single text prompt, thereby addressing the challenge of creating 3D character avatars with specified identity and artistic style that can be easily animated. It also carefully designs an optimization framework of neural implicit fields, including coarse-to-fine multi-boundary box training strategy, shape regularization and diffusion-based constraints, to generate high-quality geometry and texture, and make the character avatars animatable, thus simplifying the animation and reshaping of the generated avatars. Experiments demonstrate the effectiveness and robustness of AvatarCraft in creating character avatars, rendering new views, poses, and shapes. DreamFace [Zhang et al. 2023a] is a progressive scheme for personalized 3D facial generation guided by text. It enables ordinary users to naturally customize CG-pipe compatible 3D facial assets with desired shapes, textures and fine-grained animation capabilities. DreamFace introduces a coarse-to-fine scheme to generate a topologically unified neutral face geometry, utilizes Score Distillation Sampling (SDS) [Rombach et al. 2022] to optimize subtle translations and normals, adopts a dual-path mechanism to generate neutral appearance, and employs two-stage optimization to enhance compact priors for fine-grained synthesis, as well as to improve the animation capability for personalized deformation features. DreamFace can generate realistic 3D facial assets with physical rendering quality and rich animation capabilities from video materials, even for fashion icons, cartoons and fictional aliens in movies. Text Guided 3D Texture Generation Recently, there have been a number of works on text-to-texture, inspired by text-to-3D. This summary lists these works. TEXTure ] presents a novel text guided 3D shape texture generation, editing and transmission method. It utilizes a pre-trained deep-to-image topology model and iterative schemes to colorize the 3D models from different viewpoints, and proposes a novel detailed topology sampling procedure to generate seamless textures from different viewpoints using the three-step segmentation map. Additionally, it presents a method for transferring the generated texture maps to new 3D geometry without explicit surface-to-surface mapping and a method for extracting semantic textures from a set of images without any explicit reconstruction, and provides a way to edit and refine existing textures with text hints or user-provided doodle. TANGO proposes a novel method for programmatic rendering of realistic appearance effects on arbitrary topology given surface meshes. Based on CLIP model, the model is used to decompose the appearance style into spatially varying bidirectional reflectance distribution functions, local geometric variations, and illumination conditions. This enables realistic 3D style transfer through the automatic prediction of reflectance effects, even for bare, low-quality meshes, without the need for training on particular task-specific datasets. Numerous experiments demonstrate that TANGO outperforms the existing text-driven 3D style transfer methods in terms of realism, 3D geometry consistency, and robustness for stylizing low-quality meshes. Fantasia3D [Chen et al. 2023a] presented a novel approach for text-to-high-quality 3D content creation. The method decouples geometry and appearance modeling and learning, and uses a hybrid scene representation and Spatiallyvarying Bidirectional Reflectance Distribution Function (BRDF) learning for surface material to achieve photorealistic rendering of the generated surface. Experimental results show that the method outperforms existing approaches [Lin et al. 2022;] and supports physically plausible simulations of relit, edited, and generated 3D assets. Text Guided 3D Shape Transformation The traditional process of editing 3D models involves dedicated tools and years of training in order to manually carve, extrude, and re-texture given objects. This process is laborious and costly in terms of resources, and recently some research works have attempted to use text-guided 3D model editing, which is demonstrated in this section. Instruct-NeRF2NeRF [Haque et al. 2023] presented a novel text-instructable editing of NeRF scenes, which uses an iterative image-based diffusion model (InstructPix2Pix) [Brooks et al. 2022] to edit the input image while optimizing the underlying scene, thereby generating an optimized 3D scene that follows the edits instructions. Experimental results show that the method is capable of editing large-scale real-world scenes, achieving more realistic and targeted edits than previous works. [Kamata et al. 2023] presented a novel 3D-to-3D transformation method, Instruct 3D-to-3D, which uses a pre-trained image-to-image diffusion model to achieve 3D-to-3D transformation. Furthermore, the method also proposes dynamic scaling, as well as explicitly conditioning on the input source 3D scene, to enhance 3D consistency and controllability. Quantitative and qualitative evaluations demonstrate that Instruct 3D-to-3D achieves higher-quality 3D-to-3D transformation than the baseline methods ]. SKED [Mikaeili et al. 2023] presented a sketch-based technique for editing 3D shapes represented by NeRF. The motivation is to introduce interactive editing into the text-to-3D pipeline, enabling users to edit from a more intuitive control perspective. Results show that SKED can effectively modify the existing neural fields and generate outputs that satisfy user sketches. TextDeformer [Gao et al. 2023] proposes an automatic technique for generating input triangle mesh deformations guided entirely by text prompts. The framework is capable of generating large, low-frequency shape changes as well as small, high-frequency details, relying on differentiable rendering to connect geometry to powerful pre-trained image encoders such as CLIP ] and DINO [Caron et al. 2021]. In order to overcome the problems of artifacts, TextDeformer proposes to use the Jacobian matrix to represent the mesh deformation and encourages deep features to be computed on 2D encoded rendering to ensure shape coherence from all 3D viewpoints. Experimental results show that the method can smoothly deform various source meshes and target text prompts to achieve large modifications and add details. Text Guided 3D Scene Generation 3D scene modeling is a time-consuming task, usually requiring professional 3D designers to complete. To make 3D scene modeling easier, 3D generation should be simple and intuitive to operate while retaining enough controllability to meet users' precise requirements. Recent works [Cohen-Bar et al. 2023;Fridman et al. 2023;Höllein et al. 2023;Po and Wetzstein 2023] in text-to-3D generation has made 3D scene modeling easier. Set-the-Scene [Cohen-Bar et al. 2023] proposed an agent-based global-local training framework for synthesizing 3D scenes, thus filling an important gap from controllable text to 3D synthesis. It can learn a complete representation of each object while also creating harmonious scenes with style and lighting matching. The framework allows various editing options such as adjusting the placement of each individual object, deleting objects from scenes, or refining objects. [ Po and Wetzstein 2023] proposed a local condition diffusion-based text-to-3D scene synthesis approach, which aims to make the generation of complex 3D scenes more intuitive and controllable. By providing control on the semantic parts via text hints and bounding boxes, the method ensures seamless transitions between these parts. Experiments show that the proposed method achieves higher fidelity in the composition of 3D scenes than the related baselines Wang et al. 2022b]. SceneScape [Fridman et al. 2023] proposed a novel text-driven approach to generate permanent views, which is capable of synthesizing long videos of arbitrary scenes solely based on input texts describing the scene and camera positions. The framework combines the generative capacity of a pre-trained text-to-image model [Rombach et al. 2022] with the geometry priors learned from a pre-trained monocular depth prediction model [Ranftl et al. 2021[Ranftl et al. , 2020, to generate videos in an online fashion, and achieves 3D consistency through online testing time training to generate videos with geometry-consistent scenes. Compared with the previous works that are limited to a restricted domain, this framework is able to generate various scenes including walking through a spaceship, a cave, or an ice city. Text2Room [Höllein et al. 2023] proposed a method of generating room-scale 3D meshes with textures from given text prompts. It is the first method to generate attention-grabbing textured room-scale 3D geometries solely from text inputs, which is different from existing methods that focus on generating single objects [Lin et al. 2022; or scaling trajectories(SceneScape) [Fridman et al. 2023] from text. MAV3D (Make-A-Video3D) [Singer et al. 2023] is a method for generating 3D dynamic scenes from text descriptions. The motivation of MAV3D is to provide a method for generating dynamic videos without 3D or 4D data, thus saving a lot of time and money. This method adopts 4D dynamic NeRF and optimizes the scene appearance, density, and motion consistency by querying the Text-to-Video (T2V) [Singer et al. 2022] model based on diffusion. Quantitative and qualitative experiments show that MAV3D improves over the internal baselines established previously. MAV3D is the first method to generate 3D dynamic scenes given text descriptions. CONCLUSION This work conducts the first yet comprehensive survey on text-to-3D. Specifically, we summarize text-to-3D from three aspects: data representations, technologies and applications. We hope this survey can help readers quickly understand the field of text-to-3D and inspire more future works to explore text-to-3D. Fig. 1 . 1Voxel representation of Stanford bunny, the picture is obtained from[Shi et al. 2022]. Fig. 2 . 2Multi-view representation of Stanford bunny, the picture is obtained from[Park et al. 2016]. Fig. 3 . 3Mesh representation of Stanford bunny, the picture is obtained from[Rossi et al. 2021]. Fig. 4 . 4Point cloud representation of Stanford bunny model, the figure is obtained from [Agarwal and Prabhakaran 2009]. Fig. 5 . 5Neural field representation of Stanford bunny, the picture is obtained from[Park et al. 2019]. Fig. 7 . 7CLIP structure, picture obtained from]. Fig. 9 . 9Structure of DALLE-2, picture obtained from[Ramesh et al. 2022]. shape vectors conditioned on the CLIP embeddings, as well as a latent diffusion model conditioned on multi-class shape vectors. Experimental results show that the proposed framework outperforms existing methodsValsesia et al. 2019;Yang et al. 2019; Zhou et al. 2021a]. Fig. 10 . 10DreamFusion outcome. Picture obtained from. Fig. 12 . 123D avatar created by text guided 3D generation model, picture obtained from Fig. 14 . 14Converting input 3D scenes according to the text instructions, the picture is obtained from[Kamata et al. 2023] Fig. 15 . 15Controllable scenes generation from text prompts, the picture is obtained by[Cohen-Bar et al. 2023] ; MohammadKhalid et al. 2022;Wang et al. 2022b; Xu et al. 2022b]. However, the scarcity of 3D data makes expansion with data challenging. DreamField] and CLIP-mesh[Mohammad Khalid et al. 2022] rely on pre-trained image-text models] to optimize underlying 3D representations (RMS and meshes) in order to alleviate the training data problem, achieving high text-image alignment scores for all 2D renderings. Although these methods avoid the costly requirement of 3D training data and primarily rely on a large-scale pre-trained image-text model, they often yield less realistic 2D renderings.Recently, DreamFusion] and Magic3D[Lin et al. 2022] has demonstrated impressive capabilities in text-to-3D synthesis by leveraging a powerful pre-trained text-to-image diffusion model as a strong image prior.DreamField] proposed a method for synthesizing diverse three-dimensional objects based on natural language descriptions. This method combines neural rendering and multi-modal image and text representations, generating various categories of geometry and colors using an optimized multi-viewpoint neural radiation field without the need of three-dimensional supervision. To enhance the realism and visual quality, this paper introduces simple geometric priors including sparsity-induced transmission regularization, scene boundaries and a new MLP architecture.Experimental results show that Dream Fields can generate realistic and multi-viewpoint consistent geometry and colors from various natural language descriptions. CLIP-mesh[Mohammad Khalid et al. 2022] proposed a technique that can achieve zero-shot 3D model generation only using a target textual prompt. The motivation is that, without any 3D supervision, 3D resources corresponding to the input textual prompt can be obtained by adjusting the control shape of the restricted subdivision surface as well as its texture and normal maps, and can be easily deployed in games or modeling applications. DreamFusion] proposes a probability density distillation loss to optimize the pre-trained 2D text-to-image diffusion model as a prior for parameterized image generator for text-to-3D synthesis. It does not require 3D training data or modifications to the image diffusion model, demonstrating the effectiveness of the pre-trained image diffusion model as a prior. Magic3D[Lin et al. 2022] presents a two-stage optimization framework for Text2Tex[Chen et al. 2023b] proposed a novel approach for generating high-quality textures for 3D meshes given text prompts. The goal of this method is to address the accumulation inconsistency and stretching artifacts in text-driven texture generation. The method integrates repair and merging into a pre-trained deep-perceptual image diffusion model to synthesize high-resolution local textures progressively from multiple perspectives. Experiments show that Text2Tex significantly outperforms existing text-driven and GAN-based methods.Fig. 13. Texturing results generated by text guided 3D texture model, picture obtained byX-Mesh [Ma et al. 2023] presents a novel text-driven 3D stylization framework, containing a novel text-guided dynamic attention module (TDAM) for more accurate attribute prediction and faster convergence speed. Additionally, a new standard text-mesh benchmark, MIT-30, and two automatic metrics standards are introduced for future research to achieve fair and objective comparison. Manuscript submitted to ACM Robust blind watermarking of point-sampled geometry. Parag Agarwal, Balakrishnan Prabhakaran, IEEE Transactions on Information Forensics and Security. 4Parag Agarwal and Balakrishnan Prabhakaran. 2009. Robust blind watermarking of point-sampled geometry. IEEE Transactions on Information Forensics and Security 4, 1 (2009), 36-48. 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[]
[ "DocBank: A Benchmark Dataset for Document Layout Analysis", "DocBank: A Benchmark Dataset for Document Layout Analysis" ]
[ "Minghao Li [email protected] \nBeihang University\n\n", "Yiheng Xu \nMicrosoft Research Asia\n\n", "Lei Cui \nMicrosoft Research Asia\n\n", "Shaohan Huang [email protected] \nMicrosoft Research Asia\n\n", "Furu Wei [email protected] \nMicrosoft Research Asia\n\n", "Zhoujun Li \nBeihang University\n\n", "Ming Zhou [email protected] \nMicrosoft Research Asia\n\n" ]
[ "Beihang University\n", "Microsoft Research Asia\n", "Microsoft Research Asia\n", "Microsoft Research Asia\n", "Microsoft Research Asia\n", "Beihang University\n", "Microsoft Research Asia\n" ]
[ "Proceedings of the 28th International Conference on Computational Linguistics" ]
Document layout analysis usually relies on computer vision models to understand documents while ignoring textual information that is vital to capture. Meanwhile, high quality labeled datasets with both visual and textual information are still insufficient. In this paper, we present DocBank, a benchmark dataset that contains 500K document pages with fine-grained tokenlevel annotations for document layout analysis. DocBank is constructed using a simple yet effective way with weak supervision from the L A T E X documents available on the arXiv.com. With DocBank, models from different modalities can be compared fairly and multi-modal approaches will be further investigated and boost the performance of document layout analysis. We build several strong baselines and manually split train/dev/test sets for evaluation. Experiment results show that models trained on DocBank accurately recognize the layout information for a variety of documents. The DocBank dataset is publicly available at https: //github.com/doc-analysis/DocBank. *Equal contributions during internship at Microsoft Research Asia.This work is licensed under a Creative Commons Attribution 4.0 International License. License details: http:// creativecommons.org/licenses/by/4.0/.
10.18653/v1/2020.coling-main.82
[ "https://www.aclweb.org/anthology/2020.coling-main.82.pdf" ]
219,176,849
2006.01038
f57ab8c86e1666c76d4b924e35186b7448c22578
DocBank: A Benchmark Dataset for Document Layout Analysis OnlineCopyright OnlineDecember 8-13, 2020 Minghao Li [email protected] Beihang University Yiheng Xu Microsoft Research Asia Lei Cui Microsoft Research Asia Shaohan Huang [email protected] Microsoft Research Asia Furu Wei [email protected] Microsoft Research Asia Zhoujun Li Beihang University Ming Zhou [email protected] Microsoft Research Asia DocBank: A Benchmark Dataset for Document Layout Analysis Proceedings of the 28th International Conference on Computational Linguistics the 28th International Conference on Computational LinguisticsBarcelona, SpainOnlineDecember 8-13, 2020949 Document layout analysis usually relies on computer vision models to understand documents while ignoring textual information that is vital to capture. Meanwhile, high quality labeled datasets with both visual and textual information are still insufficient. In this paper, we present DocBank, a benchmark dataset that contains 500K document pages with fine-grained tokenlevel annotations for document layout analysis. DocBank is constructed using a simple yet effective way with weak supervision from the L A T E X documents available on the arXiv.com. With DocBank, models from different modalities can be compared fairly and multi-modal approaches will be further investigated and boost the performance of document layout analysis. We build several strong baselines and manually split train/dev/test sets for evaluation. Experiment results show that models trained on DocBank accurately recognize the layout information for a variety of documents. The DocBank dataset is publicly available at https: //github.com/doc-analysis/DocBank. *Equal contributions during internship at Microsoft Research Asia.This work is licensed under a Creative Commons Attribution 4.0 International License. License details: http:// creativecommons.org/licenses/by/4.0/. Introduction Document layout analysis is an important task in many document understanding applications as it can transform semi-structured information into a structured representation, meanwhile extracting key information from the documents. It is a challenging problem due to the varying layouts and formats of the documents. Existing techniques have been proposed based on conventional rule-based or machine learning methods, where most of them fail to generalize well because they rely on hand crafted features that may be not robust to layout variations. Recently, the rapid development of deep learning in computer vision has significantly boosted the data-driven image-based approaches for document layout analysis. Although these approaches have been widely adopted and made significant progress, they usually leverage visual features while neglecting textual features from the documents. Therefore, it is inevitable to explore how to leverage the visual and textual information in a unified way for document layout analysis. Nowadays, the state-of-the-art computer vision and NLP models are often built upon the pre-trained models (Peters et al., 2018;Radford et al., 2018;Devlin et al., 2018;Lample and Conneau, 2019;Dong et al., 2019;Raffel et al., 2019;Xu et al., 2019) followed by fine-tuning on specific downstream tasks, which achieves very promising results. However, pre-trained models not only require large-scale unlabeled data for self-supervised learning, but also need high quality labeled data for task-specific fine-tuning to achieve good performance. For document layout analysis tasks, there have been some image-based document layout datasets, while most of them are built for computer vision approaches and they are difficult to apply to NLP methods. In addition, image-based datasets mainly include the page images and the bounding boxes of large semantic structures, which are not fine-grained token-level annotations. Moreover, it is also time-consuming and labor-intensive to produce humanlabeled and fine-grained token-level text block arrangement. Therefore, it is vital to leverage weak Table , Title supervision to obtain fine-grained labeled documents with minimum efforts, meanwhile making the data be easily applied to any NLP and computer vision approaches. To this end, we build the DocBank dataset, a document-level benchmark that contains 500K document pages with fine-grained token-level annotations for layout analysis. Distinct from the conventional human-labeled datasets, our approach obtains high quality annotations in a simple yet effective way with weak supervision. Inspired by existing document layout annotations (Siegel et al., 2018;Li et al., 2019;Zhong et al., 2019), there are a great number of digital-born documents such as the PDFs of research papers that are compiled by L A T E X using their source code. The L A T E X system contains the explicit semantic structure information using mark-up tags as the building blocks, such as abstract, author, caption, equation, figure, footer, list, paragraph, reference, section, table and title. To distinguish individual semantic structures, we manipulate the source code to specify different colors to the text of different semantic units. In this way, different text zones can be clearly segmented and identified as separate logical roles, which is shown in Figure 1. The advantage of DocBank is that, it can be used in any sequence labeling models from the NLP perspective. Meanwhile, DocBank can also be easily converted into image-based annotations to support object detection models in computer vision. In this way, models from different modalities can be compared fairly using DocBank, and multi-modal approaches will be further investigated and boost the performance of document layout analysis. To verify the effectiveness of DocBank, we conduct experiments using four baseline models: 1) BERT (Devlin et al., 2018), a pre-trained model using only textual information based on the Transformer architecture. 2) RoBERTa , a robustly optimized method for pre-training the Transformer architecture. 3) LayoutLM , a multi-modal architecture that integrates both the text information and layout information. 4) Faster R- CNN (Ren et al., 2015), a high performance object detection networks depending on region proposal algorithms to hypothesize object locations. The experiment results show that the LayoutLM model significantly outperforms the BERT and RoBERTa models and the object detection model on DocBank for document layout analysis. We hope DocBank will empower more document layout analysis models, meanwhile promoting more customized network structures to make substantial advances in this area. The contributions of this paper are summarized as follows: • We present DocBank, a large-scale dataset that is constructed using a weak supervision approach. It enables models to integrate both the textual and layout information for downstream tasks. • We conduct a set of experiments with different baseline models and parameter settings, which confirms the effectiveness of DocBank for document layout analysis. • The DocBank dataset is available at https://github.com/doc-analysis/DocBank. Documents (.tex) Semantic structures with colored fonts (structure-specific colors) T oken annotations by the color to structure mapping The document layout analysis task is to extract the pre-defined semantic units in visually rich documents. Specifically, given a document D composed of discrete token set t = {t 0 , t 1 , ..., t n }, each token t i = (w, (x 0 , y 0 , x 1 , y 1 )) consists of word w and its bounding box (x 0 , y 0 , x 1 , y 1 ). And C = {c 0 , c 1 , .., c m } defines the semantic categories that the tokens are classified into. We intend to find a function F : (C, D) → S, where S is the prediction set: S = {({t 0 0 , ..., t n 0 0 }, c 0 ), ..., ({t 0 k , ..., t n k k }, c k )}(1) DocBank We build DocBank with token-level annotations that supports both NLP and computer vision models. As shown in Figure 2, the construction of DocBank has three steps: Document Acquisition, Semantic Structures Detection, Token Annotation. Meanwhile, DocBank can be converted to the format that is used by computer vision models in a few steps. The current DocBank dataset totally includes 500K document pages, where the training set includes 400K document pages and both the validation set and the test set include 50K document pages. Document Acquisition We download the PDF files on arXiv.com as well as the L A T E X source files since we need to modify the source code to detect the semantic structures. The papers contain Physics, Mathematics, Computer Science and many other areas, which is beneficial for the diversity of DocBank to produce robust models. We focus on English documents in this work and will expand to other languages in the future. Semantic Structures Detection DocBank is a natural extension of the TableBank dataset , where other semantic units are also included for document layout analysis. In this work, the following semantic structures are annotated in DocBank: {Abstract, Author, Caption, Equation, Figure, Footer, List, Paragraph, Reference, Section, Table and Title}. In TableBank, the tables are labeled with the help of the 'fcolorbox' command. However, for DocBank, the target structures are mainly composed of text, where the 'fcolorbox' cannot be well applied. Therefore, we use the 'color' command to distinguish these semantic structures by changing their font colors into structure-specific colors. Basically, there are two types of commands to represent semantic structures. Some of the L A T E X commands are simple words preceded by a backslash. The command \begin{itemize} starts an environment while the command \end{itemize} ends that environment. The real command name is declared as the parameters of the 'begin' command and the 'end' command. We insert the 'color' command to the code of the semantic structures as follows and re-compile the L A T E X documents. Meanwhile, we also define specific colors for all the semantic structures to make them distinguishable. Different structure commands require the 'color' command to be placed in different locations to take effect. Finally, we get updated PDF pages from L A T E X documents, where the font color of each target structure has been modified to the structure-specific color. Token Annotation We use PDFPlumber 1 , a PDF parser built on PDFMiner 2 , to extract text lines and non-text elements with their bounding boxes. Text lines are tokenized simply by white spaces, and the bounding boxes are defined as the most upper-left coordinate of characters and the most lower-right coordinate of characters, since we can only get the coordinates of characters instead of the whole tokens from the parser. For the elements without any texts such as figures and lines in PDF files, we use the class name inside PDFMiner and wrap it using two "#" symbols into a special token. The class names include "LTFigure" and "LTLine" that represent figures and lines respectively. The RGB values of characters and the non-text elements can be extracted by PDFPlumber from the PDF files. Mostly, a token is composed of characters with the same color. Otherwise, we use the color of the first characters as the color of the token. We determine the labels of the tokens according to the color-to-structure mapping in the Section 3.2. A structure may contain both text and not-text elements. For instance, tables consist of words and lines. In this work, both words and lines will be annotated as the "table" class, so as to obtain the layout of a table as much as possible after the elements are tokenized. Object Detection Annotation The DocBank can be easily converted to the annotation format of the object detection models, like Faster R-CNN. The object detection models accept document images, the bounding boxes of semantic structures as input. We classify all the token by the type of semantic structures on a page of the document. For the tokens of the same label, we use breadth-first search to find the Connected Component. We set an x-threshold and a y-threshold. If both of the x coordinates and the y coordinates of two tokens are within the thresholds, they are "Connected". The breadth-first search is used to find all the tokens are connected to each other, which form an object of this label. Repeat the above steps to find all the objects. The bounding box of an object is determined by the most boundary tokens. Dataset Statistics The DocBank dataset consists of 500K document pages with 12 types of semantic units. Table 1 provides the statistics of training, validation, and test set in DocBank, showing that the number of every semantic unit and the percentage of pages with it. As these document pages are randomly drawn to generate training, validation, and test set, the distribution of semantic units in different splits are almost consistent. We also show the distribution of document pages across years in Table 2. We can see that the number of papers is increasing year by year. To preserve this natural distribution, we randomly sample documents of different years to build DocBank without balancing them. Table 3 provides a comparison of the DocBank to the previous document layout analysis datasets, including Article Regions (Soto and Yoo, 2019), GROTOAP2 (Tkaczyk et al., 2014), PubLayNet (Zhong et al., 2019), and TableBank . As shown in the table, DocBank surpasses the existing datasets in both size and number of types of semantic structures. All the listed datasets are image-based while only DocBank supports both text-based and image-based models. Meanwhile, DocBank are built automatically based on the public papers, so it is extendable, which is very rare in existing datasets. Split Abstract Author Caption Equation Figure Footer List Paragraph Reference Section Table Title Train Method As the dataset was fully annotated at token-level, we consider the document layout analysis task as a text-based sequence labeling task. Under this setting, we evaluate three representative pre-trained language models on our dataset including BERT, RoBERTa and LayoutLM to validate the effectiveness of DocBank. To verify the performance of the models from different modalities on DocBank, we train the Faster R-CNN model on the object detection format of DocBank and unify its output with the sequence labeling models to evaluate. Models The BERT Model BERT is a Transformer-based language model trained on large-scale text corpus. It consists of a multi-layer bidirectional Transformer encoder. It accepts a token sequence as input and calculates the input representation by summing the corresponding token, segment, and position embeddings. Then, the input vectors pass multi-layer attention-based Transformer blocks to get the final contextualized language representation. The RoBERTa Model RoBERTa is a more powerful version of BERT, which has been proven successfully in many NLP tasks. Basically, the model architecture is the same as BERT except for the tokenization algorithm and improved training strategies. By increasing the size of the pretraining data and the number of training steps, RoBERTa gets better performance on several downstream tasks. The LayoutLM Model LayoutLM is a multi-modal pre-trained language model that jointly models the text and layout information of visually rich documents. In particular, it has an additional 2-D position embedding layer to embed the spatial position coordinates of elements. In detail, the LayoutLM model accepts a sequence of tokens with corresponding bounding boxes in documents. Besides the original embeddings in BERT, LayoutLM feeds the bounding boxes into the additional 2-D position embedding layer to get the layout embeddings. Then the summed representation vectors pass the BERT-like multilayer Transformer encoder. Note that we use the LayoutLM without image embeddings and more details are provided in the Section 4.2. The Faster R-CNN Model Faster R-CNN is one of the most popular object detection networks. It proposes the Region Proposal Network (RPN) to address the bottleneck of region proposal computation. RPN shares convolutional features with the detection network using 'attention' mechanisms, which leads to nearly cost-free region proposals and high accuracy on many object detection benchmarks. Pre-training LayoutLM LayoutLM chooses the Masked Visual-Language Model(MVLM) and Multi-label Document Classication(MDC) as the objectives when pre-training the model. For the MVLM task,its procedure is to simply mask some of the input tokens at random keeping the corresponding position embedding and then predict those masked tokens. In this case, the final hidden vectors corresponding to the mask tokens are fed into an output softmax over the vocabulary. For the MDC task, it uses the output context vector of [CLS] token to predict the category labels. With these two training objectives, the LayoutLM is pre-trained on IIT-CDIP Test Collection 1.0 3 (Lewis et al., 2006), a large document image collection. Training Samples in Reading Order We organize the DocBank dataset using the reading order, which means that we sort all the text boxes (a hierarchy level higher than text line in PDFMiner) and non-text elements from top to bottom by their top border positions. The text lines inside a text box are already sorted top-to-bottom. We tokenize all the text lines in the left-to-right order and annotate them. Basically, all the tokens are arranged top-to-bottom and left-to-right, which is also applied to all the columns of multi-column documents. Fine-tuning We fine-tune the pre-trained model with the DocBank dataset. As the document layout analysis is regarded as a sequence labeling task, all the tokens are labeled using the output with the maximum probability. The number of output class equals the number of semantic structure types. Experiment Evaluation Metrics As the inputs of our model are serialized 2-D documents, the typical BIO-tagging evaluation is not suitable for our task. The tokens of each semantic unit may discontinuously distribute in the input sequence. In this case, we proposed a new metric, especially for text-based document layout analysis methods. For each kind of document semantic structure, we calculated their metrics individually. The definition is as follows: Settings Our baselines of BERT and RoBERTa are built upon the HuggingFace's Transformers (Wolf et al., 2019) while the LayoutLM baselines are implemented with the codebase in LayoutLM's official repository 4 . We use 8 V100 GPUs with a batch size of 10 per GPU. It takes 5 hours to fine-tune 1 epoch on the 400K document pages. We use the BERT and RoBERTa tokenizers to tokenize the training samples and optimized the model with AdamW. The initial learning rate of the optimizer is 5 × 10 −5 . We split the data into a max block size of N = 512. We use the Detectron2 (Wu et al., 2019) to train the Faster R-CNN model on DocBank. We use the Faster R-CNN algorithm with the ResNeXt (Xie et al., 2017) as the backbone network architecture, where the parameters are pre-trained on the ImageNet dataset. Results Models Abstract Author Caption Equation Figure Footer List Paragraph Reference Section The evaluation results of BERT, RoBERTa and LayoutLM are shown in Table 4. We evaluate six models on the test set of DocBank. We notice that the LayoutLM gets the highest scores on the {abstract, author, caption, equation, figure, footer, list, paragraph, section, table, title} labels. The RoBERTa model gets the best performance on the "reference" label but the gap with the LayoutLM is very small. This indicates that the LayoutLM architecture is significantly better than the BERT and RoBERTa architecture in the document layout analysis task. We also evaluate the ResNeXt-101 model and two ensemble models combining ResNeXt-101 and LayoutLM. The output of the ResNeXt-101 model is the bounding boxes of semantic structures. To unify the outputs of them, we mark the tokens inside each bounding box by the label of the corresponding bounding box. After that, we calculate the metrics following the equation in Section 5.1. Case Study We visualize the outputs of pre-trained BERT and pre-trained LayoutLM on some samples of the test set in Figure 3 and Figure 4. Generally, it is observed that the sequence labeling method performs well on the DocBank dataset, where different semantic units can be identified. For the pre-trained BERT model, we can see some tokens are detected incorrectly, which illustrates that only using text information is still not sufficient for document layout analysis tasks, and visual information should be considered as well. Compared with the pre-trained BERT model, the pre-trained LayoutLM model integrates both the text and layout information. Therefore, it produces much better performance on the benchmark dataset. This is because the 2D position embeddings can model spatial distance and boundary of semantic structures in a unified framework, which leads to the better detection accuracy. The research of document layout analysis can be divided into three categories: rule-based approaches, conventional machine learning approaches, and deep learning approaches. Rule-based Approaches Most of the rule-based works (Lebourgeois et al., 1992;Ha et al., 1995a;Simon et al., 1997;Ha et al., 1995b) are divided into two main categories: the bottom-up approaches and the top-down approaches. Some bottom-up approaches (Lebourgeois et al., 1992;Ha et al., 1995a;Simon et al., 1997) first detect the connected components of black pixels as the basic computational units in document image analysis. The main part of the document segment process is combining them into higher-level structures through different heuristics methods and labeling them according to different structural features. The spatial auto-correlation approach (Journet et al., 2005;Journet et al., 2008) is a bottom-up texture-based method for document layout analysis. It starts by extracting texture features directly from the image pixels to form homogeneous regions and will auto-correlate the document image with itself to highlight periodicity and texture orientation. For the top-down strategy, (Jain and Zhong, 1996) proposed a mask-based texture analysis to locate text regions written in different languages. Run Length Smearing Algorithm converts image-background to image-foreground if the number of background pixels between any two consecutive foreground pixels is less than a predefined threshold, which is first introduced by (Wahl et al., 1982). Document projection profile method was proposed to detect document regions . (Nagy and Seth, 1984) proposed a X-Y cut algorithm that used projection profile to determine document blocks cuts. For the above work, the rule-based heuristic algorithm is difficult to process complex documents, and the Conventional Machine Learning Approaches To address the issue about data imbalance that the learning-based methods suffer from, a dynamic MLP (DMLP) was proposed to learn a less-biased machine model using pixel-values and context information . Usually, block and page-based analysis require feature extraction methods to empower the training and build robust models. The handcrafted features are developed through feature extraction techniques such as Gradient Shape Feature (GSF) or Scale Invariant Feature Transform (SIFT) (Garz et al., 2010;Garz et al., 2012;Garz et al., 2011;Wei et al., 2014a). There are several other techniques that use features extraction methods such as texture features (Chen et al., 2015;Mehri et al., 2013;Mehri et al., 2017;Mehri et al., 2015;Wei et al., 2013;Wei et al., 2014b) and geometric features (Bukhari et al., 2010;Bukhari et al., 2012). Manually designing features require a large amount of work and is difficult to obtain a highly abstract semantic context. Moreover, the above machine learning methods rely solely on visual cues and ignore textual information. Deep Learning Approaches The learning-based document layout analysis methods get more attention to address complex layout analysis. (Capobianco et al., 2018) suggested a Fully Convolutional Neural Network (FCNN) with a weight-training loss scheme, which was designed mainly for text-line extraction, while the weighting loss in FCNN can help in balancing the loss function between the foreground and background pixels. Some deep learning methods may use weights of pre-trained networks. A study by (Oliveira et al., 2018) proposed a multi-task document layout analysis approach using Convolution Neural Network (CNN), which adopted transfer learning using ImageNet. (Yang et al., 2017) treats the document layout analysis tasks as a pixel-by-pixel classification task. He proposed an end-to-end multi-modal network that contains visual and textual information. Conclusion To empower the document layout analysis research, we present DocBank with 500K high-quality document pages that are built in an automatic way with weak supervision, which enables document layout analysis models using both textual and visual information. To verify the effectiveness of DocBank, we conduct an empirical study with four baseline models, which are BERT, RoBERTa, LayoutLM and Faster R-CNN. Experiment results show that the methods integrating text and layout information is a promising research direction with the help of DocBank. We expect that DocBank will further release the power of other deep learning models in document layout analysis tasks. Figure 1 : 1Example annotations of the DocBank. The colors of semantic structure labels are: Abstract , Author , Caption , Equation , Figure , Footer , List , Paragraph , Reference , Section , Figure 2 : 2Data For instance, the section titles in L A T E X documents are usually in the format as follows: \ s e c t i o n { The t i t l e o f t h i s s e c t i o n } Other commands often start an environment. For instance, the list declaration in L A T E X documents is shown as follows: \ b e g i n { i t e m i z e } \ i t e m F i r s t i t e m \ i t e m Second i t e m \ end { i t e m i z e } Figure 3 :Figure 4 : 34Example output of pre-trained LayoutLM and pre-trained BERT on the test set Example output of pre-trained LayoutLM and pre-trained BERT on the test set 7 Related Work Figure 5 : 5Example output of Faster R-CNN on the test set applicable document types are relatively simple. 000 100.00% 50,000 100.00% 50,000 100.00% 500,000 100.00%25,387 25,909 106,723 161,140 90,429 38,482 44,927 398,086 44,813 180,774 19,638 21,688 6.35% 6.48% 26.68% 40.29% 22.61% 9.62% 11.23% 99.52% 11.20% 45.19% 4.91% 5.42% Validation 3,164 3,286 13,443 20,154 11,463 4,804 56,09 49,759 55,49 22,666 2,374 2,708 6.33% 6.57% 26.89% 40.31% 22.93% 9.61% 11.22% 99.52% 11.10% 45.33% 4.75% 5.42% Test 3,176 3,277 13,476 20,244 11,378 4,876 5,553 49,762 5,641 22,384 2,505 2,729 6.35% 6.55% 26.95% 40.49% 22.76% 9.75% 11.11% 99.52% 11.28% 44.77% 5.01% 5.46% All 31,727 32,472 133,642 201,538 113,270 48,162 56,089 497,607 56,003 225,824 24,517 27,125 6.35% 6.49% 26.73% 40.31% 22.65% 9.63% 11.22% 99.52% 11.20% 45.16% 4.90% 5.43% Table 1: Semantic Structure Statistics of training, validation, and test sets in DocBank Year Train Validation Test All 2014 65,976 16.49% 8,270 16.54% 8,112 16.22% 82,358 16.47% 2015 77,879 19.47% 9,617 19.23% 9,700 19.40% 97,196 19.44% 2016 87,006 21.75% 10,970 21.94% 10,990 21.98% 108,966 21.79% 2017 91,583 22.90% 11,623 23.25% 11,464 22.93% 114,670 22.93% 2018 77,556 19.39% 9,520 19.04% 9,734 19.47% 96,810 19.36% Total 400, Table 2 : 2Year Statistics of training, validation, and test sets in DocBankDataset #Pages #Units Image-based? Text-based? Fine-grained? Extendable? Article Regions 100 9 GROTOAP2 119,334 22 PubLayNet 364,232 5 TableBank 417,234 1 DocBank 500,000 12 Table 3 : 3Comparison of DocBank with existing document layout analysis datasets Table 4 : 4The performance of BERT, RoBERTa, LayoutLM and Faster R-CNN on the DocBank test set. https://github.com/jsvine/pdfplumber 2 https://github.com/euske/pdfminer https://ir.nist.gov/cdip/ https://aka.ms/layoutlm AcknowledgmentsThis work was supported in part by the National Natural Science Foundation of China (Grant Nos.U1636211, 61672081, 61370126), the Beijing Advanced Innovation Center for Imaging Technology (Grant No.BAICIT-2016001), and the Fund of the State Key Laboratory of Software Development Environment (Grant No.SKLSDE-2019ZX-17). Text line extraction using dmlp classifiers for historical manuscripts. Micheal Baechler, Marcus Liwicki, Rolf Ingold, 12th International Conference on Document Analysis and Recognition. 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[ "https://github.com/doc-analysis/DocBank.", "https://github.com/jsvine/pdfplumber", "https://github.com/euske/pdfminer" ]
[ "Model-Based Reinforcement Learning with Value-Targeted Regression", "Model-Based Reinforcement Learning with Value-Targeted Regression" ]
[ "Alex Ayoub \nDepartment of Computing Science\nUniversity of Alberta\n\n", "Zeyu Jia \nSchool of Mathematical Science\nPeking University\n\n", "Csaba Szepesvári \nDepartment of Computing Science\nUniversity of Alberta\n\n", "Mengdi Wang \nDepartment of Electrical Engineering\nPrinceton University\n\n\nCenter for Statistics and Machine Learning\nPrinceton University\n\n", "Lin F Yang \nDepartment of Electrical and Computer Engineering\nUniversity of California\nLos Angeles 6 DeepMind\n" ]
[ "Department of Computing Science\nUniversity of Alberta\n", "School of Mathematical Science\nPeking University\n", "Department of Computing Science\nUniversity of Alberta\n", "Department of Electrical Engineering\nPrinceton University\n", "Center for Statistics and Machine Learning\nPrinceton University\n", "Department of Electrical and Computer Engineering\nUniversity of California\nLos Angeles 6 DeepMind" ]
[]
This paper studies model-based reinforcement learning (RL) for regret minimization. We focus on finite-horizon episodic RL where the transition model P belongs to a known family of models P, a special case of which is when models in P take the form of linear mixtures: P θ = d i=1 θiPi. We propose a model based RL algorithm that is based on optimism principle: In each episode, the set of models that are 'consistent' with the data collected is constructed. The criterion of consistency is based on the total squared error of that the model incurs on the task of predicting values as determined by the last value estimate along the transitions. The next value function is then chosen by solving the optimistic planning problem with the constructed set of models. We derive a bound on the regret, which, in the special case of linear mixtures, the regret bound takes the formÕ(d √ H 3 T ), where H, T and d are the horizon, total number of steps and dimension of θ, respectively. In particular, this regret bound is independent of the total number of states or actions, and is close to a lower bound Ω( √ HdT ). For a general model family P, the regret bound is derived using the notion of the so-called Eluder dimension proposed by Russo & Van Roy (2014). * [email protected] †
null
null
219,177,541
2006.01107
175a9a3f0bb4f31fa235386aff52ad18c67275d3
Model-Based Reinforcement Learning with Value-Targeted Regression June 2, 2020 Alex Ayoub Department of Computing Science University of Alberta Zeyu Jia School of Mathematical Science Peking University Csaba Szepesvári Department of Computing Science University of Alberta Mengdi Wang Department of Electrical Engineering Princeton University Center for Statistics and Machine Learning Princeton University Lin F Yang Department of Electrical and Computer Engineering University of California Los Angeles 6 DeepMind Model-Based Reinforcement Learning with Value-Targeted Regression June 2, 2020 This paper studies model-based reinforcement learning (RL) for regret minimization. We focus on finite-horizon episodic RL where the transition model P belongs to a known family of models P, a special case of which is when models in P take the form of linear mixtures: P θ = d i=1 θiPi. We propose a model based RL algorithm that is based on optimism principle: In each episode, the set of models that are 'consistent' with the data collected is constructed. The criterion of consistency is based on the total squared error of that the model incurs on the task of predicting values as determined by the last value estimate along the transitions. The next value function is then chosen by solving the optimistic planning problem with the constructed set of models. We derive a bound on the regret, which, in the special case of linear mixtures, the regret bound takes the formÕ(d √ H 3 T ), where H, T and d are the horizon, total number of steps and dimension of θ, respectively. In particular, this regret bound is independent of the total number of states or actions, and is close to a lower bound Ω( √ HdT ). For a general model family P, the regret bound is derived using the notion of the so-called Eluder dimension proposed by Russo & Van Roy (2014). * [email protected] † Introduction Reinforcement learning (RL) enables learning to control complex environments through trial and error. It is a core problem in artificial intelligence (Russel & Norvig, 2003;Sutton & Barto, 2018) and recent years has witnessed phenomenal empirical advances in various areas such as: games, robotics and science (e.g., Mnih et al., 2015;Silver et al., 2017;AlQuraishi, 2019;Arulkumaran et al., 2019). In online RL, an agent has to learn to act in an unknown environment "from scratch", collect data as she acts, and adapt the policy to maximize the reward collected. An important problem is to design algorithms that provably achieve sublinear regret in a large class of environments. Regret minimization for RL has received considerable attention in recent years (e.g., Jaksch et al. 2010;Azar et al. 2017;Dann et al. 2017Dann et al. , 2018Agrawal & Jia 2017;Osband et al. 2017;Jin et al. 2018;Yang & Wang 2019a;Jin et al. 2019). While most of these existing works focus on the tabular or linear-factored MDP, only a handful of prior efforts have studied RL with general model classes. In particular, in a pioneering paper Strens (2000) proposed to use posterior sampling, which was later analyzed in the Bayesian setting by ; Abbasi-Yadkori & Szepesvári (2015); Theocharous et al. (2017). The reader is referred to Section 5 for a discussion of these and other related works. In this paper, we study episodic reinforcement learning in an environment where the unknown probability transition model is known to belong to a family of models, i.e., P ∈ P. The model family P is a general set of models, and it may be either finitely parametrized or nonparametric. In particular, our approach accommodates working with smoothly parameterized models (e.g., Abbasi-Yadkori & Szepesvári, 2015), and can find use in both robotics (Kober et al., 2013) and queueing systems (Kovalenko, 1968). An illuminating special case is the case of linear parametrization when elements of P take the form P θ = i θ i P i where P 1 , P 2 , . . . , P d are fixed, known basis models and θ = (θ 1 , . . . , θ d ) are unknown, real-valued parameters. Model P θ can be viewed as a mixture model that aggregates a finite family of known basic dynamical models (Modi et al., 2019). As an important special case, linear mixture models include the linear-factor MDP model of Yang & Wang (2019a), a model that allows the embedding of possible transition kernels into an appropriate space of finite matrices. The main contribution of this paper is a model-based upper confidence RL algorithm where the main novelty is the criterion to select models that are deemed consistent with past data. As opposed to standard practice where the models are selected based on their ability to predict next states or raw observations there (cf. Jaksch et al. (2010); Yang & Wang (2019a) or (Strens, 2000;Abbasi-Yadkori & Szepesvári, 2015;Ouyang et al., 2017;Agrawal & Jia, 2017) in a Bayesian setting), we propose to evaluate models based on their ability to predict the values at next states as computed using the last value function estimate produced by our algorithm. In effect, the algorithm aims to select models based on their ability to produce small losses in a value-targeted regression problem. Value-targeted regression is attractive for multiple reasons: (i) First and foremost, value-targeted regression holds the promise that model learning will focus on task-relevant aspects of the transition dynamics and can ignore aspects of the dynamics that are not relevant for the task. This is important as the dynamics can be quite complicated and modelling irrelevant aspects of the dynamics can draw valuable resources away from modelling task-relevant aspects. (ii) A related advantage is that building faithful probability models with high-dimensional state variables (or observations) can be challenging. Value-targeted regression sets up model learning as a real-valued regression problem, which intuitively feels easier than either building a model with maximum likelihood or setting up a vector-valued regression problem to model next state probabilities. (iii) Value-targeted regression aims at directly what matters in terms of the model accuracy or regret. Specifically the objective used in value-targeted is obtained from an expression that upper bounds the regret, hence it is natural to expect that minimizing this will lead to a small regret. In addition, our approach is attractive as the algorithm has a modular structure and this allows any advances on components (optimistic planning, improvements in designing confidence sets) to be directly translated into a decreased regret. One may also question whether value-targeted regression is going "too far" in ignoring details of the dynamics. Principally, one may think that since the value function used in defining the regression targets is derived based on imperfect knowledge, the model may never be sufficiently refined in a way that allows the regret to be kept under control. Secondly, one may worry about that by ignoring the rich details of observations (in our simple model, the identity of the state), the approach advocated is ignoring information available in the data, which may slow down learning. To summarize, the main question, to which we seek an answer in this paper, is the following: Is value-targeted regression sufficient and efficient for model-based online RL? Based on the theoretical and the experimental evidence that we provide in this paper, our conclusion is that the answer is 'yes'. Firstly, the regret bounds we derive conclusively show that the despite the imperfection and non-stationarity of the value targets, the algorithm cannot get "stuck" (i.e., it enjoys sublinear regret). Our results also suggest that perhaps there is no performance degradation as compared to the performance of competing algorithms. We are careful here as this conclusion is based on comparing worst-case upper bounds, which cannot provide a definitive answer. To complement the theoretical findings, our experiments also confirm that our algorithm is competitive. The experiments also allow us to conclude that it is value-targeted regression together with optimistic planning that is effective. In particular, if optimism is taken away (i.e., -greedy is applied for the purpose of providing sufficient exploration), value-targeted regression performs worse than using a canonical approach to estimate the model. Similarly, if value-targeted regression is taken away, optimism together with the canonical model-estimation approach is less sample-efficient. This still leaves open the possibility that certain combinations of value-targeted regression and canonical model building can be more effective than value-targeted regression. In fact, given the vast number of possibilities, we find this to be a quite plausible hypothesis. We note in passing that our proofs can be adjusted to deal with adding simultaneous alternative targets for constraining the set of data-consistent models. However, sadly, our current theoretical tools are unable to exhibit the tradeoffs that one expects to see here. It is interesting to note that, in an independent and concurrent work, value-targeted regression has also been suggested as the main model building tool of the MuZero algorithm. The authors of this algorithm empirically evaluated MuZero on a number of RL benchmarks, such as the 57 Atari "games", the game of "Go", chess and shogi (Schrittwieser et al., 2019). In these benchmarks, despite the fact that MuZero does not use optimistic exploration or any other "smart" exploration technique, MuZero was found to be highly competitive with its state-of-the-art alternatives, which reinforces the conclusion that training models using value-targeted regression is indeed a good approach to build effective model-based RL algorithms. The good results of MuZero on these benchmark may seem to contradict our experimental findings that value-targeted regression is ineffective without an appropriate, 'smart' exploration component. However, there is no contradiction: Smart exploration may be optional in some environments; our experiments show that it is not optional on some environments. In short, for robust performance across a wide range of environments, smart exploration is necessary but smart exploration may be optional in some environments. As to the organization of the rest of the paper, the next section (Section 2) introduces the formal problem definition. This is followed by the description of the new algorithm (Section 3) and the main theoretical results (Section 4). In particular, we first give a regret bound for the general case where the regret is expressed as a function of the "richness" of the model class P. This analysis is based on the Eluder dimension of an appropriately defined function class and its metric entropy at an appropriate scale. It is worth noting that the regret bound does not depend on either the size of the state or the size of the action space. To illustrate the strength of this general technique, we specialize the regret bound for the case of linear mixture models, for which we prove that the expected cumulative regret is at most O(d √ H 3 T ), where H is the episode length, d is the number of model parameters and T is the total number of steps that the RL algorithm interacts with its environment. To complement the upper bound, for the linear case we also provide a regret lower bound Ω( √ HdT ) by adapting a lower bound that has been derived earlier for tabular RL. After these results, we discuss the connection of our work to prior art (Section 5). This is followed by the presentation of our empirical results (Section 6), where, as it was alluded to earlier, the aim is to explore how the various parts of the algorithm interact with each other. Section 7 concludes the paper. Problem Formulation We study episodic Markov decision processes (MDPs, for short), described by a tuple M = (S, A, P, r, H, s • ). Here, S is the state space, A is the action space, P is the transition kernel, r is a reward function, H > 0 is the episode length, or horizon, and s • ∈ S is the initial state. In the online RL problem, the learning agent is given S, A, H and r but does not know P . 1 The agent interacts with its environment described by M in episodes. Each episode begins at state s • and ends after the agent made H decisions. At state s ∈ S, the agent, after observing the state s, can choose an action a ∈ A. As a result, the immediate reward r(s, a) is incurred. Then the process transitions to a random next state s ∈ S according to the transition law P (·|s, a). 2 The agent's goal is to maximize the total expected reward received over time. If P is known, the behavior that achieves this over any number of episodes can be described by applying a deterministic policy π. Such a policy is a mapping from S × [H] into A, where we use the convention that for a natural number n, [n] = {1, . . . , n}. Following the policy means that the agent upon encountering state s in stage h will choose action π(s, h). In what follows, we will use π h (s) as an alternate notation, as this makes some of the formulae more readable. We will also follow this convention when it comes to other functions whose domain is S × [H]. We will find it convenient to move the stage h into the index. In particular, for policies, we will also write π h (s) for π(s, h) but we will use the same convention for other similar objects, like the value function, defined next. The value function V π : S × [H] → R of a policy π is defined via V π h (s) = E π H i=h r(s i , π(s i )) s h = s , s ∈ S , where the subscript π (which we will often suppress) signifies that the probabilities underlying the expectation are governed by π. An optimal policy π * and the optimal value function V * are defined to be a policy and the value function such that V π h (s) achieves the maximum among all possible policies for any s ∈ S and h ∈ [H]. As noted above, there is no loss of generality in restricting the search of optimal policies to deterministic policies. In online RL, a good agent of course uses all past observations to come up with its decisions. The performance of such an agent is measured by its regret, which is the total reward the agent misses because they did not follow the optimal policy from the beginning. In particular, the total expected regret of an agent A across K episodes is given by R(T ) = K k=1 V * 1 (s k 1 ) − H h=1 r(s k h , a k h ) ,(1) where T = KH is the total number of time steps that the agent interacts with its environment, s k 1 = s • is the initial state at the start of the k-th episode, and s 1 1 , a 1 1 , . . . , s k H , a k H , . . . , s K 1 , a K 1 , . . . , s K H , a K H are the T = KH state-action pairs in the order that they are encountered by the agent. The regret is sublinear of R(T )/T → 0 as T → ∞. For a fixed T , let R * (T ) denote the worst-case regret. As is well known, no matter the algorithm used, R * (T ), grows at least as fast as √ T (e.g., Jaksch et al., 2010). In this paper, we aim to design a general model-based reinforcement learning algorithm, with a guaranteed sublinear regret, for any given family of transition models. Assumption 1 (Known Transition Model Family). The unknown transition model P belongs to a family of models P which is available to the learning agent. The elements of P are transition kernels mapping state-action pairs to signed distributions over S. That we allow signed distributions increases the generality; this may be important when one is given a model class that can be compactly represented but only when it also includes non-probability kernels (see Pires & Szepesvári 2016 for a discussion of this). An important special case is the class of linear mixture models: Definition 1 (Linear Mixture Models). We say that P is the class of linear mixture models with component models P 1 , . . . , P d if P 1 , . . . , P d are transition kernels that map state-action pairs to signed measures and P ∈ P if and only if there exists θ ∈ R d such that P (ds |s, a) = d j=1 θ j P j (ds |s, a) = P•(ds |s, a) θ * , for all (s, a) ∈ S × A. Parametric and nonparametric transition models are common in modelling complex stochastic controlled systems. For one example, robotic systems are often smoothly parameterized by unknown mechanical parameters such as friction, or just parameters that describe the geometry of the robot. The linear mixture model can be viewed as a way of aggregating a number of known basis models as considered by Modi et al. (2019). We can view each P j (·|·) as a basis latent "mode" and the actual transition is a probabilistic mixture of these latent modes. For another example, consider large-scale queueing networks where the arrival rate and job processing speed for each queue is not known. By using a discrete-time Bernoulli approximation, the transition probability matrix from time t to t + ∆t becomes increasingly close to linear with respect to the unknown arrival/processing rates as ∆t → 0. In this case, it is common to model the discrete-time state transition as a linear aggregation of arrival/processing processes with unknown parameters Kovalenko (1968). Another interesting special case is the linear-factored MDP model of Yang & Wang (2019a) where, assuming a discrete state space for a moment, P takes the form P (s |s, a) = φ(s, a) M ψ(s ) = d1 i=1 d2 j=1 M ij [ψ j (s )φ i (s, a)] , where φ(s, a) ∈ R d1 , ψ(s ) ∈ R d2 are given features for every s, s ∈ S and a ∈ A (when the state space is continuous, ψ becomes an R d2 -valued measure over S). The matrix M ∈ R d1×d2 is an unknown matrix and is to be learned. It is easy to see that the factored MDP model is a special case of the linear mixture model (2) with each ψ j (s )φ i (s, a) being a basis model (this should be replaced by ψ j (ds )φ i (s, a) when the state space is continuous). In this case, the number of unknown parameters in the transition model is d = d 1 × d 2 . In this setting, without any additional assumption, our regret bound matches the result of Yang & Wang (2019a). Upper Confidence RL with Value-Targeted Model Regression Our algorithm can be viewed as a generalization of UCRL (Jaksch et al., 2010), following ideas of . In particular, at the beginning of episode k = 1, 2, . . . , K, the algorithm first computes a subset B k of the model class P that contains the set of models that are deemed to be consistent with all the data that has been collected in the past. The new idea, value-targeted regression is used in the construction of B k . The details of how this is done are postponed to a later section. Next, the algorithm needs to find the model that maximizes the optimal value, and the corresponding optimal policy. Denoting by V * P the optimal value function under a model P , this amounts to finding the model P ∈ B k that maximizes the value V * P,1 (s k 1 ). Given the model P k that maximizes this value, an optimal policy is extracted from the model as described in the next section (this is standard dynamic programming). At the end of the episode, the data collected is used to refine the confidence set B k . The pseudocode of the algorithm can be found in Algorithm 1. Model-Based Optimistic Planning Upper confidence methods are prominent in online learning. In our algorithm, we will maintain a confidence set B k for the estimated transition model and use it for optimistic planning: P k = argmax P ∈B k V * P ,1 (s k 1 ) Algorithm 1 UCRL-VTR Observe the initial state s k 1 of episode k 6: Optimistic planning: P k = argmax P ∈B k V * P ,1 (s k 1 ) Compute Q 1,k , . . . Q H,k for P k using (3) 7: for h = 1, 2, . . . , H do 8: Choose the next action greedily with respect to Q h,k : a k h = arg max a∈A Q h,k (s k h , a) 9: Observe state s k h+1 10: Compute and store value predictions: y h,k ← V h+1,k (s k h+1 ) 11: end for 12: Construct confidence set using value-targeted regression as described in Section 3.2: B k+1 = {P ∈ P|L k+1 (P ,P k+1 ) ≤ β k } 13: end for where s k 1 is the initial state at the beginning of episode k and we use V * P ,1 to denote the optimal value function of stage one, when the transition model is P . Given model P k , the optimal policy for P k can be computed using dynamic programming. In particular, for 1 ≤ h ≤ H + 1, define Q H+1,k (s, a) = 0, V h,k (s) = max a∈A Q h,k (s, a), Q h,k (s, a) = r(s, a) + P k (·|s, a), V h+1,k ,(3) where we with a measure µ and function f over the same domain, we use µ, f to denote the integral of f with respect to µ. Then, taking the action at stage h and state s that maximizes Q h,k (s, ·) gives an optimal policy for model P k . As long as P ∈ B k with high probability, the preceding calculation gives an optimistic (that is, upper) estimate of value of an episode. Next we show how to construct the confidence set B k . Value-Targeted Regression for Confidence Set Construction Every time we observe a transition (s, a, s ) with s ∼ P (·|s, a), we receive information about the model P . Instead of regression onto fixed target like probabilities or raw states, we will refresh the model estimate by regression using the estimated value functions as target. This leads to the model P k+1 = argmin P ∈P k k =1 H h=1 P (·|s k h , a k h ), V h+1,k − y h,k 2 , where(4)y h,k = V h+1,k (s k h+1 ) , h ∈ [H], k ∈ [k] . In the above regression procedure, the regret target keeps changing as the algorithm constructs increasingly accurate value estimates. This is in contrast to typical supervised learning for building models, where the regression targets are often fixed objects (such as raw observations, features or keypoints; e.g. Jaksch et al. For a confidence set construction, we get inspiration from Proposition 5 in the paper of . The set is centered atP k+1 .Define L k+1 (P,P k+1 ) = k k =1 H h=1 P (·|s k h , a k h ) −P k+1 (·|s k h , a k h ), V h+1,k 2 . Then we let B k+1 = {P ∈ P | L k+1 (P ,P k+1 ) ≤ β k+1 } and the value of β k can be obtained using a calculation similar to that done in Proposition 5 of the paper of , which is based on the nonlinear least-squares confidence set construction from Russo & Van Roy (2014), which we describe in the appendix. It is not hard to see that the confidence set can also be written in the alternative form B k+1 = {P ∈ P |L k+1 (P ) ≤β k+1 } with a suitably definedβ k+1 and wherẽ L k+1 (P ) = k k =1 H h=1 P (·|s k h , a k h ), V h+1,k − y h,k 2 . Note that the above formulation strongly exploits that the MDP is time-homogeneous: The same transition model is used at all stages of an episode. When the MDP is time-inhomogeneous, the construction can be easily modified to accommodate that the transition kernel may depend on the stage index. Implementation of UCRL-VTR Algorithm 1 gives a general and modular template for model-based RL that is compatible with regression methods/optimistic planners. While the algorithms is conceptually simple, and the optimization and evaluation of the loss in value-targeted regression appears to be at advantage in terms of computation as compared to standard losses typically used in model-based RL, the implementation of UCRL-VTR is nontrivial in general and for now it requires a case-by-base design. Computation efficiency of the algorithm depends on the specific family of models chosen. For the linear-factor MDP model considered by Yang & Wang (2019a), the regression is linear and admits efficient implementation; further, optimistic planning for this model can be implemented in poly(d) time by using Monte-Carlo simulation and sketching as argued in the cited paper. Other ideas include loosening the confidence set to come up with computationally tractable methods, or relaxing the requirement that the same model is used in all stages. In the general case, optimistic planning is computationally intractable. However, we expect that randomized (eg Osband et al. (2017; Lu & Van Roy (2017)) and approximate dynamic programming methods (tree search, roll out, see eg Bertsekas & Tsitsiklis (1996)) will often lead to tractable and good approximations. As was mentioned above, in some special cases these have been rigorously shown to work. In similar settings, the approximation errors are known to mildly impact the regret Abbasi-Yadkori & Szepesvári (2015) and we expect the same will hold in our setting. If we look beyond methods with rigorous guarantees, there are practical deep RL algorithms that implement parts of UCRL-VTR. As mentioned earlier, the Muzero algorithm of Schrittwieser et al. (2019) is a stateof-the-art algorithm on the Atari domain and this algorithm implements both value-targeted-regression to learn a model and Monte Carlo tree search for planning based on the learned model, although it does not incorporate optimistic planning. Theoretical Analysis We will need the concept of Eluder dimension. Let F be a set of real-valued functions with domain X . To measure the complexity of interactively identify an element of F, Russo & Van Roy (2014) defines the Eluder dimension of F at scale > 0. For f ∈ F, x 1 , . . . , x t ∈ X , introduce the notation f | (x1,...,xt) = (f (x 1 ), . . . , f (x t )). We say that x ∈ X is -independent of x 1 , . . . , x t ∈ X given F if there exists f, f ∈ F such that (f − f )| (x1,...,xt) 2 ≤ while f (x) − f (x) > . Definition 2 (Eluder dimension Russo & Van Roy (2014)). The Eluder dimension dim E (F, ) of F at scale is the length of the longest sequence (x 1 , . . . , x n ) in X such that for some ≥ , for any 2 ≤ t ≤ n, x t is -independent of (x 1 , . . . , x t−1 ) given F. Let V be the set of optimal value functions under some model in P: V = {V * P : P ∈ P}. Note that V ⊂ B(S, H), where B(S, H) denotes the set of real-valued measurable functions with domain S that are bounded by H. We let X = S × A × V. In order to analyze the confidence of nonlinear regression, choose F to be the collection of functions f : X → R as F = f ∃P ∈ P s.t. for any (s, a, v) ∈ S × A × V, f (s, a, v) = P (ds |s, a)v(s ) .(5) Note that F ⊂ B(X , H). For a norm · on F and α > 0 let N (F, α, · ) denote the (α, · )-covering number of F. That is, this if m = N (F, α, · ) then one can find m points of F such that any point in F is at most α away from one of these points in norm · . Denote by · ∞ the supremum norm: f ∞ = sup x∈X |f (x)|. Now we analyze the regret of UCRL-VTR. Define the K-episode pseudo-regret as R K = K k=1 V * (s k 0 ) − V π k (s k 0 ) . Clearly, R(KH) = ER K holds for any K > 0 where R(T ) is the expected regret after T steps of interaction as defined in (1). Thus, to study the expected regret, it suffices to study R K . Our main result is as follows. Theorem 1 (Regret of Algorithm 1). Let Assumption 1 hold and let α ∈ (0, 1). For k ∈ [K], let β k be β k = 2H 2 log 2N (F, α, · ∞ ) δ + 2H(kH − 1)α 2 + log 4kH(kH − 1) δ .(6) Then, with probability 1 − 2δ, R K ≤ α+H(d ∧ K(H − 1)) + 4 dβ K K(H − 1) + H 2K(H − 1) log(1/δ) , where d = dim E (F, α) is the Eluder dimension with F given by (5). A typical choice of α is α = 1/(KH). In the special case of linear transition model, Theorem 1 implies a worst-case regret bound that depends linearly on the number of parameters. Corollary 2 (Regret of Algorithm 1 for Linearly-Parametrized Transition Model). Let P 1 , . . . , P d be d transition models, Θ ⊂ R d a nonempty set with diameter R measured in · 1 and let P = { j θ j P j : θ ∈ Θ}. Then, for any 0 < δ < 1, with probability at least 1 − δ, the pseudo-regret R K of Algorithm 1 when it uses the confidence sets given in Theorem 1 satisfies R K =Õ(d H 3 K log(1/δ)) . We also provide a lower bound for the regret in our model. The proof is by reduction to a known lower bound and is left to Appendix B. Theorem 3 (Regret Lower Bound). For any H ≥ 1 and d ≥ 8, there exist a state space S and action set A, a reward function r : S × A → [0, 1], d transition models P 1 , . . . , P d and a set Θ of diameter at most one such that for any algorithm there exists θ ∈ Θ such that for sufficiently large number of episodes K, the expected regret of the algorithm on the H-horizon MDP with reward r and transition model P = j θ j P j is at least Ω(H √ dK). Rusmevichientong & Tsitsiklis (2010) gave a regret lower bound of Ω(d √ T ) for linearly parameterized bandit with actions on the unit sphere (see also Section 24.2 of Lattimore & Szepesvári (2020)). Our regret upper bound matches this bandit lower bound in d, T . Whether the upper or lower bound is tight (or none of them) remains to be seen. The theorems validate that, in the setting we consider, it is sufficient to use the predicted value functions as regression targets. That for the special case of linear mixture models the lower bound is close to the upper bound appears to suggest that little benefit if any can be derived from fitting the transition model to predict well future observations. We conjecture that this is in fact true when considering the worst-case regret. Of course, a conclusion that is concerned with the worst-case regret has no implication for the behavior of the respective methods on particular MDP instances. We note in passing that by appropriately increasing β k , the regret upper bounds can be extended to the so-called misspecified case when P can be outside of P (for related results, see, e.g., Jin et al. 2019; Lattimore & Szepesvári 2019). However, the details of this are left for future work. Related Work A number of prior efforts have established efficient RL methods with provable regret bounds. For tabular MDPs with S states and A actions, building on the pioneering work of Jaksch et al. (2010) who studied the technically more challenging continuing setting, a number of works obtained results for the episodic setting, both with model-based (e.g., Azar et al. 2017;Dann et al. 2017Dann et al. , 2018Agrawal & Jia 2017), and model-free methods (e.g., Jin et al. 2018;Russo 2019;Zhang et al. 2020), both for the time-homogeneous case (i.e., the same transition kernel governs the dynamics in all stages of the H-horizon episode) and the time-inhomogeneous case. Results developed for the time-inhomogeneous case apply to the time-inhomogeneous case and since in this case the number of free parameters to learn is at least H times larger than for the time-homogeneous case, the regret bounds are expected to be √ H larger. As far as regret lower bounds are concerned, Jaksch et al. (2010) established a worst-case regret lower bound of Ω( √ DSAT ) for the continuing case for MDPs with diameter bounded by D (see also Chapter 38 of Lattimore & Szepesvári 2020). This lower bound can be adapted to the episodic by setting D = H. This way one obtains a lower bound of Ω( √ HSAT ) for the homogeneous, and Ω( √ H 2 SAT ) for the inhomogeneous case (because here the state space size is effectively HS). This lower bound, up to lower order terms, is matched by upper bounds both for the time-homogeneous case (Azar et al., 2017;Kakade et al., 2018) and the time inhomogeneous case (Dann et al., 2018;Zhang et al., 2020). Except the work of Zhang et al. (2020), these results are achieved by algorithms that estimate models. With a routine adjustment, the near-optimal model-based algorithms available for the homogeneous case are also expected to deliver near-optimal worstcase regret growth in the inhomogeneous case. A further variation is obtained by considering different scalings of the reward (Wang et al., 2020a). Moving beyond tabular MDP, there have been significant theoretical and empirical advances on RL with function approximation, including but not limited to Baird (1995); Tsitsiklis & Van Roy (1997); Parr et al. (2008); Mnih et al. (2013Mnih et al. ( , 2015; Silver et al. (2017); Yang & Wang (2019b); Bradtke & Barto (1996). Among these works, many papers aim to uncover algorithms that are provably efficient. Under the assumption that the optimal action-value function is captured by linear features, Zanette et al. (2019) considers the case when the features are "extrapolation friendly" and a simulation oracle is available, Wen & Van Roy (2013 tackle problems where the transition model is deterministic, Du et al. (2019) deals with a relaxation of the deterministic case when the transition model has low variance. Yang & Wang (2019b) considers the case of linear factor models, while Lattimore & Szepesvári (2019) considers the case when all the action-value functions of all deterministic policies are well-approximated using a linear function approximator. These latter works handle problems when the algorithm has access to a simulation oracle of the MDP. As for regret minimization in RL using linear function approximation, Yang & Wang (2019a) where d is the feature dimension, K is the number of basis models and d · K is their total model complexity. Very recently, Wang et al. (2020b) propose an model-free algorithm for general reward function approximation and show that the learning complexity of the function class can be bounded by the eluder dimension, which is similar to our model-based setting. As for RL with a general model class, the seminal paper Osband & Van Roy (2014) (2015) considered a smooth parameterization with a somewhat unusual definition of smoothness. To the authors' best knowledge, there are no prior works addressing the problem of designing low-regret algorithms for MDPs with a general model family. In particular, while sketch the main ideas of an optimistic model-based optimistic algorithm for a general model class, they left out the details. When the details are filled based on their approach for the Bayesian case, unlike in the present work, the confidence sets would be constructed by losses that measure how well the model predict future observations and not by the value-targeted regression loss studied here. A preliminary version of the present paper appeared at L4DC 2020, which included results for the linear transition model only. Numerical Experiments The goal of our experiments is to provide insight into the benefits and/or pitfalls of using value-targets for fitting models, both with and without optimistic planning. We run our experiments in the tabular setting as in this setup it is easy to keep all the aspects of the test environments under control and the tabular setting also lets us avoid approximate computations. Note that tabular environments are a special case of the linear model where P j (s |s, a) = I(j = f (s, a, s )), where j ∈ [S 2 A] and f is a bijection that maps its arguments to the set [S 2 A]. Thus, d = S 2 A in this case. The algorithms that we compare have a model-fitting objective which is either used to fit a nominal model or to calculate confidence sets. The objective is either to minimize mean-squared error of predicting next states (alternatively, maximize log-likelihood of observed data), which leads to standard frequency based model estimates, or it is based on minimizing the value targets as proposed in our paper. The other component of the algorithms is whether they implement optimistic planning, or planning with the nominal model and then implementing an -greedy policy with respect to the estimated model ("dithering"). We also consider mixing value targets and next state targets. In the case of optimistic planning, the algorithm that uses mixed targets uses a union bound and takes the smallest value upper confidence bounds amongst the two bounds obtained with the two model-estimation methods. These leads to six algorithms, as shown in Table 1. Results for the "mixed" variants are very similar to the variant that uses value-targeted regression. As such, the results for the "mixed" variants are shown in the appendix only, as they would otherwise make the graphs overly cluttered. In the experiments we use confidence bounds that are specialized to the linear case. For the details, see Appendix C. For -greedy, we optimize the value of in each environment to get the best results. This gives Yang & Wang (2019a) in the tabular case essentially becomes UCRL of Jaksch et al. (2010). The mixed targets use both targets. -greedy an unfair advantage; but as we shall see despite this advantage, -greedy will not fair particularly well in our experiments. Environments We compare these algorithms on the episodic RiverSwim environment due to Strehl & Littman (2008) and a novel finite horizon MDP we tentatively call WideTree. The RiverSwim environment, whose detailed description is given below in Section 6.3, is chosen because it is known that in this environment "dithering" type exploration methods (e.g., -greedy) are ineffective. We vary the number of states in the RiverSwim environment in order to highlight some of the advantages and disadvantages of Value-Targeted Regression. WideTree is designed in order it highlight the advantages of Value-Targeted Regression when compared with more tradition frequency based methods. In this environment, only one action effects the outcome thus the other actions are non-informative. The detailed description of WideTree is given in Section 6.4. Measurements We report the cumulative regret as a function of the number of episodes and the weighted model error to indicate how well the model is learned. The results are obtained from 30 independent runs for the -greedy algorithms and 10 independent runs for the UC algorithms, The weighted model error reported is as follows. Given the model estimateP , its weighted error is E(P ) = s,a s N (s, a, s ) N (s, a) |P (s | s, a) − P * (s | s, a)|,(7) where N (s, a) is the observation-count of the state-action pair (s, a), N (s, a, s ) is the count of transitioning to s from (s, a), and P * (s | s, a) is the probability of s when action a is chosen in state s, according to the true model. Here, for the algorithms that use value-targeted regression the estimated modelP is the model obtained through Eq. (4). The weighting is introduced so that an algorithm that discards a state-action pair is not penalized. This is meant to prevent penalizing good exploration algorithms that may quickly discard some state-action pairs. We are interested in this error metric to monitor whether UCLR-VTR, which is not forced to model next-state distributions, will learn the proper next state distribution. In fact, we will see one example both for the case when this and also when this does not happen. Results for RiverSwim The schematic diagram of the RiverSwim environment is shown in Figure 1. RiverSwim consists of S states arranged in a chain. The agent begins on the far left and has the choice of swimming left or right at each state. There is a current that makes swimming left much easier than swimming right. Swimming left with the current always succeeds in moving the agent left, but swimming right against the current sometimes moves the agent right (with a small probability of moving left as well), but more often than not leaves the agent in the current state. Thus smart exploration is a necessity to learn a good policy in this environment. We experiment with small environments with S ∈ {3, 4, 5} states and set the horizon to 4S for each case. Figure 1: The "RiverSwim" environment with 6 states. State s 1 has a small associated reward, state s 6 has a large associated reward. The action whose effect is shown with the dashed arrow deterministically "moves the agent" towards state s 1 . The other action is stochastic, and with relatively high probability moves the agent towards state s 6 : This represents swimming "against the current". None of these actions incur a reward. The optimal values of the initial state are 5.72, 5.66 and 5.6, respectively, in these cases. The initial state is the leftmost state (s 1 in the diagram). The value that we found to work the best for greedy is = 0.01. Results are shown in Figure 2, except for UCRL-Mixed and EG-Mixed, whose results are given in Appendix D. As noted before, the results of these algorithms are very close to those of the VTR-versions, hence, they are not included here. The columns correspond to environments with S = 3, S = 4 and S = 5, respectively, which are increasingly more challenging. The first row shows the algorithm's performance measured in terms of their respective cumulative regrets, the second row shows results for the weighted model error as defined above. The regret per episode for an algorithm that "does not learn" is expected to be in the same range as the respective optimal values. Based on this we see that 10 5 episodes is barely sufficient for the algorithms other than UCRL-VTR to learn a good policy. Looking at the model errors we see that EGRL-VTR is doing quite poorly, EG-Freq is also lacking (especially on the environment with 5 states), the others are doing reasonably well. That EG-Freq is not doing well is perhaps surprising. However, this is because EG-Freq visits more uniformly than the other methods the various state-action pairs. The results clearly indicate that (i) fitting to the state-value function alone provides enough of a signal for learning as evident by UCRL-VTR obtaining low regret as predicted by our theoretical results, and that (ii) optimism is necessary when using value targeted regression to achieve good results, as evident by UCRL-VTR achieving significantly better regret than EGRL-VTR and even in the smaller RiverSwim environment where EG-Freq performed best. It is also promising that value-targeted regression with optimistic exploration outperformed optimism based on the "canonical" model estimation procedure. We attribute this to the fact that value-targeted regression will learn a model faster that predicts the optimal values well than the canonical, frequency based approach. That value-targeted regression also learns a model with small weighted error appears to be an accidental feature of this environment. Our next experiments are targeted at further exploring this effect. WideTree We introduce a novel tabular MDP we call WideTree. The WideTree environment has a fixed horizon H = 2 but can vary in the number of states. A visualization of an eleven state WideTree environment is shown in Figure 3. In WideTree, an agent starts at the initial state s 1 . The agent then progresses to one of the many bottom terminal states and collects a reward of either 0 or 1 depending on the action selected at state s 1 . The only significant action is whether to transition from s 1 to either s 2 or s 3 . Note that the model in the second layer is irrelevant for making a good decision: Once in s 3 , all actions lead to a reward of one, and once in s 2 , all actions lead to a reward of zero. We vary the number of bottom states reachable from states s 2 and s 3 while still maintaining a reward structure depending on whether the algorithm choose to transition to either s 2 or s 3 from the initial state s 1 . We set = 0.1 in this environment, though choosing smaller but as long as > 0 then both EGRL-VTR and EG-Freq will incur linear regret dependent on the choice of . One could also change the reward function in order to make learning for a given hard. Model Error on a 15 state WideTree Figure 4: As with the RiverSwim experiments, the results for the -greedy algorithms were averaged over thirty runs and the results for the UC algorithms were averaged over ten runs. Error bars are reported for the regret plots. The results are shown in Figure 4, except for UCRL-Mixed and EG-Mixed, whose results are given in Appendix D. Both UCRL-VTR and EG-VTR learn equally poor models (their graphs are 'on the top of each other'). Yet, UCRL-VTR manages to quickly learn a good policy, as attested by its low regret. EG-Freq and EG-VTR perform equally poorly and UC-MatrixRL is even slower as it keeps exploring the environment. These experiments clearly illustrate that UCRL-VTR is able to achieve good results without learning a good model -its focus on values makes pays off swiftly in this well-chosen environment. Conclusions We considered online learning in episodic MDPs and proposed an optimistic model-based reinforcement learning method (UCRL-VTR) with the unique characteristic to evaluate and select models based on their ability to predict value functions that the algorithm constructs during learning. The regret of the algorithm was shown to be bounded by a quantity that relates to the richness of the model class through the Eluder dimension and the metric entropy of an appropriately construction function space. For the case of linear mixture models, the regret bound simplifies toÕ(H 3/2 d √ T ) where d is the number of model parameters, H is the horizon, and T is the total number of interaction steps. Our experiments confirmed that the value-targeted regression objective is not only theoretically sound, but also yields a competitive method which allows task-focused model-tuning: In a carefully chosen environment we demonstrated that the algorithm achieves low regret despite that it ignores modeling a major part of the environment. Acknowledgements Csaba Szepesvári gratefully acknowledges funding from the Canada CIFAR AI Chairs Program, Amii and NSERC. A Proof of Theorem 1 In this section, we provide the regret analysis of the UCRL-VTR Algorithm (Algorithm 1). We will explain the motivation for our construction of confidence sets for general nonlinear squared estimation, and establish the regret bound for a general class of transition models, P. A.1 Preliminaries Recall that a finite horizon MDP is M = (S, A, P, r, H, s • ) where S is the state space, A is the action space, P = (P a ) a∈A is a collection of P a : S → M 1 (S) Markov kernels, r : S × A → [0, 1] is the reward function, H > 0 is the horizon and s • ∈ S is the initial state. For a state s ∈ S and an action a ∈ A, P a (s) gives the distribution of the next state that is obtained when action a is executed in state s. For a bounded (measurable) function V : S → R, we will use P a (s), V as the shorthand for the expected value of V at a random next state s whose distribution is P a (s). Given any policy π (which may or may not use the history), its value function is V π (s) = E π,δs H i=1 r(s i , a i ) , where E π,δs is the expectation operator underlying the probability measure P π,δs induced over sequences of state-action pairs of length H by executing policy π starting at state s in the MDP M and s h is the state visited in stage h and action a h is the action taken in that stage after visiting s h . For a nonstationary Markov policy π = (π 1 , . . . , π H ), we also let V π h (s) = E π h:H ,δs H−h+1 i=1 r(s i , a i ) be the value function of π from stage h to H. Here, π h:H denotes the policy (π h , . . . , π H ). The optimal value function V * = (V * 1 , . . . , V * H ) is defined via V * h (s) = max π V π h (s), s ∈ S. For simplicity assume that r is known. To indicate the dependence of V * on the transition model P , we will write V * P = (V * P,1 , . . . , V * P,H ). For convenience, we define V * P,H+1 = 0. Algorithm 1 is an instance of the following general model-based optimistic algorithm: Algorithm 2 Generic Algorithm 1-Schema for finite horizon problems 1: Input: P -a set of transition models, K -number of episodes, s 0 -initial state 2: Set B 1 = P 3: for k = 1, . . . , K do 4: Construct B k+1 based on (s k 1 , a k 1 , . . . , s k H , a k H ) 12: end for Specific instances of Algorithm 2 differ in terms of how B k+1 is constructed. In particular, UCRL-VTR uses the construction described in Section 3.2. P k = argmax{V * P (s 0 ) :P ∈ B k } 5: V k = V * Recall that V k = (V 1,k , . . . , V H,k , V H+1,k ) (with V H+1,k = 0) in Algorithm 2. Let π k be the nonstationary Markov policy chosen in episode k by Algorithm 2. Let R K = K k=1 V * 1 (s k 1 ) − V π k 1 (s k 1 ) be the pseudo-regret of Algorithm 1 for K episodes. The following standard lemma bounds the kth term of the expression on the right-hand side. Lemma 4. Assuming that P ∈ B k , we have V * 1 (s k 1 ) − V π k 1 (s k 1 ) ≤ sup P ∈B k H−1 h=1 P a k h (s k h ) − P a k h (s k h ), V h,k + H−1 h=1 ξ h+1,k , where ξ h+1,k = P a k h (s k h ), V h+1,k − V π k h+1 − V h+1,k (s k h+1 ) − V π k h+1 (s k h+1 ) . Note that (ξ 2,1 , ξ 3,1 , . . . , ξ H,1 , ξ 2,2 , ξ 3,2 , . . . , ξ H,2 , ξ 2,3 , . . . ) is a sequence of martingale differences. Proof. Because P ∈ B k , V * 1 (s k 1 ) ≤ V 1,k (s k 1 ) by the definition of the algorithm. Hence, V * 1 (s k 1 ) − V π k 1 (s k 1 ) ≤ V 1,k (s k 1 ) − V π k 1 (s k 1 ) . Fix h ∈ [H]. In what follows we bound V h,k (s k h ) − V π k h (s k h ) . By the definition of π k , P k and a k h , we have V h,k (s k h ) = r(s k h , a k h ) + P k a k h (s k h ), V h+1,k and V π k h (s k h ) = r(s k h , a k h ) + P a k h (s k h ), V π k h+1 . Hence, V h,k (s k h ) − V π k h (s k h ) = P k a k h (s k h ), V h+1,k − P a k h (s k h ), V π k h+1 = P k a k h (s k h ) − P a k h (s k h ), V h+1,k + P a k h (s k h ), V h+1,k − V π k h+1 . Therefore, by induction, noting that V H+1,k = 0, we get that V * 1 (s k 1 ) − V π k 1 (s k 1 ) ≤ H−1 h=1 P k a k h (s k h ) − P a k h (s k h ), V h+1,k + H−1 h=1 ξ h+1,k ≤ sup P ∈B k H−1 h=1 P a k h (s k h ) − P a k h (s k h ), V h+1,k + H−1 h=1 ξ h+1,k . A.2 The confidence sets for Algorithm 1 The previous lemma suggests that at the end of the kth episode, the model could be estimated usinĝ P k = argminP ∈P k k =1 H−1 h=1 P a k h (s k h ), V h+1,k − V h+1,k (s k h+1 ) 2 (8) For a confidence set construction, we get inspiration from Proposition 5 in the paper of . The set is centered atP k : B k = {P ∈ P : L k (P k ,P ) ≤ β k } ,(9) where L k (P ,P ) = k k =1 H−1 h=1 P a k h (s k h ) −P a k h (s k h ), V h+1,k 2 . Note that this is the same confidence set as described in Section 3.2. To obtain the value of β k , we now consider the nonlinear least-squares confidence set construction from Russo & Van Roy (2014). The next section is devoted to this construction. A.3 Confidence sets for general nonlinear least-squares Let (X p , Y p ) p=1,2,... be a sequence of random elements, X p ∈ X for some measurable set X and Y p ∈ R. Let F be a subset of the set of real-valued measurable functions with domain X . Let F = (F p ) p=0,1,... be a filtration such that for all p ≥ 1, (X 1 , Y 1 , . . . , X p−1 , Y p−1 , X p ) is F p−1 measurable and such that there exists some function f * ∈ F such that E[Y p | F p−1 ] = f * (X p ) holds for all p ≥ 1. The (nonlinear) least-squares predictor given (X 1 , Y 1 , . . . , X t , Y t ) isf t = argmin f ∈F t p=1 (f (X p ) − Y p ) 2 . We say that Z is conditionally ρ-subgaussian given the σ-algebra F if for all λ ∈ R, log E[exp(λZ)|F] ≤ 1 2 λ 2 ρ 2 . For α > 0, let N α be the · ∞ -covering number of F at scale α. That is, N α is the smallest integer for which there exist G ⊂ F with N α elements such that for any f ∈ F, min g∈G f − g ∞ ≤ α. For β > 0, define F t (β) = {f ∈ F : t p=1 (f (X p ) −f t (X p )) 2 ≤ β} . We have the following theorem, the proof of which is given in Section A.6. Theorem 5. Let F be the filtration defined above and assume that the functions in F are bounded by the positive constant C > 0. Assume that for each s ≥ 1, (Y p − f * (X p )) p is conditionally σ-subgaussian given F p−1 . Then, for any α > 0, with probability 1 − δ, for all t ≥ 1, f * ∈ F t (β t (δ, α)), where β t (δ, α) = 8σ 2 log(2N α /δ) + 4tα C + σ 2 log(4t(t + 1)/δ) . The proof follows that of Proposition 6, Russo & Van Roy (2014), with minor improvements, which lead to a slightly better bound. In particular, with our notation, Russo & Van Roy stated their result with β RvR t (δ, α) = 8σ 2 log(2N α /δ) + 2tα 8C + 8σ 2 log(8t 2 /δ) . While β t (δ, α) ≤ β RvR t (δ, α), the improvement is only in terms of smaller constants. A.4 The choice of β k in Algorithm 1 To use this result in our RL problem recall that P is the set of transition probabilities parameterized by θ ∈ Θ. We index time t = 1, 2, . . . in a continuous fashion. Episode k = 1, 2, . . . and stage h = 1, . . . , H − 1 corresponds to time t = (k − 1)(H − 1) + h: Let V (t) be the value function used by Algorithm 1 at time t (V (t) is constant in periods of length H − 1), while let (s (t) , a (t) ) be the state-action pair visited at time t. episode (k) 1 1 . . . 1 2 2 . . . Let V be the set of optimal value functions under some model in P: V = {V * P : P ∈ P}. Note that V ⊂ B(S, H), where B(S, H) denotes the set of real-valued measurable functions with domain S that are bounded by H. Note also that for all t, V (t) ∈ V. Define X = S × A × V. We also let X t = (s (t) , a (t) , V (t) ), Y t = V (t) (s (t+1) ) when t + 1 ∈ {H + 1, 2H + 1, . . . } and Y t = V (t) (s k H+1 ), and choose F = f : X → R : ∃P ∈ P s.t. f (s, a, v) = P a (ds |s)v(s ) .(10) Note that F ⊂ B ∞ (X , H). Let φ : P → F be the natural surjection to F: φ(P ) = f where f (s, a, v) = P a (ds |s)v(s ) for (s, a, v) ∈ X . We know show that φ is in fact a bijection. If P = P , this means that for some (s, a) ∈ S × A and U ⊂ S measurable, P a (U |s) = P a (U |s). Choosing v to be the indicator of U , note that (s, a, v) ∈ X . Hence, φ(P )(s, a, v) = P a (U |s) = P a (U |s) = φ(P )(s, a, v), and hence φ(P ) = φ(P ): φ is indeed a bijection. For convenience and to reduce clutter, we will write f P = φ(P ). Choose F = (F t ) t≥0 so that F t−1 is generated by (s (1) , a (1) , V (1) , . . . , s (t) , a (t) , V (t) ). Then E[Y t |F t−1 ] = P a (t) (ds |s (t) )V (t) (s ) = f P (X t ) and by definition f P ∈ F. Now, Y t ∈ [0, H], hence, Z t = Y t − f P (X t ) is conditionally H/2-subgaussian given F t−1 . Let t = k(H − 1) for some k ≥ 1. Thus, this time step corresponds to finishing episode k and thus V (t) = V k . Furthermore, lettingf t = argmin f ∈F t p=1 (f (X p ) − Y p ) 2 , since φ is an injection, we see that f t = fP k whereP k is defined using (8). For P , P ∈ P, we have L k (P , P ) = t p=1 (f P (X p ) − f P (X p )) 2 and thus B k = {P ∈ P : L k (P k ,P ) ≤ β k } = {P ∈ P : t p=1 (f t (X p ) − fP (X p )) 2 ≤ β k } = {φ −1 (f ) : f ∈ F and t p=1 (f t (X p ) − f (X p )) 2 ≤ β k } = φ −1 (F t (β k )) . Corollary 6. For α > 0 and k ≥ 1 let β k = 2H 2 log 2N (F, α, · ∞ ) δ + 2H(kH − 1)α 2 + log 4kH(kH − 1) δ . Then, with probability 1 − δ, for any k ≥ 1, P ∈ B k where B k is defined by (9). A.5 Regret of Algorithm 1 Recall that X = S ×A×V where V ⊂ B ∞ (S, H) is the set of value functions that are optimal under some model in P. We will abbreviate (x 1 , . . . , x t ) ∈ X t as x 1:t . Further, we let F| x1:t = {(f (x 1 ), . . . , f (x t )) : f ∈ F}(⊂ R t ) and for S ⊂ R t , let diam(S) = sup u,v∈S u − v 2 be the diameter of S. We will need the following lemma, extracted from Russo & Van Roy (2014): Lemma 7 (Lemma 5 of Russo & Van Roy (2014) ). Let F ⊂ B ∞ (X , C) be a set of functions bounded by C > 0, (F t ) t≥1 and (x t ) t≥1 be sequences such that F t ⊂ F and x t ∈ X hold for t ≥ 1. Then, for any T ≥ 1 and α > 0 it holds that T t=1 diam(F t | xt ) ≤ α + C(d ∧ T ) + 2δ T √ dT , where δ T = max 1≤t≤T diam(F t | x1:t ) and d = dim E (F, α). Let W k = sup P ∈B k H−1 h=1 P a k h (s k h ) − P a k h (s k h ), V h,k . From Lemma 4, we get R K ≤ K k=1 W k + K k=1 H−1 h=1 ξ h+1,k .(11) Lemma 8. Let α > 0 and d = dim E (F, α) where F is given by (10). Then, for any nondecreasing sequence (β 2 k ) K k=1 , on the event when P ∈ ∩ k∈[K] B k , K k=1 W k ≤ α + H(d ∧ K(H − 1)) + 4 dβ K K(H − 1) . Proof. Let P ∈ ∩ k∈[K] B k holds. Using the notation of the previous section, lettingF t = F t (β k ) for (k − 1)(H − 1) + 1 ≤ t ≤ k(H − 1), we have K k=1 W k ≤ K k=1 sup P ∈B k H−1 h=1 fP (s k h , a k h , V h+1,k ) − f P (s k h , a k h , V h+1,k ) ≤ K(H−1) t=1 diam(F t | Xt ) (because P ∈ ∩ k∈[K] B k ) ≤ α + H(d ∧ K(H − 1)) + 2δ K(H−1) dK(H − 1) , where X t is defined in Section A.4 and where the last inequality is by Lemma 7, which is applicable because F ⊂ B ∞ (X , H) holds by choice, and δ K(H−1) = max 1≤t≤K(H−1) diam(F t | X1:t ). Thanks to the definition of F t , δ K(H−1) ≤ 2 √ β K . Plugging this into the previous display finishes the proof. A.5.1 Proof of Theorem 1 Proof. Note that for any k ∈ [K] and h ∈ [H − 1], ξ h+1 , k ∈ [−H, H]. As noted beforehand, ξ 2,1 , ξ 3,1 , . . . , ξ H,1 , ξ 2,2 , ξ 3,2 , . . . , ξ H,2 , ξ 2,3 , . . . is a martingale difference sequence. Thus, with probability 1 − δ, K k=1 H−1 h=1 ξ h+1,k ≤ H 2K(H − 1) log(1/δ). Consider the event when this inequality holds and when P ∈ ∩ k∈[K] B k . By using Corollary 6 and a union bound, this event holds with probability at least 1 − 2δ. On this event, by (11) and Lemma 8, we obtain R K ≤ α + H(d ∧ K(H − 1)) + 4 dβ K K(H − 1) + H 2K(H − 1) log(1/δ) . Using α ≤ 1, which holds by assumption, finishes the proof. A.5.2 Proof of Corollary 2 Proof. Note that For α > 0 let N (P, α, · ∞,1 ) denote the (α, · ∞,1 )-covering number of P. Then we have N (F, α, · ∞ ) ≤ N (P, α/H, · ∞,1 ). f P − f P ∞ = sup s,a,v | (P a (ds |s) − P a (ds |s))v(s )| ≤ H sup Then, by Corollary 6, β K = 2H 2 log(2N (F, α, · ∞ )/δ) + C ≤ 2H 2 log(2N (P, α/H, · ∞,1 )/δ) + C with some universal constant C > 0. Let f : (Θ, · ) → (P, · ∞,1 ) be defined by θ → j θ j P j . Note that f (θ) − f (θ ) ∞,1 ≤ sup s,a j (θ j − θ j )P j,a (s) 1 = j |θ j − θ j | = θ − θ 1 . Hence, any ( , · 1 ) covering of Θ induces an ( , · ∞,1 )-covering of P and so N (P, α/H, · ∞,1 ) ≤ N (Θ, α/H, · 1 ) ≤ C (RH/α) d with some universal constant C > 0. Now, choose 1/α = K log(KH/δ). Hence, β K ≤ 2H 2 (log(2C /δ) + d log(RH/α)) + C . Suppressing log factors (e.g., log(RH)), log log terms and constants, we have β K = H 2 (d + log(1/δ)). Let F be given by (10). We now bound dim E (F, α). Let X = S × A × B(S) as before. Define z : S × A × B(S) → R d using z(s, a, v) j = P j,a (s), v and note that if x ∈ X is ( , F)-independent of x 1 , . . . , x k ∈ X then z(x) ∈ R d is ( , Θ)-independent of z(x 1 ), . . . , z(x k ) ∈ R d . This holds because if P = j θ j P j ∈ P then f P (s, a, v) = θ, z(s, a, v) for any (s, a, v) ∈ X . Hence, dim E (F, α) ≤ dim E (Lin(Z, Θ), α), where Lin(Z, Θ) is the set of linear maps with domain Z = {z(x) : x ∈ X } ⊂ R d and parameter from Θ: Lin(Z, Θ) = {h : h : Z → R s.t. ∃θ ∈ Θ : h(z) = θ, z , z ∈ Z}. Now, by Proposition 11 of Russo & Van Roy (2014), dim E (Lin(Z, Θ), α) = O(d log(1 + (Sγ/α) 2 ) where S is the · 2 diameter of Θ and γ = sup z∈Z z 2 . We have z 2 2 = j ( P j,a (s), v ) 2 ≤ H 2 d , hence γ ≤ H √ d. By the relation between the 1 and 2 norms, the 2-norm diameter of Θ is at most √ dR. Dropping log terms, dim E (F, α) =Õ(d). Plugging into Theorem 1 gives the desired result. A.6 Proof of Theorem 5 Recall the following: Definition 3. A random variable X is σ-subgaussian if for all λ ∈ R, it holds that E[exp(λX)] ≤ exp λ 2 σ 2 /2 . The proof of the next couple of statements is standard and is included only for completeness. Theorem 9. If X is σ-subgaussian, then for any λ > 0, with probability at least 1 − δ, X < 1 λ log 1 δ + λ σ 2 2 .(12) Proof. Let λ > 0. We have, {X ≥ } = {exp(λ(X − )) ≥ 0}. Hence, Markov's inequality gives P(X ≥ ) ≤ exp(−λ )E[exp(λX)] ≤ exp(−λ + 1 2 λ 2 σ 2 ). Equating the right-hand side with δ and solving for , we get that log(δ) = −λ + 1 2 λ 2 σ 2 . Solving for gives = log(1/δ)/λ + σ 2 2 λ, finishing the proof. Choosing the λ that minimizes the right-hand side of the bound gives the usual form: P(X ≥ 2σ 2 log(1/δ)) ≤ δ .(13) Lemma 10 (Lemma 5.4 of Lattimore & Szepesvári (2020)). Suppose that X is σ-subgaussian and X 1 and X 2 are independent and σ 1 and σ 2 -subgaussian, respectively, then: 1. E[X] = 0. 2. cX is |c|σ-subgaussian for all c ∈ R. 3. X 1 + X 2 is σ 2 1 + σ 2 2 -subgaussian. Let (Z p ) p be an F = (F p ) p -adapted process. Recall that (Z p ) p is conditionally σ-subgaussian given F if for all p ≥ 1, log E[exp(λZ p )|F p−1 ] ≤ 1 2 λ 2 σ 2 , for all λ ∈ R . A standard calculation gives that S t = t p=1 Z p is √ tσ-subgaussian (essentially, a refinement of the calculation that is need to show Part (3) of Lemma 10) and thus, in particular, for any t ≥ 1 and λ > 0, with probability 1 − δ, S t < 1 λ log 1 δ + λ tσ 2 2 . In fact, by slightly strengthening the argument, one can show that the above inequality holds simultaneously for all t ≥ 1: Theorem 11 (E.g., Lemma 7 of Russo & Van Roy (2014)). Let F be a filtration and let (Z p ) p be an F-adapted, conditionally σ-subgaussian process. Then for any λ > 0, with probability at least 1 − δ, for all t ≥ 1, S t < 1 λ log 1 δ + λ tσ 2 2 ,(14) where S t = t p=1 Z p . Proof of Theorem 5 Let us introduce the following helpful notation: For vectors x, y ∈ R t , let x, y t = t p=1 x p y p , x 2 t = x, x t , and for f : X → R, f 2 t = t p=1 f 2 (X p ). More generally, we will overload addition and subtraction such that for x ∈ R t , x + f ∈ R t is the vector whose pth coordinate is x p + f (X p ) (x p and X p both appear on purpose here). We also overload ·, · t such that x, f t = f, x t = t p=1 x p f (X p ). Define Z p using Y p = f * (X p ) + Z p and collect (Y p ) t p=1 and (Z p ) t p=1 into the vectors Y and Z. As in the statement of the theorem, let F = (F p ) p=0,1,... be such that for any s ≥ 1, (X 1 , Y 1 , . . . , X p−1 , Y p−1 , X p ) is F p−1 -measurable. Note that for any p ≥ 1, Z p = Y p − f * (X p ) is F p -measurable, hence (Z p ) p≥1 is F-adapted. With this, elementary calculation gives t , we get 1 2 f * −f t 2 t ≤ E(f t ) .(16) Thus, it remains to bound E(f t ). For this fix some α > 0 to be chosen later and let G(α) ⊂ F be an α-cover of F in · ∞ . Let g ∈ G(α) be a random function, also to be chosen later. We have E(f t ) = E(f t ) − E(g) + E(g) ≤ E(f t ) − E(g) + max g∈G(α) E(g)(17) We start by bounding the last term above. A simple calculation gives that for any fixed f ∈ F, w.p. 1 − δ, 2 Z, f − f * t is 2σ f − f * t -subgaussian. Hence, with probability 1 − δ, simultaneously for all t ≥ 1, E(f ) ≤ − 1 2 f * − f 2 t + 1 λ log 1 δ + λ 4σ 2 f − f * 2 t 2 = 4σ 2 log 1 δ , where the equality follows by choosing λ = 1/(4σ 2 ) (which makes the first and last terms cancel). (Note how splitting f − f * and θ i = P (s |s, a) if σ(s, a, s ) = i, we will have P (s |s, a) = d i=1 θ i P i (s |s, a). Therefore P can be parametrized using (2). Therefore, the known lower bound Ω( √ HSAT ) implies a worst-case lower bound of Ω( H · d/2 · T ) = Ω( √ HdT ) for our model. C Implementation C.1 Analysis of Implemented Confidence Bounds In the implementation of UCRL-VTR used in Section 6, we used different confidence intervals then the ones stated in the paper. The confidence intervals used in our implementation are the ones introduced in Abbasi- Yadkori et al. (2011). These confidence intervals are much tighter in the linear setting than the ones introduced in Section 3 and thus have better practical performance. The purpose of this section is to formally introduce the confidence intervals used in our implementation of UCRL-VTR as well as show how these confidence intervals were adapted from the linear bandit setting to the linear MDP setting. C.1.1 Linear MDP Assumptions For our implementation of UCRL-VTR we used different confidence then was introduced in the paper. These are the tighter confidence bounds from the seminal work done by Abbasi-Yadkori et al. (2011) and further expanded upon in Chapter 20 of Lattimore & Szepesvári (2020). Now we will state some assumptions in the MDP setting, then we will state the equivalent assumptions from the linear bandit setting, and lastly we will make the connections between the two that allow us to use the confidence bounds from the linear bandit setting in the RL setting. .4. Also note that in this section (·) * denotes the true parameter or model, (·) MDP denotes something derived or used in the linear MDP setting, and (·) LIN denotes something derived or used in the linear bandit setting. Now, under 1-3 of C.1.1 we hope to construct a confidence set C MDP t such that 1. P * (s | s, a) = d i=1 (θ MDP * ) i P i (s | s, a) 2. s k h+1 ∼ P * (· | s k h , a k h ) 3. C MDP t = {θ MDP ∈ R d : θ MDP −θ MDP t M k ≤ β t } where t is defined in the table of Aθ MDP ∈ ∞ t=1 C MDP t with high probability. Now the choice of how to choose both C MDP t and β t comes from the linear bandit literature. We will introduce the necessary theorems and assumptions to derive both C LIN t and β t in the linear bandit setting and then adapt the results from the linear bandit setting to the linear MDP setting. C.1.2 Tighter Confidence Bounds for Linear Bandits The following results are introduced in the paper by Abbasi-Yadkori et al. (2011) and are further explained in Chapter 20 of the book by Lattimore & Szepesvári (2020). In this section, we will introduce the theorems and lemmas that allows us to derive tighter confidence intervals for the linear bandit setting. Then we will carefully adapt the confidence intervals to the linear bandit setting. Now supposed a bandit algorithm has chosen actions A 1 , ..., A t ∈ R d and received rewards X LIN 1 , ..., X LIN t with X LIN s = A t , θ LIN * + η s where η s is some zero mean noise. The least squares estimator of θ LIN * is the minimizer of the following loss function L t (θ LIN ) = t s=1 (X LIN s − A t , θ LIN ) 2 + λ θ LIN 2 2 where λ > 0 is the regularizer. This loss function is minimized bŷ θ LIN t = W −1 t t s=1 X LIN s A s with W t = λI + t s=1 A s A s notice how this linear bandit problem is very similar to the linear MDP problem introduced in section 3 of our paper. In our linear MDP setting, it is convenient to think of M and W as serving equivalent purposes (storing rank one updates) thus it is also convenient to think of A t and X MDP t as serving equivalent purposes (the features by which we use to make our predictions), where X MDP t is defined in section 3 of our paper with some added notation to distinguish it from the X LIN t used here in the linear bandit setting. We will now build up some intuition by making some simplifying assumptions. 1. No regularization: λ = 0 and W t is invertible. 2. Independent subgaussian noise: (η s ) s are independent and σ-subgaussian 3. Fixed Design: A 1 , ..., A t are deterministically chosen without the knowledge of X LIN 1 , ..., X LIN t finally it is also convenient to think of X LIN t and V t+1 (s t+1 ) as serving equivalent purposes (the target of our predictions). Thus the statements we prove in the linear bandit setting can be easily adapted to the linear MDP setting. While none of the assumptions stated above is plausible in the bandit setting, the simplifications eases the analysis and provides insight. Comparing θ LIN * andθ LIN t in the direction x ∈ R d , we have θ LIN t − θ LIN * , x = x, W −1 t t s=1 A s X LIN s − θ LIN * = x, W −1 t t s=1 A s (A s θ LIN * + η s ) − θ LIN * = x, W −1 t t s=1 A s η s = t s=1 x, W −1 t A s η s Since (η s ) s are independent and σ-subgaussian, by Lemma 5.4 and Theorem 5.3 (need to be stated), P   θ LIN t − θ LIN * , x ≥ 2σ 2 t s=1 x, W −1 t A s 2 log 1 δ   ≤ δ A little linear algebra shows that t s=1 x, W −1 t A s 2 = x 2 W −1 t and so, P θ LIN t − θ LIN * , x ≥ 2σ 2 x 2 W −1 t log 1 δ ≤ δ(18) We now remove the limiting assumptions we stated above and use the newly stated assumptions for the rest of this section 1. There exists a θ LIN * ∈ R d such that X LIN t = θ LIN * , A t + η t for all t ≥ 1. 2. The noise is conditionally σ-subgaussian: for all α ∈ R and t ≥ 1, E[exp(αη t ) | F t−1 ] ≤ exp ασ 2 2 a.s. where F t−1 is such that A 1 , X LIN 1 , ..., A t−1 , X LIN t−1 are F t−1 -measurable. 3. In addition, we now assume λ > 0. The inclusion of A t in the definition of F t−1 allows the noise to depend on past choices, including the most recent action. Since we want exponentially decaying tail probabilities, one is tempted to try the Cramer-Chernoff method: P( θ LIN t − θ LIN * 2 Wt ≥ u 2 ) ≤ inf α>0 E exp α θ LIN t − θ LIN * 2 Wt − αu 2 . Sadly, we do not know how to bound this expectation. Can we still somehow use the Cramer-Chernoff method? We take inspiration from looking at the special case of λ = 0 one last time, assuming that W t = t s=1 A s A s is invertible. Let S t = t s=1 η s A s Recall thatθ LIN t = W −1 t t s=1 X LIN s A s = θ LIN * + W −1 t S t . Hence, 1 2 θ LIN t − θ LIN * 2 Wt = 1 2 S t 2 W −1 t = max x∈R d x, S t − 1 2 x 2 Wt . The next lemma shows that the exponential of the term inside the maximum is a supermartingale even when λ ≥ 0. Lemma 12. For all x ∈ R d the process D t (x) = exp( x, S t − 1 2 x W 2 t ) is an F-adapted non-negative supermartingale with D 0 (x) ≤ 1. The proof for this Lemma can be found in Chapter 20 of the book by Lattimore & Szepesvári (2020). For simplicity, consider now again the case when λ = 0. Combining the lemma and the linearisation idea almost works. The Cramer-Chernoff method leads to P 1 2 θ LIN t − θ LIN * 2 Wt ≥ log(1/δ) = P exp max x∈R d x, S t − 1 2 x 2 Wt ≥ log(1/δ) (19) ≤ δE exp max x∈R d x, S t − 1 2 x 2 Wt = δE max x∈R d D t (x)(20) Now Lemma 12 shows that E[D t (x)] ≤ 1. Now using Laplace's approximation we write max x D t (x) ≈ R d D t (x)dh(x), where h is some measure on R d chosen so that the integral can be calculated in closed form. This is not a requirement of the method, but it does make the argument shorter. The main benefit of replacing the maximum with an integral is that we obtain the following lemma Lemma 13. Let h be a probability measure on R d ; then;D t = R d D t (x)dh(x) is an F-adapted non-negative supermartingale withD 0 = 1. The proof of Lemma 13 can, again, be found in Chapter 20 of the book by Lattimore & Szepesvári (2020). Now the following theorem is the key result from which the confidence set will be derived. Theorem 14. For all λ > 0, and δ ∈ (0, 1) P exists t ∈ N : S t 2 W −1 t ≥ 2σ 2 log 1 δ + log det W t λ d ≤ δ Furthermore, if θ LIN * 2 ≤ m 2 , then P(exists t ∈ N + : θ LIN * / ∈ C LIN t ) ≤ δ with C LIN t = θ ∈ R d : θ LIN t−1 − θ Wt−1 < m 2 √ λ + 2σ 2 log 1 δ + log W t−1 λ d . The proof of Theorem 14 can be found in Chapter 20 of the book by Lattimore & Szepesvári (2020). C.2 UCRL-VTR In the proceeding subsections we discuss the implementation of the algorithms studied in Section 6 of the paper. The first algorithm we present is the algorithm used to generate the results for UCRL-VTR. Algorithm 3 UCRL-VTR with Tighter Confidence Bounds 1: Input: MDP, d, H, T = KH; 2: Initialize: M 1,1 ← I, w 1,1 ← 0 ∈ R d×1 , θ 1 ← M −1 1,1 w 1,1 for 1 ≤ h ≤ H, d 1 = |S| × |A|; 3: Initialize: δ ← 1/K, and for 1 ≤ k ≤ K, 4: Compute Q-function Q h,1 using θ 1,1 according to (3); 5: for k = 1 : K do 6: Obtain initial state s k 1 for episode k; The iterative Q-update for Algorithm 3 is V h+1,k (s) = 0 Q h,k (s, a) = r(s, a) + X h,k θ k + β h,k X h,k M −1 1,k+1 X h,k V h,k (s) = max a Q h,k (s, a) The choice of the confidence bounds used in Algorithm 3 comes from the tight bounds derived in Abbasi-Yadkori et al. (2011) for linear bandits and further expanded upon in Chapter 20 of Lattimore & Szepesvári (2020). The details of which are shown and stated in C.1. We slightly tighten the values for the noise at each stage by using the fact that for each stage in the horizon, h ∈ [H], the value V k h (·) is capped as to never be greater than H − h + 1. The appearance of the √ d 1 comes from the fact that θ * 2 ≤ √ d 1 for all θ * ∈ R d in the tabular setting since θ * in the tabular setting is equal to the true model of the environment. C.3 EGRL-VTR In this section we discuss the algorithm EGRL-VTR. This algorithm is very similar to UCRL-VTR expect it performs ε-greedy value iteration instead of optimistic value iteration and acts ε-greedy with respect to Q h,k . Algorithm 4 EGRL-VTR 1: Input: MDP, d, H, T = KH, ε > 0; 2: Initialize: M 1,1 ← I, w 1,1 ← 0 ∈ R d×1 , θ 1 ← M −1 1,1 w 1,1 for 1 ≤ h ≤ H; 3: Compute Q-function Q h,1 using θ 1,1 according to (22); 4: for k = 1 : K do 5: Obtain initial state s k 1 for episode k; Compute Q h,k+1 for h = H, . . . , 1, using θ k+1 according to (22) Computing Q functions 15: end for The iterative value update for EGRL-VTR is C.4 EG-Frequency In this section we discuss the algorithm EG-Frequency. This algorithm is the ε-greedy version of UC-MatrixRL Yang & Wang (2019a If we compare the results of Figure 5 with the results of Figure 2 from Section 6.3 we see that the cumulative regret of UCRL-MIX is almost identical to the cumulative regret of UCRL-VTR. The model errors of both the VTR and the canonical models are almost identical to the model errors of UCRL-VTR and UC-MatrixRL respectively. From Figure 6, we see that on the RiverSwim environment, UCRL-MIX almost always uses the VTR-model for planning. We calculate this frequency by counting the number of times Step 7 of Alg 6 was observed up until episode k and by counting the number of times Step 9 of Alg 6 was observed up until episode k. We then divide these counts by the sum of the counts to get a percentage. We believe the reason the algorithm overwhelming chose the VTR-model was due to the fact that the confidence intervals for the VTR-model shrink much faster than the confidence intervals for the canonical model. The canonical model is forced to explore much longer than the VTR-model as its objective is to learn a globally optimal model rather than a model that yields high reward. Thus, the canonical model is forced to explore all state-action-next state tuples, even ones that do not yield high reward, in order to meet its objective of learning a globally optimal model while the VTR-model is only forced to explore state-action-next state tuples that fall in-line with its objective of accumulating high reward. The set of all state-action-next state tuples is much larger then the set of state-action-next state tuples that yield high reward which means the confidence intervals for the canonical model shrink slower than the confidence sets of the VTR-model on the RiverSwim environment. assumed the transition model admits a matrix embedding of the form P (s |s, a) = φ(s, a) M ψ(s ), and proposed a model-based MatrixRL method with regret boundsÕ(H 2 d √ T ) with stronger assumptions andÕ(H 2 d 2 √ T ) in general, where d is the dimension of state representation φ(s, a). Jin et al. (2019) studied the setting of linear MDPs and constructed a model-free least-squares action-value iteration algorithm, which was proved to achieve the regret boundÕ( √ H 3 d 3 T ). (Modi et al., 2019) considered a related setting where the transition model is an ensemble involving state-action-dependent features and basis models and proved a sample complexity d 3 K 2 H 2 2 provided a general posterior sampling RL method that works for any given classes of reward and transition functions. It established a Bayesian regret upper bound O( √ d K d E T ), where d K and d E are the Kolmogorov and the Eluder dimensions of the model class. In the case of linearly parametrized transition model (Assumption 2 of this paper), this Bayesian regret becomes O(d √ T ), and our worst-case regret result matches with the Bayesian one. Abbasi-Yadkori & Szepesvári (2015); Theocharous et al. (2017) also considered the Bayesian regret and in particular Abbasi-Yadkori & Szepesvári Figure 2 : 2The results for the -greedy algorithms were averaged over thirty runs and the results for the UC algorithms were averaged over ten runs. Error bars are only reported for the regret plots. Figure 3 : 3An eleven state WideTree MDP. The algorithm starts in the initial state s 1 . From the initial state s 1 the algorithm has a choice of either deterministically transitioning to either state s 2 or state s 3 . Finally from either state s 2 or state s 3 the algorithm picks one of two possible actions and transitions to one of the terminal states e i . Depending on which state the algorithm transitioned to from the initial state s 1 , determines which delayed reward the algorithm will observe. The delayed reward is observed at the final stage h = 2 of this MDP. . . . H − 1 1 2 . . . H − 1 1 . . . time step (t) 1 2 . . . H − 1 H H + 1 . . . 2H − 2 2H − 1 . . . Note that the transitions at stage h = H are skipped and the time index at the end of episode k ≥ 1 is k(H − 1). |P a (ds |s) − P a (ds |s)| = H sup s,a P a (s) − P a (s) 1 =: H P − P ∞,1 . ,k (s, a) = r(s, a) + X h,k θ k V h,k (s) = (1 − ε)Π [0,H] max a Q h,k (s, a) + ε |A| a∈A Q h,k (s, a) Figure 5 : 5In the plots for the model error we include model error for both the VTR-model and the canonical model. Even though only one is used during planning both are updated at the end of each episode. Figure 6 : 6UCRL-MIX rarely, if ever, chooses the canonical model for planning on the RiverSwim environments. 7 : 7for h = 1 : H do Compute the predicted value vector:Evaluate the expected value of next stateX h,k ← E•[V h+1,k (s)|s k h , a k h ] = M h+1,k ← M h,k + X h,k X h,k 12: w h+1,k ← w h,k + y h,k · X h,kUpdate at the end of episode: Update Model ParametersM 1,k+1 ← M H+1,k , w 1,k+1 ← w H+1,k , θ k+1 ← M −1 1,k+1 w 1,k+1 ;Compute Q h,k+1 for h = H, . . . , 1, using θ k+1 according to (21) using8: Choose action greedily by a k h = arg max a∈A Q h,k (s k h , a) and observe the next state s k h+1 . 9: s∈S V h+1,k (s) · P•(s|s k h , a k h ). 10: y h,k ← V h+1,k (s k h+1 ) Update regression parameters 11: 13: end for 14: 15: β h,k ← d 1 + H − h + 1 2 2 log 1 δ + log det(M 1,k+1 ); Computing Q functions 16: end for 6 : 6for h = 1 : H do else pick a uniform random action a k h ∈ A. Observe the next state s k h+1 .Compute the predicted value vector: Evaluate the expected value of next stateX h,k ← E•[V h+1,k (s)|s k h , a k h ] = M h+1,k ← M h,k + X h,k X h,k 11: w h+1,k ← w h,k + y h,k · X h,kUpdate at the end of episode: Update Model Parameters M 1,k+1 ← M H+1,k , w 1,k+1 ← w H+1,k , θ k+1 ← M −1 1,k+1 w 1,k+1 ;7: With probability 1 − ε do a k h = arg max a∈A Q h,k (s k h , a) 8: s∈S V h+1,k (s) · P•(s|s k h , a k h ). 9: y h,k ← V h+1,k (s k h+1 ) Update regression parameters 10: 12: end for 13: 14: ).0 1 2 3 4 5 1e4 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Regret 1e5 Cumulative Regret for a 3 state RiverSwim UCRL-VTR UCRL-MIX 0 1 2 3 4 5 1e4 0.0 0.2 0.4 0.6 0.8 1.0 1e5 Cumulative Regret for a 4 state RiverSwim UCRL-VTR UCRL-MIX 0 1 2 3 4 5 1e4 0.0 0.2 0.4 0.6 0.8 1.0 1e5 Cumulative Regret for a 5 state RiverSwim UCRL-VTR UCRL-MIX 0 1 2 3 4 5 Episode 1e4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Weighted L1 norm Model Error on a 3 state RiverSwim VTR-model Freq-model 0 1 2 3 4 5 Episode 1e4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Model Error on a 4 state RiverSwim VTR-model Freq-model 0 1 2 3 4 5 Episode 1e4 0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Model Error on a 5 state RiverSwim VTR-model Freq-model Our results are easy to extend to the case when r is not known. The precise definitions require measure-theoretic concepts(Bertsekas & Shreve, 1978), i.e., P is a Markov kernel, mapping from S × A to distributions over S, hence, all these spaces need to be properly equipped with a measurability structure. For the sake of readability and also because they are well understood, we omit these technical details. t ≤ Y − f * t into two halves allowed us to bound the "error term" E(f ) independently of t.) Now, by a union bound, it follows that with probability at least 1 − δ, the second term is bounded by 4σ 2 log(|G(α)|/δ).Let us now turn to bounding the first term. We calculate Splitting f * − f 2 t and rearranging giveswhereRecall thatf t = argmin f ∈F Y − f 2 t . Pluggingf t into 15 in place of f and using that thanks to f * ∈ F,where for the last inequality we chose g = argming ∈G(α) f t −g ∞ so that f t − g t ≤ α √ t and used Cauchy-Schwartz, together with that g t , f t t , f * t ≤ C √ t, which follows from g,f t , f * ∈ F and that by assumption all functions in F are bounded by C.It remains to bound Z t . For this, we observe that with probability 1 − δ, simultaneously for all t ≥ 1,Indeed, this follows because with probability 1 − δ, simultaneously for any s ≥ 1, |Z p | 2 ≤ 2σ 2 log(2s(s + 1)/δ) holds because of a union bound and Eq. (13). Therefore, for the above choice g, with probability 1 − δ, simultaneously for all t ≥ 1, it holds thatMerging this with Eqs.(16)and(17)and with another union bound, we get that with probability 1 − δ, for any t ≥ 1,where N α is the (α, · ∞ )-covering number of F.B Proof of Theorem 3In this section we establish a regret lower bound by reduction to a known result for tabular MDP.Proof. We assume without loss of generality that d is a multiple of 4 and d ≥ 8. We set S = 2 and A = d/4 ≥ 2. According toAzar et al. (2017),Osband & Van Roy (2016), there exists an MDP M(S, A, P, r, H) with S states, A actions and horizon H such that any algorithm has regret at least Ω( √ HSAT ). In this case, we have |S × A × S| = d. We use σ(s, a, s ) to denote the index of (s, a, s ) in S × A × S. Letting P i (s |s, a) = 1 if σ(s, a, s ) = i, 0 otherwise,C.1.3 Adaptation of the Confidence Bounds to our Linear MDP SettingNow with the Lemmas and Theorems introduced in the previous section we are ready to derive the confidence bounds used in our implementation of UCRL-VTR. Now using the notation from the linear bandit setting we set.., s t−1 , a t−1 ), which just means the filtration is set to be the sigma-algebra generated by all past states and actions observed.is the true model of the MDP.5. M t in the linear MDP setting is defined equivalently to W t in the linear bandit setting, i.e. they are both the sums of a regularizer term and a bunch of rank one updates.it can be seen that our the noise in our system η t has zero mean E[η t | F t−1 ] = 0 finally the noise in our system has variance H/2 thus our system in H/2-subgaussian.Thus ψ (α) can be interpreted as the variance of the random variable Z under the probability measure dQ = exp(αZ) E[exp(αZ)] dP, but since Z ∈ [a, b] almost surely, we have, under any probabilityThe fundamental theorem of calculus yields, the noise η t in our linear MDP setting is H/2-subgaussian. This result is also stated in a proof from A.4.Putting this all together we can derive the tighter confidence set for UCRL-VTR in the linear setting,where here in the linear MDP setting M t replaces W t from the linear bandit setting and θ MDP * 2 ≤ m 2 . The justification of using these bounds in the linear MDP setting follow exactly from the justification given above for using these bounds in the linear bandit setting.Algorithm 5 EG-Frequency 1: Input: MDP, Features φ : S × A → R |S||A| and ψ : S → R |S| , ε > 0, and the total number of episodes K; 2: Initialize: A 1 ← I ∈ R |S||A|×|S||A| , M 1 ← 0 ∈ R |S||A|×|S| , and K ψ ← s ∈S ψ(s )ψ(s ) ; 3: for k = 1 : K do 4:Let Q h,k be given in(23)With probability (1−ε) play action a k h = arg max a∈A Q h,k (s k h , a) else pick a uniform random action a k h ∈ A.8:Record the next state s k h+1 9:end for 10:The iterative Q-update for EG-Frequency is Q h+1,k (s, a) = 0 andNote that Ψ is a |S| × |S| whose rows are the features ψ(s ) and Φ is a |S||A| × |S||A| whose rows are the features φ(s, a). In the tabular RL setting both Ψ and Φ are the identity matrix which is what we used in our numerical experiments. In the tabular RL setting, EG-Frequency stores the counts of the number of times it transitioned to next state s from the state-action pair (s, a) and fits the estimated model M k accordingly.C.4.1 Futher Implementation NotesIn this section, we include some further details on how we implemented Algorithms 3, 4, and 5. All code was written in Python 3 and used the Numpy and Scipy libraries. All plots were generated using MatPlotLib. In Algorithm 3, Numpy's logdet function was used to calculate the determinate in step 15 for numerical stability purposes. No matrix inversion was performed in our code, instead a Sherman-Morrison update was performed for each matrix in which a matrix inversion is performed at each (k, h) in order to save on computation. To read more about the Sherman Morrison update in the context of RL, we refer to the reader to Eqn (9.22) ofSutton & Barto (2018). When computing the weighted L1-norm, we added a small constant to each summation in the denominator to avoid dividing by zero. Finally, when computing UC-MatrixRL we also used the self-normalize bounds introduced in the beginning of this section. Some pseudocode for using self-normalized bounds with UC-MatrixRL can be found in step 5 of Alg 6.D Mixture ModelIn this section, we introduce, analyze, and evaluate a Linear model-based RL algorithm that used both the canonical model and the VTR model for planning. We call this algorithm UCRL-MIX.D.1 UCRL-MIXBelow a meta-algorithm for UCRL-MIX for s ∈ |S| and a ∈ |A| do 5:Compute the confidence set bonuses as followsk+1 ); V h,k (s) = max a Q h,k (s, a) 12:end for 13: end forWe are now using multiple models instead of a single model, we must adjust our confidence sets accordingly. By using a union bound we replace δ with δ/2 for our confidence parameter. This updated confidence parameter changes the term inside the logarithm. 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[]
[ "Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting", "Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting" ]
[ "Giorgos Bouritsas \nImperial College London\nUK\n", "Fabrizio Frasca \nImperial College London\nUK\n\nTwitterUK\n", "Stefanos Zafeiriou \nImperial College London\nUK\n", "Michael M Bronstein \nImperial College London\nUK\n\nTwitterUK\n" ]
[ "Imperial College London\nUK", "Imperial College London\nUK", "TwitterUK", "Imperial College London\nUK", "Imperial College London\nUK", "TwitterUK" ]
[]
While Graph Neural Networks (GNNs) have achieved remarkable results in a variety of applications, recent studies exposed important shortcomings in their ability to capture the structure of the underlying graph. It has been shown that the expressive power of standard GNNs is bounded by the Weisfeiler-Leman (WL) graph isomorphism test, from which they inherit proven limitations such as the inability to detect and count graph substructures. On the other hand, there is significant empirical evidence, e.g. in network science and bioinformatics, that substructures are often intimately related to downstream tasks. To this end, we propose "Graph Substructure Networks" (GSN), a topologically-aware message passing scheme based on substructure encoding. We theoretically analyse the expressive power of our architecture, showing that it is strictly more expressive than the WL test, and provide sufficient conditions for universality. Importantly, we do not attempt to adhere to the WL hierarchy; this allows us to retain multiple attractive properties of standard GNNs such as locality and linear network complexity, while being able to disambiguate even hard instances of graph isomorphism. We perform an extensive experimental evaluation on graph classification and regression tasks and obtain state-of-the-art results in diverse realworld settings including molecular graphs and social networks. The code is publicly available at https: //github.com/gbouritsas/graph-substructure-networks.
10.1109/tpami.2022.3154319
[ "https://arxiv.org/pdf/2006.09252v3.pdf" ]
219,708,613
2006.09252
659597b1699ba5b73da9a8628bf7e4ad9bebd242
Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting Giorgos Bouritsas Imperial College London UK Fabrizio Frasca Imperial College London UK TwitterUK Stefanos Zafeiriou Imperial College London UK Michael M Bronstein Imperial College London UK TwitterUK Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting While Graph Neural Networks (GNNs) have achieved remarkable results in a variety of applications, recent studies exposed important shortcomings in their ability to capture the structure of the underlying graph. It has been shown that the expressive power of standard GNNs is bounded by the Weisfeiler-Leman (WL) graph isomorphism test, from which they inherit proven limitations such as the inability to detect and count graph substructures. On the other hand, there is significant empirical evidence, e.g. in network science and bioinformatics, that substructures are often intimately related to downstream tasks. To this end, we propose "Graph Substructure Networks" (GSN), a topologically-aware message passing scheme based on substructure encoding. We theoretically analyse the expressive power of our architecture, showing that it is strictly more expressive than the WL test, and provide sufficient conditions for universality. Importantly, we do not attempt to adhere to the WL hierarchy; this allows us to retain multiple attractive properties of standard GNNs such as locality and linear network complexity, while being able to disambiguate even hard instances of graph isomorphism. We perform an extensive experimental evaluation on graph classification and regression tasks and obtain state-of-the-art results in diverse realworld settings including molecular graphs and social networks. The code is publicly available at https: //github.com/gbouritsas/graph-substructure-networks. Introduction The field of graph representation learning has undergone a rapid growth in the past few years. In particular, Graph Neural Networks (GNNs), a family of neural architectures designed for irregularly structured data, have been successfully applied to problems ranging from social networks and recommender systems [1] to bioinformatics [2,3], chemistry [4,5,6] and physics [7,8], to name a few. Most GNN architectures are based on message passing [5], where the representation of each node is iteratively updated by aggregating information from its neighbours. A crucial difference from traditional neural networks operating on grid-structured data is the absence of canonical ordering of the nodes in a graph. To address this, the aggregation function is constructed to be invariant to neighbourhood permutations and, as a consequence, to graph isomorphism. This kind of symmetry is not always desirable and thus different inductive biases that disambiguate the neighbours have been proposed. For instance, in geometric graphs, such as 3D molecular graphs and meshes, directional biases are usually employed in order to model the positional information of the nodes [9,10,11,12,13]; for proteins, ordering information is used to disambiguate amino-acids at different positions in the sequence [14]; in multi-relational knowledge graphs, a different aggregation is performed for each relation type [15]. The structure of the graph itself does not usually explicitly take part in the aggregation function. In fact, most models rely on multiple message passing steps as a means for each node to discover the global structure of the graph. However, since message-passing GNNs are at most as powerful as the Weisfeiler Leman test (WL) [16,17], they are limited in their abilities to adequately exploit the graph structure, e.g. by counting substructures [18,19]. This uncovers a crucial limitation of GNNs, as substructures have been widely recognised as important in the study of complex networks. For example, in molecular chemistry, functional groups and rings are related to a plethora of chemical properties, while cliques are related to protein complexes in Protein-Protein Interaction networks and community structure in social networks, respectively [20,21]. Therefore, three major questions arise when designing GNN architectures: (a) How to go beyond isotropic, i.e., locally symmetric, aggregation functions ? (b) How to ensure that GNNs are aware of the structural chatacteristics of the graph? (c) How to achieve the above two without sacrificing invariance to isomorphism and hence the ability of GNNs to generalise? In this work we attempt to simultaneously provide an answer to the above. We propose to break local symmetries by introducing structural information in the aggregation function, hence addressing (a) and (b). In particular, the contribution of each neighbour (message) is transformed differently depending on its structural relationship with the central node. This relationship is expressed by counting the appearance of certain substructures. Since substructure counts are vertex invariants, i.e. they are invariant to vertex permutations, it is easy to see that the resulting GNN will be invariant to isomorphism, hence also addressing (c). Moreover, by choosing the substructures, one can provide the model with different inductive biases, based on the graph distribution at hand. We characterise the expressivity of our message-passing scheme, coined as Graph Substructure Network (GSN), showing that GSN is strictly more expressive than traditional GNNs for the vast majority of substructures, while retaining the locality of message passing, as opposed to higher-order methods [22,23,24,17] that follow the WL hierarchy (see Section 2). In the limit, our model can yield a unique representation for every isomorphism class and is thus universal. We provide an extensive experimental evaluation on hard instances of graph isomorphism testing (strongly regular graphs), as well as on real-world networks from the social and biological domains, including the recently introduced large-scale benchmarks [25,26]. We observe that when choosing the structural inductive biases based on domain-specific knowledge, GSN achieves state-of-the-art results. Preliminaries Let G = (V G , E G ) be a graph with vertex set V G and edge set E G , directed or undirected. A subgraph G S = (V G S , E G S ) of G is any graph with V G S ⊆ V G , E G S ⊆ E G . When E G S includes all the edges of G with endpoints in V G S , i.e., E G S = E G ∩ V G S × V G S , the subgraph is said to be induced. Isomorphism & Automoprhism Two graphs G, H are isomorphic (denoted H G), if there exists an adjacency-preserving bijective mapping (isomorphism) f : V G → V H , i.e., (v, u) ∈ E G iff (f (v), f (u)) ∈ E H . Given some small graph H, the subgraph isomorphism problem amounts to finding a subgraph G S of G such that G S H. An automorphism of H is an isomorphism that maps H onto itself. The set of all the unique automorphisms forms the automorphism group of the graph, denoted as Aut(H), which contains all the possible symmetries of the graph. The automorphism group yields a partition of the vertices into disjoint subsets of V H called orbits. Intuitively, this concept allows us to group the vertices based on their structural roles, e.g. the endpoint vertices of a path, or all the vertices of a cycle (see Figure 1). Formally, the orbit of a vertex v ∈ V H is the set of vertices to which it can be mapped via an automorphism: Orb(v) = {u ∈ V H | ∃g ∈ Aut(H) s.t. g(u) = v}, and the set of all orbits H \ Aut(H) = {Orb(v) | v ∈ V H } is usually called the quotient of the automorphism when it acts on the graph H. We are interested in the unique elements of this set that we will denote as {O Analogously, we define edge structural roles via edge automorphisms, i.e., bijective mappings from the edge set onto itself, that preserve edge adjacency (two edges are adjacent if they share a common endpoint). In particular, every vertex automorphism g induces an edge automorphism by mapping each edge (u, v) to (g(u), g(v)). 1 In the same way as before, we construct the edge automorphism group, from which we deduce the partition of the edge set in edge orbits {O Weisfeiler-Leman tests The Weisfeiler-Leman graph-isomorphism test [28], also known as naive vertex refinement, 1-WL, or just WL), is a fast heuristic to decide if two graphs are isomorphic or not. The WL test proceeds as follows: every vertex v is initially assigned a colour c 0 v that is later iteratively refined by aggregating neighbouring information: c t+1 v = HASH c t v , c t u u∈Nv ,(1) where · denotes a multiset (a set that allows element repetitions) and N (v) is the neighbourhood of v. The WL algorithm terminates when the colours stop changing, and outputs a histogram of colours. Two graphs with different histograms are non-isomorphic; if the histograms are identical, the graphs are possibly, but not necessarily, isomorphic. Note that the neighbour aggregation in the WL test is a form of message passing, and GNNs are the learnable analogue. A series of works on improving GNN expressivity mimic the higher-order generalisations of WL, known as k-WL and k-Folklore WL (WL hierarchy) and operate on k-tuples of nodes (see Appendix B.1). The (k + 1)-FWL is strictly stronger than k-FWL, k-FWL is as strong as (k + 1)-WL and 2-FWL is strictly stronger than the simple 1-WL test. Graph Substructure Networks Graphs consist of nodes (or edges) with repeated structural roles. Thus, it is natural for a neural network to treat them in a similar manner, akin to weight sharing between local patches in CNNs for images [29] or positional encodings in language models for sequential data [30,31,32]. Nevertheless, GNNs are usually unaware of the nodes' different structural roles, since all nodes are treated equally when performing local operations. Despite the initial intuition that the neural network would be able to discover these roles by constructing deeper architectures, it has been shown that GNNs are ill-suited for this purpose and are blind to the existence of structural properties, e.g. triangles or larger cycles [19,18]. To this end, we propose to explicitly encode structural roles as part of message passing, in order to capture richer topological properties. Our method draws inspiration from [33], where it was shown that GNNs become universal when the nodes in the graph are uniquely identified, i.e when they are equipped with different features. However, it is not clear how to choose these identifiers in a permutation equivariant way. Structural roles, when treated as identifiers, although not necessarily unique, are not only permutation equivariant, but also more amenable to generalisation due to their repetition across different graphs. Thus, they can constitute a trade-off between uniqueness and generalisation. : Node (left) and edge (right) induced subgraph counting for a 3-cycle and a 3-path. Counts are reported for the blue node on the left and for the blue edge on the right. Different colors depict orbits. Structural features x V H,i (v) = G S H v ∈ V G S , f (v) ∈ O V H,i } .(2) Note that there exist |Aut(H)| different functions f that can map a subgraph G S to H, but any of those can be used to determine the orbit mapping of each node v. By combining the counts from different substructures in H and different orbits, we obtain the feature vector x V v = [x V H1 (v), . . . , x V H K (v)] ∈ N D×1 of dimension D = Hi∈H d Hi . Similarly, we can define edge structural features x E H,i (u, v) by counting occurrences of edge automorphism orbits: x E H,i (u, v) = G S H (u, v) ∈ E G S , (f (u), f (v)) ∈ O E H,i ,(3) and the combined edge features x E u,v = [x E H1 (u, v), . . . , x E H K (u, v) ]. An example of vertex and edge structural features is illustrated in Figure 1. Structure-aware message passing The key building block of our architecture is the graph substructure layer, defined in a general manner as a Message Passing Neural Network (MPNN) [5], where now the messages from the neighbouring nodes also contain the structural information. In particular, each node v updates its state h t v by combining its previous state with the aggregated messages: h t+1 v = UP t+1 h t v , m t+1 v (4) m t+1 v =            M t+1 (h t v , h t u , x V v , x V u , e u,v ) u∈N (v) (GSN-v) or M t+1 (h t v , h t u , x E u,v , e u,v ) u∈N (v) (GSN-e),(5) where UP t+1 is an arbitrary function approximator (e.g. a MLP), M t+1 is the neighborhood aggregation function, i.e. an arbitrary function on multisets (e.g., of the form u∈N (v) MLP(·)) and e u,v are the edge features. The two variants, named GSN-v and GSN-e, correspond to vertex-or edge-counts, respectively, which are analogous to absolute and relative positional encodings in language models [34,35]. It is important to note here that contrary to identifier-based GNNs [33,36,37] that obtain universality at the expense of permutation equivariance (since the identifiers are arbitrarily chosen with the sole requirement of being unique), GSNs retain this property, hence they are by construction ivariant to isomorphism. This stems from the fact that the process generating our structural identifiers (i.e. subgraph isomorphism) is permutation equivariant itself (proof provided in the Appendix A.1). Figure 2: (Left) Decalin and Bicyclopentyl : Non-isomorphic molecular graphs than can be distinguished by GSN, but not the by the WL test [38] (nodes represent carbon atoms and edges chemical bonds). (Right) Rook's 4x4 graph and the Shrikhande graph: the smallest pair of strongly regular non-isomorphic graphs with the same parameters SR (16,6,2,2). GSN can distinguish them with 4-clique counts, while 2-FWL fails. How powerful are GSNs? We now turn to the expressive power of GSNs in comparison to MPNNs and the WL tests, a key tool for the theoretical analysis of the expressivity of graph neural networks so far. Since GSN is a generalisation of MPNNs, it is easy to see that it is at least as powerful. Importantly, GSNs have the capacity to learn functions that traditional MPNNs cannot learn. The following observation derives directly from the analysis of the counting abilities of the 1-WL test [18] and its extension to MPNNs [19] (for proofs, see Appendices A.2-A.4). Theorem 3.1. GSN is strictly more powerful than MPNN and the 1-WL test when one of the following holds: • H is any graph except for star graphs of any size, and structural features are inferred by subgraph matching, i.e. we count all subgraphs G S ∼ = H for which it holds that E G S ⊆ E G . Or, • H is any graph except for single edges and single nodes, and structural features are inferred by induced subgraph matching, i.e. we count all subgraphs G S ∼ = H for which it holds that E G S = E G ∩ V G S × V G S . Proof. It is easy to see that GSN model class contains MPNNs, and is thus at least as expressive. We can also show that GSN is at least as expressive as the 1-WL test by repurposing the proof of Theorem 3 in [16] (see Appendix A.2) Given the first part of the proposition, in order to show that GSNs are strictly more expressive than the 1-WL test, it suffices to show that GSN can distinguish a pair of graphs that 1-WL deems isomorphic. [18] showed that 1-WL, and consequently MPNNs, can count only forests of stars. Thus, if the subgraphs are required to be connected, then they can only be star graphs of any size (note that this contains single nodes and single edges). In addition, [19] showed that 1-WL, and consequently MPNNs, cannot count any connected induced subgraph with 3 or more nodes, i.e. any connected subgraph apart from single nodes and single edges. If H is a substructure that 1-WL cannot learn to count, i.e. the ones mentioned above, then there is at least one pair of graphs with different number of counts of H, that 1-WL deems isomorphic. Thus, by assigning counting features to the nodes/edges of the two graphs based on appearances of H, a GSN can obtain different representations for G 1 and G 2 by summing up the features. Hence, G 1 , G 2 are deemed non-isomorphic. An example is depicted in Figure 2 (left), where the two non-isomorphic graphs are distinguishable by GSN via e.g. cycle counting, but not by 1-WL. Universality. A natural question that emerges is what are the sufficient conditions under which GSN can solve graph isomorphism. This would entail that GSN is a universal approximator of functions defined on graphs [37,39]. To address this, we can examine whether there exists a specific substructure collection that can completely characterise each graph. As of today, we are not aware of any results in graph theory that can guarantee the reconstruction of a graph from a smaller collection of its subgraphs. However, the Reconstruction Conjecture [40,41], states that a graph with size n ≥ 3 can be reconstructed from its vertex-deleted subgraphs (proven for n ≤ 11 [42]). Consequently, (proof in the Appendix A.3): Computational complexity for real-life graphs is significantly better than the worst case. Corollary 3.2. If the Reconstruction Conjecture holds and the substructure collection H contains all graphs of size k = n − 1, then GSN can distinguish all non-isomorphic graphs of size n and is therefore universal. GSN-v vs GSN-e. We can also examine the expressive power of the two proposed variants. A crucial observation that we make is that for each graph H in the collection, the vertex structural identifiers can be reconstructed by the corresponding edge identifiers. Thus, we can show that for every GSN-v there exists a GSN-e that can simulate the behaviour of the former (proof provided in the Appendix A.4). Theorem 3.3. For a given subgraph collection H, let C V the set of functions that can be expressed by a GSN-v with arbitrary depth and with, and C E the set of functions that can be expressed by a GSN-e with the same properties. Then, it holds that C E ⊇ C V , or in other words GSN-e is at least as expressive as GSN-v. Comparison with higher-order WL tests. Finally, the expressive power of GSN can be compared to higher-order versions of the WL test. In particular, for each k-th order Folklore WL test in the hierarchy, it is known that there exists a family of graphs that will make the test fail. These are known in the literature as k-isoregular graphs [43], and the most well-known example is the Strongly Regular (SR) graph family, for k = 2 (more details can be found in Appendix B). Hence, if we can find a substructure collection that allows GSN to distinguish certain pairs from these families, then this guarantees that the corresponding k-FWL test is no stronger than GSN. In this work, we identify such counterexamples for the 2-FWL test. Formally: Proposition 3.4. There exist substructure families with O(1) size, i.e., independent of the size of the graph n, such that 2-FWL is no stronger than GSN. We provide numerous counterexamples that prove this claim. Figure 2 (right) provides a typical pair of SR graphs that can be distinguished with a 4-clique, while in section 5.1 this is extended to a large-scale study, where other constant size substructures (paths, cycles and cliques) can achieve similar results. Although it is not clear if there exists a certain substructure collection that results in GSNs that align with the WL hierarchy, we stress that this is not a necessary condition in order to design more powerful GNNs. In particular, despite the increase in expressivity, k-WL tests are not only more computationally involved, but they also process the graph in a non-local fashion. However, locality is presumed to be a strong inductive bias of GNNs and key to their excellent performance in real-world scenarios. How to choose the substructures? Expressivity. The Reconstruction Conjecture provides a sufficient, albeit impractical condition for universality. This motivates us to analyse the constant size case k = O(1) for practical scenarios, similar to the argument put forward for hard instances of graph isomorphism (Proposition 3.4). In particular, one can count only the most discriminative subgraphs, i.e. the ones that can achieve the maximum possible vertex disambiguation, similarly to identifier-based approaches. Whenever these subgraph counts can provide a unique identification of the vertices, then universality will also hold (Corollary 3.1. in [33]). We conjecture, that in real-world scenarios the number of subgraphs needed for unique, or near-unique identification, are far fewer than those dictated by Corollary 3.2. This is consistent with our experimental findings, where we observed that certain small substructures such as paths and trees, significantly improve vertex disambiguation, compared to the initial vertex features (see Figure 5 (left) and Table 2 in the appendix). As expected this allows for better fitting of the training data, which validates our claim that GNN expressivity improves. Generalisation. However, none of the above claims can guarantee good generalisation in unseen data. For example, in Figure 5, we observe that the test set performance does not follow the same trend with train performance when choosing substructures with strong vertex disambiguation. Aiming at better generalisation, it is desirable to make use of substructures for which there is prior knowledge of their importance in certain network distributions and have been observed to be intimately related to various properties. For example, small substructures (graphlets) have been extensively analysed in protein-protein interaction networks [44], triangles and cliques characterise the structure of ego-nets and social networks in general [20], simple cycles (rings) are central in molecular distributions, directed and temporal motifs have been shown to explain the working mechanisms of gene regulatory networks, biological neural networks, transportation networks and food webs [45,46,47]. In Figure 5 (right), we showcase the importance of these inductive biases: a cycle-based GSN predicting molecular properties achieves smaller generalisation gap compared to a traditional MPNN, while at the same time generalising better with less training data. Choosing the best substructure collection is still an open problem that does not admit a straightforward solution due to its combinatorial nature. Alternatively, various heuristics can be used, e.g., motif frequencies or feature selection strategies. Answering this question is left for future work. Complexity The complexity of GSN comprises two parts: precomputation (substructure counting) and training/testing. The key appealing property is that training and inference are linear w.r.t the number of edges, O(|E|), as opposed to higher-order methods [24,17] with O(n k ), and [48] with O(n 2 ) training complexity and relational pooling [49] with O(n!) training complexity in absence of approximations. The worst-case complexity of subgraph isomorphism of fixed size k is O(n k ), by examining all the possible k-tuples in the graph. However, for specific types of subgraphs, such as paths and cycles, the problem can be solved even faster (see e.g. [50]). Approximate counting algorithms are also widely used, especially for counting frequent network motifs [51,52,53,54], and can provide a considerable speed-up. Furthermore, recent neural approaches [55,56] provide fast approximate counting. In our experiments, we performed exact counting using the common isomorphism algorithm VF2 [57]. Although its worst case complexity is O(n k ), it scales better in practice, for instance when the candidate subgraph is infrequently matched or when the graphs are sparse, and is also trivially parallelisable. In Figure 3, we show a quantitative analysis of the empirical runtime of the counting algorithm against the worst case, for three different graph distributions: molecules, protein contact maps, social networks. It is easy to see that when the graphs are sparse (for the first two cases) and the number of matches is small, the algorithm is significantly faster than the worst case, while it scales better with the size of the graph n. Even, in the case of social networks, where several examples are near-complete graphs, both the runtime and the growth w.r.t both n and k are better than the worst case. Overall, the preprocessing computational burden in most of the cases remains negligible for relatively small and sparse graphs, as it is the case of molecules. 4 Related Work 4.1 Expressive power of GNNs WL hierarchy. The seminal results in the theoretical analysis of the expressivity of GNNs [16] and k-GNNs [17] established that traditional message passing-based GNNs are at most as powerful as the 1-WL test. [39] showed that graph isomorphism is equivalent to universal invariant function approximation. Higher-order Invariant Graph Networks (IGNs) have been studied in a series of works [22,23,24,39,58,59], establishing connections with the WL hierarchy, similarly to [17,60]. The main drawbacks of these methods are the training and inference time complexity and memory requirements of O(n k ) and the super-exponential number of parameters (for linear IGNs) making them impractical, as well as their non-local nature making them more prone to overfitting. Finally, [61] also analysed the expressive power of MPNNs and other more powerful variants and provided generalisation bounds. Unique identifiers. From a different perspective, [62] and [33] showed the connections between GNNs and distributed local algorithms [63,64,65] and suggested more powerful alternatives based on either local orderings or unique global identifiers (in the form of random features in [36]) that make GNNs universal. Similarly, [37] propose to use random colorings in order to uniquely identify the nodes. However, these methods lack a principled permutation equivariant way to choose orderings/identifiers. To date this is an open problem in graph theory called graph canonisation and it is at least as hard as solving graph isomorphism itself. A possible workaround is proposed in [49], where the authors take into account all possible vertex permutations. However, obviously this quickly becomes intractable (O(n!)) even when considering small-sized graphs. More expressive permutation equivariant GNNs. Concurrently with our work, other more expressive GNNs have been proposed using equivariant message passing. In [66], the authors propose to linearly transform each message with a different kernel based on the local isomorphism class of the corresponding edge (similar to our definition of structural roles). However, as also noted by the authors, taking into account all possible local isomorphism classes leads to insufficient weight sharing and hence to overfitting. In contrast, in GSN, usually the substructure collection is small (∼5-10 graphs) and the substructures are repetitive in the graph distribution, and as a result generalisation improves. Vignac et al. [48] propose a message passing scheme where matrices of order equal to the size of the graph are propagated instead of vectors. This can be perceived as a practical unique identification scheme, but the neural network complexity becomes quadratic in the number of nodes. Finally, [67] and [68] enhance the aggregation function with distance encodings (a strategy more relevant for vertex-level tasks) and graph eigenvectors respectively as alternative symmetry breaking mechanisms. In the experimental section, GSN is compared against these methods in real-world scenarios. Quantifying expressivity. Solely quantifying the expressive power of GNNs in terms of their ability to distinguish non-isomorphic graphs does not provide the necessary granularity: even the 1-WL test can distinguish almost all (in the probabilistic sense) non-isomorphic graphs [69]. As a result, there have been several efforts to analyse the power of k-WL tests in comparison to other graph invariants [70,71,18,72], while recently [19] approached GNN expressivity by studying their ability to count substructures. Substructures in Complex Networks. The idea of analysing complex networks based on small-scale topological characteristics dates back to the 1970's and the notion of triad census for directed graphs [73]. The seminal paper of [45] coined the term network motifs as over-represented subgraph patterns that were shown to characterise certain functional properties of complex networks in systems biology. The closely related concept of graphlets [44,74,75,76], different from motifs in being induced subgraphs, has been used to analyse the distribution of real-world networks and as a topological signature for network similarity. Our work is similar in spirit with the graphlet degree vector (GDV) [74], a node-wise descriptor based on graphlet counting. Substructures have been also used in the context of ML. In particular, subgraph patterns have been used to define Graph Kernels (GKs) [77,78,79,80,81], with the most prominent being the graphlet kernel [78]. Motif-based node embeddings [82,83] and diffusion operators [84,85,86] that employ adjacency matrices weighted according to motif occurrences, have recently been proposed for graph representation learning. Our formulation provides a unifying framework for these methods and it is the first to analyse their expressive power. Finally, GNNs that operate in larger induced neighbourhoods [87,88] or higher-order paths [89] have prohibitive complexity since the size of these neighbourhoods typically grows exponentially. Experimental Evaluation In the following section we evaluate GSN in comparison to the state-of-the-art in a variety of datasets from different domains. We are interested in practical scenarios where the collection of subgraphs, as well as their size, are kept small. Depending on the dataset domain we experimented with typical substructure families (cycles, paths and cliques) and maximum substructure size k (note that for each setting, our substructure collection consists of all the substructures of the family with size ≤ k). We also experimented with both graphlets and motifs and observed similar performance in most cases. To showcase that structural features can be used as an off-the-shelf strategy to boost GNN performance, we usually choose a base message passing architecture and minimally modify it into a GSN. Unless otherwise stated, the base architecture is a generalpurpose MPNN with MLPs used in the message and update functions. Additional implementation details can be found in the Appendix C. Synthetic Graph Isomorphism test We tested the ability of GSNs to decide if two graphs are non-isomorphic on a collection of Strongly Regular graphs of size up to 35 nodes, attempting to disambiguate pairs with the same number of nodes (for different sizes the problem becomes trivial). As we are only interested in the bias of the architecture itself, we use GSN with random weights to compute graph representations. Two graphs are deemed isomorphic if the Euclidean distance of their representations is smaller than a predefined threshold . Figure 4 shows the failure percentage of our isomorphism test when using different graphlet substructures (cycles, paths, and cliques) of varying size k. Interestingly, the number of failure cases of GSN decreases rapidly as we increase k; cycles and paths of maximum length k = 6 are enough to tell apart all the graphs in the dataset. Note that the performance of cliques saturates, possibly because the largest clique in our dataset has 5 nodes. Observe also the discrepancy between GSN-v and GSN-e. In particular, vertex-wise counts do not manage to distinguish all graphs, although missing only a few instances, which is in accordance with Theorem 3.3. Finally, 1-WL [16] and 2-FWL [24] equivalent models demonstrate 100% failure, as expected from theory. TUD Graph Classification Benchmarks We evaluate GSN on datasets from the classical TUD benchmarks. We use seven datasets from the domains of bioinformatics and computational social science and compare against various GNNs and Graph Kernels. The base architecture that we used is GIN [16]. We follow the same evaluation protocol of [16], performing 10-fold cross-validation and then reporting the performance at the epoch with the best average accuracy across the 10 folds. Table 1 lists all the methods evaluated with the split of [90]. We select our model by tuning architecture and optimisation hyperparameters and substructure related parameters, that is: (i) k, (ii) motifs against graphlets. Following domain evidence we choose the following substructure families: cycles for molecules, cliques for social networks. Best performing substructures both for GSN-e and GSN-v are reported. As can be seen, our model obtains state-of-the-art performance in most of the datasets, with a considerable margin against the main GNN baselines in some cases. ZINC Molecular graphs We evaluate GSN on the task of regressing the "penalized water-octanol partition coefficient -logP" (see [98,99,100] for details) of molecules from the ZINC database [101,25]. We use structural features obtained with k-cycle counting and report the result of the best performing substructure w.r.t. the validation set. As dictated by the evaluation protocol of [25], the total number of parameters of the model is approximately 100K, which is achieved by selecting an appropriate network width. 2 The data split is obtained from [25] and the evaluation metric is the Mean Absolute Error (MAE). We compare against a variety of baselines, ranging from traditional message passing NNs to recent more expressive architectures [102,68,48,103] and a molecular-specific one which is based on the junction tree molecular decomposition [104]. Wherever possible, [111] 0.7858 ± 0.0117 0.8427 ± 0.0063 HIMP [104] 0.7880 ± 0.0082 -GCN+GraphNorm [97] 0.7883 ± 0.0100 0.7904 ± 0.0115 PNA [102] 0.7905 ± 0.0132 0.8519 ± 0.0099 PHC-GNN [112] 0.7934 ± 0.0116 0.8217 ± 0.0089 DeeperGCN+FLAG [113] 0.7942 ± 0.0120 0.8425 ± 0.0061 DGN + eigenvectors [68] 0.7970 ± 0.0097 0.8470 ± 0.0047 P-WL [114] 0.8039 ± 0. OGB-MOLHIV We use ogbg-molhiv from the Open Graph Benchmark -OGB - [26] as a graph-level binary classification task, where the aim is to predict if a molecule inhibits HIV replication or not. We obtain baseline results from a plethora of different methods 3 , ranging from algorithms specific to molecular graphs to general purpose architectures. Following the same rationale as in the previous experiments, we choose a base architecture and modify it into a GSN variant by introducing structural features in the aggregation function (cycle counts, similar to other molecular datasets). Here we use the following two base architectures: (a) GIN-VN, a variation of GIN that allows for edge features and is extended with a virtual node, i.e. a node connected to every node in the graph. (b) Directional Graph Networks (DGN), a GNN that propagates messages in an anisotropic manner, based on a predefined graph vector field. Observe that the vector field is an alternative way to break local symmetries. The authors of DGN use vector fields defined by the eigenvectors of the graph, while in our case the vector field is defined by graph substructures. More information can be found in the supplementary material. Using the evaluator provided by the authors, we report the ROC-AUC metric at the epoch with the best validation performance (substructures are also chosen based on the validation set). By examining the results in Table 3 the following observations can be made, (a) general purpose GNNs benefit from symmetry breaking mechanisms, either in the form of eigenvectors or in the form of substructures. (b) Cyclical substructures are a good inductive bias when learning on molecules (e.g. P-WL is a graph kernel based on topological features that contain information of graph cycles, similar to GSN). (c) Further evidence for that is provided by observing the performance of molecular fingerprints methods. In specific, a method based on the extended-connectivity fingerprints [105], which mainly focuses on the structure of the molecule reports 0.8060 ± 0.0010 test performance, while one that additionally uses the MACCS fingerprints [106], which mainly encode the presence of certain functional groups (i.e. both structure and attributes), reports 0.8232 ± 0.0047 test performance. These methods although not directly comparable to ours, currently achieve the best results in this dataset, thus this further motivates research on making GNNs structure-aware. Ablation Studies Comparison between substructure collections In Figure 5 (left), we compare the training and test error for different substructure families (cycles, paths and non-isomorphic trees -for each experiment we use all the substructures of size ≤ k in the family). Additionally, we measure the "uniqueness" of the identifiers each substructure yields as follows: for each graph G in the dataset, we measure the number of unique vertex features u G (input vertex features concatenated with vertex structural identifiers for GSN-v). Then, we sum them up over the entire training set and divide by the total number of nodes, yielding the disambiguation score δ = G u G G |V G | . The disambiguation scores for the different types of substructures are illustrated as horizontal bars in Figure 5 (the exact values can be found in Appendix C.2, Table 2). A first thing to notice is that the training error is tightly related to the disambiguation score. As identifiers become more discriminative, the model gains expressive power. On the other hand, the test error is not guaranteed to decrease when the identifiers become more discriminative. For example, although cycles have smaller disambiguation scores, they manage to generalise much better than the other substructures, the performance of which is similar to the baseline architecture (MPNN with MLPs). This is also observed when comparing against [36] (MPNN-r method in Table 2), where, akin to unique identifiers, random features are used to strengthen the expressivity of GNN architectures. This approach also fails to improve the baseline architecture in terms of the performance in the test set. This validates our intuition that unique identifiers can be hard to generalise when chosen in a non-permutation equivariant way and motivates once more the importance of choosing the identifiers not only based on their discriminative power, but also in a way that allows incorporating the appropriate inductive biases. Finally, we observe a substantial jump in performance when using GSN with cycles of size k ≥ 6. This is not surprising, as cyclical patterns of such sizes (e.g. aromatic rings) are very common in organic molecules. Generalisation We repeat the experimental evaluation on ZINC using different fractions of the training set and compare the vanilla MPNN model against GSN. In Figure 5 (right), we plot the training and test errors of both methods. Regarding the training error, GSN consistently performs better, following our theoretical analysis on its expressive power. More importantly, GSN manages to generalise much better even with a small fraction of the training dataset. Observe that GSN requires only 20% of the samples to achieve approximately the same test error that MPNN achieves when trained on the entire training set. Structural Features & Message Passing: We perform an ablation study on the abilities of the structural features to predict the task at hand, when given as input to a graph-agnostic network. In particular, we compare our best performing GSN with a DeepSets model [115] that treats the input features and the structural identifiers as a set. For fairness of evaluation the same hyperparameter search is performed for both models (see Appendix C.5. Interestingly, as we show in Table 4, our baseline attains particularly strong performance across a variety of datasets and often outperforms other traditional message passing baselines. This demonstrates the benefits of these additional features and motivates their introduction in GNNs, which are unable to compute them. As expected, we observe that applying message passing on top of these features, brings performance improvements in the vast majority of the cases, sometimes considerably, as in the ZINC dataset. Conclusion In this paper, we propose a novel way to design structure-aware graph neural networks. Motivated by the limitations of traditional GNNs to capture important topological properties of the graph, we formulate a message passing scheme enhanced with structural features that are extracted by subgraph isomorphism. We show both theoretically and empirically that our construction leads to improved expressive power and attains state-of-the-art performance in real-world scenarios. In future work, we will further explore the expressivity of GSNs as an alternative to the k-WL tests, as well as their generalisation capabilities. Another important direction is to infer prominent substructures directly from the data and explore the ability of graph neural networks to compose substructures. A Deferred Proofs A.1 GSN is permutation equivariant Proof. Let A ∈ R n×n the adjacency matrix of the graph, H 0 ∈ R n×d V in the input vertex features, E ∈ R n×n×d E in the input edge features and E i ∈ R n×n the edge features at dimension i. Let S V (A) ∈ N n×d V , S E (A) ∈ N n×n×d E the functions generating the vertex and edge structural identifiers respectively. It is obvious that subgraph isomorphism is invariant to the ordering of the vertices, i.e. we will always obtain the same matching between subgraphs G S and graphs H in the subgraph collection. Thus, each vertex v (edge (v, u)) in the graph will be assigned the same structural identifiers x V v (x E v,u ) regarless of the vertex ordering, and S V and S E are permutation equivariant, i.e., for any permutation matrix P ∈ {0, 1} n×n it holds that Overall, a GSN network is permutation equivariant as composition of permutation equivariant functions, or permutation invariant when composed with a permutation invariant layer at the end, i.e. the READOUT function. S V (PAP T ) = PS V (A) S E (PAP T ) = P[S E 1 (A); . . . ; S E d E (A)]P T , A.2 Proof of Theorem 3.1: GSN is at least as powerful as the 1-WL test Proof. To show that GSN it is at least as expressive as the 1-WL test, we will repurpose the proof of Theorem 3 in [16] and demand the injectivity of the update function (w.r.t. both the hidden state h t v and the message m t+1 v ), and the injectivity of the aggregation w.r.t. the multiset of the hidden states of the neighbours h t u u∈N (v) . It suffices then to show that if injectivity is preserved then GSNs are at least as powerful as the 1-WL. We will show the above statement for vertex-labelled graphs, since traditionally the 1-WL test does not take into account edge labels. 4 We can rephrase the statement as follows: If GSN deems two graphs G 1 , G 2 as isomorphic, then also 1-WL deems them isomorphic. Given that the graph-level representation is extracted by a readout function that receives the multiset of the vertex colours in its input (i.e. the graphlevel representation is the vertex colour histogram at some iteration t), then it suffices to show that if for the two graphs the multiset of the vertex colours that GSN infers is the same, then also 1-WL will infer the same multiset for the two graphs. Consider the case where the two multisets that GSN extracts are the same: i.e. h t v v∈V G 1 = h t u u∈V G 2 . Then both multisets contain the same distinct colour/hidden representations with the exact same multiplicity. Thus, it further suffices to show that if two vertices v, u (that may belong to the same or to different graphs) have the same GSN hidden representations h t v = h t u at any iteration t, then they will also have the same colours c t v = c t u , extracted by 1-WL. Intuitively, this means that GSN creates a partition of the vertices of each graph that is at least as fine-grained as the one created by 1-WL. We prove by induction (similarly to [16]) that GSN model class contains a model where this holds (w.l.o.g. we show that for GSN-v; same proof applies to GSN-e). For t = 0 the statement holds since the initial vertex features are the same for both GSN and 1- WL, i.e. h 0 v = c 0 v , ∀v ∈ V G1 ∪ V G2 . Suppose the statement holds for t − 1, i.e. h t−1 v = h t−1 u ⇒ c t−1 v = c t−1 u . Then we show that it also holds for t. Every vertex hidden representation at step t is updated as follows: h t v = UP t h t−1 v , m t v . Assuming that the update function UP t is injective, we have the following: if h t v = h t u , then: • h t−1 v = h t−1 u , which from the induction hypothesis implies that c t−1 v = c t−1 u . • m t v = m t u , where the message function is defined as in Eq. (5) of the main paper. Additionally here we require M t to be injective w.r.t. the multiset of the hidden representations of the neighbours: 5 m t v = m t u ⇒ h t−1 w w∈Nv = h t−1 z z∈Nu From the induction hypothesis we know that h t−1 w = h t−1 z implies that c t−1 w = c t−1 z for any w ∈ N v , z ∈ N u , thus c t−1 w w∈Nv = c t−1 z z∈Nu . Concluding, given the update rule of 1-WL: c t v = HASH c t−1 v , c t−1 w w∈Nv , it holds that c t v = c t u . A.3 Proof of Corollary 3.2 Proof. In order to prove the universality of GSN, we will show that when the substructure collection contains all graphs of size n − 1, then there exists a parametrisation of GSN that can infer the isomorphism classes of all vertex-deleted subgraphs of the graph G (the deck of G). The reconstruction conjectures states that two graphs with at least three vertices are isomorphic if and only if they have the same deck. Thus, the deck is sufficient to distinguish all non-isomorphic graphs. The deck can be defined as follows: Let H = {H 1 , H 2 , · · · , H K } the set of all possible graphs of size n − 1. The vertex-deleted subgraphs of G are by definition all the induced subgraphs of G with size n − 1, which we denote as G n−1 = {G S : induced subgraph of G with |V G S | = n − 1}. Then, the deck D(G) can be defined as a vector of size |H|, where at the j-th dimension D j (G) = G S ∈ G n−1 | G S H j = G S ∈Gn−1 1[G S H j ], 5 Lemma 5 from [16] states that such a function always exists assuming that the elements of the multiset originate from a countable domain where 1[·] the indicator function. The structural feature x V Hj ,i (v) for each substructure H j and orbit i are computed as follows: x V Hj ,i (v) = G S H j | v ∈ V G S , f (v) ∈ O V Hj ,i = G S ∈Gn−1 1[v ∈ V G S ]1[G S H j ]1[f G S (v) ∈ O V Hj ,i ] where f G S (·) = ∅ if G S H j , otherwise it is the bijective mapping function. The deck can be inferred as follows: v∈V G d H j i=1 x V Hj ,i (v) = v∈V G d H j i=1 G S ∈Gn−1 1[v ∈ V G S ]1[G S H j ]1[f G S (v) ∈ O V Hj ,i ] = v∈V G G S ∈Gn−1 1[v ∈ V G S ]1[G S H j ] d H j i=1 1[f G S (v) ∈ O V Hj ,i ] = G S ∈Gn−1 1[G S H j ] v∈V G 1[v ∈ V G S ] = G S ∈Gn−1 (n − 1)1[G S H j ] = (n − 1)D j (G) where we used that d H j i=1 1[f G S (v) ∈ O V Hj ,i ] = 1, since each vertex can be mapped to a single orbit only. Thus, the deck can be inferred by a simple GSN-v parametrisation (a linear layer with depth equal to |H| that performs orbit-wise summation and division by the constant n − 1 for each vertex separately, followed by a sum readout). Since GSN-v can be inferred by GSN-e (Theorem 3.3), then GSN-e is also universal. A.4 Proof of Theorem 3.3 Proof. Without loss of generality we will show Theorem 3.3 for the case of a single substructure H. In order to show that GSN-e can express GSN-v, we will first prove the following: the vertex identifier of a vertex v can be inferred by the edge identifiers of its incident edges. To simplify notation we define the following: For an orbit Orb V (v), denote the orbit neighbourhood as the multiset of the orbits of the neighbours of v and the orbit degree as the degree of any vertex in Orb V (v) : N Orb V (v) = Orb V (u) | u ∈ N (v) and deg(v) = deg(Orb V (v)) = |N Orb V (v) |. For brevity we will use the following notation: vertex orbits are indexed as follows O V 1 , O V 2 , . . . O V d V and edge orbits O E 1,1 , O E 1,2 , . . . O E d V ,d V with O E i,j = O V i , O V j . Structural features are denoted accordingly: x V i (v) and x E ij (v, u) Let's assume that there exists only one matched subgraph G S H and the bijection between V G S and H is denoted as f . Then, for an abitrary vertex v and a vertex orbit O V i , one of the following holds: • v ∈ V G S . Then x V i (v) = 0 and x E ij (v, u) = 0, ∀u ∈ N (v), • v ∈ V G S and Orb(f (v)) = O V i . Then, x V i (v) = 0 and x E ij (v, u) = 0, ∀u ∈ N (v). Note that here the directionality of the edge is important, otherwise we cannot determine the value of x E ij (v, u) unless we also know the orbit of f (u). • v ∈ V G S and Orb(f (v)) = O V i . Then, x V i (v) = 1 and since f (v) has exactly deg(O V i ) neighbours in H, then v has exactly deg(O V i ) neighbours in G S with vertex orbits N O V i . In other words u∈N (v) j:O V j ∈N (O V i ) x E ij (u, v) = deg(O V i ) Thus, by induction, for m matched subgraphs G S H with v ∈ V G S and Orb(f (v)) = O V i , it holds that x V i (v) = m and u∈N (v) j:O V j ∈N (O V i ) x E ij (u, v) = m * deg(O V i ). Then it follows that: x V i (v) = 1 deg(O V i ) u∈N (v) j:O V j ∈N (O V i ) x E ij (u, v)(6) The rest of the proof is straightforward: we will assume a GSN-v using substructure counts of the graph H, with L layers and width w defined as in the main paper (Eq. (5)) Then, there exists a GSN-e with L + 1 layers, where the first layer has width d in V +d V and implements the following function: UP t+1 h t v , m t+1 v = [h t v ; m t+1 v ], where: m t+1 v = M t+1 (h t v , h t u , x E v,u , e u,v ) u∈N (v) = [ 1 deg(O V 1 ) u∈N (v) j:O V j ∈N (O V 1 ) x E 1j (u, v); . . . ; 1 deg(O V d V ) u∈N (v) j:O V j ∈N (O V d V ) x E d V ,j (u, v)] = x V v Note that since M is a universal multiset function approximator, then there exist parameters of M with which the above function can be computed. The next L layers of GSN-e can implement a traditional MPNN where now the input vertex features are [h t v ; x V v ] (which is exactly the formulation of GSN-v) and this concludes the proof. B Comparison with higher-order Weisfeiler-Leman tests B.1 The WL hierarchy Following the terminology introduced in [24], we describe the so-called Folklore WL family (k-FWL). Note that, in the majority of papers on GNN expressivity [17,24,19] another family of WL tests is discussed, under the terminology k-WL with expressive power equal to (k − 1)-FWL. In contrast, in most graph theory papers on graph isomorphism [116,71,18] the k-WL term is used to describe the algorithms referred to as k-FWL in GNN papers. Here, we follow the k-FWL convention to align with the work mostly related to ours. The k-FWL operates on k-tuples of vertices v = (v 1 , v 2 , . . . , v k ) to which an initial colour c 0 v is assigned based on their isomorphism types (see section B.2), which can loosely be thought of as a generalisation of isomorphism that also preserves the ordering of the vertices in the tuple. Then, at each iteration the colour is refined as follows: c t+1 v = HASH c t v , c t vu,1 , c t vu,2 , . . . , c t v u,k u∈V ,(7) where v u,j = (v 1 , v 2 , . . . , v j−1 , u, v j+1 , . . . , v k ). The multiset c t vu,1 , c t vu,2 , . . . , c t v u,k u∈V can be perceived as a form of generalised neighbourhood. Observe that all possible tuples in the graph store information necessary for the updates, thus each k-tuple receives information from the entire graph, contrary to the local nature of the 1-WL test. B.2 Why does 2-FWL fail on strongly regular graphs? Proof. We first formally define what an isomorphism type is and what are the properties of the SR family: Definition 1 (Isomorphism Types). Two k-tuples v a = {v a 1 , v a 2 , . . . , v a k }, v b = {v b 1 , v b 2 , . . . , v b k } will have the same isomorphism type iff: • ∀ i, j ∈ {0, 1, . . . , k}, v a i = v a j ⇔ v b i = v b j • ∀ i, j ∈ {0, 1, . . . , k}, v a i ∼ v a j ⇔ v b i ∼ v b j , where ∼ means that the vertices are adjacent. Note that this is a stronger condition than isomorphism, since the mapping between the vertices of the two tuples needs to preserve order. In case the graph is employed with edge and vertex features, these need to be preserved as well (see [19]) for the extended case). Definition 2 (Strongly regular graph). A SR(n,d,λ,µ)-graph is a regular graph with n vertices and degree d, where every two adjacent vertices have always λ mutual neighbours, while every two non-adjacent vertices have always µ mutual neighbours. Now we can proceed to the details of the proof. For the 2-FWL test, when working with simple undirected graphs without self-loops, we have the following 2-tuple isomorphism types: • v = {v 1 , v 1 }: vertex type. Mapped to the colour c 0 = c α • v = {v 1 , v 2 } and v 1 ∼ v 2 : non-edge type. Mapped to the colour c 0 = c β • v = {v 1 , v 2 } and v 1 ∼ v 2 : edge type. Mapped to the colour c 0 = c γ For each 2-tuple v = {v 1 , v 2 }, a generalised "neighbour" is the following tuple: (v u,1 , v u,2 ) = (u, v 2 ), (v 1 , u) , where u is an arbitrary vertex in the graph. Now, let us consider a strongly regular graph SR(n,d,λ,µ). We have the following cases: • generalised neighbour of a vertex type tuple: (v u,1 , v u,2 ) = (u, v 1 ), (v 1 , u) . The corresponding neighbour colour tuples are: -(c α , c α ) if v 1 = u, -(c β , c β ) if v 1 ∼ u , -(c γ , c γ ) if v 1 ∼ u. The update of the 2-FWL is: c 1 v = HASH c α , (c α , c α ) 1 time , (c β , c β ) n − 1 − d times , (c γ , c γ ) d times same for all vertex type 2-tuples. • generalised neighbour of a non-edge type tuple: u) . The corresponding neighbour colour tuples are: (v u,1 , v u,2 ) = (u, v 2 ), (v 1 ,-(c α , c β ) if v 2 = u, -(c β , c α ) if v 1 = u, -(c γ , c β ) if v 2 ∼ u and v 1 ∼ u, -(c β , c γ ) if v 1 ∼ u and v 2 ∼ u, -(c β , c β ) if v 1 ∼ u and v 2 ∼ u, -(c γ , c γ ) if v 1 ∼ u and v 2 ∼ u. The update of the 2-FWL is: c 1 v = HASH c β , (c α , c β ) 1 time , (c β , c α ) 1 time , (c γ , c β ) d − µ times , (c β , c γ ) d − µ times , (c β , c β ) n − 2 − (2d − µ) times , (c γ , c γ ) µ times same for all nonedge type 2-tuples. • generalised neighbour of an edge type tuple: -(c α , c γ ) if v 2 = u, -(c γ , c α ) if v 1 = u, -(c γ , c β ) if v 2 ∼ u and v 1 ∼ u, -(c β , c γ ) if v 1 ∼ u and v 2 ∼ u, -(c β , c β ) if v 1 ∼ u and v 2 ∼ u, -(c γ , c γ ) if v 1 ∼ u and v 2 ∼ u. The update of the 2-FWL is: c 1 v = HASH c γ , (c α , c γ ) 1 time , (c γ , c α ) 1 time , (c γ , c β ) d − λ times , (c β , c γ ) d − λ times , (c β , c β ) n − 2 − (2d − λ) times , (c γ , c γ ) λ times same for all edge type 2-tuples. From the analysis above, it is clear that all 2-tuples in the graph of the same initial type are assigned the same colour in the 1st iteration of 2-FWL. In other words, the vertices cannot be further partitioned, so the algorithm terminates. Therefore, if two SR graphs have the same parameters n,d,λ,µ then 2-FWL will yield the same colour distribution and thus the graphs will be deemed isomorphic. C Experimental Settings -Additional Details In this section, we provide additional implementation details of our experiments. All experiments were performed on a server equipped with 8 Tesla V100 16 GB GPUs, except for the Collab dataset where a Tesla V100 GPU with 32 GB RAM was used due to larger memory requirements. Experimental tracking and hyperparameter optimisation were done via the Weights & Biases platform (wandb) [117]. Our implementation is based on native PyTorch sparse operations [118] in order to ensure complete reproducibility of the results. PyTorch Geometric [119] was used for additional operations (such as preprocessing and data loading). In each one of the different experiments we aim to show that structural identifiers can be used off-theshelf and independently of the architecture. At the same time we aim to suppress the effect of other confounding factors in the model performance, thus wherever possible we build our model on top of a baseline architecture. For more details, please see the relevant subsections. Interestingly, we observed that in most of the cases it was sufficient to replace only the first layer of the baseline architecture with a GSN layer, in order to obtain a boost in performance. Throughout the experimental evaluation the structural identifiers x V v and x E u,v are one-hot encoded, by taking into account the unique count values present in the dataset. Other more sophisticated methods can be used, e.g. transformation to continuous features via a normalisation scheme or binning. However, we found that the number of unique values in our datasets were usually relatively small (which is a good indication of recurrent structural roles) and thus such methods were not necessary. C.1 Synthetic Experiment For the Strongly Regular graphs dataset (available from http://users.cecs.anu.edu.au/~bdm/data/graphs. html) we use all the available families of graphs with size of at most 35 vertices: improvement in the validation loss. Training is stopped when the learning rate reaches the minimum learning rate value of 10 −5 . Validation and test metrics are inferred using the model at the last training epoch. We select our best performing substructure related parameters based on the performance in the validation set in the last epoch. We search cycles with k = 3, . . . , 10, graphlets vs motifs, and GSN-v vs GSN-e. The chosen hyperparameters for GSN are: GSN-e, cycle graphlets of 10 vertices and for GSN-EF : GSN-v, cycle motifs of 8 vertices. Once the model is chosen, we repeat the experiment 10 times with different seeds and report the mean and standard deviation of the test MAE in the last epoch. Disambiguation Scores: In Table 6, we provide the disambiguation scores δ as defined in section 5.5.1 of the main paper for different types of substructures. These are computed based on vertex structural identifiers (GSN-v). C.4 Graph Classification on ogbg-molhiv The ogbg-molhiv dataset contains ≈ 41K graphs, with 25.5 vertices and 27.5 edges on average. As most molecular graphs, the average degree is small (2.2) and they exhibit a tree-like structure (average clustering coefficient 0.002). The average diameter is 12 (more details in [26]). Below we describe how we extend the two base architectures: GIN+VN. We follow the design choices of the authors of [26] and extend their architectures to include structural identifiers. Initial vertex and edge features are multi-hot encodings passed through linear layers that project them in the same embedding space, i.e. h 0 v = W 0 h · h in v , e t v,u = W t e · e in u,v . The baseline model is a modification of GIN that allows for edge features: for each neighbour, the hidden representation is added to an embedding of its associated edge feature. Then the result is passed through a ReLU non-linearity which produces the neighbour's message. Formally, the aggregation is as follows: h t+1 v = UP t+1 h t v + u∈Nv σ h t u + e t v,u(10) In order to allow global information to be broadcasted to the vertices, a virtual node takes part in the message passing. The virtual node representation, denoted as G t , is initialised as a zero vector G 0 and then Message Passing becomes:h t v = h t v + G t , h t+1 v = UP t+1 h t v + u∈Nv σ h t u + e t v,u , G t+1 = M LP t+1 G t + u∈Nvh t u(11) We modify this model, as follows: first the substructure counts are embedded into the same embedding space as the rest of the features. Then, for GSN-v, they are added to the corresponding vertex embeddings: h t v = h t v + W t V · x V v , or for GSN-e, they are added to the edge embeddingsé t v,u = e t v,u + W t E · x E u,v . DGN + substructures. We use the directional average operator as defined in [68]: m t+1 v = [ u∈N (v) α 1 v,u h t u ; . . . ; u∈N (v) α D v,u h t u ],(12) where α i v,u are weighting average coefficients. In our case, each orbit i induces a separate set of averaging coefficients. For example, for GSN-e α v,u = |x E v,u | + u∈N (v) |x E v,u | , where x E v,u denotes edge-wise substructure counts (the index of the orbit i was dropped to simplify notation). Similarly, for GSN-v, α v,u = |x V v −x V u | + u∈N (v) |x V v −x V u | . Subsequently, the vertex representation is updated as follows: h t+1 v = MLP t+1 (m t+1 v ). Observe that this model is simpler than the aforementioned, in terms of both its parameter count and its expressive power. Since the MOLHIV dataset poses a significant challenge w.r.t. generalisation (the data splits reflect different molecular distributions), architectures biased towards simpler solutions usually perform better, sincey the mitigate the risk of overfitting. In both cases we use the the same hyperparameters as the ones provided by the authors, and only select the substructure related parameters based on the highest validation ROC-AUC (choosing the best scoring epoch as in [26]). We search cycles with k = 3, . . . , 12, graphlets vs motifs, and GSN-v vs GSN-e. The chosen hyperparameters are: GSN-e, cycle graphlets of 6 vertices. We repeat the experiment 10 times with different seeds and report the mean and standard deviation of the train, validation and test ROC-AUC, again by choosing the best scoring epoch w.r.t the validation set. C.5 Structural Features & Message Passing The baseline architecture treats the input vertex and edge features, along with the structural identifiers, as a set. In particular, we consider each graph as a set of independent edges (v, u) endowed with the features of the endpoint vertices h v , h u , the structural identifiers x V v , x V u and the edge features e(v, u), and we implement a DeepSets universal set function approximator [115] to learn a prediction function: f hv, hu, x V v , x V u , ev,u (v,u)∈E G = ρ (v,u)∈E G φ hv, hux V v , x V u , ev,u ,(13) with E G the edge set of the graph and ρ, φ MLPs. This baseline is naturally extended to the case where we consider edge structural identifiers by replacing (x V v , x V u ) with x E v,u . For fairness of evaluation, we follow the exact same parameter tuning procedure as the one we followed for our GSN models for each benchmark, i.e. for the TUD datasets we first tune network and optimisation hyperaparameters (network width was set to be either equal to the ones we tuned for GSN, or such that the absolute number of learnable parameters was equal to those used by GSN; depth of the MLPs was set to 2) and subsequently we choose the substructure related parameters based on the evaluation protocol of [16]. For ZINC and ogbg-molhiv we perform only substructure selection, based on the performance on the validation set. Using the same widths as in GSN leads to smaller baseline models w.r.t the absolute number of parameters, and we interestingly observed this to lead to particularly strong performance in some cases, especially Proteins and MUTAG, where our DeepSets implementation attains state-of-art results. This finding motivated us to explore 'smaller' GSNs (with either reduced layer width or a single message passing layer). These GSN variants exhibited a similar trend, i.e. to perform better than their 'larger' counterparts over these two datasets. We hypothesise this phenomenon to be mostly due to the small size of these datasets, which encourages overfitting when using architectures with larger capacity. In Table 4 in the main paper, we report the result for the best performing architectures, along with the number of learnable parameters. Figure 1 1Figure 1: Node (left) and edge (right) induced subgraph counting for a 3-cycle and a 3-path. Counts are reported for the blue node on the left and for the blue edge on the right. Different colors depict orbits. Figure 3 : 3Empirical (solid) vs worst case (dashed) runtime for different graph distributions (in seconds, log scale). For each distribution we count the best performing (and frequent) substructures of increasing sizes k. Figure 4 : 4SR graphs isomorphism test (log scale, smaller values are better). Different colours indicate different substructure sizes. two variants, one that does not take edge features into account and one that does. In both cases, GSN achieves state-of-the-art results outperforming all the baseline architectures. Figure 5 : 5(Left) Train (dashed) and test (solid) MAEs for Path-, Tree-and Cycle-GSN-EF as a function of the maximum substructure size k. Vertical bars indicate standard deviation; horizontal bars depict disambiguation scores δ. (Right) Train (dashed) and test (solid) MAEs for GSN-EF (blue) and MPNN-EF (red) as a function of the dataset fraction used for training where the permutation is applied at each slice S E i (A) ∈ N n×n of the tensor S E (A). Let f (A, H, E) ∈ R n×dout a GSN layer. We will show that f is permutation equivariant. We need to show that if Y = f (A, H, E) the output of the GSN layer, then PY = f (PAP T , PH, P[E 1 ; . . . ; E d E ]P T ) for any permutation matrix P. It is easy to see that GSN-v can can be expressed as a traditional MPNN g(A, H, E) ∈ R n×dout (similar to Eq. (5) of the main paper) by replacing the vertex features H with the concatenation of the input vertex features and the vertex structural identifiers, i.e. Y = f (A, H, E) = g(A, [H; S V (A)], E). Thus, f (PAP T , PH, P[E 1 ; . . . ; E d E ]P T , ) = = g(PAP T , [PH; S V (PAP T )], P[E 1 ; . . . ; E d E ]P T ) = g(PAP T , P[H; S V (A)], P[E 1 ; . . . ; E d E ]P T ) = Pg(A, [H; S V (A)], E) = PY where in the last step we used the permutation equivariant property of MPNNs. Similarly, we can show that a GSN-e layer is permutation equivariant, since it can be expressed as a traditional MPNN layer by replacing the edge features with the concatenation of the original edge features and the edge structural identifiers f (A, H, E) = g(A, H, [E; S E (A)]). where d H is the cardinality of the quotient.V H,1 , O V H,2 , . . . , O V H,d H }, Structural roles are encoded into features by counting the appearance of certain substructures. Define a set of small (connected) graphs H = {H 1 , H 2 . . . H K }, for example cycles of fixed length or cliques. For each graph H ∈ H, we first find its isomorphic subgraphs in G denoted as G S . For each node v ∈ V G S we infer its role w.r.t. H by obtaining the orbit of its mapping f (v) in H, Orb H (f (v)). By counting all the possible appearances of different orbits in v, we obtain the vertex structural feature x V H (v) of v, defined as follows. For all i ∈ {1, . . . , d H }: Table 1 : 1Graph classification accuracy on TUD Dataset. First, Second, Third best methods are highlighted. For GSN, the best performing structure is shown. * Graph Kernel methods.Dataset MUTAG PTC Proteins NCI1 Collab IMDB-B IMDB-M RWK* [91] 79.2±2.1 55.9±0.3 59.6±0.1 >3 days N/A N/A N/A GK* (k=3) [78] 81.4±1.7 55.7±0.5 71.4±0.31 62.5±0.3 N/A N/A N/A PK* [92] 76.0±2.7 59.5±2.4 73.7±0.7 82.5±0.5 N/A N/A N/A WL kernel* [93] 90.4±5.7 59.9±4.3 75.0±3.1 86.0±1.8 78.9±1.9 73.8±3.9 50.9±3.8 GNTK* [94] 90.0±8.5 67.9±6.9 75.6±4.2 84.2±1.5 83.6±1.0 76.9±3.6 52.8±4.6 DCNN [95] N/A N/A 61.3±1.6 56.6±1.0 52.1±0.7 49.1±1.4 33.5±1.4 DGCNN [90] 85.8±1.8 58.6±2.5 75.5±0.9 74.4±0.5 73.8±0.5 70.0±0.9 47.8±0.9 IGN [22] 83.9±13.0 58.5±6.9 76.6±5.5 74.3±2.7 78.3±2.5 72.0±5.5 48.7±3.4 GIN [16] 89.4±5.6 64.6±7.0 76.2±2.8 82.7±1.7 80.2±1.9 75.1±5.1 52.3±2.8 PPGNs [24] 90.6±8.7 66.2±6.6 77.2±4.7 83.2±1.1 81.4±1.4 73.0±5.8 50.5±3.6 Natural GN [66] 89.4±1.60 66.8±1.79 71.7±1.04 82.7±1.35 N/A 74.8±2.01 51.3±1.50 WEGL [96] N/A 67.5±7.7 76.5±4.2 N/A 80.6±2.0 75.4±5.0 52.3±2.9 GIN+GraphNorm [97] 91.6 ± 6.5 64.9 ± 7.5 77.4 ± 4.9 82.7 ± 1.7 80.2 ± 1.0 76.0 ± 3.7 N/A GSN-e 90.6±7.5 68.2±7.2 76.6±5.0 83.5± 2.3 85.5±1.2 77.8±3.3 54.3±3.3 6 (cycles) 6 (cycles) 4 (cliques) 15 (cycles) 3 (triangles) 5 (cliques) 5 (cliques) GSN-v 92.2±7.5 67.4±5.7 74.59±5.0 83.5±2.0 82.7±1.5 76.8±2.0 52.6±3.6 12 (cycles) 10 (cycles) 4 (cliques) 3 (triangles) 3 (triangles) 4 (cliques) 3 (triangles) Table 2 : 2MAE in ZINCMethod MAE MAE (EF) GCN [107] 0.469±0.002 - GIN [16] 0.408±0.008 - GraphSage[108] 0.410±0.005 - GAT [109] 0.463±0.002 - MoNet[10] 0.407±0.007 - GatedGCN [110] 0.422±0.006 0.363±0.009 MPNN 0.254±0.014 0.209±0.018 MPNN-r 0.322±0.026 0.279±0.023 PNA[102] 0.320±0.032 0.188±0.004 DGN[68] 0.219±0.010 0.168±0.003 GNNML[103] 0.161±0.006 - HIMP[104] - 0.151±0.006 SMP[48] 0.219± 0.138± GSN 0.140±0.006 0.115±0.012 Table 3 : 3Test and Validation ROC-AUC in OGB-MOLHIV.Method Test ROC-AUC Validation ROC-AUC GIN+VN[16] 0.7707 ± 0.0149 0.8479 ± 0.0068 DeeperGCN Table 4 : 4Comparison between DeepSets and GSN with the same structural featuresDataset DeepSets # params GSN # params MUTAG 93.3±6.9 3K 92.8±7.0 3K PTC 66.4±6.7 2K 68.2±7.2 3K Proteins 77.8±4.2 3K 77.8±5.6 3K NCI1 80.3 ±2.4 10K 83.5± 2.0 10K Collab 80.9 ±1.6 30K 85.5±1.2 52K IMDB-B 77.1 ±3.7 51K 77.8±3.3 65K IMDB-M 53.3 ±3.2 68K 54.3±3.3 66K ZINC 0.288 ±0.003 366K 0.108 ±0.018 385K ogbg-molhiv 77.34±1.46 3.4M 77.99±1.00 3.3M Table 5 : 5Chosen hyperparameters for each of the two GSN variants for each dataset. Table 6 : 6Disambiguation scores δ on ZINC for different substructure families and their maximum size k. Size k = 0 refers to the use of the original vertex features only.k Cycles Paths Trees 0 0.196 0.196 0.196 3 0.199 0.540 0.540 4 0.200 0.746 0.762 5 0.256 0.866 0.875 6 0.327 0.895 0.897 7 0.330 0.900 0.900 8 0.330 0.901 0.901 9 0.330 0.901 0.901 10 0.330 0.901 0.901 Note that the edge automorphism group is larger than that of induced automorphisms, but strictly larger only for 3 trivial cases[27]. However, induced automorphisms provide a more natural way to express edge structural roles. A larger version of GSN using 500K parameters attains 0.101 ± 0.010 test MAE. the most representative ones from the OGB public leaderboard: https://ogb.stanford.edu/docs/leader_graphprop/ #ogbg-molhiv if one considers a simple 1-WL extension that concatenates edge labels to neighbour colours, then the same proof applies. more details on the description of the datasets and the corresponding tasks can be found at[16]. AcknowledgementsThis research was partially supported by the ERC Consolidator Grant No. 724228 -LEMAN (GB and MB). The work of GB is partially funded by a PhD scholarship from the Department of Computing, Imperial College London. SZ acknowledges support from the EPSRC Fellowship DEFORM: Large Scale Shape Analysis of Deformable Models of Humans (EP/S010203/1) and a Google Faculty award. MB acknowledges support from Google Faculty awards and the Royal Society Wolfson Research Merit award.C.2 TUD Graph Classification BenchmarksFor this family of experiments, due to the usually small size of the datasets, we choose a parameter-efficient architecture, in order to reduce the risk of overfitting. In particular, we follow the simple GIN architecture[16]and we concatenate structural identifiers to vertex or edge features depending on the variant. Then for GSN-v, the hidden representation is updated as follows:and for GSN-e:where x E v,v is a dummy variable (also one-hot encoded) used to distinguish self-loops from edges. Empirically, we did not find training the parameter used in GIN to make a difference.We implement an architecture similar to GIN[16], i.e. message passing layers: 4 , jumping knowledge from all the layers[120](including the input), transformation of each intermediate graph-level representation: linear layer, readout: sum for biological and mean for social networks. Vertex features are one-hot encodings of the categorical vertex labels. Similarly to the baseline, the hyperparameters search space is the following: batch size in {32, 128} (except for Collab where only 32 was searched due to GPU memory limits), dropout in {0,0.5}, network width in {16,32} for biological networks, 64 for social networks, learning rate in {0.01, 0.001}, decay rate in {0.5,0.9} and decay steps in {10,50} (number of epochs after which the learning rate is reduced by multiplying with the decay rate). For social networks, since they are not attributed graphs, we also experimented with using the degree as a vertex feature, but in most cases the structural identifiers were sufficient.Model selection is done in two stages. First, we choose a substructure that we perceive as promising based on indications from the specific domain: triangles for social networks and Proteins, and 6-cycles (motifs) for molecules. Under this setting we tune model hyperparameters for a GSN-e model. Then, we extend our search to the parameters related to the substructure collection: i.e. the maximum size k and motifs vs graphlets. In all the molecular datasets we search cycles with k = 3, . . . , 12, except for NCI1, where we also consider larger sizes due to the presence of large rings in the dataset (macrocycles[121]). For social networks, we searched cliques with k = 3, 4, 5. InTable 5we report the hyperparameters chosen by our model selection procedure, including the best performing substructures.The seven datasets 6 we chose are the intersection of the datasets used by the authors of our main baselines: the Graph Isomorphism Network (GIN)[16], a simple, yet powerful GNN with expressive power equal to the 1-WL test, and the Provably Powerful Graph Network (PPGN)[24], a polynomial alternative to the Invariant Graph Network[22], that increases its expressive power to match the 2-FWL. We also compare our results to other GNNs as well as Graph Kernel approaches. Our main baseline from the GK family is the Graph Neural Tangent Kernel (GNTK)[94], which is a kernel obtained from a GNN of infinite width. 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PMLR, 2019. • Sr, 15 graphs • SR. 622741 graphs • SR• SR(16,6,2,2): 2 graphs • SR(25,12,5,6): 15 graphs • SR(26,10,3,4): 10 graphs • SR(28,12,6,4): 4 graphs • SR(29,14,6,7): 41 graphs • SR(35,16,6,8): 3854 graphs • SR(35,18,9,9): 227 graphs We used a simple 2-layer architecture with width 64. The message aggregation was performed as in the general formulation of Eq. (5) of the main paper, where the update and the message functions are MLPs. The prediction is inferred by applying a sum readout function in the last layer and then passing the output through a MLP. Regarding the substructures, we use graphlet counting, as certain motifs (e.g. cycles of length up to 7) are known to be unable to distinguish strongly regular graphs. The total number of non-isomorphic pairs of the same size is ≈ 7 * 10 7. since they can be counted by the 2-FWL [71, 18The total number of non-isomorphic pairs of the same size is ≈ 7 * 10 7 . We used a simple 2-layer architecture with width 64. The message aggregation was performed as in the general formulation of Eq. (5) of the main paper, where the update and the message functions are MLPs. The prediction is inferred by applying a sum readout function in the last layer and then passing the output through a MLP. Regarding the substructures, we use graphlet counting, as certain motifs (e.g. cycles of length up to 7) are known to be unable to distinguish strongly regular graphs (since they can be counted by the 2-FWL [71, 18]). Given the adversities that strongly regular graphs pose in graph isomorphism testing, it would be interesting to see how this method can perform in other categories of hard instances, such as the classical CFI counterexamples for k-WL proposed in [116], and explore further its expressive power and combinatorial properties. We leave this direction to future workGiven the adversities that strongly regular graphs pose in graph isomorphism testing, it would be interesting to see how this method can perform in other categories of hard instances, such as the classical CFI counter- examples for k-WL proposed in [116], and explore further its expressive power and combinatorial properties. We leave this direction to future work.
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[ "A Novel Beam-Dump Measurement with the LHC General-Purpose Detectors", "A Novel Beam-Dump Measurement with the LHC General-Purpose Detectors" ]
[ "Bhaskar Dutta \nDepartment of Physics and Astronomy\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A&M University\n77843College StationTXUSA\n", "Doojin Kim \nDepartment of Physics and Astronomy\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A&M University\n77843College StationTXUSA\n", "Hyunyong Kim \nDepartment of Physics and Astronomy\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A&M University\n77843College StationTXUSA\n" ]
[ "Department of Physics and Astronomy\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A&M University\n77843College StationTXUSA", "Department of Physics and Astronomy\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A&M University\n77843College StationTXUSA", "Department of Physics and Astronomy\nMitchell Institute for Fundamental Physics and Astronomy\nTexas A&M University\n77843College StationTXUSA" ]
[]
We propose a novel scheme for performing a beam-dump-like experiment with the general-purpose detectors (ATLAS and CMS) at the LHC. Collisions of high-energy protons result in jets containing a number of energetic hadrons and electromagnetic objects that are essentially "dumped" to hadronic and electromagnetic calorimeters, respectively, and induce the production of secondary hadrons, electrons, and photons in calorimetric showers. We envision a situation where new physics particles are produced by the interactions of these secondary particles inside the calorimeters. For proof of principles, we consider the axion-like particles (ALPs) produced via the Primakoff process in the presence of their interaction with photons at CMS. We argue that the drift tube chambers and the ME0 module of the muon system can serve as detectors to record the photons from the ALP decay, demonstrating that the resulting sensitivity reach is competitive due to their close proximity to the signal source points. We further show that the LHC does not suffer from a barrier, dubbed beamdump "ceiling", that typical beam-dump experiments hardly surpass, carrying the great potential for exploring a wide range of parameter space in increasing statistics. This analysis can be extended to investigate various types of light mediators with couplings to the Standard Model leptons and quarks.
null
[ "https://export.arxiv.org/pdf/2305.16383v1.pdf" ]
258,947,669
2305.16383
65356fff70b024253c3070a675b76b2e39eb3d6a
A Novel Beam-Dump Measurement with the LHC General-Purpose Detectors Bhaskar Dutta Department of Physics and Astronomy Mitchell Institute for Fundamental Physics and Astronomy Texas A&M University 77843College StationTXUSA Doojin Kim Department of Physics and Astronomy Mitchell Institute for Fundamental Physics and Astronomy Texas A&M University 77843College StationTXUSA Hyunyong Kim Department of Physics and Astronomy Mitchell Institute for Fundamental Physics and Astronomy Texas A&M University 77843College StationTXUSA A Novel Beam-Dump Measurement with the LHC General-Purpose Detectors We propose a novel scheme for performing a beam-dump-like experiment with the general-purpose detectors (ATLAS and CMS) at the LHC. Collisions of high-energy protons result in jets containing a number of energetic hadrons and electromagnetic objects that are essentially "dumped" to hadronic and electromagnetic calorimeters, respectively, and induce the production of secondary hadrons, electrons, and photons in calorimetric showers. We envision a situation where new physics particles are produced by the interactions of these secondary particles inside the calorimeters. For proof of principles, we consider the axion-like particles (ALPs) produced via the Primakoff process in the presence of their interaction with photons at CMS. We argue that the drift tube chambers and the ME0 module of the muon system can serve as detectors to record the photons from the ALP decay, demonstrating that the resulting sensitivity reach is competitive due to their close proximity to the signal source points. We further show that the LHC does not suffer from a barrier, dubbed beamdump "ceiling", that typical beam-dump experiments hardly surpass, carrying the great potential for exploring a wide range of parameter space in increasing statistics. This analysis can be extended to investigate various types of light mediators with couplings to the Standard Model leptons and quarks. Introduction. While the existence of dark matter in the universe is clearly indicative of new physics beyond the Standard Model (SM), the experimental effort in the search for dark-matter candidates, especially, GeV-scale weakly interacting massive particles, via their hypothetical non-gravitational interactions is not yet fruitful. As an alternative possibility, MeV-scale dark-matter candidates are receiving great attention, as they can be thermally produced as well and are less constrained by the existing searches. For them to reproduce the observed dark-matter relic abundance, their interaction strengths to the SM are likely to be feeble and MeV-scale mediators often come into play. Therefore, the dark-matter search program is now promoted to a more generic darksector search program including the mediators and their feebly interacting nature motivates the intensity-frontier facilities such as beam-based neutrino experiments. By contrast, the general-purpose detectors (ATLAS and CMS) of the LHC, an energy-frontier facility, has been designed to be optimally sensitive to GeV-to-TeVscale physics. The LHC is (indirectly) capable of probing MeV-scale physics by placing additional forward-physics facilities distant from the primary interaction point (e.g., FASER [1] and SND [2]). On the contrary, the exploration of MeV-scale physics in the central region is still seemingly unpromising and irrelevant. We point out that the hadronic calorimeters (HCAL) and electromagnetic calorimeters (ECAL) of the detectors can be considered as dumps of particles inside a * [email protected] † [email protected] ‡ [email protected] jet produced by a proton collision, sourcing MeV-scale new physics particles as in typical beam-dump experiments. We then infer the production of a new physics particle when it decays to SM particles inside the muon system. The large production cross-section of jets ensures the copious production of feebly-interacting new physics particles and the large angular coverage (i.e., almost 4π coverage in solid angle) of the muon system allows for a large fiducial signal flux. Furthermore, highenergy beam protons allow new physics particles to be energetic, hence significantly boosted, and the muon system is extremely close to the calorimeters. Therefore, it is possible to access the region of parameter space toward larger couplings and larger masses (henceforth denoted by prompt-decay regime) that typical beam-dump experiments would hardly reach. We illustrate the main idea with an axion-like particle (ALP) interacting with the SM photon: −L int ⊃ 1 4 g aγγ aF µνF µν ,(1) where a and F µν (F µν ) denote the ALP field and the (dual) field strength tensor of the SM photon and where g aγγ parameterizes the interaction strength between the ALP and the SM photon. We further assume that an ALP decays to a pair of photons that leave experimental signatures at the CMS muon system. Proposal outline. The proposed measurement begins with the production of jets at the interaction point. A number of hadrons and electromagnetic particles inside a jet are essentially "dumped" and absorbed to HCAL and ECAL, respectively. They eventually end up with electromagnetic showers creating photons (plus electrons and positrons) copiously, while potentially inducing the arXiv:2305.16383v1 [hep-ph] 25 May 2023 production of secondary hadrons in the midst of absorption. These photons can undergo the Primakoff process and convert to ALPs in the presence of a non-zero g aγγ . The produced ALPs then enter a drift tube (DT) chamber [3,4] in the muon system and decay to photons therein. We also consider ME0, a projected new station in the muon system available at Run4 [4], as it may readily allow for dedicated triggers for signal detection. We will elaborate on the production of ALP events, the detection principle at DT chambers and ME0, potential backgrounds, our sensitivity study, and the trigger aspects in the next sections. Signal production. A precise estimate of the photon flux inside HCAL and ECAL is a key factor for a precise estimate of the sensitivity reach. In our study, we first utilize Pythia8.2 [5] to simulate jet production. Longlived hadrons (e.g., π ± and K ± ) and stable particles (e.g., e ± and γ) are fed, respectively, into the simplified HCAL and ECAL modules built under the GEANT4 [6] code package with the QGSP BERT physics list which simulates their subsequent interactions including electromagnetic showering. We take simplified specifications that the HCAL and ECAL modules are mostly Cu-based and PbWO 4based, respectively, and their effective average lengths are set to be 1.3 m and 0.6 m, respectively. [7] Denoting the photon energy by E γ and the angle of a photon with respect to the beam axis by θ γ , we show the E γ − θ γ correlations of the photons inside ECAL (left) and HCAL (right) that are predicted by our GEANT4 simulations in FIG. 1. We choose the photons satisfying E γ > 1 MeV and ∆R γ = (∆θ γj ) 2 + (∆ϕ γj ) 2 < 0.5 with ∆θ γj and ∆ϕ γj being the polar and azimuthal angle distances between a given photon and the center of the associated jet. By restricting photons to within the jet radius, we isolate signals from potential backgrounds arising randomly outside the jet cone. The number of photons is normalized to an integrated luminosity of 500 fb −1 at √ s = 13.6 TeV and the total numbers of photons are 1.1 × 10 19 and 9.3 × 10 18 for ECAL and HCAL, respectively. While the majority of photons are soft (≲ 10 MeV) and forward-moving, a sizable fraction of photons can be found in the central region. Also, we observe that the forward-moving photons have a tendency to be harder than the central-region photons. As mentioned earlier, a produced photon can convert to an ALP via the Primakoff process with an atomic target (denoted by N ) inside the calorimeters, i.e., γ + N → a + N , and the differential production cross-section σ P in θ a , angle of the outgoing ALP with respect to the incoming photon is given by dσ P dθ a = 1 4 g 2 aγγ α em Z 2 |F Helm (t)| 2 p 4 a sin 3 θ a t 2(2) where α em , Z, and p a are the electromagnetic fine structure constant, the atomic number of the target material, and the momentum of the outgoing ALP, and momentum transfer t = m 2 a − 2E γ (E a − p a cos θ a ). In the collinear limit where the momentum transfer to the atomic system is negligible, E a ≈ E γ and θ a → 0. We take this limit in our data analysis, i.e., the produced ALPs inherit the energy and momentum direction from the incoming photons. Finally, F Helm describes the usual Helm form factor, F Helm (t) = 3j 1 ( |t|R 1 ) |t|R 1 exp − |t|s 2 2 ,(3) where j 1 is the spherical Bessel function, s = 0.9 fm, and R 1 = (1.23A 1/3 − 0.6) 2 + 2.18 fm with A being the atomic mass number of the target material [8]. Basically, a photon may go through either the Primakoff process or the other SM processes (e.g., pair production and photoelectric absorption). Therefore, the ALP production probability P prod is given by P prod = σ P σ SM + σ P ≈ σ P σ SM ,(4) where we use the total cross-section of SM interactions σ SM reported in Ref. [9] and the approximation holds for σ SM ≫ σ P . Detection principle. Once an ALP is produced, it should first travel to a detector module of interest and decay therein. The detection probability, say P det , is essentially governed by the exponential decay law and we have P det = exp − L l 1 − exp − ∆L l ,(5) where L is the distance between the ALP production point and the detector of interest and ∆L is the travel length of the ALP within the detector of interest. Herẽ l denotes the mean decay length that can be calculated with the ALP boost factor γ a [= (1 − β 2 a ) −1/2 ] and the ALP decay width Γ a :l = β a c · γ a /Γ a with Γ a given by Γ a = 1 64π g 2 aγγ m 3 a .(6) As mentioned earlier, we consider two types of detector modules in the muon system for our sensitivity estimates: • DT chambers [3,4]: DT chambers are located in |η| < 1.2 of the barrel region. Each of them consists of 8 − 12 layers each of which is further segmented into long aluminum drift cells filled with gas. • ME0 [4]: ME0 will be introduced to the muon system in the endcap region of CMS to maintain its performance at the high-luminosity LHC. It has a 6-layer structure and is based on the Gas Electron Multiplier technology and is intended to cover the far-forward region of 2.0 < |η| < 2.8. Both modules are gas-based. Basically, gaseous detectors use the ionization, drift, and diffusion processes to amplify signals generated by charged particles passing through the gas volume. The chamber system can detect photons, although it is primarily designed for detecting muons. When an ALP enters a DT chamber, it can decay into two photons inside a drift cell. Each of them then splits into an electron-positron pair knocking electrons off the atoms of the gas. If such an electron (or positron) is energetic enough, it can pass through several layers and fire multiple drift cells, followed by getting absorbed into an iron yoke without firing the drift cells inside the next DT chamber. Therefore, the expected experimental signature is successively fired multiple drift cells belonging to a series of layers, which collectively shows a diphoton pattern aligning with the associated jet. We perform GEANT4 simulations, taking a simplified geometry of multi-layered aluminum drift cells filled with argon gas and injecting photons. The efficiency of photon detection depends on the energy of the photons. Our simulation study suggests that photons with energies ranging from a few MeV to GeV can pass through a single DT chamber and generate multiple hits in the drift cells of consecutive layers. The basic concept of signal detection at ME0 is essentially the same as that of DT chambers. The photons from an ALP decay split into electron-positron pairs which subsequently ionize the atoms of the gas and eventually result in an electron avalanche recorded by some of the finely spaced readout strips. Again if the initial electron (or positron) is energetic enough, it can pass through several layers. Therefore, the expected experimental signature is a set of fired readout strips in multiple layers, collectively forming a diphoton pattern. Background consideration. Given the detection principle, we identify two possible background sources [10]: escaping hadrons and neutron gas. Their collective rate is estimated to be ∼ 10 −8 /(dijet event) at DT chambers [10]. Although HCAL is designed to capture and absorb hadrons, a small fraction of them may escape from HCAL and continue to move into the muon system. However, these hadrons are typically energetic enough to fire the associated drift cells from the beginning of a given DT chamber to the end and even multiple DT chambers. Likewise, they fire the associated readout strips in every layer of ME0. The experimental signatures of this sort are clearly distinguished from the signatures that typical ALP events would leave. Therefore, given the small escape rate and the difference in the detector signatures, we expect the escaping-hadron-induced backgrounds to be negligible. Neutron gas originates from radioactivated detector components and such neutrons can induce spontaneous photons in a temporally and spatially random manner that can further fire DT chambers or ME0. Since the typical energy of these photons is within 0.1 − 10 MeV [10], it is challenging for them to fire more than a single drift cell in the detector modules of our interest. A neutroninduced photon of a few MeV can fire a handful of drift cells, but it is hard for it to accidentally create a diphoton pattern in association with the jet of interest. Since expected signal events can fire multiple drift cells and create an outward diphoton pattern, we expect that the neutron-gas-induced backgrounds can be significantly suppressed to a negligible level. Beam-dump "ceiling" and the LHC potential. Before presenting our main results, we discuss the general features of sensitivity reaches in the beam-dump-type experiments and the great potential of the LHC, especially focusing on the prompt-decay regime. Typically in this regime, L, ∆L ≫l and Eq. (5) is approximated to P det ≈ exp(−L/l). [11] Suppose that N γ photons set the sensitivity at g aγγ with N sig ALP events. [12] If x-times larger data acquisition were expected, the new reach g ′ aγγ under a zero-background assumption would be g ′ aγγ ≈ g aγγ 1 + log x log(N γ ⟨P prod ⟩/N sig ) − 1 ,(7) where ⟨P prod ⟩ denotes the average value of P prod at g aγγ ; N γ ⟨P prod ⟩ ≫ N sig in typical situations. For a sufficiently large background, log x is replaced with log x 1/2 . This implies that the sensitivity is effectively no longer improved even with much larger statistics due to the logarithmic behavior shown above. [13] Therefore, most of the beam-dump experiments encounter this "ceiling" in probing the prompt-decay regime. This argument is generically relevant to decay channels of long-lived particles (e.g., dark photon, Higgs-portal scalar, etc). However, this does not hold for the proposed beamdump measurements at the LHC. The reason is that as we will show shortly, for the (m a , g aγγ ) pairs defining the sensitivity lines in the prompt-decay regime, the L ≫l(∝ γ a ) condition is hardly satisfied; L ∼ O(1−2 m) and typical γ a is very large under the LHC environment. Therefore, we expect that our proposal does not suffer from the aforementioned beam-dump "ceiling" and possesses great potential for exploring a wide range of the prompt-decay regime in increasing statistics. This essentially motivates the development of dedicated trigger algorithms to maximize signal statistics. Results. We are now in the position to report our sensitivity estimates. Four different scenarios are considered here: i) Run2 (150 fb −1 ) with the zero-bias trigger [14], ii) Run2+3 (150 + 250 fb −1 ) with all high-level trigger [15], iii) Run4 (500 fb −1 ) with a dedicated trigger, and iv) Run4 (500 fb −1 ) at ME0 with a dedicated trigger. Integrated luminosities of 250 fb −1 during Run3 and 500 fb −1 during Run4 are expected in 2025 and 2030, respectively [16]. In i)-iii), DT chambers are the main detection modules while in iv) only ME0 is considered. Regarding the trigger, in i), we conservatively consider events collected by the existing zero-bias trigger with a prescale factor of 1.1 × 10 −6 [14]. In ii), we consider all highlevel triggers most of which involve energetic jets and/or electromagnetic objects. Our GEANT study suggests that the spectral behavior of shower photons is similar modulo the overall normalization, so we assume an effective zero-bias trigger with a prescale factor enhanced by an order of magnitude that corresponds to the total highlevel trigger rate 1 kHz. Finally, in iii) and iv), we assume that dedicated trigger algorithms are developed to maximize the acceptance of the signal flux. For all scenarios, we further restrict ourselves to ALPs converted from photons whose energy is greater than 5 MeV to ensure hits in multiple layers of the detection modules (i.e., DT chambers and ME0). Regarding the photon detection efficiency, based on the aforementioned GEANT study, we assume idealized detection modules for simplicity, but our final results are not appreciably affected by moderate degradation (i.e., a few factors) of detection efficiency. The 90% C.L. sensitivity reaches of the above-listed four scenarios are displayed in FIG. 2 in the (m a , g aγγ ) plane. Since we expect negligible backgrounds, we require the number of signal events to be 2.3 on the basis of the expected statistical error only. We also show the current constraints from various existing experiments (e.g., e + + e − → γ + inv. [18][19][20][21][22], Belle-II [23], CCM [24], CHARM [25], E137 [26], E141 [27], LEP [28], NA64 [29], νCal [30], and PrimEx [31]), based on the limits compiled in e.g., Refs. [32][33][34]. The gray-colored region is based on the laboratory-produced ALP searches, and the boundaries are set by various lepton collider and beamdump experiments. By contrast, the regions constrained by astrophysical considerations [35][36][37][38][39] (e.g., HB stars, supernova, etc.) are shown in yellow. [17] Our sensitivity study suggests that CMS can explore the regions beyond the current beam-dump limits which none of the existing (in)direct experiments have ever probed because ALPs belonging here would decay too promptly to reach their detector system. Note that the detector modules of interest, i.e., DT chambers and ME0, are within just a few meters from the signal production points, i.e., HCAL and ECAL. Moreover, due to the high-energy nature of the LHC, a sizable fraction of high-energy photons are produced. The ALPs from such photons are significantly boosted and thus they can be long-lived enough to reach the closely located detector modules despite fairly large values of m a and g aγγ . It is remarkable that the existing Run2 data collected with the zero-bias trigger (red line) allows for sensitivity into the unexplored regions of parameter space. Adding [18][19][20][21][22][23][24][25][26][27][28][29][30][31] are shown in gray, while astrophysics-based limits [35][36][37][38][39] are shown in yellow. Run3 data with all high-level triggers (blue line) further extends the regions that can be probed. Regarding Run4, as mentioned earlier, we assume that dedicated trigger algorithms are implemented to capture the full signal flux although designing the triggers is beyond the scope of this letter. As expected, larger luminosity and acceptance significantly (green line) improve the sensitivity reaches and expand explorable regions, not suffering from the beamdump "ceiling" as predicted earlier. Finally, Run4 data collected by ME0 (brown line) allows us to explore the large-mass regime above m a ∼ 1 GeV. The ME0 module will be placed in the endcap region so that it will accept more energetic signals stemming from higher-energetic photons, as also suggested by the energy-angle correlations shown in FIG. 1. We further emphasize that the smallness of the ME0 module facilitates the implementation of dedicated triggers as they are less likely to affect the total trigger rate. Discussion and conclusions. In this letter, we proposed a paradigm-shifting idea about performing a beamdump-like measurement at the two LHC general-purpose detectors, ATLAS and CMS. Under the proposed measurement scheme, the two calorimeters, HCAL and ECAL, can behave as dumps to which hadrons and electromagnetic objects inside jets produced by high-energy proton collisions are dumped. An enormous number of secondary particles including photons are produced therein while the aforementioned jet constituents are being absorbed. For illustration purposes, we investigated ALP production via the Primakoff process of the secondary photons and detection prospects at the muon system of CMS. In this regard, we carefully assessed the feasibility of DT chambers and ME0 as signal "detectors", identifying the expected experimental signatures and related potential backgrounds. Our sensitivity study sug-gests that CMS is capable of probing a broad range of unexplored parameter space even with the existing Run2 data collected with the zero-bias trigger. Our study is predicated upon a few simple assumptions and approximations. While we performed a simplified GEANT-based simulation study to assess the possibility to utilize DT chambers and ME0 as signal detectors, a more dedicated simulation study in line with careful hardware-level consideration is highly encouraged to assess the feasibility of the proposed "beam-dump" measurement at the LHC and identify and estimate related backgrounds more precisely. Along the line, we expect that signal-background identification will benefit from the machine-learning-based techniques, as it is deeply related to pattern recognition. A trigger is another important ingredient to enhance the signal rate to be recorded. In preparation for future Run4, we strongly encourage experimental collaborations to develop dedicated trigger algorithms. Finally, we emphasize that similar ideas are applicable to other new physics scenarios, e.g., ALPs interacting with leptons and models of dark photons decaying into leptons which we plan to investigate in a forthcoming publication. Also, this analysis in the context of light mediators (vectors and scalars) can be extended to investigate the parameter spaces associated with various anomalies, e.g., ATOMKI [41], g µ − 2 [42], etc. Using this new scheme, we hope that ATLAS and CMS will discover new physics in the near future. ACKNOWLEDGMENTS We thank Carlo Battilana, Francesca Romana Cavallo, Karri DiPetrillo, Teruki Kamon, and Jason Sang Hun Lee for useful discussions. This article was supported by the computing resources of the Global Science Experimental Data Hub Center (GSDC) at the Korea Institute of Science and Technology Information (KISTI). The work of BD, DK, and HK is supported by the U.S. Department of Energy Grant DE-SC0010813. SUPPLEMENT Sensitivity in the delayed-decay regime. If the (laboratory-frame) lifetime of a produced ALP is long enough, L, ∆L ≪l holds. The detection probability of the ALP traveling toward the detection module of interest is approximated to P det ≈ ∆L l ∝ g 2 aγγ .(8) Since the ALP production rate is also proportional to g 2 aγγ for any small g aγγ , if a x-times larger data collection were expected, the new reach g ′ aγγ would be approximately g ′ aγγ ≈      g aγγ x 1/4 for zero backgrounds, g aγγ x 1/8 for large backgrounds, where we determine N sig , which is required to have the sensitivity, with statistical error only. Therefore, unlike the case of L, ∆L ≫l, the increase of data allows to continue the exploration of more parameter space toward the delayed-decay regime, i.e., smaller m a and g aγγ . FIG. 1 . 1Energy-angle (Eγ − θγ) correlations of the photons inside ECAL (left) and HCAL (right) satisfying Eγ > 1 MeV and ∆Rγ < 0.5. The number of photons is normalized to an integrated luminosity of 500 fb −1 at √ s = 13.6 TeV. See the text for more details. FIG. 2 . 2Expected 90% C.L. sensitivity reaches of the ALP decay signal at the detector modules of interest at CMS, under the assumption of negligible backgrounds. Existing laboratory-based limits Technical Proposal for FASER: ForwArd Search ExpeRiment at the LHC. A Ariga, arXiv:1812.09139FASER). physics.ins-detA. Ariga et al. (FASER), Technical Proposal for FASER: ForwArd Search ExpeRiment at the LHC, (2018), arXiv:1812.09139 [physics.ins-det]. G Acampora, arXiv:2210.02784SND@LHC), SND@LHC: The Scattering and Neutrino Detector at the LHC, (2022). hep-exG. Acampora et al. (SND@LHC), SND@LHC: The Scat- tering and Neutrino Detector at the LHC, (2022), arXiv:2210.02784 [hep-ex]. . CMS, the Compact Muon Solenoid. 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[ "Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor", "Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor" ]
[ "Naman K Gupta \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n", "Ronny Sutarto \nCanadian Light Source\nUniversity of Saskatchewan\nS7N 2V3SaskatoonSaskatchewanCanada\n", "Rantong Gong \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n", "Stefan Idziak \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n", "Hiruy Hale \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n", "Young-June Kim \nDepartment of Physics\nUniversity of Toronto\nM5S 1A7TorontoCanada\n", "David G Hawthorn \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n" ]
[ "Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada", "Canadian Light Source\nUniversity of Saskatchewan\nS7N 2V3SaskatoonSaskatchewanCanada", "Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada", "Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada", "Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada", "Department of Physics\nUniversity of Toronto\nM5S 1A7TorontoCanada", "Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada" ]
[]
Unidirectional spin and charge density wave order in the cuprates is known to compete with superconductivity. In the stripe order (La,M)2CuO4 family of cuprates, spin and charge order occur as unidirectional order that can be stabilized by symmetry breaking structural distortions, such as the low temperature tetragonal (LTT) phase. Here we examine the interplay between structure and the formation of charge density wave (CDW) order in the LTT phase of La1.475Nd0.4Sr0.125CuO4 by applying uniaxial stress to distort the structure and influence the formation of CDW order. Using resonant soft x-ray scattering to measure both the CDW order and (0 0 1) structural-nematic Bragg peaks, we find that the application of uniaxial stress along the Cu-O bond direction suppresses the (0 0 1) peak and has the net effect of reducing CDW order, but does so only for CDW order propagating parallel to the applied stress. We connect these observations to previous work showing an enhanced superconducting transition temperature under uniaxial stress; providing insight into how CDW, superconductivity, nematicity, and structure are related and can be tuned relative to one another in cuprates.
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[ "https://export.arxiv.org/pdf/2305.16499v1.pdf" ]
258,947,740
2305.16499
2464622f1ea566de05f90432ff3adc05a3f9f97c
Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor 25 May 2023 Naman K Gupta Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Ronny Sutarto Canadian Light Source University of Saskatchewan S7N 2V3SaskatoonSaskatchewanCanada Rantong Gong Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Stefan Idziak Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Hiruy Hale Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Young-June Kim Department of Physics University of Toronto M5S 1A7TorontoCanada David G Hawthorn Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor 25 May 2023 Unidirectional spin and charge density wave order in the cuprates is known to compete with superconductivity. In the stripe order (La,M)2CuO4 family of cuprates, spin and charge order occur as unidirectional order that can be stabilized by symmetry breaking structural distortions, such as the low temperature tetragonal (LTT) phase. Here we examine the interplay between structure and the formation of charge density wave (CDW) order in the LTT phase of La1.475Nd0.4Sr0.125CuO4 by applying uniaxial stress to distort the structure and influence the formation of CDW order. Using resonant soft x-ray scattering to measure both the CDW order and (0 0 1) structural-nematic Bragg peaks, we find that the application of uniaxial stress along the Cu-O bond direction suppresses the (0 0 1) peak and has the net effect of reducing CDW order, but does so only for CDW order propagating parallel to the applied stress. We connect these observations to previous work showing an enhanced superconducting transition temperature under uniaxial stress; providing insight into how CDW, superconductivity, nematicity, and structure are related and can be tuned relative to one another in cuprates. In the cuprate superconductors, superconductivity, spin and charge density wave orders [1][2][3], and electronic nematicity [4][5][6][7][8] are intertwined, often co-existing and competing [9][10][11]. How these orders manifest and relate to each other depends on the crystalline structure (tetragonal or orthorhombic, single layer or bilayer,..) and symmetries that result from the structure [12]. A powerful avenue to explore and tune these relationships is via applying uniaxial stress to controllably manipulate the structure and examine the response of key properties related to superconductivity present in different families of cuprate superconductors. Application of uniaxial stress in the cuprates has been shown to result in significant changes in the superconducting transition temperature and charge density wave order [13][14][15][16][17][18][19][20]. In the stripe-ordered cuprate, La 1.74 Eu 0.2 Sr 0.16 CuO 4 , Takeshita et al. showed that uniaxial stress applied in the ab plane enhances the superconducting transition temperature, T C [13]. More recent work in La 1.885 Ba 0.115 CuO 4 (LBCO), showed a similar enhancement of T C , along with an accompanying suppression of magnetic order, as measured from µSR [15,20]. This result is indicative of a competition between magnetic order and superconductivity that can be tuned by uniaxial stress. Other studies have examined the impact of uniaxial stress on charge density wave order. In YBa 2 Cu 3 O 6+x (YBCO), ∼1% compressive uniaxial strain enhances both 2D CDW order and results in a unidirectional 3D CDW order, characterized by long in-plane and c-axis correlation lengths along with a strong competition with superconductivity [16,17]. In La 1.875 Sr 0.125 CuO 4 (LSCO), uniaxial stress was found to enhance CDW order propagating perpendicular to the applied stress and sup-press CDW order propagating parallel to the applied stress [18,21]. How uniaxial stress impacts CDW order depends on the structure of the crystal. For instance, in LSCO, the structure is in the low-temperature orthorhombic ( [22,23]. Consequently, the application strain along [100] was interpreted to detwin the CDW order to favour only domains of unidirectional CDW order propagating along [100] for stress applied along [010] [18]. However, in many stripe-ordered cuprates, the application of strain may play a different role than in LSCO. In the (La,M) 2 CuO 4 , substituted with larger rare earth ions Nd, Eu or Ba, a low temperature tetragonal (LTT) phase can be induced that is characterized by octahedral tilts parallel to the Cu-O bond [24,25]. Notably, it is in these materials that spin and charge stripe order was first observed [1] and where, in 1/8th doped LBCO, stripe order is found to be most robust and compete strongly with superconductivity [26]. Early on it was recognized that there was a strong relationship between the LTT structure and stabilizing stripe order, with several compounds exhibiting stripe order to onset at the LTO to LTT phase transition [1,12,26]. The reason for this association is that in these compounds, the LTT structure, shown in Figure 1A and B, induces anisotropy in the electronic structure, such as the hopping between Cu and O (t pd ), the nearest neighbour exchange interaction, J, or the charge transfer energy ∆ pd . [ 1. A The crystal structure of (La,M)2CuO4 in the high temperature tetragonal (HTT) phase. Two CuO2 planes at z = 0 and z = 0.5 are present within a single unit cell. B In the low temperature tetragonal (LTT) phase CuO6 octahedra tilt along the Cu-O bond direction, with the tilt axis alternating between a and b for neighbouring layers. Uniaxial stress along a will be parallel to the tilt axis for half the layers and perpendicular to the tilt axis for the other half of the layers. C The orientation of unidirectional stripe order alternates between neighbouring layers. Uniaxial stress along a will be parallel or perpendicular to the CDW propagation wavevector, depending on the layer. D Schematic of the uniaxial stress device. Rotation of a differential screw is used to apply uniaxial stress to a sample that is mounted to a BeCu plate. The device can be actuated in vacuum and at low temperatures in order to change the applied stress or to rotate the device azimuthally to orient the strain parallel or perpendicular to the horizontal scattering plane of the diffractometer. The sample shown in the inset is set in position using silver epoxy. this anisotropy that is thought to stabilize unidirectional stripe order that within an individual layer is oriented parallel or perpendicular to the octahedral tilt axis. Importantly, although each individual layer breaks C 4 rotational symmetry due to the octahedral tilts, the axis by which the octahedra tilt rotates by 90 • between neighbouring CuO 2 planes such that the crystal structure remains globally tetragonal, as depicted in Figure 1B. Consequently, unidirectional spin and charge order in neighbouring layers is alternates between propagating along [100] and [010] [1,28], as depicted in Figure 1C. Unlike in LSCO without Nd, Eu or Ba substitution, the structural distortion of LTT phase serves to detwin the CDW order within an individual layer. Consequently, the application of modest uniaxial stress may be used to explore the role of modifying the anisotropy of the electronic structure on CDW order, rather than the balance of [100] and [010] domains within a layer. As shown in Figure 1(A-C), the application of a uniaxial stress along the [100] direction will act along the tilt axis for half of the layers and perpendicular to the tilt axis for the other half of the layers, affecting the anisotropy of the electronic structure differently for the two orientations of layers. The conjugate (strain) field provides a local as well as a global orientating potential to the free energy stabilizing the electronic symmetry breaking phases. In this study we utilize resonant soft x-ray scatter-ing to study the impact of uniaxial strain on both the structural phase transition and the CDW order, in La 1.475 Nd 0.4 Sr 0.125 CuO 4 (LNSCO). We find that the application of strain along [100] reduces the intensity of the CDW peak at Q CDW = (-0.24 0 1.5), propagating parallel to applied stress, by a factor of ∼ 2, while having little impact on its correlation length or temperature dependence. In contrast, the applied strain has only modest impact on the intensity of the CDW peak at (0 -0.24 1.5), propagating perpendicular to applied stress. The overall suppression of CDW order due to uniaxial stress is consistent with competition between CDW order and superconductivity, namely that uniaxial stress along [100] modifies the anisotropy of the electronic structure in a manner that suppresses CDW order, which in turn enables the enhancement of the superconducting transition temperature [13,15,20]. Resonant x-ray scattering measurements were performed at the REIXS beamline at the Canadian Light Source [29], on a sample of La 1.475 Nd 0.4 Sr 0.125 CuO 4 , fragments of which have been studied in past reports of x-ray absorption spectroscopy [30] and resonant x-ray scattering [6,31]. The CDW (-0.24 0 1.5) and (0 -0.24 1.5) Bragg peaks were investigated at a photon energy corresponding to the peak of the Cu-L 3 absorption edge (931.3 eV). Note, measurements of the CDW peaks are shown as raw data, normalized only to the incident beam . Scattering intensity is normalized at base temperature (20 K). The inset shows a theta-2theta scan of the (0 0 1) Bragg peak on-resonance at the apical O-K edge, comparing the strained (red and green) and the unstrained (black) case. Notably, the peak intensity drops upon applying uniaxial stress. intensity, but otherwise free of any background subtraction. In addition, to examine the LTO to LTT structural phase transition, we measured the (0 0 1) Bragg peak beak at the Cu-L edge (931.3 eV) and at an energy associated with the apical oxygen (533.3 eV). The sample was orientated by measuring the (0 0 4), (-1 0 3) and (0 -1 3) structural Bragg peaks at 2200 eV, with all Bragg peaks indexed to the high temperature tetragonal phase. The sample was cut and polished to form a rectangular ab face sample of dimensions 2 x 0.5 x 0.2 mm. Unixial stress was applied parallel to the longest axis of the sample, which is oriented parallel to [100]. Subsequent to polishing, the sample's surface was etched with bromine. The application of uniaxial stress was applied by mounting the sample on a custom mechanically-actuated stage, depicted in Figure 1D. The sample is mounted on a BeCu plate that has been machined with notch flexures to enable motion along a single axis, but provide a degree of stiffness to bending or tilting. The sample is positioned in a groove in the BeCu plate, with the edges encapsulated above and below with EPO-TEK H2OE silver epoxy in an effort to apply uniform stress. The BeCu plate is affixed to two Cu blocks that can be translated relative to one another by actuating a differential screw that enables small controlled displacements between the plates, resulting in uniaxial compressive or tensile stress on the sample. A novel feature of the strain device is the capability to rotate the device azimuthally under stress, in-vacuum and at low temperature in order to orient the direction of the applied stress either parallel or perpendicular to the scattering plane of the diffractometer. As such, stress can be applied to the sample along the a-axis and CDW peaks can be investigated along both (-0.24 0 1.5) and (0 -0.24 1.5) without needing to thermally cycle or remount the sample. Unfortunately, our device does not measure the magnitude of the applied stress. However, by measuring the structural Bragg peaks before and after the application of uniaxial stress, we can identify that the peak positions do not shift by amounts larger than the accuracy of our measurements. This limits the in-plane (or caxis) compressive strain to be less than 0.2%. Given the C 11 = 267 GPa in related materials [32], this strain limit corresponds to a uniaxial stress along [100] of < 0.5 GPa. As shown in Figure 2, the temperature dependence of the (0 0 1) peaks measured at the apical O-K edge, exhibits a sharp transition to the LTT phase at 70 K, consistent with previous unstrained measurements on this sample [6,33]. The application of stress along [100] results in a decrease in the (0 0 1) peak intensity at low temperatures, as well as, a more gradual temperature dependence between 70 K and 40 K, with a plateau occurring below 40 K. At this photon energy, the (0 0 1) peak results from a difference in the orbital symmetry of apical O atoms between neighbouring (La,M) 2 O 2 layers [6,34,35]. Consequently, the more gradual temperature dependence of the (0 0 1) peak under applied stress is understood to be associated with a reduction in the difference in orbital symmetry between neighbouring layers. This reduction may result from changes in the angles by which CuO 6 octahedra tilt out of the ab plane and/or changes in the octahedral tilt axis away from the Cu-O bond direction. Notably, the strain dependence of the (0 0 1) Bragg peak in Figure 2 is in qualitative agreement with recent reports hard x-ray scattering study of the LTO-LTT transition under unaxial stress in another stripeordered cuprate La 1.885 Ba 0.115 CuO 4 [20]. In that material, Guguchia et al. show the LTT transition to be completely suppressed for compressive uniaxial stress above ∼0.06 GPa. However, for lower stress values (∼0.017 GPa), the LTT phase remains, but with the LTT Bragg peak intensity suppressed, the onset temperature is only weakly dependent on stress, and the LTO-LTT transition is broadened [20], qualitatively similar to the dependence on uniaxial stress dependence of the (0 0 1) we observe in LNSCO for the range of pressure applied. Regarding CDW order, in the unstrained situation, we observe comparable peak intensities for the (-0.24 0 1.5) and (0 -0.24 1.5) peaks. As shown in Figure 3, the response of CDW order to uniaxial stress is asymmetric -having a different impact of the (-0.24 0 1.5) and (0 -0.24 1.5) peaks. For the largest strain applied, the (-0.24 0 1.5) Bragg peak, corresponding to CDW order . A) and B) Intensity at 20 K and at 90 K, above the CDW ordering temperature, for the unstrained and strained case. In A) measurements are shown through the Bragg peak (-0.24 0 1.5) peak position, corresponding to CDW order propagating parallel to the applied stress. In B measurements are shown through the Bragg peak (0 -0.24 1.5) peak position, corresponding to CDW order propagating perpendicular to the applied stress. C) Measurements through (-0.24 0 1.5) in the unstrained configuration, before (UNS 1) and after releasing (UNS 2) applied stress to the sample. D) The temperature dependence at (-0.24 0 1.5) with and without strain. The onset temperature of the CDW order is similar in the strained and unstrained configurations. All panels show raw data, only normalized to incident beam intensity, free from any background subtractions. propagating parallel to the applied stress, is reduced in intensity by a factor of ∼ 2 at 20 K. However, as shown in Figure 3B, this same strain has minimal to no impact on the intensity of the (0 -0.24 1.5) Bragg peak, corresponding to CDW order propagating perpendicular to the applied stress. Moreover, although the intensity of the (-0.24 0 1.5) decreases with the application of stress, the width of the Bragg peak, associated with the CDW correlation length along the a-axis, is unaffected. In both cases, the application of uniaxial stress does not change the background measured at 90 K, which is dominated by x-ray fluorescence from the sample. The temperature dependence of the intensity at (-0.24 0 1.5) in strained and unstrained configurations is shown in Figure 3D. This shows that the onset temperature is not strongly impacted by strain, with the main impact of strain being the reduction in peak intensity in the region where CDW order exists. This reduction in intensity without a substantive change in the CDW order onset temperature matches well with how the spin order measured by muon spin rotation responds to compressive uniaxial stress in LBCO [20]. To verify that the strain-induced suppression in the (-0.24 0 1.5) peak intensity is not associated with an irreversible degradation of the sample (cracking, buckling, mosaic domains...) upon straining, we released the strain and remeasured the sample in an unstrained configuration (UNS 2). These measurements show a recovery of the original CDW peak intensity (see UNS 1 in Figure 3C), as well as a recovery of the (0 0 1) peak temperature dependence. Notably, our results with uniaxial compressive stress differ from reported measurements in LNSCO with uniaxial tensile stress by Boyle et al. [19]. They also observe a reduction in the CDW peak intensity at low T under strain. However, where we find T LT T and T CDW are unchanged by the compressive strain we applied, they report that a tensile strain of ε a = +0.046 ± 0.026% reduced both T CDW and T LT T by 29 K. Moreover, they do not observe an appreciable broadening of the LTT transition or reduction in the (0 0 1) peak intensity at low temperatures, as we observed in our study. This may indicate a marked difference in the impact of tensile versus compressive strain or the magnitudes of strain applied, warranting further investigation. The compressive strain dependence observed in our study is also in contrast to that of reports in LSCO, where CDW order occurs within the LTO structural phase. In LSCO, CDW propagating along the a and b axes would be degenerate in an unstrained crystal, giving rise to domains of unidirectional CDW order that run along both a and b within an individual layer [18]. As such, even a small uniaxial strain along a or b can break the degeneracy, detwinning the CDW such that only a or b oriented CDW order occur, consistent with the measurements of Choi et al. [18]. Notably, in LSCO application of strain beyond that required to detwin the CDW order did not further enhance the CDW order [18]. Connecting the observations to our results in LNSCO, the application of uniaxial strain here does not have a similar effect of detwinning CDW order -enhancing CDW order along b and suppressing CDW order along a. This is likely due to octahedral tilts along a and b in the LTT structure already being effective at detwinning the CDW order within individual layers. As such, the strain dependence CDW order observed here is likely associated with how anisotropic strain impacts the in-plane anisotropy of the electronic structure, such as the nearest-neighbour hopping and exchange interactions. This electronic anisotropy, in turn, can affect the amplitude of CDW order. Structural refinements of CDW order in the LTT phase of LBCO have identified that CDW order within an individual layer propagates along a direction parallel to the bent O-Cu-O bonds, as opposed to the straight O-Cu-O bonds, as depicted in Figure 1 [28]. While it is not clear how the electronic anisotropy of the individual layers changes in response to anisotropic strain, one might expect that the electronic anisotropy increases for the layers with the applied stress parallel to the straight O-Cu-O bonds and decreases for layers with the applied stress perpendicular to the straight O-Cu-O bonds. If this is indeed the case, our findings of a suppression of CDW order along H (parallel to the applied stress) indicate that a reduction in electronic anisotropy of a layer results in a reduction of CDW order, but the converse need not be the case. Ultimately, the net effect of applying uniaxial stress along the Cu-O bond direction is to suppress CDW order. This is in accordance with observed increases in the superconducting transition temperature by applying stress in the [100] direction in related systems La [15,20] that share the LTT structure with LNSCO. As such, at least part of the increase in T c may be attributed to the competition of CDW order with superconductivity. Namely, uniaxial stress affects the anisotropy of the electronic structure in a manner that suppresses CDW order, resulting in an enhancement of superconductivity. In addition to investigating CDW order, we also probe the temperature dependence of the Q x = Q y = 0 electronic anisotropy by measuring the temperature dependence of the (0 0 1) Bragg peak under uniaxial strain at the Cu-L edge, in addition to O-K edge (at an energy corresponding to apical oxygen). As shown in Figure 4, the temperature dependence of the (0 0 1) peak inten-sity is different and more gradual when measured at the Cu L relative to the O-K edge, consistent with previous studies without strain [6,33]. However, uniaxial stress appears to eliminate this difference in the T dependence. Achkar et al. [6] argued that this difference in temperature dependence is due to electronic nematic order that is coupled to CDW order and is in addition electronic asymmetry that directly results from the structure distortions. Moreover, although the (0 0 1) peak measures the difference in orbital symmetry between neighbouring layers, in the unstrained case the symmetry of the crystal structure is such that this difference in the interlayer orbital asymmetry maps to a measure of nematicity within a single layer. However, because of the inequivalence of neighbouring layers under anisotropic strain, this mapping is not valid, complicating the interpretation of the (0 0 1) peak T -dependence under uniaxial stress. The lack of a difference in the T -dependence between Cu-L and apical O-K measurements may indicate that uniaxial stress suppresses electronic nematic order within individual layers. However, it may also indicate that nematic order remains strong and is perhaps saturated. In this scenario, uniaxial stress may align electronic nematic order between neighbouring layers, such that signatures of it cancel in measurements of the (0 0 1) Bragg peak. Future work will be required to differentiate the origin of this anomalous result. In conclusion, we find that in stripe-ordered La 1.475 Nd 0.4 Sr 0.125 CuO 4 , uniaxial stress along [100] suppresses the magnitude of CDW order parallel to the applied stress, but has little impact on CDW order propagating perpendicular to the applied stress. We attribute this suppression to modification of the electronic asymmetry within the CuO 2 planes rather than a detwinning of CDW order, seen in LSCO. Signatures of nematic order, as observed via the relative temperature dependence of the (0 0 1) peak at the Cu-L edge and O-K edge, are also observed to be suppressed by uniaxial strain. This suppression of CDW order is likely linked to the enhancement of superconductivity under uniaxial stress along [100] direction. We gratefully acknowledge Christopher Keegan, Erica Carlson, Steve Kivelson, and Louis Taillefer for useful discussions. We also acknowledge assistance from Rafael Mirabal and Michael Boulaine during the sample etching process. This research was supported by the Natural Sciences and Engineering Research Council (NSERC). N.K. Gupta acknowledges support from the Waterloo Institute of Nanotechnology (WIN). Part of the research described in this paper was performed at the Canadian Light Source, a national research facility of the University of Saskatchewan, which is supported by the Canada Foundation for Innovation (CFI), the Natural Sciences and Engineering Research Council (NSERC), the National Research Council Canada (NRC), the Canadian Institutes of Health Research (CIHR), the Government FIG. 2 . 2Temperature dependence of the (0 0 1) Bragg peak at apical O-K edge under varying strain (increasing as shown by an arrow) FIG. 3 . 3Response of CDW Bragg peaks to applied uniaxial stress along [100] FIG. 4 . 4Temperature dependence of the structural-nematic (0 0 1) Bragg peaks for planar and apical atoms. A Temperature evolution of the theta-2theta scans of the (0 0 1) Bragg peak at the Cu-L edge under in-plane, compressive uniaxial strain. Raw data, only normalized to incident beam intensity, free from any background subtractions. B Comparison of temperature dependence of the (0 0 1) peak measured at the Cu-L (931.3 eV) and apical O-K edge (533.3 eV) energies for unstrained and strained case. Scattering intensity is normalized at base temperature(20 K). The temperature evolution of the (0 0 1) peak at the two energies differs in the unstrained case,[36], but behaves similarly under uniaxial stress. LTO) phase, characterized by tilts of CuO 6 octahedra about an axis diagonal to the Cu-O bond. 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[ "Study experimental time resolution limits of recent ASICs at Weeroc with different SiPMs and scintillators", "Study experimental time resolution limits of recent ASICs at Weeroc with different SiPMs and scintillators" ]
[ "Tasneem Saleem [email protected] ", "Salleh Ahmad ", "Jean-Baptiste Cizel ", "Christophe De ", "La Taille ", "Maxime Morenas ", "Vanessa Nadig ", "Florent Perez ", "Volkmar Schulz ", "Stefan Gundacker ", "Julien Fleury ", "\nDepartment of Physics of Molecular Imaging Systems\nOmega laboratory\nEcole Polytechnique CNRS-IN2P3\n4 avenue de la baltique, BP 51591140Weeroc, Villebon sur Yvette, PalaiseauFrance, France\n", "\nRWTH Aachen University\nPauwelsstrasse 1752074AachenGermany\n" ]
[ "Department of Physics of Molecular Imaging Systems\nOmega laboratory\nEcole Polytechnique CNRS-IN2P3\n4 avenue de la baltique, BP 51591140Weeroc, Villebon sur Yvette, PalaiseauFrance, France", "RWTH Aachen University\nPauwelsstrasse 1752074AachenGermany" ]
[]
Medical applications, such as positron emission tomography (PET), and space application, such as Light Detection and Ranging (LIDAR), are in need for highly specialized ASICs. Weeroc, in collaboration with different partners, is highly involved in developing a new generation of front-end ASICs. In the context of a joined LIDAR project among Weeroc, CNES, and Airbus, Weeroc is working on the development of Liroc, an ASIC for space LIDAR application. Weeroc is also working on advancing ASICs for medical applications, hence, another ASIC, Radioroc, is under development and intended to be used for PET applications.This study experimentally evaluates the time resolution limits of these ASICs in different configurations, with some of the most recent silicon photomultipliers (SiPMs) technologies available on the market coupled with different scintillation crystals. The best single-photon time resolution (SPTR) was achieved using FBK NUV-HD SiPMs with an FWHM of 79 ps with Liroc and 73 ps with Radioroc. Furthermore, the best coincidence time resolution (CTR) of Radioroc was determined to 83 ps (FWHM) with Broadcom Near UltraViolet -Metal Trench (NUV-MT) SiPMs coupled to TAC LYSO:Ce,Ca (2×2×3 mm 3 ).
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[ "https://export.arxiv.org/pdf/2305.14855v2.pdf" ]
258,947,826
2305.14855
61a36e57723c00a75027b6cd7293ffbe7a8d47b2
Study experimental time resolution limits of recent ASICs at Weeroc with different SiPMs and scintillators Tasneem Saleem [email protected] Salleh Ahmad Jean-Baptiste Cizel Christophe De La Taille Maxime Morenas Vanessa Nadig Florent Perez Volkmar Schulz Stefan Gundacker Julien Fleury Department of Physics of Molecular Imaging Systems Omega laboratory Ecole Polytechnique CNRS-IN2P3 4 avenue de la baltique, BP 51591140Weeroc, Villebon sur Yvette, PalaiseauFrance, France RWTH Aachen University Pauwelsstrasse 1752074AachenGermany Study experimental time resolution limits of recent ASICs at Weeroc with different SiPMs and scintillators Prepared for submission to JINSTfront-end electronics for detector readouttiming detectorphoton detectors for UV, visible and IR photons (solid-state) Si-PMTsscintillators, scintillation, and light emission processes (solid, gas and liquid scintillators)on-board space electronics, space instrumentationgamma camera, SPECT, PET PET/CT, coronary CT angiography (CTA) Medical applications, such as positron emission tomography (PET), and space application, such as Light Detection and Ranging (LIDAR), are in need for highly specialized ASICs. Weeroc, in collaboration with different partners, is highly involved in developing a new generation of front-end ASICs. In the context of a joined LIDAR project among Weeroc, CNES, and Airbus, Weeroc is working on the development of Liroc, an ASIC for space LIDAR application. Weeroc is also working on advancing ASICs for medical applications, hence, another ASIC, Radioroc, is under development and intended to be used for PET applications.This study experimentally evaluates the time resolution limits of these ASICs in different configurations, with some of the most recent silicon photomultipliers (SiPMs) technologies available on the market coupled with different scintillation crystals. The best single-photon time resolution (SPTR) was achieved using FBK NUV-HD SiPMs with an FWHM of 79 ps with Liroc and 73 ps with Radioroc. Furthermore, the best coincidence time resolution (CTR) of Radioroc was determined to 83 ps (FWHM) with Broadcom Near UltraViolet -Metal Trench (NUV-MT) SiPMs coupled to TAC LYSO:Ce,Ca (2×2×3 mm 3 ). Introduction Solid-state photo-detectors like silicon photomultipliers (SiPMs) are playing an important role in the digitization of optical photons in medical imaging, life sciences, and high-energy physics. The use of high-performance application-specific integrated circuits (ASICs) has been a key element for getting the best performance of SiPMs and has allowed their successful use in large detectors. Novel high-performance readout electronics need to be evaluated regarding their single-photon time resolution (SPTR) achievable with state-of-the-art analog SiPMs and scintillation materials. Weeroc has developed several multi-channel chips such as Liroc and Radioroc which allow the usage of compact photodetectors applied in several large experiments for time measurement. Some of the targeted applications are PET and LIDAR. For both applications, the ASICs are required to have a high intrinsic timing performance in order to optimize the overall timing performance of the detection system incorporating the ASICs. The analog performance of each chip is the key element, which is required to have sufficient bandwidth accommodating the SiPM's fast signal component, low noise for reducing the signal jitter, and reasonable power consumption in order to comply with the power budget in a multi-channel detection system. LIDAR is a laser imaging method used to create high-resolution three-dimensional representations of scanned surfaces or to determine distances to objects by targeting an object or a surface with a laser and measuring the two-way travel time of a laser light pulse. For instance, the space industry uses LIDAR to scan planet surfaces before landing missions, measure distances between spacecraft, and for many other applications. Having the most sensitive detector is critical to measure long distances, particularly in space applications. The development of readout ASICs for SiPM-based LIDAR is relatively new and Liroc is one of the first products available on the market. It should be mentioned that other ASICs such as TOFLAR [1] for similar applications are expected to be available on the market later. In time-of-flight (TOF) PET applications, a gain in the signal-to-noise ratio (SNR) is directly dependent on the timing performance of the PET system, i.e., the coincidence time resolution (CTR) [2]. Current state-of-the-art TOF-PET achieves a CTR of 228 ps [3]. The electronic front end is one of the three key contributors to the overall timing performance [4], which is why special care needs to be taken with respect to the electronic noise and bandwidth of the front end. Emerging front-end designs employing high-frequency (HF) readout electronics show recordbreaking timing performance [5,6], but are far from being transferred to system level due to a high power consumption and large form factor [7,8]. Hence, highly integrated front-end designs in the form of ASICs are required. It has been shown that current system-applicable solutions available on the market show a non-negligible contribution of the front end on the measured CTR [9]. For PET, several ASIC solutions have been developed over the past two decades, amongst others: NINO [10], TOFPET2 [11], STIC [12], and FlexTOT [10]. Apart from NINO, all the cited ASICs are mixed-signal ASICs with embedded digital processing. Weeroc's new ASICs lineup will have both analog and mixed-signal ASICs in order to provide flexibility to the users. This paper presents the latest results obtained for the ASIC SPTR and CTR. As the time resolution is also dependent on the photosensor technology and not only on the electronics, several measurements have been conducted with several detectors (different SiPM technologies and scintillators) to provide a comprehensive overview. The best achieved SPTR and CTR values are obtained with FBK SiPM of the Near UltraViolet -High-Density type with a metallic mask (NUV-HD-LF M3) and with the Broadcom Near UltraViolet -Metal Trench (NUV-MT) SiPMs coupled to TAC LYSO:Ce,Ca 2×2×3 mm 3 crystal. The timing performance of different measured CTR and SPTR of the various scintillators/SiPM/ASIC configurations are summarized and discussed. ASICs development at Weeroc: Liroc and Radioroc Currently, at Weeroc, different ASICs are available for reading out SiPMs, among them: Liroc and Radioroc. Other ASICs are also in development at Weeroc, which will incorporate signal conversion and data readout on-chip. All photo-detector readout ASICs are done using TSMC 130 nm process. First LIDAR-dedicated SiPMs are getting on the market among other detectors such as photomultiplier tubes (PMTs). However, other available multi-channel readout electronics do not allow for an SPTR precise enough for applications of interest. The main read-out requirements of LIDAR read-out are an excellent timing resolution and a few ns double-peak separation. None of the ASICs on the market allows such a fast response. Weeroc has designed Liroc ASIC, a LIDAR dedicated multi-channel read-out chip prototype, focusing on bandwidth and fast return to the baseline to fulfill fast counting and time measurement requirements. In scientific instrumentation, medical imaging, and radiation detection in general, SiPM has been quite prominent in terms of usage for the detection system. Weeroc has been supplying ASICs in the applications such as cosmic ray observation telescopes, Compton cameras, and PET scanners; such as the trimodal PET/MRI/EEG TRIMAGE scanner [13] based on the Triroc ASIC [14]. Most of these applications require a low-noise analog performance better than a single photo-electron and excellent intrinsic timing jitter in the order of a few ps. Additionally, most cited applications are in need of charge measurement with an energy cut-off feature in order to reproduce the input signal spectrum. The Radioroc ASIC has been developed to fulfill these requirements and also will be replacing Weeroc's previous generation ASICs such as Petiroc2A [15], Triroc1A [16] and Citiroc1A [17]. This chip also incorporates a versatile output system so that users would be able to select the outputs based on their applications' needs, such as photon counting and energy measurement with charge integration. ASICs characterization: Materials and methods To evaluate the timing performance in terms of CTR and SPTR of the two ASICs Liroc and Radioroc, a set of measurements has been carried out at RWTH Aachen University. Table 1 summarizes the different SiPM technologies used to carry out these measurements. The NUV-UHD SiPM type is a new SiPM technology developed at FBK which implements a new microcell border structure. The NUV-HD-LF M3 is another structure from FBK with a metallic mask to isolate the subcells. The Broadcom NUV-MT is the newest development from Broadcom and it has a special metallic trench structure in the inter-pixel region. Trench BRCM AFBR-S4N33C013 3×3 NUV-HD HPKS13361-2050-08 2×2 8×8 array Moreover, table 2 lists all scintillators that have been used in this study. In general, two crystal categories have been used: scintillating (LYSO) and non-scintillating (PbF 2 ). The effect of codoping with Calcium (LYSO:Ce,Ca) has been studied in previous publications [18]. Co-doped scintillators are particularly interesting because of their faster scintillation rise times and decay times leading to improved timing [19,20]. Additionally, non-scintillating crystals, e.g. PbF 2 , with sole prompt photon emission due to the Cherenkov effect, are interesting to test the single photon response of the SiPMs and readout electronics. The Cherenkov effect has already been studied in several publications and its potential influence on the CTR was discussed [21,22]. Devices and characteristics Experimental setup The setup used for the CTR measurement is composed of the ASIC, SiPM, and glued crystals on each side. The radioactive source is placed at the center to irradiate both at the same time. SPTR measurement is done with a similar setup but a non-scintillating crystal (PbF 2 ) is used at one side. In addition, an HF electronics [5,7] were used to perform the SPTR measurement as a reference against the ASIC under test. The HF readout of SiPMs significantly improves the measured SPTR, allowing the evaluation of the intrinsic performance of large-area devices, e.g. SiPMs, and ASCIs under test. Figure 1 shows the schematic of the experimental setup used for these measurements. The SiPM is biased with a DC power supply and the output is fed into two channels -timing and voltage amplification. The ASIC preamplifier discriminator was used to determine the time stamp of the leading edge of the signal when it first crosses a set amplitude threshold. In addition, the time over threshold signal, which contains the charge information, is obtained and used for the energy resolution measurements. The output signals were acquired by a Lecroy oscilloscope (WaveRunner 9404, 4 GHz, 40 GS/s) which is capable of performing mathematical operations on the data in real-time. This enabled us to save the processed information such as the signal charge, and time delay (time difference between the two electronic triggers) directly during the acquisition. All the electronic instruments that control the voltages of the SiPM and the acquisition system were centrally controlled by a LabVIEW program set up on a PC. The ASICs threshold was controlled by dedicated software developed at Weeroc. Figure 2 shows the experimental setup used for SPTR and CTR measurements. Analysis methodology This section describes the methods used in this study to perform analysis and extrapolate the SPTR and CTR values. Time delay (i.e. time difference for single photo-electron (p.e) events only) between two triggers or two leading edges time stamps is estimated in this analysis as described below. The two triggers are the trigger of the ASIC under study (Liroc, Radioroc) and the trigger of the HF electronics in the case of SPTR measurement. Whereas for CTR measurement, both triggers are triggers of each ASIC. The events contained within the single p.e peak (corresponding to single-photon triggers) are selected with respect to their energy value and the histogram of the time delay signal is plotted. The full width at half maximum (FWHM) of the time delay histogram yields the SPTR or CTR. The right end of the spectrum (i.e. delay histogram) has a tail that can be a result of delayed signals generated by photons that get converted deeper in the junction as described in [23,24]. A Gaussian fit does not account for tails. Hence, a Gaussian convoluted with an exponential probability distribution (an exponentially modified Gaussian function) is used to fit histograms (described through equation 3.1 and 3.2). ( ; , , ) = 2 2 (2 + 2 −2 ) erfc + 2 − √ 2 ,(3.1) where erfc is the complementary error function defined as erfc( ) = 1 − erf( ) = 2 √ ∫ ∞ − 2 . (3.2) All reported values of SPTR and CTR in this study are the FWHM of the fitted function. The FWHM is estimated directly from the fitted function and not from the fit parameter sigma since it bears to have no direct relationship with FWHM, unlike the Gaussian function. In the case of the measurement of an ASIC against the reference, the SPTR of ASICs (SPTR ASIC ) depends on the SPTR value of the reference (SPTR Ref ) and the time difference of the whole system; measured SPTR (SPTR measured ). In this case, the SPTR is calculated as follows: SPTR ASIC = √︄ SPTR 2 measured − SPTR 2 Ref 2 (3.3) where SPTR = 65 ps measured using a Broadcom AFBR-S4N33C013 SiPM coupled to LYSO:Ce,Ca 2×2×3 mm 3 crystal. While for the CTR measurement, for real coincidence, the CTR is measured using two identical boards (i.e. Radioroc against Radioroc) unless otherwise stated. The analysis is performed similarly, except for the last step where the CTR is corrected for the reference detector in the case of measurement against the HF reference readout. The CTR of the ASIC (CTR ASIC ) is calculated according to equation 3.4. CTR ASIC = √︃ 2CTR 2 measured − CTR 2 Ref (3.4) The corrected CTR value takes into account the correction for the reference HF and a Broadcom AFBR-S4N33C013 coupled to a 2×2×3 mm 3 LYSO:Ce,Ca crystal with a CTR of 65 ps. Measurements and results Single-photon Time Resolution (SPTR) measurement SPTR measurements of Radioroc and Liroc have been performed against a reference readout, the HF, using several SiPM from various manufacturers. Exposed to Cherenkov radiation by coupling the SiPM to a black painted EPIC PbF 2 2×2×3 mm 3 , the timing difference between the two readout chip triggers on single-photon events is studied. Measurement settings are listed in Table 3. The HF readout is measured with a Broadcom AFBR-S4N33C013 SiPM coupled to a TAC LYSO crystal of 2×2×3 mm 3 . All SPTR values stated in this study have been corrected for the HF reference readout. Figure 3 shows the SPTR Measurement of Radioroc with FBK NUV-HD-LF M3 SiPM against HF. Figure 3 (a) shows the time delay histogram, where total events are presented in blue, single p.e selected events in orange, and Gaussian fit in red. Figure 3 (b) shows the measured SPTR of Radioroc as a function of SiPM bias voltage (HV). The best SPTR achieved is 73 ps (FWHM) at a bias voltage of 39.5 V. It is important to note that the values of SPTR as a function of bias voltage are plotted for a constant discriminating threshold selected by optimizing it for each bias voltage setting that gives the best timing performance for each device. Such a chosen discrimination threshold value corresponds to about half the p.e level at that setting. Figure 4 (a) shows the time delay histogram, where total events are presented in blue, single p.e selected events in orange, and Gaussian fit in red. Figure 4 (b) shows the measured SPTR of Liroc as a function of SiPM bias voltage (HV). The best SPTR achieved is 102 ps (FWHM) at a bias voltage of 45 V. Figure 5 shows the SPTR Measurement of Liroc with FBK NUV-HD-LF M3 SiPM against HF. Figure 5 (a) shows the time delay histogram, where total events are presented in blue, single p.e selected events in orange, and Gaussian fit in red. Figure 5 Figure 6 shows the SPTR Measurement of Liroc with Hamamatsu (S13361-2050-08) SiPM against HF. Figure 6 (a) shows the time delay histogram, where total events are presented in blue, single p.e selected events in orange, and Gaussian fit in red. Figure 6 (b) shows the measured SPTR of Liroc as a function of SiPM bias voltage (HV). The best SPTR achieved is 169 ps (FWHM) at a bias voltage of 57 V. Coincidence Time Resolution (CTR) measurement The coincidence between two SiPM-Crystal coupled detector blocks using Radioroc ASIC is studied in this section. The energy and trigger outputs of both SiPMs are sent to a high-resolution oscilloscope. Measurement settings are listed in Table 4. Figure 7 shows the CTR Measurement of Radioroc with Broadcom NUV-MT SiPM against HF. An LYSO crystal of 3×3×20 mm 3 was coupled to the SiPM. Figure 7 (a) shows the time delay histogram, where total events are presented in blue, single p.e selected events in orange, and Gaussian fit in red. Figure 7 (b) shows the measured CTR of Radioroc as a function of SiPM bias voltage (HV). The best CTR achieved is 137 ps (FWHM) at bias voltage 48 V. While using smaller crystal exhibit better CTR. Figure 8 shows the CTR Measurement of Radioroc against identical ASIC (i.e. Radioroc), with two Broadcom NUV-MT SiPMs are measured in coincidence. An LYSO crystal of 2×2×3 mm 3 was coupled to the SiPM. Figure 8 (a) shows the time delay histogram, where total events are presented in blue, single p.e selected events in orange, and Gaussian fit in red. Figure 8 (b) shows the measured CTR of Radioroc as a function of SiPM bias voltage (HV). The best CTR achieved is 83 ps (FWHM) at bias voltage 49 V. Energy measurement with Radioroc in ToT mode Fast timing and energy-resolving capabilities are crucial for a SiPM detector readout. Time-overthreshold (ToT) offers an attractive method for combining timing and energy encoding in SiPM readout. In the ToT method, one measures the signal pulse width at a selected threshold, which can be used to estimate the signal's charge. And as mentioned before, Radiroc is a multi-purpose readout chip for SiPM detectors. It can measure time and energy simultaneously by employing time-of-arrival (ToA) and time-over-threshold (ToT) techniques. In this section, an example of the energy measurement performed using Radioroc is presented and energy resolution is estimated. Figure 9 (a) shows the energy spectrum before calibration. An energy calibration, that takes into account the statistical saturation effects of the SiPM on the evaluated energy resolution, was performed based on the method explained in [25]. This method makes use of the two peaks at 511 keV and 1275 keV present in the Na-22 spectrum. The two peaks are fitted with a Gaussian to determine their centroid in time units (ns). Assigning these positions to their respective value in keV and assuming that 0 keV is located at zero, a logarithmic function is fit to these data points. The resulting parameters are used to correct the spectrum for the saturation of the SiPM and convert the time units (ns) in keV. Using a Gaussian fit, the energy resolution can then be extracted as the FWHM of the photopeak in the saturation-corrected spectrum. Figure 9 (b) shows the saturation-corrected energy spectrum. This energy spectrum corresponds to a ToT measurement performed with Radioroc and a Broadcom SiPM (NUV-MT) coupled to a TAC LYSO:Ce,Ca crystal of 3×3×20 mm 3 at 40 V (equivalent to 10 V overvoltage). After calibration, the energy resolution at 511 keV is 9.2%. Tables 5 and 6 below summarize all measurements carried out during this campaign providing the best time resolution value obtained. Radioroc and Liroc show an improved performance compared to other state-of-the-art circuits [26]. Figure 11 and Figure 10 show, respectively, all single-photon and coincidence time resolution measurements for Liroc and Radioroc with different devices in different configurations. The SPTR measurements here have been carried out using the small LYSO:Ce,Ca (TAC) crystal of 2×2×3 mm 3 on the reference side and a black-painted PbF 2 crystal on the ASIC side. The best SPTR values obtained using small LYSO:Ce,Ca crystal of 2×2×3 mm 3 coupled to FBK NUV-HD-LF M3 SiPM: 73 ps (FWHM) for Radioroc and 79 ps (FWHM) for Liroc. The best SPTR measured with the HF readout for this SiPM (shown also in Figure 10) is 27 ps. In addition, it was observed that FBK's technology: FBK NUV-HD-LF M3 and FBK NUV-HD-RH UHD-DA behave differently. FBK NUV-HD-RH UHD-DA best SPTR value achieved is 102 ps (Liroc), while the best SPTR for the HF reference readout found with this SiPM is 40 ps. On the other hand, the best CTR value of 83 ps (FWHM) obtained with Radioroc in a real coincidence (Radioroc against Radioroc) using a small LYSO:Ce,Ca (TAC) of 2×2×3 mm 3 coupled to Broadcom NUV-MT SiPM (at bias voltage 49 V). Coupling Broadcom NUV-MT SiPM to larger LYSO:Ce,Ca of 3×3×20 mm 3 increases CTR values from 83 ps (FWHM) (at bias voltage 45 V) up to 137 ps (FWHM) at the same configuration. This CTR value has been measured for Radioroc against reference and it has been corrected to reference HF readout. Scintillator-based detectors are able to achieve very good timing approaching a CTR of 100 ps FWHM, as shown in several publications [27,28]. For instance, Broadcom AFBR-S4N33C013 was reported in [29] with intrinsic SPTR lower than 80 ps at an overvoltage of 11 V. The state-of-the-art devices have an FWHM time resolution of around 70 ps [29]. Although the ASIC's time resolution can be limited when coupled with SiPM, the obtained time resolution limits obtained in this study are promising and the current Weeroc's ASICs are achieving at least as best as reference values. At the time of writing this article, new versions of these ASICs, with improved functionalities, are being fabricated and will be available soon. In view of this work, Weeroc collaborates with different industrial partners such as CAEN, TAC, and other research institutes to study and optimize a sub-module (ASIC, SiPM, and crystal) for different applications. FBK NUV-HD-RH UHD-DA 1×1 UHD: Ultra High Density FBK NUV-HD-RH UHD-DE 1×1 UHD: Ultra High Density FBK NUV-HD-LF M3 1×1 M: Metal Mask, LF: Low Field BRCM NUV-MT 4×4 MT: Metal Figure 1 : 1A schematic of the experimental setup used for CTR and SPTR measurements. Figure 2 : 2Experimental setup used for SPTR measurements (a) and CTR measurements (b) Figure 3 :Figure 4 : 34SPTR Measurement of Radioroc with FBK NUV-HD-LF M3 SiPM coupled to Epic PbF 2 crystal of 2×2×3 mm 3 against HF. (a) Histogram of the time delay between the two triggers: total events (blue), single p.e selected events (orange), and Gaussian fit (red). (b) Measured SPTR of Radioroc as a function of SiPM bias voltage (HV). The best SPTR achieved is 73 ps (FWHM) at bias voltage 39.5 V. SPTR values corrected to reference. SPTR Measurement of Liroc with FBK NUV-HD-RH UHD-DA SiPM coupled to Epic PbF 2 crystal of 2×2×3 mm 3 against HF. (a) Histogram of the time delay between the two triggers: total events (blue), single p.e selected events (orange), and Gaussian fit (red). (b) Measured SPTR of Liroc as a function of SiPM bias voltage (HV). The best SPTR achieved is 102 ps (FWHM) at bias voltage 45 V. SPTR values corrected to reference. Figure 4 4shows the SPTR Measurement of Liroc with FBK NUV-HD-RH UHD-DA SiPM against HF. Figure 5 :Figure 6 : 56(b) shows the measured SPTR of Liroc as a function of SiPM bias voltage (HV). The best SPTR achieved is 79 ps (FWHM) SPTR Measurement of Liroc with FBK NUV-HD-LF M3 SiPM coupled to Epic PbF 2 crystal of 2×2×3 mm 3 against HF. (a) Histogram of the time delay between the two triggers: total events (blue), single p.e selected events (orange), and Gaussian fit (red). (b) The best SPTR achieved is 79 ps (FWHM) at bias voltage 39.5 V. SPTR values corrected to reference. SPTR Measurement of Liroc with Hamamatsu (S13361-2050-08) SiPM coupled to Epic PbF 2 crystal of 2×2×3 mm 3 against HF. (a) Histogram of the time delay between the two triggers: total events (blue), single p.e selected events (orange), and Gaussian fit (red). (b) Measured SPTR of Liroc as a function of SiPM bias voltage (HV). The best SPTR achieved is 169 ps (FWHM) at bias voltage 57 V. SPTR values corrected to reference. Figure 7 : 7CTR Measurement of Radioroc with BRCM NUV-MT SiPM coupled to LYSO crystal of 3×3×19 mm 3 against HF with BRCM AFBR-S4N33C013 coupled with a 2×2×3 mm 3 TAC LYSO crystal. (a) Histogram of the time delay between the two triggers: total events (blue), single p.e selected events (orange), and Gaussian fit (red). (b) Measured CTR of Radioroc as a function of SiPM bias voltage (HV). The best CTR of 137 ps (FWHM) at bias voltage 48 V. CTR values corrected to reference. Figure 8 : 8CTR Measurement of Radioroc against Radioroc with two Broadcom NUV-MT SiPMs are measured in coincidence. Both BRCM NUV-MT SiPM coupled to LYSO crystal of 2×2×3 mm 3 . (a) Histogram of the time delay between the two triggers: total events (blue), single p.e selected events (orange), and Gaussian fit (red). (b) Measured CTR of Radioroc as a function of SiPM bias voltage (HV). The best CTR of 83 ps (FWHM) at bias voltage 49 V. Figure 9 : 9An example of a ToT measurement performed with Radioroc with Broadcom NUV-MT SiPM coupled to LYSO:Ce,Ca TAC crystal of 3×3×19 mm 3 at 40 V. (a) Energy spectrum before calibration. (b) Energy spectrum after calibration with an energy resolution of 9.2 %. 5 Discussion and summary Radioroc and Liroc time resolution (SPTR & CTR) have been evaluated by exploring different SiPM and crystals in different configurations. Figure 10 : 10Measured SPTR of Radioroc, Liroc and HF with different SiPMs coupled to PbF 2 crystal of 2×2×3 mm 3 . Figure 11 : 11Measured CTR of Radioroc with different SiPMs and scintillation crystals. Table 1 : 1List of SiPM Technologies used in this studySiPM Producer & Type SiPM Size [mm 2 ] Technology Specification Table 2 : 2List of scintillator crystals coupled to SiPM used in this study.Measurement Types Crystal Type Crystal Size [mm 3 ] SPTR Measurement EPIC PbF 2 (black painted) 2×2×3 CTR Measurement TAC LYSO:Ce,Ca (PMI1×050/1) 2×2×3 TAC LYSO:Ce,Ca (PMI1×025/6) 3×3×20 Table 3 : 3SPTR measurement settings.# Measurement Types ASICs Sensor Breakdown Voltage [V] 1 SPTR Radioroc vs. HF FBK NUV-HD-LF M3, Single, 1x1 mm 2 32.5 2 SPTR Liroc vs. HF FBK NUV-HD-RH UHD-DA, Single, 1x1 mm 2 32.5 3 SPTR Liroc vs. HF FBK NUV-HD-LF M3, Single, 1x1 mm 2 32.5 4 SPTR Liroc vs. HF HPK S13361-2050-08, 8x8 Array, 2x2 mm 2 53 Table 4 : 4CTR measurement settings# Measurement Types ASICs Sensor Breakdown Voltage [V] Crystal 1 CTR Radioroc vs. HF BRCM NUV-MT, Single, 4×4 mm 2 44 TAC LYSO:Ce,Ca 3×3×20 mm 3 2 CTR Radioroc vs. Radioroc BRCM NUV-MT, Single, 4×4 mm 2 44 TAC LYSO:Ce,Ca 2×2×3 mm 3 Table 5 : 5Summary of best obtained SPTR for Liroc and Radioroc in different configurations# ASIC (1) Scintillator Crystal ASIC (2) Scintillator Crystal SiPM Array Specifications SPTR (FWHM) [ps] 1 Radioroc PbF 2 -Black -2×2×3 HF TAC2×2×3 -LYSO:Ce,ca FBK NUV-HD-LF M3 39.5 V 73 2 Liroc PbF 2 -Black -2×2×3 HF TAC2×2×3 -LYSO:Ce,ca FBK NUV-HD-RH UHD-DA 45 V 102 3 Liroc PbF 2 -Black -2×2×3 HF TAC2×2×3 -LYSO:Ce,ca FBK NUV-HD-LF M3 39.5 V 79 4 Liroc PbF 2 -Black -2×2×3 HF TAC2×2×3 -LYSO:Ce,ca HPKS13361-2050-08 57 V 169 Table 6 : 6Summary of best obtained CTR for Liroc and Radioroc in different configurations Radioroc PMI1X025-LYSO:Ce,ca -3×3×20 Radioroc PMI1X026-LYSO:Ce,ca -3×3×20 BRCM-NUV-MT 50.5 V 114 4 Radioroc PMI1X025-LYSO:Ce,ca -3×3×20 Radioroc PMI1X026-LYSO:Ce,ca -3×3×20 BRCM-AFBR-S4N33C013 38 V 148# ASIC (1) Scintillator Crystal ASIC (2) Scintillator Crystal SiPM Array Specifications CTR (FWHM) [ps] 1 Radioroc PMI1X026-LYSO:Ce,ca -3×3×20 HF TAC2×2×3 -LYSO:Ce,ca BRCM-NUV-MT 48 V 137 2 Radioroc PMI1X050-LYSO:Ce,ca -2×2×3 Radioroc PMI1X051-LYSO:Ce,ca -2×2×3 BRCM-NUV-MT 49 V 83 3 AcknowledgmentsThis project is the result of collaborative work between Weeroc and OMEGA and has received funding from the EC-funded project ATTRACT under grant agreement 777222. 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[ "Do RESTful API Design Rules Have an Impact on the Understandability of Web APIs? A Web-Based Experiment with API Descriptions", "Do RESTful API Design Rules Have an Impact on the Understandability of Web APIs? A Web-Based Experiment with API Descriptions" ]
[ "Justus Bogner [email protected] ", "Sebastian Kotstein [email protected] ", "Timo Pfaff ", "Justus Bogner ", "Sebastian Kotstein ", "\nInstitute of Software Engineering\nUniversity of Stuttgart\nGermany\n", "\nReutlingen University\nHerman Hollerith ZentrumGermany\n" ]
[ "Institute of Software Engineering\nUniversity of Stuttgart\nGermany", "Reutlingen University\nHerman Hollerith ZentrumGermany" ]
[]
Context: Web APIs are one of the most used ways to expose application functionality on the Web, and their understandability is important for efficiently using the provided resources. While many API design rules exist, empirical evidence for the effectiveness of most rules is lacking.Objective: We therefore wanted to study 1) the impact of RESTful API design rules on understandability, 2) if rule violations are also perceived as more difficult to understand, and 3) if demographic attributes like RESTrelated experience have an influence on this.Method: We conducted a controlled Web-based experiment with 105 participants, from both industry and academia and with different levels of experience. Based on a crossover design, we studied 12 design rules using API snippets in two complementary versions: one that adhered to a rule and one that was a violation of this rule. Participants answered comprehension questions and rated the perceived difficulty.Results: For 11 of the 12 rules, we found that violation performed significantly worse than rule for the comprehension tasks. Regarding the subjective ratings, we found significant differences for 9 of the 12 rules, meaning that most violations were subjectively rated as more difficult to understand. Demographics played no role in the comprehension performance for violation.
10.48550/arxiv.2305.07346
[ "https://export.arxiv.org/pdf/2305.07346v1.pdf" ]
258,676,457
2305.07346
59a299c134bd355123f834344b5e2b6ba40e6eb3
Do RESTful API Design Rules Have an Impact on the Understandability of Web APIs? A Web-Based Experiment with API Descriptions 12 May 2023 Justus Bogner [email protected] Sebastian Kotstein [email protected] Timo Pfaff Justus Bogner Sebastian Kotstein Institute of Software Engineering University of Stuttgart Germany Reutlingen University Herman Hollerith ZentrumGermany Do RESTful API Design Rules Have an Impact on the Understandability of Web APIs? A Web-Based Experiment with API Descriptions 12 May 202310.5281/zenodo.7381500Received: date / Accepted: dateEmpirical Software Engineering manuscript No. (will be inserted by the editor) Timo Pfaff Independent researcher, Germany Research conducted while at University of Stuttgart, Institute of Software Engineering, Germany 2 Bogner, Kotstein, Pfaff Conclusions: Our results provide first empirical evidence for the impor-tance of following design rules to improve the understandability of Web APIs, which is important for researchers, practitioners, and educators.Web API design · understandability · controlled experiment Context: Web APIs are one of the most used ways to expose application functionality on the Web, and their understandability is important for efficiently using the provided resources. While many API design rules exist, empirical evidence for the effectiveness of most rules is lacking.Objective: We therefore wanted to study 1) the impact of RESTful API design rules on understandability, 2) if rule violations are also perceived as more difficult to understand, and 3) if demographic attributes like RESTrelated experience have an influence on this.Method: We conducted a controlled Web-based experiment with 105 participants, from both industry and academia and with different levels of experience. Based on a crossover design, we studied 12 design rules using API snippets in two complementary versions: one that adhered to a rule and one that was a violation of this rule. Participants answered comprehension questions and rated the perceived difficulty.Results: For 11 of the 12 rules, we found that violation performed significantly worse than rule for the comprehension tasks. Regarding the subjective ratings, we found significant differences for 9 of the 12 rules, meaning that most violations were subjectively rated as more difficult to understand. Demographics played no role in the comprehension performance for violation. Introduction Technologies for Web Application Programming Interfaces (APIs) like WSDL, SOAP, and HTTP are the technical foundations for realizing modern Web applications (Jacobson et al., 2011). These technologies allow developers to independently implement smaller software components and share their functionality on the Internet. Successfully reusing an existing component, however, depends on the ability to understand its purpose and behavior, especially its API, which hides internal logic and complexity. In cases where neither additional documentation nor developers of the original component can be consulted, the Web API might even be the first and only point of contact with the exposed functionality. Therefore, understandability is an important quality attribute for the design of Web APIs (Palma et al., 2017). Over the last two decades, HTTP combined with other well-established Web standards like URI has become a popular choice for realizing Web APIs that expose their functionality through Web resources (Schermann et al., 2016;Bogner et al., 2019). In these resource-oriented Web APIs, the role of HTTP has shifted from a transport mechanism for XML-based messages to an application-layer protocol for interacting with the respective API (Pautasso et al., 2008). With Representational State Transfer (REST) (Fielding and Taylor, 2002), there exists an architectural style that formalizes the proper use of Web technologies like HTTP and URIs in Web applications. REST is considered a foundation for high-quality, so-called RESTful API design, and it describes a set of constraints for the recommended behavior of Web applications, e.g, HTTP-based Web APIs. However, it does not instruct developers how to implement this behavior (Rodríguez et al., 2016). Since there exist various interpretations and (mis-)understandings among practitioners how RESTful API design looks like, users and integrators of these services are confronted with a multitude of heterogeneous interface designs, which can make it difficult to understand a given Web API (Palma et al., 2021). Therefore, several works have proposed design rules and best practices to complement the original REST constraints and to guide developers when designing and implementing Web APIs, e.g., Pautasso (2014), Palma et al. (2017), Richardson and Ruby (2007), and Massé (2011). In most cases, however, we do not have sufficient empirical evidence for the effectiveness of these RESTful API design rules, i.e., if they really have a positive impact on the quality of Web APIs. Additionally, multiple studies analyzed the degree of REST compliance in practice by systematically comparing real-world Web APIs against proposed design rules. Many of these works, e.g., Neumann et al. (2018), Renzel et al. (2012), and Rodríguez et al. (2016), concluded that only a small degree of real-world Web APIs are truly RESTful. This suggests that many practitioners perceive proposed design rules differently in terms of their importance. We have provided confirmation for this in previous work (Kotstein and Bogner, 2021). In a Delphi study, we confronted industry practitioners with 82 RESTful API design rules by Massé (2011) to find out which ones they perceived as important and how they perceived their impact on software quality. Only 45 out of 82 rules were rated with high or medium importance, and maintainability and usability were the most associated quality attributes. Both of these attributes are closely related to understandability. To confirm these opinion-based results with additional empirical evidence, we conducted a controlled Web-based experiment, in which we presented 12 Web API snippets to 105 participants with at least basic REST-related experience. Each API snippet existed in two versions, one adhering to a design rule and one violating the rule (see, e.g., Fig. 1). The participants' task was to answer comprehension questions about each snippet, while we measured the required time. Furthermore, participants also had to rate the perceived difficulty to understand an API snippet. In this paper, we present the design and results of our controlled experiment on the understandability impact of RESTful API design rules. Background and Related Work We start with a discussion of terminology around Web APIs and REST and explain how we use these terms in the paper. Furthermore, we mention existing works that propose rules and best practices for RESTful API design, and present existing studies about Web API quality. Terminology In this paper, we focus on resource-oriented HTTP-based Web APIs. In distinction to SOAP-/WSDL-based APIs (Pautasso et al., 2008), we use the term Web API for any resource-oriented API that exposes its functionality via HTTP and URIs at the application level. We consider a Web API as RESTful, i.e., a so-called RESTful API, if the respective Web API satisfies all mandatory REST constraints defined by Fielding and Taylor (2002). As a consequence, a Web API that implements only some RESTful API design rules cannot automatically be considered as RESTful. Best Practices for REST in Practice To combat the potentially harmful heterogeneity of interface designs among Web APIs, several works tried to translate the REST constraints into more concrete guidelines to instruct developers how to achieve good RESTful design with HTTP. RESTful API design rules and best practices have been proposed in scientific articles, e.g., by Pautasso (2014), Petrillo et al. (2016), Palma et al. (2014), and Palma et al. (2017), but also in textbooks, e.g., by Richardson and Ruby (2007), Massé (2011), andWebber et al. (2010). Moreover, Leonard Richardson developed a maturity model (Martin Fowler, 2010) allowing Web developers to estimate the degree of REST compliance of their Web APIs. Incorporating the principles of REST, the model defines four levels of maturity. -Level 0: Web APIs offer their functionality over a single URI and use HTTP solely as a transport protocol for tunneling requests through this endpoint by using POST. Examples of level 0 are SOAP-and XML-RPCbased services. -Level 1: Web APIs use the concept of resources, i.e., expose different URIs for different resources. However, operations are still identified via URIs or specified in the request payload rather than by different HTTP methods. -Level 2: Web APIs use HTTP mechanisms and semantics, including different HTTP methods for different operations and semantically correct status codes. Level 2 partially aligns with the Uniform Interface constraint. -Level 3: Web APIs additionally conform to the HATEOAS constraint by embedding hypermedia controls into responses to advertise semantic relations between resources and to offer navigational support to clients. Despite the existence of these design rules and guidelines, multiple studies, e.g., by Neumann et al. (2018), Renzel et al. (2012), and Rodríguez et al. (2016), revealed that only a small number of existing Web APIs are indeed RESTful, although many Web APIs claimed to be RESTful (Neumann et al., 2018). This suggests that there is still no common understanding of how a RESTful API should look like in industrial practice. Providing empirical evidence may therefore help to identify the effective and important rules from the large collection of existing guidelines. Related Work Several works investigated the quality of real-world Web APIs, mainly by analyzing interfaces, their descriptions, and exchanged HTTP messages. Rodríguez et al. (2016) analyzed more than 78 GB of HTTP traffic to gain insights into the use of best practices in Web API design. They applied APIspecific heuristics to extract API-related messages from the whole data set and validated these extracted API requests and responses against 18 heuristics aligned with REST design principles and best practices. For a few heuristics, they described their negative effect on maintainability and evolvability when violating associated design principles and best practices. Moreover, they mapped the heuristics to the levels of the Richardson maturity model to estimate the level of REST compliance of investigated Web APIs. The paper concluded that only a few APIs reach level 3 of the maturity model, but the majority of investigated APIs complied with level 2. The governance of RESTful APIs was the focus for Haupt et al. (2018): using a framework developed by Haupt et al. (2017), they conducted a structural analysis of 286 real-world Web APIs. In detail, the framework takes an interface description, converts it into a canonical metamodel, and calculates several metrics to support API governance. As a usage example, they demonstrated how calculated metrics can be used to estimate the user-perceived complexity of an API, which is related to API understandability. For this, they randomly selected 10 of their 286 APIs and let 9 software developers rank them based on their perceived complexity. Based on their knowledge and experience, the authors then defined several metrics for user-perceived complexity and calculated these metrics for the 10 APIs. As a result, some calculated metrics coincided with the developers' judgments and were proposed for automatic complexity estimation. A follow-up confirmatory study with a larger sample size to substantiate this exploratory approach is missing so far. An approach similar to Haupt et al. (2018) was used in a study by Bogner et al. (2019). They proposed a modular framework, the RESTful API Metric Analyzer (RAMA), which calculates maintainability metrics from interface descriptions and enables the automatic evaluation of Web APIs. More precisely, RAMA converts an interface description into a hierarchical model and calculates 10 service-based maintainability metrics. In a benchmark run, the authors applied RAMA to a set of 1,737 real-world APIs, and calculated quartile-based thresholds for the metrics. However, relating the metrics to other software quality correlates is missing to evaluate their effectiveness. The impact of good and poor Web API design on understandability and reusability has been investigated in a series of publications by Palma et al.: in (Palma et al., 2017), the authors defined 12 linguistic patterns and antipatterns focusing on URI design in Web APIs, which may impact understandability and reusability of such APIs. Moreover, they proposed algorithms for their detection and implemented them as part of the Service-Oriented Framework for Antipatterns (SOFA). They used SOFA to detect linguistic (anti-)patterns in 18 real-world APIs, with the result that most of the investigated APIs used appropriate resource names and did not use verbs within URI paths. However, URI paths often did not convey hierarchical structures. In another study, Palma et al. (2021) tried to answer whether a welldesigned RESTful API also has good linguistic quality and, vice versa, whether poorly designed Web APIs have poor linguistic quality. For this, they used SOFA to analyze 8 Google APIs and to detect 9 design patterns and antipatterns, as well as 12 linguistic patterns and antipatterns. However, their statistical tests revealed only negligible relationships between RESTful design and linguistic design qualities. Subsequently, they extended SOFA with further linguistic (anti-)patterns, improved approaches for their detection, and applied the linguistic quality analysis on Web APIs from the IoT domain (Palma et al., 2022b) or compared the quality between public, partner, and private APIs (Palma et al., 2022a). In summary, existing studies assessed the quality of real-world Web APIs by collecting metrics and detecting (anti-)patterns. The latter are somewhat related to RESTful API design rules and best practices that should, in theory, improve several quality aspects of an API. However, in many cases, there is no empirical evidence for the effectiveness of the impact of these design rules and best practices on software quality, especially on understandability. Haupt et al. (2018) and Kotstein and Bogner (2021) used subjective ratings, but no studies in which human participants solve comprehension or maintenance tasks have been conducted. To the best of our knowledge, our experiment is the first study that investigated the understandability impact of violating RESTful API design rules from the perspective of human API consumers. Research Design In this section, we describe the details of our methodology. We roughly follow the reporting structure for software engineering experiments proposed by Jedlitschka et al. (2008). Inspired by the experiment characteristics discussed by Wyrich et al. (2022), Table 1 provides a quick overview of the most important characteristics of the study. For transparency and reproducibility, we publish our experiment artifacts on Zenodo 1 . Research Questions We investigated three different research questions in this study. RQ1: Which design rules have a significant impact on the understandability of Web APIs? Our hypothesis for this central, confirmatory RQ was that each selected rule should improve understandability, i.e., the effectiveness and efficiency of grasping the functionality and intended purpose of a Web API endpoint. RQ2: Which design rules have a significant impact on software professionals' perceived difficulty while understanding Web APIs? For the confirmatory part of this RQ, we hypothesized that API snippets with rule violations are rated as more difficult to understand. Additionally, we analyzed the correlation between actual and perceived understandability in an exploratory part to identify potential differences. RQ3: How do participant demographics influence the effectiveness and perception of design rules for understanding Web APIs? We did not have strong hypotheses for this exploratory RQ. Nonetheless, we had some intuitions about attributes that may be interesting to analyze. For example, it could be possible that adhering to the design rules mostly has an influence on experienced professionals but not on students (or vice versa). Furthermore, some rules might require the participant to know about the Richardson maturity model or some rule violations might be perceived as more critical by participants from academia or from industry. During the study design phase, we selected some general demographic attributes plus several specific ones for the experiment context. Participants and Sampling The only requirements for participation were basic knowledge of REST and HTTP, as well as the ability to understand English. Our goal was to attract participants from diverse backgrounds and experience levels, e.g., both students and professionals, both participants from industry and academia, etc. We used convenience sampling mixed with referral-chain sampling, i.e., we distributed the call for participation within our personal networks via email, and kindly asked for forwarding to relevant circles (Baltes and Ralph, 2022). A similar message was displayed after the experiment to encourage sharing. Students were recruited via internal mailing lists of several universities. Moreover, we advertised the study via social media, such as Twitter 2 , LinkedIn 3 , XING 4 , and in several technology-related subreddits 5 . As additional motivation, we pledged to donate 1e for each of the first 100 participants to UNICEF 6 . Experiment Objects In this experiment, the objects under study were 12 design rules for RESTful APIs that have been proposed in the literature. They are summarized in Table 2 together with short identifiers that we use throughout the rest of the paper. Rule selection was guided by the results of our previous Delphi study (Kotstein and Bogner, 2021), i.e., we focused on rules from Massé (2011) that were perceived as very important by industry experts, with an influence on maintainability or usability. Additionally, we included three instances of the PathHierarchy rule proposed by Richardson and Ruby (2007). While this rule was not part of our previous study, it has strong relationships to Massé's rules Variable path segments may be substituted with identity-based values and Forward slash separator (/) must be used to indicate a hierarchical relationship. Both of these rules fulfill the above criteria, i.e., high importance plus influence on maintainability or usability. In the following, the rule descriptions are taken from Massé (2011) and Richardson and Ruby (2007) respectively. For each rule, we also present the concrete endpoint pair that was used in the experiment, one version for following the rule and one for violating it. These concrete Web API examples and the violation versions are based on our industry experience, but also on existing public APIs that we identified via the APIs Guru repository 7 . For our experiment, the chosen real-world examples were adapted to simplify them and to avoid that participants are already familiar with the presented HTTP endpoints. During the pilot, we discussed the created pairs of rule and violation with external experts to validate if the violation snippets were not strongly exaggerated. Several snippets were adapted based on this feedback, and one task was dropped entirely. URI Design The three rules in this category are concerned with the concrete design of URI paths in an API. PluralNoun: Massé (2011) defines this rule for both collections and stores, which we merge into a single rule for simplicity. In both cases, it prescribes to VerbController: A controller provides an action that cannot be easily mapped to a typical CRUD operation on a resource. In relation to function names in source code, Massé (2011) proposes to always use a verb or verb phrase for controller resources. Using a noun instead would be a violation. Rule: POST /servers/{serverId}/backups/{backupId}/restore Violation: POST /servers/{serverId}/backups/{backupId}/restoration CRUDNames: Based on the invoked HTTP method, a RESTful API selects the semantically equivalent CRUD operation to perform. Therefore, Massé (2011) prescribes not to use CRUD function names like "create" or "update" in URIs, especially not with incorrect HTTP verbs. Adhering to this rule means solely relying on the HTTP verb to indicate the wanted CRUD operation. Our chosen example also includes the rule "DELETE must be used to remove a resource from its parent" (Massé, 2011) for the violation. Rule: DELETE /messaging-topics/{topicId}/queues/{queueId} Violation: GET /messaging-topics/{topicId}/delete-queue/{queueId} Hierarchy Design A prominent rule from Richardson and Ruby (2007) prescribes the use of path parameters to encode the hierarchy of resources in a URI. Since there are several possibilities to apply and interpret this, we created three specific rules based on this idea (PathHierarchy1 to PathHierarchy3). PathHierarchy1 (path params vs. query params): Version 1 explores the difference between using path parameters (rule) and query parameters (violation) for retrieving a hierarchically structured resource. Rule: GET /shops/{shopId}/products/{productId} Violation: GET /shops/products?shopId={shopId}&productId={productId} PathHierarchy2 (top-down vs. bottom-up): In version 2, the difference between structuring the hierarchy top-down / from left to right (rule) and bottom-up / from right to left (violation) is tested. Rule: GET /companies/{companyId}/employees Violation: GET /employees/companies/{companyId} PathHierarchy3 (hierarchical path vs. short path): Lastly, version 3 analyzes differences during the creation of a resource when either using a long hierarchical path with parameters (rule) or a short path with parameters in the request body (violation). Rule: POST /customers/{customerId}/environments/{environmentId}/servers Violation: POST /servers Request Methods In accordance with level 2 of the Richardson maturity model, Massé (2011) proposes several rules that prescribe that each HTTP method should exclusively be used for its semantically equivalent operation. We selected three of these rules for our experiment. NoTunnel: One of these rules states that GET and POST must not be used to tunnel other request methods, which might seem tempting for the sake of simplicity. In our example, an API correctly uses PUT to update a resource (rule), whereas the violation always uses POST and tunnels the update operation via an additional query parameter. Rule: PUT /trainings/{trainingId}/organizers/{organizerId} Violation: POST /trainings/{trainingId}/organizers/{organizerId}?operation=update GETRetrieve: In similar fashion, another rule from Massé (2011) states that the HTTP method GET must be used to retrieve a representation of a resource. Since GET requests have no request body, it may seem tempting to use POST in some cases to be able to use a JSON object instead of overly complex query parameters. A typical example of this is a search resource. Adhering to the rule requires using GET plus query parameters, while reverting to POST plus a request body with the search options is a violation. Rule: GET /events?date=2022-10-03&category=music Violation: POST /events/search POSTCreate: Lastly, we tested the complementary rule for POST, namely that this method must be used to create resources in a collection (Massé, 2011). A typical violation of this rule is the use of PUT to create a resource, as seen in our chosen example. Rule: POST /customers/{customerId}/orders Violation: PUT /customers/{customerId}/orders HTTP Status Codes Another important theme for RESTful API design is the correct usage of HTTP status codes with response messages. Massé (2011) provides a number of rules in this area, from which we selected three in total, namely the rules for 200 (OK), 401 (Unauthorized), and 415 (Unsupported Media Type). Contrary to the previous three categories, the two versions of our used examples (rule and violation) do not differ in the displayed endpoint, but only for the displayed response including the response code. NoRC200Error: The first rule in this category states that the response code 200 (OK) must not be used in case of error. Instead, client-side (4XX) or serverside (5XX) error codes must be used accordingly, based on the nature of the error. Rule: A required parameter is missing in the request body. The server indicates there was a problem and correctly responds with 400 (Bad Request). Violation: A required parameter is missing in the request body. The server indicates there was a problem, but incorrectly responds with 200 (OK). RC401: A similar rule prescribes the use of the status code 401 (Unauthorized) in case of issues with client credentials, e.g., during a login attempt. Using a different client-side response code like 400 (Bad Request) or 403 (Forbidden) should be avoided for such errors. Since the nuances between 401 (not logged in or login failed) and 403 (logged-in user does not have the required privileges) might not be fully clear to all participants, we opted for the more generic code 400 (Bad Request) in the violation example. Rule: A secured resource is accessed with an empty Bearer token in the Authorization header. The server indicates there was a problem and correctly responds with 401 (Unauthorized). Violation: A secured resource is accessed with an empty Bearer token in the Authorization header. The server indicates there was a problem, but incorrectly responds with 400 (Bad Request). RC415: The final rule in this category focuses on the correct usage of the status code 415 (Unsupported Media Type), which must be returned if the client uses a media type for the request body that cannot be processed by the server. A typical example is a request body in XML when the server only supports JSON. Returning a different client-side error code, e.g., 400 (Bad Request), should be avoided in this case. Rule: A request body contains XML, even though the server only accepts JSON. The server indicates there was a problem and correctly responds with 415 (Unsupported Media Type). Violation: A request body contains XML, even though the server only accepts JSON. The server indicates there was a problem, but incorrectly responds with 400 (Bad Request). Material To incorporate the created API rule examples into our experiment, we relied on the OpenAPI specification format 8 , one of the most popular ways to document Web APIs many practitioners and researchers should be familiar with. Using the Swagger editor 9 , we created two OpenAPI documents, one with the examples following the rules and one with the violation examples. Each document contained 12 endpoints, one per rule. We then created screenshots of the graphical representation of each resource, with the purpose of showing them to participants with each task (see, e.g., Fig. 1). To reach a larger and more diverse audience, we decided to conduct an online experiment via a web-based tool. We selected the open-source survey tool LimeSurvey 10 for this purpose, as it provides all the features we need. Additionally, we had access to an existing LimeSurvey instance via one of our universities. It supports several types of questions, is highly customizable, and also allows measuring the duration per task, i.e., survey question, which we needed for our experiment. Lastly, random assignment of participants to sequences is also possible. This setup meant that participants exclusively used LimeSurvey for the experiment via a computing device and web browser of their choice. All necessary information was provided this way. Tasks The participants' main task was to inspect and understand several Web API snippets that were presented in the graphical representation of the Swagger editor. To evaluate understanding, participants had to answer one comprehension question per snippet. Each of these questions was a single choice question with five different options presented in a random order, with exactly one of them being correct. The options were the same regardless of whether the version adhering to the rule or the one violating the rule was displayed. Opting for single choice comprehension questions had several advantages. While freetext answers might reflect a more in-depth understanding of participants, they are much harder to correct. Additionally, writing free-text answers takes more time and effort for participants, which may increase the drop-out rate. Lastly, participants may specify answers with different level of details. This not only complicates the grading, but it also influences the time to answer, which we include in our comprehension measures. Depending on the shown snippet, exactly one of three different types of comprehension questions was asked. Return Value: Participants had to determine the return value of an endpoint, i.e., the type of entity and if a single object or a collection was returned. This question type was used for snippets with GET requests, namely PluralNoun, PathHierarchy1, and PathHierarchy2. An example is shown in Fig. 1. Endpoint Purpose: Participants had to determine the purpose of the shown endpoint, i.e., what operation or functionality was executed on invocation. This question type was used for snippets which do not always return entities, e.g., POST, PUT, and DELETE requests, namely VerbController, CRUDNames, PathHierarchy3, NoTunnel, GETRetrieve, and POSTCreate. An example is presented in Fig. 2. Response Code Reason: Participants had to determine why a certain request failed or what the outcome of a request was. This question type was exclusively used for snippets of the category HTTP Status Codes, namely NoRC200Error, RC401, and RC415. An example is presented in Fig. 3. After each comprehension question, participants also had to rate the perceived difficulty of understanding the API snippet on a 5-point ordinal scale, ranging from very easy (1) to very hard (5). For this purpose, the last snippet was shown again, so participants did not have to rely solely on memory. Variables and Hypotheses Two dependent variables were used in this experiment. For RQ1, we needed a metric for understandability. To operationalize this quality attribute, we collected both the correctness and duration for each task per participant. Since every task had a single correct answer, correctness was a binary variable, with 0 for false and 1 for correct. The required duration per task was documented in seconds. To combine these two measures into a single variable, we adapted an aggregation procedure from Scalabrino et al. (2021), namely Timed Actual Understandability (TAU). In our experiment, TAU for a participant p and task t was calculated as follows: T AU p,t = correctness p,t × (1 − duration p,t max(durations t ) )(1) TAU produces values between 0 and 1, with values closer to 1 indicating a higher degree of understandability. For an incorrect answer, TAU is always 0. For a correct answer, the task duration is set in relation to the maximum duration that was recorded for this task. This is then inverted by subtracting it from 1, meaning the faster the correct answer was found, the greater is TAU. As such, TAU represents a pragmatic aggregation of correctness and duration that respects differences between participants in the sample and leads to easily interpretable values. Even though it leads to unusual distributions (see, e.g., Fig. 8), we therefore chose TAU as the dependent variable for RQ1. For RQ2, the dependent variable was the perceived difficulty, i.e., the rating that participants had to give after each task using a 5-point ordinal scale, ranging from very easy (1) to very hard (5). As the directly controlled independent variable, we used the version number of the respective snippet. This was either version 1 (rule) that followed the rule or version 2 (violation) that violated the rule. Additionally, we had uncontrollable independent variables that we collected for RQ3, namely various demographic attributes of participants such as their current role, years of experience with REST, or knowledge of the Richardson maturity model. Based on these variables, we formulated hypotheses for the confirmatory questions RQ1 (difference in actual understandability between rule and violation) and RQ2 (difference in perceived understandability). They are displayed in Table 3. In both cases, we expected that the version following the design rule would lead to significantly better results in comparison to the version violating the rule. While we only list two hypotheses, each of the 12 rules was tested individually to clearly identify which rules have an impact and which do not. Finally, the exploratory RQ3 did not have a clear hypothesis. Experiment Design To reduce the impact of potential expertise or motivation differences between participants, we used a crossover design (Vegas et al., 2016), a special form of a within-subjects design. In such a design, each participant receives each treatment at least once, but the order in which participants receive the treatments differs. Our participants worked on six tasks with the rule version and six tasks with violation. While this means that only half of our participants worked on each version of a specific task, it also avoids familiarization effects. Additionally, we still have many observations to overall compare rule vs. violation. This design is fairly robust regarding inter-participant differences. Letting each participant work on both treatments for the same rule would be even more robust in this regard, but would also create problems with familiarization effects. It also would increase the experiment duration considerably. To avoid suboptimal task orderings regarding treatments or task categories that could cause carryover or order effects, we did not randomize the order of tasks. Instead, we consciously designed two sequences with counterbalanced task orders (see Table 4), which is common for crossover designs (Vegas et al., 2016). Our goals were to spread out the tasks of the same category, to ensure the same number of rule and violation tasks, to avoid having the same treatment too often in a row, and to ensure that rule and violation for the same snippet appear at the same position per sequence. Participants were randomly assigned to one of these sequences at the start of the experiment. This experiment design was the result of several iterations with internal discussion, followed by a pilot with external reviewers. Based on the pilot results, we refined the tasks and survey text. Furthermore, we removed a few previously planned tasks to keep the required participation time below 15 minutes, leading to the total of 12 tasks. Experiment Execution Participation in the online experiment via LimeSurvey was open for a period of approximately three weeks. At the start of and during this period, we actively promoted the experiment URL within in our network. Participants starting the experiment were first presented with a welcome page containing some general information about the study. We explained the study goal, how much time it would approximately take, and what the requirements for participation were. Afterwards, we described the tasks to be solved. We highlighted that time would be measured for the comprehension questions (not for the rating questions), but that answering correctly had priority. To familiarize participants with the used API snippet visualization, we also presented an example snippet with additional explanations about the different elements (see Fig. 4). The last part of the welcome page contained our privacy policy. We explained what data would be requested and what we would do with it. Additionally, we emphasized that participation was strictly voluntary and anonymous, and that participants could choose to abandon the experiment at any time, resulting in the deletion of the previously entered data. Participants had to consent to these terms before being able to start the experiment. Upon accepting our privacy policy, participants started working on the tasks of their randomly assigned sequence (see Table 4). For each of the 12 tasks, participants first analyzed the presented API snippet and answered the comprehension question. Time was measured for this part. Afterwards, participants had to rate the difficulty of understanding the studied API snippet. The snippet was displayed again for this purpose. No time was measured for these rating questions. For the experiment part, participants answered a total of 24 questions, one comprehension and one rating question for each task. After finalizing the experiment part, participants were forwarded to a page with demographic questions, which consisted of their country of origin, current role, technical API perspective (either API user / client developer, API developer / designer, or both), years of professional experience with REST, knowledge of the Richardson maturity model, and (if they knew about the model) their opinion about the minimal required maturity level a Web API should possess to be considered RESTful. We also provided an optional freetext field for any final remarks or feedback participants wanted to give. Finally, an outro page was presented, where we thanked participants and kindly asked them to forward the experiment URL to suitable colleagues. Experiment Analysis To analyze the experiment results, we first exported all responses as a CSV file. We then performed the following data cleaning and transformation steps: -Removing 48 incomplete responses, i.e., participants that aborted the experiment at some point (rationale: avoid including responses without demographics, participants not fully committed, or participants who realized themselves that their background was not suitable) -Resolving and harmonizing free-text answers for current role and technical API perspective ("Other:") -Harmonizing country names -Adding binary variables (1 or 0) for is Student, is Academia (an academic professional), and is Germany; if both is Student and is Academia were 0, the participant was an industry practitioner No substantial ambiguity was identified during the harmonization, and no final participant comments had to be considered for adapting the assigned correctness score. However, since TAU is sensitive to outliers of the measured duration, we also analyzed the durations for all comprehension questions. If answering the question took less than 5 seconds or more than 3 minutes, we judged this as invalid and removed the individual answer for all three dependent variables (correctness, time, and subjective rating). The rationale for this was that we did not want to consider any responses (not even the subjective rating) where it was very likely that the participant had not concentrated fully on the task, e.g., by answering before reading everything or leaving the experiment for more than a few seconds to do something else. Since questions of the type HTTP Status Codes were more verbose, the allocated thresholds for these questions were 10 seconds and 4 minutes. If these duration thresholds were triggered three or more times for the same participant, we removed their complete response instead of only the individual answers. Due to this filtering, 2 complete responses (initially, we had 107 complete responses) and 12 individual answers were removed. The number of valid responses considered in the analysis is displayed for each individual task and treatment in Table 5. The cleaned CSV file was then imported by an analysis script written in R 11 , which had also been tested and refined during the pilot study. The script performs basic data transformation, calculates TAU, and provides general descriptive statistics as well as diagrams to visualize the data, such as box plots. To select a suitable hypothesis test for RQ1 and RQ2, we first analyzed the data distributions with the Shapiro-Wilk test (Shapiro and Wilk, 1965). For both dependent variables, the test resulted in a p-value 0.05, i.e., the data did not follow a normal distribution, which called for a test without the assumption of normality. We therefore opted for the non-parametric Wilcoxon-Mann-Whitney test (Neuhäuser, 2011), which has a mature R im- plementation in the stats package 12 . To combat the multiple comparison problem (in our case, the testing of 12 rules), we applied the Holm-Bonferroni correction (Shaffer, 1995). We used the p.adjust() function from the stats package 13 to adjust the computed p-values. When an adjusted p-value was less than our targeted significance level of 0.05, we rejected the null hypothesis and accepted the alternative. To judge the effect size of accepted hypotheses, we additionally calculated Cohen's d (Cohen, 1988) For the exploratory RQ3, we first used a correlation matrix for visual exploration. Correlations of identified variable pairs were then further analyzed with Kendall's Tau (Kendall, 1938), as it is more robust and permissive regarding assumptions about the data than other methods. Lastly, we used linear regression 15 to further analyze combined effects of demographic attributes and to explore the potential for predictive modelling. 12 https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/wilcox. test 13 https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/p. adjust 14 https://www.rdocumentation.org/packages/effectsize/versions/0.8.2 15 https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/lm Results and Discussion In this section, we first present some general statistics about our participants. Afterwards, we provide the results for each RQ, starting with descriptive statistics and then presenting the hypothesis testing or correlation results. Participant Demographics After the cleaning procedure, we were left with 105 valid responses, which is more than most software engineering experiments have. 16 96 of these were complete, while we had to exclude one or two answers for 9 of them. 52 participants were randomly assigned to sequence 1 and 53 to sequence 2. Table 6 compares the attributes for the two sequences, which are pretty similar in most cases. Overall, our 105 participants had between 1 and 15 years of experience with REST, with a median of 4 years. 59 participants were from industry (56%), 18 were professionals from academia (17%), and 28 were students (27%). Most of our participants, namely 70 of 105 (67%), were located in Germany, followed by Portugal (12), the US (6), and Switzerland (5). The remaining 12 responses were distributed across 7 countries with between 1 and 3 participants. Regarding the roles of the 59 industry participants, the majority of them were software engineers (41). Eight were consultants, two were software architects, and two were team managers. The remaining six each had a role that was only mentioned once, e.g., test engineer. Concerning the technical API perspective, most participants reported to be both active in API development and usage (76), while 16 exclusively were API users / client developers, 8 exclusively were API developers, and 5 did not provide an answer to this optional question. Interestingly, only 28 participants (27%) reported knowing the Richardson maturity model. When these 28 were then asked their opinion about the minimal maturity level that a Web API should have to be RESTful, 22 chose level 2 (79%) and 6 level 3 (21%). No one selected level 0 and 1. This may be an indication that the Richardson maturity model is not very well known and that HATEOAS is not perceived as an important requirement for RESTfulness by most professionals. Especially the latter is in line with previous findings (Kotstein and Bogner, 2021). Impact on Understandability (RQ1) For our central research question, we wanted to analyze which design rules have a significant impact on understandability, as measured via comprehension tasks and TAU. We visualize the results for the percentage of correct answers (Fig. 5), the required time to answer (Fig. 6), and TAU (Fig. 7) per individual rule and treatment. For a more detailed comparison, Tables 8, 9, and 10 in the appendix list the descriptive statistics for this per task. For 11 of the 12 tasks, participants with the rule version performed better than participants with the violation version, i.e., mean TAU was higher for rule. Correctness was often the deciding factor, e.g., for the tasks CRUDNames (96% vs. 38%), GETRetrieve (100% vs. 60%), or NoRC200Error (98% vs. 58%). In other cases, correctness was much closer between the two treatments, but participants with violation required more time. This is, e.g., visible for the tasks PluralNoun ( Mean time to answer in seconds To further visualize the experiment performance, we created strip plots of TAU for all rules, which makes it easier to understand its unusual distribution and to compare the two treatments. Fig. 8 shows the TAU distributions for the rules in the categories URI Design and Hierarchy Design. Incorrect answers are displayed as dots at the bottom (TAU = 0), with the median value being displayed as a black diamond. For all three URI Design tasks (1-3), it immediately becomes apparent that violation (red) performed worse, i.e., the median of violation is below rule in all cases. This difference is especially large for the task CRUDNames, which constitutes the worst performance for violation. Additionally, the values are more spread out for violation in all three tasks. For the rules of the Hierarchy Design category (4-6), we see a similarly improved performance for rule in the case of PathHierarchy1 and PathHierachy2. However, the exception is PathHierarchy3, which shows a visibly better performance and less spread for violation. A potential explanation could be that the general difference in path length might have been the deciding factor, i.e., POST /customers/{customerId}/environments/{environmentId}/servers vs. the much simpler violation POST /servers. Even though following the rule provides richer details about the resource hierarchies ("there are customers, who have IT environments like Production, in which servers are placed"), it seems to make the endpoint purpose more difficult to understand, in this case that a new server is created. Additionally, from the five response options, three were about creating a new resource, which fits to the POST of the shown endpoint. However, it was probably clearer to exclude the options "Create multiple new servers..." and "Create an environment..." for violation, as "environment" did not occur in the path. The needed time was not that different (42.69 s vs. 37.51 s), but only 83% answered correctly for rule, while 96% did the same for violation. Extrapolating from the other PathHierarchy rules (1 and 2), we might state that using a path hierarchy correctly, i.e., from left to right with sequential pairs of {collectionName}/{id}, is significantly better than using it incorrectly, but not automatically better than not using it at all. However, more research is needed to confirm this, e.g., by analyzing other variants of PathHierarchy3, e.g., with GET instead of POST, and identifying potential thresholds below which the path length might become irrelevant. Fig. 9 presents the TAU distributions for the categories Request Methods and HTTP Status Codes. For Request Methods (7-9), the rule version shows again better performance in all three tasks, with substantial distance between treatment medians. However, in the NoTunnel task, the spread of TAU values is more similar between rule and treatment than for other tasks. Overall, NoTunnel was the worst performance for rule, with only 66% answering it correctly. One explanation may be that several practitioners may not see it as a violation to use PUT to create a new resource. In the final HTTP Status Codes category (10-12), the performance of rule is also better in all three cases, however, to varying degrees. For NoRC200Error, the differences are substantial, but for RC401 and especially RC415, the median values are not as far apart. In the case of RC401, the spread in TAU values is also much more similar between the two treatments. This was the second-worst performance for rule, with only 75% providing the correct answer. To verify if the visually identified differences between rule and violation were statistically significant, we continued with hypothesis testing. Despite the applied Holm-Bonferroni correction, 11 of our 12 hypothesis tests produced significant results, i.e., 11 of 12 rules had significant impact on understandability. The only outlier was PathHierarchy3, where violation had performed better. Additionally, we calculated the effect size for the significant tests via Cohen's d. We visualize the d values in Fig. 10. For a more detailed comparison of the test results, please refer to Table 11 in the appendix. The rules NoTunnel, RC401, and RC415 produced a small effect (0.2 < d < 0.5) and PluralNoun and VerbController a medium one (0.5 < d < 0.8). The remaining six rules showed even stronger effects, namely a large one for PathHierarchy1 (0.8 < d < 1.2), a very large one for NoRC200Error, GETRetrieve, POSTCreate, and PathHierarchy2 (1.2 < d < 2.0), and even a huge one for CRUDNames (d > 2.0). It is difficult to determine the most impactful category of rules, as the six rules with a Cohen's d > 1.0 cover all four categories. For URI Design, violating CRUDNames had by far the largest effect (d = 2.17), but VerbController (d = 0.75) and PluralNoun (d = 0.72) were in the bottom half. The categories Request Methods and Hierarchy Design also had two impactful rules, but NoTunnel produced only a small effect and PathHierarchy3 none at all. Lastly, HTTP Status Codes was the least impactful category. Even though NoRC200Error had a Cohen's d = 1.25, both RC401 and RC415 resulted in small effects. Results for RQ1: For 11 of the 12 tasks (PathHierarchy3 being the exception), adhering to the design rule resulted in significantly better comprehension performance. Effect sizes were between 0.24 and 2.17, with 6 rules resulting in a Cohen's d > 1.0 (large up to huge effects). Violating the rules CRUDNames, PathHierarchy2, and POSTCreate had the strongest negative impact on understandability. Impact on Perceived Difficulty (RQ2) After showing that the majority of design rules led to significantly better comprehension performance, we analyzed if the results were similar for the subjective perceived understandability. Table 7 summarizes the results for the perceived difficulty ratings and sets them in relation to the corresponding TAU values. The mean difficulty rating is lower for rule in all 12 tasks, while the median difficulty rating is only lower for 10, the exceptions being RC415 and again PathHierarchy3 (median rating of 2 for both treatments). In four tasks, namely VerbController, CRUDNames, GETRetrieve, and NoRC200Error, the difference in median rating is 1 point, while it is even higher for the remaining six tasks (between 1.5 and 3 points). To further analyze differences, we visualized the results with a comparative bar plot of the ratings 1 (very easy) and 2 (easy) in Fig. 11. For a full Likert plot, please refer to Fig. 14 in the appendix. The difference between rule and violation becomes apparent for many of the tasks here, e.g., for CRUDNames, PathHierarchy1, PathHierarchy2, and RC401. However, in some tasks, the treatment ratings also appear to be decently close to each other, despite the median rating being different, e.g., for VerbController or GETRetrieve. In general, the violations perceived as the least difficult to understand were PathHierarchy3, RC415, and PluralNoun. To confirm if the differences between rule and violation are significant, we again applied hypothesis testing. This time, we only found significant differences for 9 of the 12 tasks in the perception of rule and violation. In addition to PathHierarchy3 and RC415 with equal medians, the difference between treatments for VerbController was also not significant. We visualize the values for Cohen's d in Fig. 12. For a more detailed comparison, please refer to Fig. 11 Bar plots of perceived difficulty (RQ2) for the ratings 1 (very easy) and 2 (easy), rule ratings on the left, violation ratings on the right Fig. 12 Effect sizes for the 9 significant differences between perceived difficulty for treatments rule and violation, ordered by Cohen's d, insignificant PathHierarchy3, RC415, and VerbController not included Especially for PathHierarchy2 with Cohen's d = 2.06 (huge), we see that the constructed URI for violation (GET /employees/companies/{companyId}) was a very extreme case leading to much confusion. Mixing the URI path segments in this way seems to have made it much more difficult for participants to identify the cardinalities of the original domain model (all employees be-longing to a specific company). For PathHierachy3, the missing effect was to be expected and complementary to the insignificant results for TAU. However, VerbController and RC415 are not as easy to explain, as both of them had a significant impact on the actual understandability. While violation was perceived as slightly more difficult than rule in both tasks, this difference was not statistically significant. For RC415, this might be explained by the fairly small effect size for the TAU difference (d = 0.28) and the similar levels of correctness per treatment. If participants only needed a bit more time but were still fairly certain to have found the correct answer, they might not have directly associated this task with a high difficulty. For VerbController, however, this explanation is not applicable, as both correctness and time differed per treatment, and it produced a medium effect (d = 0.75). This makes it an especially dangerous rule to violate because it has considerable impact, but many people might not notice that something is ambiguous or unclear with the endpoint. Fig. 13 Adjusted R 2 values for the regression between TAU and perceived difficulty ratings for violation, ordered by adjusted R 2 , higher is better Other rule violations that may be especially critical can be identified by analyzing the correlations between TAU and perceived difficulty for the violation treatment. Ideally, we would like to have significant negative correlations between the two dependent variables for all violation API snippets, i.e., the worse the comprehension performance of a rule violation, the higher the perceived difficulty ratings should be. Rule violations where this is not the case may indicate that participants felt confident in their performance despite answering incorrectly or slowly because they unknowingly misunderstood the API snippet. For nine tasks, there was no significant negative correlation between TAU and perceived difficulty, the exceptions being PathHierarchy3, VerbController, and GETRetrieve. For the detailed correlation results, please refer to Table 13 in the appendix. As a measure of explanatory power, we visualize the adjusted R 2 values for the regression between TAU and the perceived difficulty ratings for all violation API snippets in Fig. 13. For many of them, the values are close to zero, meaning that the perceived difficulty ratings cannot explain any variation in TAU. Among them are also rule violations that have a substantial impact on understandability, like CRUDNames, PathHierarchy2, or POSTCreate, which makes these rule violations especially problematic. Results for RQ2: For 9 of the 12 tasks (except for PathHierarchy3, RC415, and VerbController), adhering to the API design rule resulted in significantly smaller perceived difficulty ratings than violating it. Effect sizes were between 0.79 and 2.06, with 6 rules resulting in a Cohen's d > 1.0 (large up to huge effects). Additionally, there was no significant relationship between perceived difficulty and TAU for many violation tasks, making these rule violations especially dangerous. Relationships with Demographic Attributes (RQ3) For RQ3, we analyzed if there were any relationships between the dependent variables (TAU and the perceived difficulty ratings) and demographic attributes of our participants. We explored this question separately for each treatment (rule and violation), and compared the results. Studied predictors were being from Germany, being from academia (vs. industry), being a student, being an API developer / designer, years of professional experience with REST, having knowledge of the Richardson maturity model, and the preferred minimal maturity level. Even though this was an exploratory RQ, we used Holm-Bonferroni adjusted p-values and a significance level of α = 0.05. Overall, we did not find many significant relationships, and no strong ones at all. Years of experience with REST had a small positive correlation with TAU for rule (Kendall's τ = 0.1956, p = 0.0311), i.e., for the API snippets adhering to the rules, participants with more experience had a slight tendency to perform better. However, this was not the case for API snippets violating the rules. Similarly, participants knowing the Richardson maturity model also tended to perform slightly better for rule, with a Kendall's τ = 0.1941. However, after adjusting the original p-value (p = 0.0158), this correlation was no longer significant (p = 0.0791). The deciding factor here should be years of experience, though, as it was also positively correlated with knowing the Richardson maturity model in our sample (Kendall's τ = 0.3760, p < 0.001). Conversely, knowledge of the Richardson maturity model was also positively correlated with the perceived difficulty ratings for violation, i.e., if participants knew about this model, they tended to rate the API snippets violating the rules as slightly more difficult to understand (Kendall's τ = 0.3035, p = 0.0014). This correlation was absent for rule. All other demographic attributes did not produce any significant relationships. While the identified correlations were small, they still seem to highlight differences between the two treatments that may provide an explanation. Experience and knowledge about REST is only linked to better experiment performance if no design rules are violated. If rules are violated, it does not matter much if people are more experienced: their performance still suffers. However, people who know advanced REST-related concepts are at least more likely to notice that something is wrong with the rule-violating API snippets, even though this does not help to understand them better. This theory also seems to be supported by our results of trying to build linear regression models to predict TAU per treatment based on the demographic attributes. While both models (rule and violation) were unable to provide reliable predictions for the majority of our sample, the model for violation performed considerably worse. The model for rule was able to explain a decent percentage of variability (adjusted R 2 = 0.2171, p = 0.0618), while the one for violation had no explanatory power at all (adjusted R 2 = −0.1792, p = 0.9674). Results for RQ3: Participants with more REST-related experience and knowledge of the Richardson maturity model performed slightly better within rule, but not within violation. Conversely, people knowing the Richardson maturity model perceived the violation API snippets as slightly more difficult to comprehend. This seems to indicate that violating design rules decreases understandability regardless of demographic attributes like experience. Threats to Validity This section describes how we tried to mitigate potential threats to validity, and which threats and limitations to our results remain. We discuss these issues mainly through the perspectives provided by Wohlin et al. (2012). Construct validity is concerned with relating the experiment and especially its collected measures to the studied concepts. This includes whether our dependent variables were adequate representations of the constructs. Our measure for understandability, namely TAU as a combination of correctness and time (Scalabrino et al., 2021), has been used in several studies before. It provides the advantage of increasing the information density in a single statistical test. However, TAU's trade-off between time and correctness is not ideal when correctness is binary. TAU therefore definitely has limitations, but looking at time and correctness separately to interpret the results allows reducing these threats. Regarding the perceived understandability, it is an accepted practice to use ordinal scales that are balanced and symmetrical for subjective ratings. To avoid hypothesis guessing (Wohlin et al., 2012), participants were told that the goal of the experiment was to analyze the understandability of different Web API designs. We see it as very unlikely that participants figured out the true goal of the experiment and then deliberately gave worse responses for violation tasks. Lastly, it is possible that our experiment may be slightly impacted by a mono-operation bias (Wohlin et al., 2012), i.e., the underrepresentation of the construct by focusing on a single treatment. While we used different categories with several rules each in the experiment, we still only compared one exemplary rule implementation to its complementary violation in isolation. We did not combine several rules or involved other RESTful API concepts, which could have led to a richer theory. Internal validity can suffer from threats that may impact the dependent variables without the researchers' knowledge, i.e., confounders (Wohlin et al., 2012). In general, a crossover design is fairly robust against many confounders by reducing the impact of inter-participant differences, e.g., large expertise or experience differences between participants. The relatively short experiment duration of 10-15 minutes also made most history (Wohlin et al., 2012) or maturation (Wohlin et al., 2012) effects unlikely, e.g., noticeable changes in performance due to tiredness or boredom. However, due to the similar task structures, learning effects are very likely. The fixed sequences nonetheless ensured that both treatments for the same task appeared at the same position, thereby ensuring a similar level of maturation. Additionally, randomization was used to assign participants to the two sequences and to display the different answers per comprehension question. This study was conducted as an online experiment, with considerably less control over the experiment environment. While random irrelevancies in the experimental setting (Wohlin et al., 2012) might have occurred in some cases, such as reduced concentration due to loud noise or an interruption by colleagues or family members, this potential increase in variance still did not impact our hypothesis tests. Moreover, we also had no means to prevent people from participating several times, as responses were anonymous. Due to no reasonable incentive for this, we deem this threat as very unlikely. External validity is the extent to which the results are generalizable to other settings or parts of the population. Our sample was fairly diverse, with participants with different levels of experience from both industry and academia. With 105 participants, our sample size was also decently large when compared to many other software engineering experiments. Based on this, we see it as unlikely that the interaction of selection and treatment threat (Wohlin et al., 2012) could impact our results. Even though most of our sample was located in Germany (67%), we do not believe that country-specific differences might have a noticeable influence on the results. To combat the threat interaction of setting and treatment (Wohlin et al., 2012), we ensured using realistic API concepts and violations inspired by real-world examples. The Swagger editor is also a very popular tool in the area of Web APIs. Finally, we need to emphasize that the understandability of Web APIs was tested in a consciously constructed setting, without a hosted implementation for manual exploration or additional API documentation. Participants could only rely on the provided API snippets in the visual notation from the Swagger editor. This is obviously different from a real-world environment, e.g., in industry, where a software engineer trying to use a Web API may have ac-cess to the API documentation or even a running API instance for manual testing. However, in many cases, access to more documentation or the API itself is also not available in the real world. Additionally, using these artifacts also requires additional time, i.e., being able to understand the purpose of an endpoint without having to consult other materials is still preferable. Moreover, it is also plausible that some rules might even have a stronger effect in a real-world setting, e.g., RC401 and RC415: getting access to HTTP header information requires a lot more effort there, while we conveniently presented the headers in the experiment. All in all, we believe that our results are decently transferrable to less controlled real-world environments. To analyze the full degree of this generalization, follow-up research is necessary, e.g., based on repository mining or industrial case studies. Conclusion and Future Work In this paper, we presented the design and results of our controlled Web-based experiment on the understandability impact of 12 RESTful API design rules from Massé (2011) and Richardson and Ruby (2007). In detail, we presented 12 Web API snippets to 105 participants, asked them comprehension questions about each snippet, and let them rate the perceived understandability. For 11 of 12 rules, we identified a significant negative impact on understandability for the violation treatment. Effect sizes ranged from small to huge, with Cohen's d between d = 0.24 and d = 2.17. Furthermore, our participants also rated 9 of 12 rule violations as significantly more difficult to understand. All in all, our results indicate that violating commonly accepted design rules for RESTful APIs has a negative impact on understandability, regardless of REST-related experience or other demographic factors. For several rule violations, we could also show that they are prone to misinterpretation and misunderstandings, making them especially dangerous. Practitioners should therefore respect these rules during the design of Web APIs, as understandability is linked with important quality attributes like maintainability and usability. This becomes especially important for publicly available APIs that are meant to be used by many external people. Providing comprehensive API documentation may partly mitigate understandability problems caused by rule violations, but is still an insufficient solution. In the future, additional experiments should try to replicate these findings with other samples of the population or other violations of the same rules, and potentially extend the evidence to other rules not tested in our study. To enable such studies, we publicly share our experiment materials 17 . Additionally, toolsupported approaches to automatically identify these rule violations will be helpful for practitioners trying to ensure the quality of their Web APIs. Such tool support would also pave the way for large-scale studies of these rule violations in less controlled environments, such as software repository mining or industrial case studies. Fig. 1 1Example of a return value question based on the rule PluralNoun (rule on the left, violation on the right, correct answer is checked) Fig. 2 Example of an endpoint purpose question based on the rule VerbController (rule on the left, violation on the right, correct answer is checked) Fig. 3 Example of a response code reason question based on the rule RC401 (rule on the left, violation on the right, correct answer is checked) Fig. 4 4Example API snippet from the introduction page, explanations were provided for the numbered elements Fig. 6 Fig. 7 67Comparison of duration between treatments rule and violation (lower is better) Comparison of TAU between treatments rule and violation (higher is better) Fig. 10 10Effect sizes for the 11 significant differences between TAU for treatments rule and violation, ordered by Cohen's d, insignificant PathHierarchy3 not included Table 1 1Experiment overviewGoal Study the impact of design rules on the understandability of Web APIs Study objects 12 design rules compiled from Massé (2011) and Richard- son and Ruby (2007), two functionally equivalent API snippets per rule (one follows the rule, one violates it) Participants 105 people with at least basic REST-related experience (both students and professionals, both from academia and industry) Setting Online experiment via LimeSurvey Tasks Answering comprehension question about API snippets for RQ1 (12 per participant), rating the difficulty to un- derstand an API snippet for RQ2 (12 per participant) Dependent variables Timed actual understandability (TAU) for RQ1, per- ceived difficulty rating for RQ2 Treatments API snippet follows a rule (version 1) or API snippet contains a violation of a rule (version 2) Other independent variables Demographic attributes like REST-related experience or current role (RQ3) Design Crossover design (2 counterbalanced sequences, each with 6 tasks for rule and 6 for violation) Table 2 2Selected design rules for RESTful APIs use a plural noun as the name in the URI. A violation of this rule would be to use a singular noun for a collection or store name instead.ID Rule Source Category PluralNoun A plural noun should be used for collection and store names Massé (2011) URI Design VerbController A verb or verb phrase should be used for controller names Massé (2011) URI Design CRUDNames CRUD function names should not be used in URIs Massé (2011) URI Design PathHierarchy (3 versions) Use path variables to encode hierarchy Richardson and Ruby (2007) Hierarchy Design NoTunnel GET and POST must not be used to tunnel other request methods Massé (2011) Request Methods GETRetrieve GET must be used to retrieve a representation of a resource Massé (2011) Request Methods POSTCreate POST must be used to create a new resource in a collection Massé (2011) Request Methods NoRC200Error 200 (OK) must not be used to communicate errors in the response body Massé (2011) HTTP Status Codes RC401 401 (Unauthorized) must be used when there is a problem with the client's credentials Massé (2011) HTTP Status Codes RC415 415 (Unsupported Media Type) must be used when the media type of a request's payload cannot be processed Massé (2011) HTTP Status Codes Rule: GET /groups/{groupId}/members Violation: GET /groups/{groupId}/member Table 3 3Null hypotheses with their alternatives for the two confirmatory RQs Web API snippets adhering to design rules are rated as less difficult to understand than snippets violating rules.RQ Metric Null Hypothesis Alternative Hypothesis RQ1 Timed actual understand- ability (TAU) H 1 0 : Web API snippets adher- ing to design rules are equally or less understandable than snippets violating rules. H 1 1 : Web API snippets adher- ing to design rules are more un- derstandable than snippets vi- olating rules. RQ2 Perceived dif- ficulty H 2 0 : Web API snippets adher- ing to design rules are rated as equally or more difficult to un- derstand than snippets violat- ing rules. H 2 1 : Table 4 4Task orders for the two experiment sequences# Rule Category Sequence 1 Sequence 2 1 VerbController URI Design violation rule 2 RC415 HTTP Status Codes violation rule 3 PathHierarchy2 Hierarchy Design rule violation 4 GETRetrieve Request Methods violation rule 5 PathHierarchy3 Hierarchy Design rule violation 6 NoRC200Error HTTP Status Codes violation rule 7 NoTunnel Request Methods rule violation 8 PluralNoun URI Design rule violation 9 POSTCreate Request Methods violation rule 10 PathHierarchy1 Hierarchy Design violation rule 11 CRUDNames URI Design rule violation 12 RC401 HTTP Status Codes rule violation Table 5 5Number of valid responses per individual task and treatment# Rule # of responses for rule # of responses for violation 1 VerbController 53 52 2 RC415 53 52 3 PathHierarchy2 52 53 4 GETRetrieve 53 52 5 PathHierarchy3 52 52 6 NoRC200Error 53 50 7 NoTunnel 50 52 8 PluralNoun 52 52 9 POSTCreate 53 51 10 PathHierarchy1 53 50 11 CRUDNames 51 53 12 RC401 51 53 using the effectsize package 14 . FollowingSawilowsky (2009), the values can be interpreted as follows:-d < 0.2: very small effect -0.2 ≤ d < 0.5: small effect -0.5 ≤ d < 0.8: medium effect -0.8 ≤ d < 1.2: large effect -1.2 ≤ d < 2.0: very large effect -d ≥ 2.0: huge effect Table 6 6Participant demographics by sequence (RMM: Richardson Maturity Model)Sequence # REST exp. in years (median) # of industry participants # of academic participants # of students # with RMM knowledge 1 52 3.5 31 (60%) 9 (17%) 12 (23%) 9 (17%) 2 53 4 28 (53%) 9 (17%) 16 (30%) 19 (36%) Total 105 4 59 (56%) 18 (17%) 28 (27%) 28 (27%) 100% vs. 94%, but 20.82 s vs. 32.23 s) or RC401 (75% vs. 70%, but 43.42 s vs. 67.21 s). The only surprising exception was the task PathHierarchy3, where participants with violation performed notably better (mean TAU of 0.6141 vs. 0.7505).96% 94% 75% 70% 98% 58% 98% 57% 100% 60% 66% 54% 83% 96% 100% 83% 98% 80% 96% 38% 94% 73% 100% 94% RC415 RC401 NoRC200Error POSTCreate GETRetrieve NoTunnel PathHierarchy3 PathHierarchy2 PathHierarchy1 CRUDNames VerbController PluralNoun 0% 25% 50% 75% 100% Correct answers Rule rule violation Fig. 5 Comparison of correctness between treatments rule and violation (higher is better) 50.1 58 43.4 67.2 41.4 58.4 25 43.9 28.4 45.3 44.8 52.6 42.7 37.5 23.6 46.6 22.9 44.2 27.1 41.6 33.1 45.2 20.8 32.2 RC415 RC401 NoRC200Error POSTCreate GETRetrieve NoTunnel PathHierarchy3 PathHierarchy2 PathHierarchy1 CRUDNames VerbController PluralNoun 0 20 40 60 Table 7 7Descriptive statistics for RQ2, perceived difficulty rating ranging from 1 (very easy) to 5 (very difficult), mean TAU provided for comparisonTask Median difficulty Mean difficulty Mean TAU rule violation rule violation rule violation PluralNoun 1 2.5 1.25 2.40 0.8369 0.7046 VerbController 2 3 2.13 2.50 0.7617 0.5345 CRUDNames 2 3 1.73 3.36 0.7392 0.1982 PathHierarchy1 1 3 1.34 2.94 0.8457 0.5678 PathHierarchy2 1 4 1.23 3.36 0.7916 0.4913 PathHierarchy3 2 2 2.31 2.42 0.6141 0.7505 NoTunnel 2 4 2.34 3.27 0.4820 0.3396 GETRetrieve 2 3 1.83 2.77 0.8095 0.4337 POSTCreate 1 3 1.45 2.75 0.8427 0.4201 NoRC200Error 2 3 1.75 2.80 0.7268 0.3776 RC401 2 4 1.80 3.45 0.6058 0.5170 RC415 2 2 2.00 2.35 0.7654 0.7044 Table 12 12in the appendix. The effect sizes for the significant tasks ranged from 0.79 (medium) to 2.06 (huge), with 8 rules producing a Cohen's d > 0.8 (large and higher). Violating the rules PathHierarchy2, RC401, PathHierarchy1, and CRUDNames had the strongest impact on difficulty perception.23 15 19 13 24 18 9 3 19 28 8 14 33 17 11 13 24 16 14 9 14 16 8 6 12 20 17 13 42 8 6 8 37 14 10 10 20 26 14 4 15 21 14 10 41 10 8 18 RC415 RC401 NoRC200Error POSTCreate GETRetrieve NoTunnel PathHierarchy3 PathHierarchy2 PathHierarchy1 CRUDNames VerbController PluralNoun 50 40 30 20 10 0 10 20 30 # of ratings per difficulty level Rule Very easy (rule) Easy (rule) Very easy (violation) Easy (violation) Table 10 10Extended descriptive statistics for response time (RQ1)Table 11 Hypothesis testing results for TAU (RQ1), Holm-Bonferroni adjusted p-values, α = 0.05, sorted by effect sizeFig. 14 Likert plots of perceived difficulty ratings (RQ2)Task Min duration (s) Max duration (s) Variance of duration rule violation rule violation rule violation Table 12 12Hypothesis testing results for perceived difficulty ratings (RQ2), Holm-Bonferroni adjusted p-values, α = 0.05, sorted by effect sizeTask Test statistic (U) p-value Cohen's d Accepted PathHierarchy2 2462.0 < 0.001 2.06 yes RC401 2297.5 < 0.001 1.57 yes PathHierarchy1 2203.0 < 0.001 1.53 yes CRUDNames 2254.5 < 0.001 1.52 yes POSTCreate 2076.0 < 0.001 1.18 yes PluralNoun 2069.5 < 0.001 1.18 yes NoRC200Error 1851.0 < 0.001 0.94 yes GETRetrieve 1972.5 < 0.001 0.87 yes NoTunnel 1839.5 < 0.001 0.79 yes VerbController 1676.0 0.0701 - no RC415 1642.5 0.0774 - no PathHierarchy3 1388.0 0.4051 - no Table 13 13Correlation between TAU and perceived difficulty for the treatment violation, sorted by correlation strength, Holm-Bonferroni adjusted p-values, α = 0.05, insignificant correlations are marked with †, difficulty ratings and TAU provided for comparison https://doi.org/10.5281/zenodo.7381500 https://www.twitter.com 3 https://www.linkedin.com 4 https://www.xing.com 5 https://www.reddit.com 6 https://www.unicef.org 7 https://apis.guru https://www.openapis.org 9 https://editor.swagger.io 10 https://www.limesurvey.org https://www.r-project.org See, e.g., the mapping study on code comprehension experiments byWyrich et al. (2022): median number of participants was 34 (for journals alone, it was 61) https://doi.org/10.5281/zenodo.7381500 Acknowledgements We kindly thank all our experiment participants for their valuable time! We also thank the experts participating in our pilot for their detailed feedback! Lastly, we thank Dr. Daniel Graziotin (University of Stuttgart) for discussing the experiment design and its terminology with us.Appendix Sampling in software engineering research: A critical review and guidelines. S Baltes, P Ralph, https:/link.springer.com/10.1007/s10664-021-10072-8Empirical Software Engineering. 2747764Baltes S, Ralph P (2022) Sampling in software engineering research: A critical review and guidelines. 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Berlin, HeidelbergSpringer Berlin HeidelbergWebber J, Parastatidis S, Robinson I (2010) REST in Practice: Hypermedia and Systems Architecture, 1st edn. O'Reilly Media, Inc., Sebastopol, USA Wohlin C, Runeson P, Höst M, Ohlsson MC, Regnell B, Wesslén A (2012) Planning. In: Experimentation in Software Engineering, Springer Berlin Hei- delberg, Berlin, Heidelberg, pp 89-116, DOI 10.1007/978-3-642-29044-2 8 40 years of designing code comprehension experiments: A systematic mapping study. M Wyrich, J Bogner, S Wagner, 10.48550/ARXIV.2206.11102Wyrich M, Bogner J, Wagner S (2022) 40 years of designing code compre- hension experiments: A systematic mapping study. DOI 10.48550/ARXIV. 2206.11102
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[ "Sparse random matrices and Gaussian ensembles with varying randomness", "Sparse random matrices and Gaussian ensembles with varying randomness" ]
[ "Takanori Anegawa \nDepartment of Physics\nOsaka University Toyonaka\n560-0043OsakaJAPAN\n", "Norihiro Iizuka \nDepartment of Physics\nOsaka University Toyonaka\n560-0043OsakaJAPAN\n", "Arkaprava Mukherjee \nDepartment of Physics\nThe Ohio State University Columbus\n43210OHUSA\n\nDepartment of Theoretical Physics\nTata Institute of Fundamental Research Colaba\n400 005MumbaiINDIA\n", "Sake ♠,♦Sunil Kumar \nDepartment of Physics\nOsaka University Toyonaka\n560-0043OsakaJAPAN\n\nDepartment of Theoretical Physics\nTata Institute of Fundamental Research Colaba\n400 005MumbaiINDIA\n", "Sandip P Trivedi \nDepartment of Theoretical Physics\nTata Institute of Fundamental Research Colaba\n400 005MumbaiINDIA\n" ]
[ "Department of Physics\nOsaka University Toyonaka\n560-0043OsakaJAPAN", "Department of Physics\nOsaka University Toyonaka\n560-0043OsakaJAPAN", "Department of Physics\nThe Ohio State University Columbus\n43210OHUSA", "Department of Theoretical Physics\nTata Institute of Fundamental Research Colaba\n400 005MumbaiINDIA", "Department of Physics\nOsaka University Toyonaka\n560-0043OsakaJAPAN", "Department of Theoretical Physics\nTata Institute of Fundamental Research Colaba\n400 005MumbaiINDIA", "Department of Theoretical Physics\nTata Institute of Fundamental Research Colaba\n400 005MumbaiINDIA" ]
[]
We study a system of N qubits, at large N , with a random Hamiltonian obtained by drawing coupling constants from Gaussian distributions in various ways. This results in a rich class of systems which include the GUE and the fixed q SYK theories. Starting with the GUE, we study the resulting behaviour as the randomness is decreased. While in general the system goes from being chaotic to being more ordered as the randomness is decreased, the changes in various properties, including the density of states, the spectral form factor, the level statistics and out-of-time-ordered correlators, reveal interesting patterns. Subject to the limitations of our analysis which is mainly numerical, we find some evidence that the behaviour changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in N . We also study the opposite limit of much reduced randomness obtained in a local version of the SYK model where the number of couplings scales linearly in N , and characterise its behaviour. Our investigation suggests that a more complete theoretical analysis of this class of systems will prove quite worthwhile. arXiv:2305.07505v1 [hep-th] 12 May 2023
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[ "https://export.arxiv.org/pdf/2305.07505v1.pdf" ]
258,676,510
2305.07505
f385a3422c53a9938cd2a793083cdc3ca70ef200
Sparse random matrices and Gaussian ensembles with varying randomness Takanori Anegawa Department of Physics Osaka University Toyonaka 560-0043OsakaJAPAN Norihiro Iizuka Department of Physics Osaka University Toyonaka 560-0043OsakaJAPAN Arkaprava Mukherjee Department of Physics The Ohio State University Columbus 43210OHUSA Department of Theoretical Physics Tata Institute of Fundamental Research Colaba 400 005MumbaiINDIA Sake ♠,♦Sunil Kumar Department of Physics Osaka University Toyonaka 560-0043OsakaJAPAN Department of Theoretical Physics Tata Institute of Fundamental Research Colaba 400 005MumbaiINDIA Sandip P Trivedi Department of Theoretical Physics Tata Institute of Fundamental Research Colaba 400 005MumbaiINDIA Sparse random matrices and Gaussian ensembles with varying randomness OU-HET-1149 TIFR/TH/23-6 We study a system of N qubits, at large N , with a random Hamiltonian obtained by drawing coupling constants from Gaussian distributions in various ways. This results in a rich class of systems which include the GUE and the fixed q SYK theories. Starting with the GUE, we study the resulting behaviour as the randomness is decreased. While in general the system goes from being chaotic to being more ordered as the randomness is decreased, the changes in various properties, including the density of states, the spectral form factor, the level statistics and out-of-time-ordered correlators, reveal interesting patterns. Subject to the limitations of our analysis which is mainly numerical, we find some evidence that the behaviour changes in an abrupt manner when the number of non-zero independent terms in the Hamiltonian is exponentially large in N . We also study the opposite limit of much reduced randomness obtained in a local version of the SYK model where the number of couplings scales linearly in N , and characterise its behaviour. Our investigation suggests that a more complete theoretical analysis of this class of systems will prove quite worthwhile. arXiv:2305.07505v1 [hep-th] 12 May 2023 Introduction and Motivation Random Matrix theory (RMT) is well known to have interesting connections with two-dimensional gravity. See for review [1,2]. More recently an interesting and new version of this connection has been discovered in the study of two-dimensional Jackiw-Teitelboim (JT) [3,4] gravity. It has been shown that the key features of JT gravity are correctly reproduced by the low-energy limit of the Sachdev-Ye-Kitaev (SYK) model, which is a quantum mechanical model of N flavors of Majorana fermions with random couplings [5][6][7][8]. Furthermore it was shown in [9] that the partition function of JT gravity on surfaces with an arbitrary number of boundaries and handles is correctly reproduced by Random Matrix theory in a suitable double scaling limit. Since JT gravity arises quite universally as a description of the low-energy dynamics for a wide class of higher dimensional near extremal black holes [10,11], these fascinating results promise to hold general lessons for the study of quantum gravity and black holes in higher dimensions as well. The Hamiltonian of the SYK model with a ψ q coupling (with q being an even number) is given by H SYKq = i q/2 1≤i 1 <i 2 <···iq≤N j i 1 i 2 ···iq ψ i 1 ψ i 2 · · · ψ iq ,(1.1) Here ψ i , i = 1, · · · N, are the N flavors of Majorana fermions and j i 1 i 2 ···iq are real random couplings drawn from a Gaussian ensemble with variance j i 1 i 2 ···iq , j j 1 j 2 ···jq = (q − 1)!J 2 N q−1 δ i 1 ,j 1 δ i 2 ,j 2 · · · δ iq,jq (1.2) We refer to this model as SYK q model. Quite remarkably, this model reproduces many aspects of JT gravity, as was mentioned above. In particular, the low-energy dynamics of the SYK model exhibits a characteristic pattern of symmetry breaking, tied to how time reparametrisation symmetry is realised. This gives rise to soft Goldstone-like modes whose dynamics is governed by the Schwarzian action. The behaviour of this action accounts for the thermodynamics and low-energy density of states and the resulting dynamics gives rise to Out-of-Time-Ordered Correlators (OTOCs) in the system saturating the chaos bound. These features -the pattern of symmetry breaking, the Schwarzian action for the time reparametrizations, the resulting thermodynamics and the behaviour of OTOCs -are all shared by JT gravity. As we will discuss in detail below, the fermions ψ i can be realised as operators in an L = 2 N/2 dimensional Hilbert space obtained by taking a tensor product of Hilbert spaces of N/2 qubits and the Hamiltonian H can then be thought of as a L × L dimensional Hermitian matrix acting on this Hilbert space. Such a matrix in general has L 2 real independent elements. Taking all these to be independent Gaussian random variables with zero mean and equal variance, gives rise to the Gaussian Unitary Ensemble (GUE) of Random Matrix Theory (RMT). Its behaviour, even at low-energies, is quite different from that of JT gravity. In contrast, in the SYK model, even though the random variables present are independent Gaussian variables, as was mentioned above, their total number is only O(N q ) which is much smaller than L 2 -the number of random variables present in the GUE. These observations raise several interesting questions: How much randomness is needed for agreement with gravity? What happens when we start from the GUE and begin reducing the randomness, by decreasing the number of Gaussian random variables? Is the resulting behaviour, at low-energies, dependent on only the number of random variables or also on which variables have been retained? When do we get the behaviour at low-energies to agree with JT gravity? Etc. The current investigation is prompted by questions such as these and will attempt to address some of them. Briefly, we follow two lines of investigation here. Our analysis is mostly numerical. In the first investigation, starting with the GUE we reduce the randomness by retaining only n < L 2 of the matrix elements to be non-zero and taking each of these n elements to be independent Gaussian random variables. We find that the behaviour changes in a fairly pronounced manner as the randomness reduces and n becomes smaller. Subject to the limitations of our numerical analysis, we also find that this change occurs in quite a small interval, when n reaches a value, n c ∼ L (1.5) , so that the fraction of random matrix elements n c /L 2 ∼ L (−0.5) → 0, as L → ∞. We study several properties of the system as n is varied, including the density of states, thermodynamics, the nearest neighbour spacing distribution, spectral form factors and Out of Time Correlators and find that these reveal interesting difference with both the GUE and the SYK cases. For example, initially, starting from the GUE, as n is reduced, we find that the density of states is well fitted to a rescaled Wigner semi-circle distribution. After the transition though, this is no longer true. The second line of investigation involves going to the other limit, and now having very little randomness. In this analysis we study a "local" version of the SYK model. In the Hamiltonian eq.(1.1) the Hamiltonian connects all the N fermions to each other through the ψ q couplings which take fluctuating values. In the altered system we study -which we often refer to as the local SYK model below -only sets of q nearest neighbour fermions will be coupled together by random couplings. This reduces the number of random couplings to O(N ), instead of the O(N q ) in the SYK model. To differentiate our model with the more conventional one described above, eq.(1.1), we will refer to the latter sometimes as the "Non-local" SYK model. Once again we study several properties of the local system, along the lines of the first investigation mentioned above, and find these compare and contrast in interesting ways with other cases. Two comments are worth making before we proceed. In the double scaling limit considered in [9], which gives agreement with JT gravity, we have to fine-tune the potential V (M ), and furthermore "zoom in" on the edge of the matrix theory potential. This means while all L 2 random variables in the Hermitian matrix M are retained, its matrix elements are not independent Gaussian Random variables. In contrast in our study, as was mentioned above, we reduce the number of random variables, but those which are retained are taken to be independent Guassian random variables. Second, there are other variations of the SYK model which have also been considered in the literature. These involve taking q → ∞, along with N → ∞. The case where N → ∞ first, then q → ∞, was studied in [7]. In [28,30] a double scaled limit was considered in which q 2 /N is held fixed while q, N → ∞. These limits are also quite revealing and we will make some comments about them below. More on the Relation Between the SYK Model and RMT: It is helpful at this stage to give some more details on how the N -flavor SYK model can be realised in the Hilbert space obtained from the tensor product of N/2 qubits; this will help clarify the relation between the SYK model and RMT. The Majorana fermions ψ i , i = 1, · · · N of the SYK model satisfy the Clifford algebra {ψ i , ψ j } = δ ij . (1.3) On the Hilbert space obtained by taking the tensor product of N/2 qubits, this algebra can be realised by taking the following representation for the fermions in terms of tensor products of Pauli matrices: ψ 2i−1 = 1 √ 2 σ z 1 ⊗ σ z 2 ⊗ . . . σ z i−1 ⊗ σ x i ⊗ 1 ⊗ 1 . . . ⊗ 1N 2 , ψ 2i = 1 √ 2 σ z 1 ⊗ σ z 2 ⊗ . . . σ z i−1 ⊗ σ y i ⊗ 1 ⊗ 1 . . . ⊗ 1 N 2 ,(1.4) where σ x , σ y , σ z are the usual Pauli matrices and 1 is 2 × 2 identity matrix. Note that in this representation the ψ i are L × L matrices where L = 2 N/2 . (1.5) It follows then that a general Hermitian matrix acting on the Hilbert space of the N/2 qubits can be written as the sum over various ψ q terms, with q taking both even and odd values: H = N q=0 α q i 1 <i 2 <···<iq j i 1 i 2 ···iq ψ i 1 ψ i 2 · · · ψ iq (1.6) where α q = e iπq(q−1) 4 (1.7) and j i 1 ···iq is real. The matrix eq.(1.6) has N q=0 N C q = 2 N = 2 N/2 × 2 N/2 = L 2 (1.8) number of real independent couplings due to binomial theorem, which is equal to the total number of independent elements in an L × L Hermitian matrix. It is also easy to see that if we now draw all the couplings j i 1 ···iq which appear on the RHS in eq.(1.6) from a Gaussian random ensemble with equal variance j i 1 i 2 ···iq , j j 1 j 2 ···jq = σ 2 δ i 1 ,j 1 δ i 2 ,j 2 , · · · δ iq,jq , (1.9) i.e., with a variance σ 2 independent of q and the coupling j i 1 i 2 ,···iq , then the resulting ensemble for H is the GUE. This follows from noting that the set of matrices α q ψ i 1 ψ i 2 · · · ψ iq form a basis in which any Hermitian matrix can be expanded. Furthermore, this basis is orthonormal with respect to the inner product, M 1 , M 2 = 1 L Tr(M † 1 M 2 ). Restating the GUE in this way allows for a better comparison with the SYK model. As was mentioned above, with fixed value of q, the SYK model has only N C q ∼ O(N q ) random couplings. In addition we note that in the SYK model, eq.(1.2) the variance (for a fixed value of q) is taken to scale as 1/N q−1 , to ensure that the model has a good large N limit. This is very different from the GUE where, as mentioned above, besides including all powers of q we also take the variance to be independent of q. A key difference in the resulting behaviour as was emphasised above, between the GUE and the SYK model at fixed q, is in the low energy dynamics, which for the ψ q case arises from the dynamics of the Schwarzian mode. More recently, as has been discussed in [28,30], even when q is not fixed but instead scales like N a , where a ≤ 1/2, as N → ∞, one continues to get the Schwarzian mode and its low-energy dynamics. At q = N 1/2 there are O(N √ N 2 ) number of random couplings. It is not clear using analytic techniques what happens when the number of random couplings becomes even bigger. Here we are finding that when the number n becomes sufficiently big, n > n c ∼ L 1.5 ∼ 2 0.75N , the behaviour is of GUE type with no Schwarzian dynamics. For n < n c , but still much bigger that O(N √ N 2 ), the behaviour changes and is qualitatively different from the GUE. Unfortunately, with the limitations of our analysis, we are not able to establish the precise nature of the low-energy behaviour, in particular whether or not a Schwarzian mode is present for such values of n. At the other extreme, in terms of having less randomness, one can take the ψ q theory, for fixed q, as discussed in [14] and reduce the number of couplings by setting some of them to vanish at random. It was found, as we will also review in appendix A, that as long as the number of couplings present still grows like N α , with α > 1, i.e., the number of couplings grows faster than a linear power of N , the low-energy theory will be due to the Schwarzian mode. Together, these results imply that the Schwarzian dynamics at low energies is in fact quite universal -arising for any fixed value of q, or in double scaling limits as long as q ≤ √ N , and persisting in any of these cases even with vastly reduced randomness as long as the total number of couplings present are of order N (1+ ) , for > 0. What happens when the randomness is reduced even further, so that the number of couplings grows only linearly with N ? The local SYK model is an example of such a situation. Our analysis suggests that interesting differences set in, in the linear case. In particular, our analysis suggests that the Schwarzian mode is absent. Many of our comments have focussed on the low energy dynamics and the presence of a Schwarzian theory, because they are of primary interest from the point of view of connecting with a theory of gravity. However, it is worth mentioning that the detailed spectrum, beyond the low-energy limit, and many other properties, are of course much less universal and will change. This is true as n, the number of non-zero matrix elements in H, is varied. Or as the power q in the SYK model is varied, or in SYK models as the number of non-vanishing couplings for fixed q is varied. Even when the Schwarzian mode is present, the detailed nature of the conformal fixed point at low energies changes as q is varied, with the fermion having anomalous dimension 1/q. Interestingly though, for fixed q, when we reduce the number of random couplings, keeping the total number to grow faster than O(N ), the anomalous dimension of the fermion fields is still given by 1/q and does not change. One more comment is in order. We had mentioned above that among the properties we study are the OTOCs and the growth of operator size, which tell us about the scrambling behaviour. Using the mapping above, eq.(1.4), we see that the fermionic operators are defined in general for any system in an L = 2 N/2 dimensional Hilbert space, including RMT. One can therefore use these fermionic operators, for example, to study the growth of OTOC, or the growth of operator size. In particular, we will often consider the OTOC which involves a sum over all flavours and is given by, [6][7][8], N i,j=1,i =j ψ i (t)ψ j (0)ψ i (t)ψ j (0) ∼ 1 − 1 N e λt , λ = 2π β (1.10) This saturates the chaos bound with a maximal Lyapunov exponent in the fixed q SYK model. In contrast, the behaviour of the same correlator in the GUE is very different, and is that of a "hyperfast scrambler" as discussed further below. Organisation: The paper is organised as follows. In section 2 we review some aspects of the SYK model, the behaviour of the GUE and the connection between these two systems. In section 3 we report on our investigation of L × L matrices with reduced randomness and in section 4 on the local version of the SYK model. We end with discussion in section 5. Before we end this introduction section, let us comment on some additional related literature. Our local SYK model belongs to the so-called "sparse" SYK models, where the number of random couplings is reduced and which have been actively studied recently [13][14][15][16]. Our local SYK model is based on the locality of fermion interactions. The question of how universal these sparse SYK models are, compared with usual SYK model, is interesting and needs to be investigated more 1 . Convention in figures: In all figures, whenever we denote log x, we choose its basis 10, i.e., log x = log 10 x. On the other hand, for ln x, its basis is e, i.e., ln x = log e x. 2 The GUE and SYK q Theories: We briefly summarise the behaviour of the GUE and the SYK ψ q models here. For concreteness in the numerical plots we take q = 4. Partition functions, eigenvalue densities and thermodynamics The partition function for the GUE, Z GU E is given by Z GU E = 1 Z 0 i dM ii i<j dR[M ij ]dI[M ij ] exp − 1 2σ 2 TrM 2 ,(2.1) Here M are L × L Hermitian Matrices, Z 0 is a normalisation constant, and σ is given by σ = 1 √ L . (2.2) The matrix integral in eq.(2.1) is invariant under the U (N ) symmetry M → U † M U . Using this, we can obtain the density of eigenvalues ρ L (λ), Z GU E = L dλ ρ L (λ) , where ρ L (λ) = 1 √ 2L K L L 2 λ ,(2.3) and K L (x) is given in terms of Hermite polynomials H j as K L (x) = e −x 2 2 L (L − 1)! √ π (H L (x)H L−1 (x) − H L−1 (x)H L (x)) . (2.4) The expectation values of single trace operators, Tr(M n ), can be computed using ρ L (λ) as Tr(M n ) = dλ ρ L (λ)λ n dλ ρ L (λ) . (2.5) In the L → ∞ limit, K L (λ) -the Kernel function -becomes the famous Wigner semi-circle distribution, lim L→∞ K L (λ) = 1 π √ 2L − λ 2 (2.6) leading to ρ W (λ) ≡ lim L→∞ ρ L (λ) = 1 2π √ 4 − λ 2 , (2.7) which is normalised to unity, The eigenvalue density ρ L (λ) for the GUE with L = 4096, N = 24 and the density of states for SYK model with q = 4, J = 1, eq. (1.2), are shown in Fig 1. Qualitatively the eigenvalues are more "clumped" together in the SYK model towards the centre and the density is correspondingly smaller towards the edges, compared to the GUE case. One can heuristically understand this as arising in the SYK model due to the reduced randomness which causes smaller eigenvalue repulsion. Reducing the rank L for the GUE gives rise to oscillations, which grow as L becomes smaller. Thermodynamics As we have discussed in the introduction, regarding the random matrix M as Hamiltonian H H = M , (2.9) one can interpret λ as energy eigenvalue, and ρ W (λ) as the density of states. Throughout the paper we will refer to the normalised density of states as ρ, where it satisfies the condition dλρ(λ) = 1. Then the thermal partition function 2 of the GUE in the L → ∞ limit is Z(T ) = L 2 −2 dλ ρ W (λ)e −βλ = Lβ −1 I 1 (2β) (2.10) ∼ L T 3/2 exp 2 T (1 + O(T )) ,(2.11) where I 1 is the modified Bessel function of the first kind and the second line is the T → 0 behaviour, and the resulting entropy at low temperature is given by S = 1 + T ∂ ∂T ln Z(T ) ln L + 3 2 ln(T ) + O(T 0 ) . (2.12) The T → 0 features are due to the density of states going like ρ W → 1 π √ 2 + λ near the lower edge of the spectrum at λ = −2. Note that the temperaturedependent part of the entropy is negative indicating that thermodynamics is not a good approximation in the very low temperature limit. In fact one can show that thermodynamics is not a good approximation at any temperature in this model. Thermodynamics corresponds to a saddle point in the evaluation of the partition function (2.10), which here takes the form, Z(β) = L 2 −2 dλ 2π e [ln( √ 4−λ 2 )−βλ] . (2.13) It is easy to see that there is no good saddle point approximation to this integral, since L is an overall factor and we do not have large N parameter here. In contrast, the behaviour of the SYK ψ q model is very different. Denoting the energy difference from the ground state by E we have at large N that the density of states for energies E J is given by [7,17,18,29] ρ(E) ∝ e S 0 sinh cN E J (2.14) where c is a positive O(1) number (for q = 4, c ≈ 0.396). Near E → 0, ρ ∝ E/J which is analogous to the Wigner distribution near the edge λ → −2. However, in the range 1 E J 1 N (2.15) the partition function has a good saddle point approximation (for N 1) Z(β) = e S 0 dEρ(E)e −βE ∝ e S 0 1 β 3/2 e cN 4βJ (2. 16) and is given by the second line on the RHS above. As a result thermodynamics is a good approximation in this regime. One can also see this from the entropy S = S 0 + cN T 2J + 3 2 ln T (2.17) which is positive and with a resultant positive specific heat which is linear in T . The constant S 0 scales with N , S 0 = c 0 N (2.18) where c 0 is also an O(1) constant. The presence of this constant is a characteristic feature of the SYK model. Although for any finite N the ground state degeneracy goes to zero, in the temperature range eq. (2.15), the system behaves as if, in effect, it has a big "ground state" degeneracy, since eigenvalue differences are O(e −S 0 ) which is very tiny compared to the range eq. (2.15). Most of the above features arise from the Schwarzian action. A plot of the entropy S as a function of T for the GUE and SYK models is shown in Spectral form factor (SFF) for RMT and SYK model This subsection briefly reviews the work of [29]. The spectral form factor is an important quantity to study from the perspective of chaos. It is defined to be , SFF ≡ g(t, β) ≡ Z(β + it)Z(β − it) Z(β) 2 (2.19) and it can be divided into a disconnected and connected piece, g d (t, β) = Z(β, t) Z(β, t) * Z(β) 2 (2.20) g c (t, β) = g(t, β) − g d (t, β) (2.21) where Z(β, t) = Z(β + it). In the rest of the paper, we call g(t, β) as total SFF, g d (t, β) as disconnected SFF, and g c (t, β) as connected SFF respectively. We note that the SFF can be expressed in terms of the two point correlator ρ(λ 1 )ρ(λ 2 ) Z(β + it)Z(β − it) = dλ 1 dλ 2 e −β(λ 1 +λ 2 ) e −it(λ 1 −λ 2 ) ρ(λ 1 )ρ(λ 2 ) (2.22) This shows that as t increases the SFF is sensitive to the two point density correlator at smaller separations in the eigenvalues. Interestingly the SFF of the SYK model agrees with RMT [29]. Depending on whether N (mod 8) is 2 or 6, 0, 4, respectively, the SFF of SYK model agrees with that of the GUE, GOE or GSE [29]. As a function of time t there are three basic features seen in the SFF: an initial decay, dominated by the disconnected part; a subsequent ramp due to the connected part and a final plateau. The disconnected part in GUE is given by Z(β + it) ≡ Tr e −βH−iHt = L β + it I 1 (2(β + it)) (2.23) as a result, at late times, t → ∞, (2.24) so that both the real and imaginary part of Z(β +it) decay with time, leading to Z ∼ L 2π(β + it) 3/2 e β+it ∼ L t 3/2 e β+itg d (β = 0, t) ∼ 1 t 3 (2.25) In contrast to the disconnected piece, the connected part grows with time, g c (β = 0, t) ∼ t 2πL 2 (2.26) for 1 t < 2L The connected and disconnected contributions therefore become equal at a time of order t dip ∼ √ L (2.27) which is called dip time. Thereafter the connected part dominates and the SFF exhibits a ramp-like character growing linearly with t. Finally at time t ∼ 2L, g c reaches a plateau attaining a constant value g c (β = 0, t) = 1 πL (2.28) The plateau region arises due to nearest neighbour eigenvalues repelling each other. Compared with the plateau region, the ramp region is sensitive to interactions between eigenvalues which are further apart. As time increases, the SFF becomes more sensitive to the eigenvalue interactions between the closer ones. All of these features can be seen in Fig 4 which also show, [29] that the SFF in RMT and SYK are very similar. We also note that the behaviour of the disconnected piece of the SFF and the ramp region in the connected piece, for the SYK model, are in good agreement (for sufficiently large β) with the JT theory where they correspond to the behaviour of the disk and double trumpet partition functions, respectively [21]. Nearest Neighbour Level Spacing for RMT and SYK model The nearest neighbour level spacing is also well known to be a good diagnostic of chaos. For GOE this spacing takes the form of the well known Wigner surmise. The probability of the spacing taking value s, P (s) is given by P (s) = π 2 se − πs 2 4 (2.29) In Fig 5 we plot this probability distribution for the GOE and the SYK q = 4, N = 24, models. We see very good agreement with the Wigner surmise, [29]. Note that these plots have been obtained after unfolding the spectrum, as described in [27], see also section 3.2 below. The comparison is made with the GOE since the number of flavours, N = 24, is divisible by 8, see [27]. OTOC Finally we turn to OTOCs. These are defined in eq.(1.10) above. More precisely, we consider the normalized OTOCs F ij (t) = Tr(yψ i (t)yψ j (0)yψ i (t)yψ j (0)) Z ψ j (0)ψ j (0) ψ i (t)ψ i (t) , y = e − βH 4 (2.30) where Z is the partition function, Z = Tre −βH and ψ j (0)ψ j (0) = Tr(e −βH ψ j (0)ψ j (0)) Z , ψ i (t)ψ i (t) = Tr(e −βH ψ i (t)ψ i (t)) Z . (2.31) In the non-local SYK model, the OTOCF ij (t) is same for all pairs as far as i = j, since all the flavour of fermions are connected to each other with random couplings j ijkl of the same strength. Numerically, we calculate the OTOC by averaging over the pairs (i, j) for i = j and denote this quantity asF (t); since the OTOC is independent of the choice of flavour, this average should be the same as that for any particular values of (i, j). Below we plot the function The plots for the non-local SYK model are obtained as follows, see also appendix E. The dots in the left hand panel are the data, obtained with an ensemble of 2 elements. The solid curves are obtained by fitting the initial region (up to a maximum time t max ) for each β with a curve of the form A + Be λt . The best fit value of λ obtained in this manner is then plotted against β in the right hand panel. The maximum time t max up to which we fit the OTOC with the exponential form is obtained by starting with an initial range for t where this form is a good fit and then extending this range to the largest time, t max , where the exponential fit continues to be a good one. This was done by first fitting the data naively with "the eye" and then, more systematically, by using a chi-square distribution to estimate the goodness of the fit, as discussed in appendix E. In the right side panel the error bars for each data point corresponds to the 1σ errors obtained by fitting the OTOC up to t max . The red curve in this panel corresponds to λ = 2π β , and the black curve is obtained by fitting the data with a curve of the form λ = A β + B. The best fit gives B = 0 and A = 6.5, which is in reasonable agreement with the chaos bound exponent 2π β . Strictly speaking, the Lyapunov exponent is defined in the large N limit. Obtaining it in this limit numerically though is quite challenging, see [31]. From the right hand panel above we see that while working at a moderately large value of N , our analysis still allows for a fairly reliable estimate of the Lyapunov exponent. Theoretically, one can show in the large N limit that G 4 takes the form G 4 = A N e 2πt β (2.33) where A is a coefficient of order unity determined by the fermionic fields. This shows that G 4 reaches its asymptotic value of order unity at a scrambling time t ∼ β 2π ln N. (2.34) G 4 is suppressed by a factor of 1/N because the non-local SYK model has a saddle point in the large N limit, and the value of the action at the saddle point scales like N . We see that as a result the scrambling time is parametrically larger than β by ln N . Random matrix OTOC, L=4096 The OTOC for RMT is plotted in Fig 7. In appendix B, eq. (B.13), we show that in the L → ∞ limit, G 4 is given by G 4 = 1 − F (t) F (0) (2.35) where In Fig. 7, we show this function as a solid curve for three different values of β and also the numerically obtained data for L = 4096, which is in reasonable agreement. F (t) = J 1 2 t − i β 4 t − i β4 The important point to note is that G 4 reaches its asymptotic value 1 in a time scale of order unity, not scaling with L. This shows that the GUE exhibits "hyperfast" scrambling. As was noted in [32] the thermalisation time scale is also of order unity and therefore comparable to this scrambling time. This in contrast to what we saw above for the non-local SYK model where the scrambling time scales like β ln N . The underlying reason for this hyperfast scrambling is the U (L) symmetry which is preserved by the GUE, see appendix B for further discussion and also in [32]. There is no notion of "simple operators" which is invariant under this symmetry. The ψ i operators under a general U (L) transformation can be turned into arbitrarily complicated operators involving up to N factors of the ψ's. States prepared by acting with such complicated operators are already scrambled at the get-go, resulting in the hyperfast scrambling in the GUE. Sparse Random Matrices In this section, we shall explore the consequences of considering random matrices which are not fully random in the sense that not all elements of the matrix are randomly drawn numbers. We refer to this more general class of matrices as sparse random matrices. The reduction of randomness is achieved as follows. We set the diagonal matrix elements M ii to vanish. We then pick randomly and uniformly, n off-diagonal matrix elements, M ij , with i < j. The value of each of these matrix elements is chosen independently with its real and imaginary parts being drawn from a Gaussian distribution with variance 1 2L . All other off-diagonal matrix elements of M are also set to zero. Finally the matrix is made Hermitian by taking it to be 1 2 (M + M † ). The behaviour of the resulting random ensemble we obtain in this way, as a function of the number of random elements n, is then studied below numerically. We now present results of numerical analysis for various quantities such as density of states, specific heat, level spacing and spectral form factor for the resulting ensemble. The density of states In the plots below, the density of states (DoS) is normalized to unity unless specified otherwise, i.e., dλρ(λ) = 1 . (3.1) The plot for a matrix of size L = 4096, as n (≤ 1.6×10 7 ) is varied, is shown below. The fully random case corresponds to the GUE with L = 4096. We see that there is a general tendency for the eigenvalues to get more concentrated towards the centre, as n decreases, i.e., the sparseness is increased, indicating that the repulsion between eigenvalues decreases. However starting from the GUE there is an interesting change in behaviour as n continues to decrease. Initially, when n is not too small, corresponding to the values, n ≥ 1.5 × 10 5 , i.e., , n L 2 ≥ 9 × 10 −3 , in Fig. 8 (a), the density of states can be fitted to a Wigner semicircle, after rescaling the energy. We find, ρ(E) = 2 B ρ W 2E B , (3.2) where ρ W is the Wigner semicircle, eq. (2.7). The parameter B scales with n in a power law fashion, B ∼ n L 2 0.45 , as can be seen in Fig. 9. However once the randomness becomes very small, n ≤ 1.5 × 10 4 , i.e., n L 2 ≤ 9 × 10 −4 , in Fig. 8 (b), there is an important change. The effect of the repulsion among eigenvalues decreases to a point where the eigenvalues now like to clump much more at the center, near E = 0, resulting in a sort of cusp at E = 0. This is reminiscent of a condensate forming in bosonic systems. As n decreases further, the clumping and resulting cusp at E = 0 get more pronounced. The transition to the regime where the Wigner distribution stops describing ρ occurs over a fairly narrow range in n. From Fig 8 we see that the value n = 1.5 × 10 5 is well fitted with ρ W , but this is no longer true for n = 1.5 × 10 4 . It would be very interesting to find out how sudden is this transition, as L → ∞, but this is not easy to analyse numerically. Let us denote by n c the approximate value of n which separates the two regimes. One would like to understand how n c behaves as L → ∞; a natural ratio to consider is n c /L 2 which is the fraction of total matrix elements that are non-zero. This ratio is plotted in Fig. 9 below as a function of 1/L. We find that, subject to the limitations of our numerical analysis, an approximate fit is given by Note that in Fig. 10 the solid line is the best fit obtained after discarding the last point (at the value of log L = 4.623). n c ∝ L 1.52 , n c L 2 ∝ L −0.48 (3.3) We leave a better understanding of this transition for the future. Comments on Various Properties Here we discuss some of the properties of ensemble obtained above with reduced randomness. Thermodynamics The density of states determines the finite temperature behaviour of the thermal partition function. Since for n > n c , ρ is well fitted by ρ W , in this case the thermodynamics will agree with that of the GUE, after an appropriate rescaling of energy or temperature. However once n < n c this will no longer be true. The resulting behaviour of the entropy S at low temperatures is shown in Fig. 11. We have not been able to obtain a good analytical fit to this result. Note that in this case n c lies in between 1.5 × 10 4 and 1.5 × 10 5 . It is interesting to contrast the behaviour of entropy, S, for the random sparseness case with that of the GUE and the SYK 4 theory shown in Fig 3. In particular, for the SYK 4 model, in the large N limit, we saw that there is a range of temperatures, eq.(2.15), where a saddle point approximation is valid resulting in the entropy behaving linearly with temperature, eq. (2.17). One would like to study if something similar happens for the random sparseness case, for n < n c , L → ∞. This is however difficult to do numerically. Figure 11: The Entropy at low temperature for varying values of n. Here we plotted only the temperature dependent part of the entropy, after subtracting log L. Spectral Form Factor (SFF) We now turn to the spectral form factor, defined in eq. (2.19), with connected and disconnected pieces given in eq. (2.20), (2.21). We shall analyse how the SFF changes with changing randomness, n. We recall, eq. (2.22), that the SFF is determined by the two-point density correlator and therefore is sensitive to the interactions between the eigenvalues. The numerical results for the random sparseness case are in Fig. 12, again for L = 4096. We take β = 5 and consider varying values for n. The plots show much richer behaviour, compared to the GUE, Fig.4. In Fig. 12 besides the fully random case, the SFF is plotted for 5 other values of n. Of these n = 138240, is roughly comparable to n c , all other values of n lie in the range n < n c , where ρ W is no longer a good fit. Generally speaking, differences in the SFF compared with the GUE grow as n decreases. However, it is interesting to note that the n = 138240 case already shows differences with the GUE. Figure 12: Left: disconnected, middle: connected, right: total spectral form factor with changing number of random elements, parametrised by n, for a matrix of rank 2 12 = 4096. We have taken an average over 300 ensembles for each cases. Here n c ∼ 1.5 × 10 5 . While the general features of a dip, ramp and plateau are present, there are interesting differences: • The top panel has n going down till 9000. The bottom panel has lower values of n ≤ 5760. In each case we plot the disconnected (left), connected (middle) and total (right) SFF. We see that in the top panel the ramp region in the connected SFF has much more structure, and unless n becomes very small and we go to the bottom panel (n ≤ 5760), it is not monotonic, and exhibits an intermediate local maximum, which we call a "peak", followed by the local minimum and then grows until the final plateau. This indicates that while the nearest neighbour eigenvalues continue to repel, leading to a plateau at late times; at intermediate separation, beyond nearest neighbours, in fact an attraction develops between eigenvalues. • The height of the plateau decreases with decreasing n suggesting that the repulsion between the nearest neighbours also reduces. This also agrees with what we saw in the density of states, Fig. 8, which get more clumped in the centre as n decreases. Some more comments are as follows: • The disconnected piece itself has interesting structure. For the GUE and SYK models, the disconnected piece is known to decay and be self averaging, meaning that a single instantiation of the Hamiltonian (for big enough N ), reproduces the ensemble average. Here we see that the disconnected piece has an intermediate 'spike' followed by a slower decay, with the intermediate spike being more pronounced for in-between values of n. See Fig.12, left panels, n = 13824, 10000, 9000 around log(t) = 2 for spike. • The connected piece becomes smaller in magnitude as n decreases, and the full SFF is increasingly dominated, for smaller n, by the disconnected term. Level Spacing Next we turn to the nearest neighbour level spacing. There are a few points to keep in mind. It is important to compute the level spacing after unfolding the spectrum to obtain the Wigner surmise for the Gaussian ensembles or SYK q theory 3 . The eigenvalue spacing after unfolding is given by s = ∆λ i = ρ(λ i ) (λ i+1 − λ i ) . (3.4) we see that it is the local average value of the eigenvalue difference. The histogram plot of the quantity ∆λ for Gaussian ensembles or the SYK q theory then gives the standard Wigner Surmise. Alternatively, another quantity which avoids this procedure is the ratio of eigenvalue differences, since the dependence of the density of states in eq. (3.4) cancels out in the ratio; r i = ∆λ i ∆λ i+1 . (3.5) Some more information on how the distribution of r i behaves for various Gaussian ensembles is given in appendix D. If the eigenvalues are uncorrelated then the statistics will be that of the Poisson distribution which corresponds to p(r) = 1 (1 + r) 2 (3.6) As in [27], it will actually be convenient to plot the distribution of the logarithmic ratio ln r, which is, P (ln r) = rp(r) (3.7) Also, we need to be careful about degeneracies in the spectrum. There will be a degeneracy in the spectrum if the model under consideration possesses symmetries. For example, if the Hamiltonian has fermion parity symmetry, then the eigenvalues will be 2-fold degenerate. Therefore in order to compute the level statistics, one has to first separate the eigenvalues according to the parity sector and plot the differences in eigenvalues corresponding to any one sector. As mentioned in [27], for N = 0 (mod) 8, which is the case for N = 24 for which we present the analysis, there are no degeneracies after we do this. Let us now turn to the analysis of the plots. We have considered matrix of size L = 4096 and an ensemble size of 1000 in obtaining these plots. The left panel is a plot of the probability P (s) versus the unfolded level spacing s given by eq. (3.4), for varying values of the randomness parameter in the range n ≥ n c . The Wigner distribution is given by the black curve and corresponds for the GUE to the function, P (s) = 32 π 2 s 2 e − 4s 2 π (3.8) The fully random case is the data for the GUE, and other curves are obtained by reducing n until it reaches a value close to n c . In the right hand panel we plot P (ln(r)), defined in eq.(3.7), vs ln(r). Again the black curve is the result for the Wigner surmise, the fully random case is the data for the GUE, and other cases correspond to varying values of n, which for n ≤ 5760 are less than n c . Note that for the Wigner surmise p(r) is given by p(r) = N β r β (1 + r) β (1 + r + r 2 ) 1+ 3β 2 , (3.9) from which it follows from eq. (3.7) that P (ln r) = N β r β+1 (1 + r) β (1 + r + r 2 ) 1+ 3β 2 , (3.10) where N β=1 = 27 8 for GOE(β = 1) , N β=2 = 81 √ 3 4π for GUE(β = 2). (3.11) For the Poisson distribution it is given by P (ln r) = r (1 + r) 2 (3.12) The result in eq. (3.9) is derived in appendix D for the simple case of 3 × 3. We see from the left panel of fig. (13) that up till n = 1.5 × 10 5 , when the density of states fits a rescaled Wigner semi-circle, the level statistics is also well fitted by the Wigner surmise. However when n decreases further past n c , then differences set it, as we can see from the right panel. These grow, as n decreases and eventually for n = 576, when n/L 2 ∼ 3 × 10 −5 and the randomness is very sparse, the statistics becomes Poisson. It is not unexpected that for vastly reduced randomness the statistics becomes close to Poisson. Ideally one would have liked to obtain the behaviour with varying n in the limit when L → ∞, but that is difficult to do numerically and left for the future. OTOCs Finally, we consider the behaviour of the OTOC, G 4 = 1−F (t), eq. (2.32), as a function of time t and analyse it as we vary n, the number of random elements 4 . In each case we work with L = 4096, and an ensemble size of 2. Four figures are given below for different values of n, 10 6 ,10 5 ,10 4 , 4.5 × 10 3 . In the left panel of each figure the points are the data. Note that the range of t is different for n = 10 6 , 10 5 and n = 10 4 , 4500. Comparing the left panels for various n, we can see that in general, the growth of the OTOCs slows down as n decreases. Furthermore, there is a noticeable slowing down once n crosses the value n c , which is somewhat smaller than 10 5 . For example, the OTOCs in Fig. 14, with n = 10 6 , exhibit a saturation time scale, at which they approach their asymptotic value, which is quite similar to that for the GUE, Fig. 7. for example, for β = 10, Fig. 14, this time scale is t ∼ 10, which is comparable to the time scale in fig. 7. Whereas for n = 10 4 the saturation time scale, for β = 10 is t ∼ 50 − 60. The solid curves in the left panel are obtained by fitting an initial region, t ≤ t max , of the OTOCs to a functional form A + Be λt , as in the previous discussion of OTOCs above. The value of t max for each curve has been estimated by choosing the maximum range in t for which such a functional form is a good fit. The resulting estimate for the Lyapunov exponent is then plotted in the right panel of each figure. We have done the estimate for t max in a preliminary manner in this analysis and our corresponding estimate for λ should also be taken to be preliminary one. In particular, we recall that the GUE exhibits hyperfast scrambling with the OTOCs saturating in a time scale of order unity, Fig 7. One would expect this to continue to be true for sufficiently large values of n ∼ L 2 . As n beginning to decrease it is possible that exponential growth sets in and a Lyapunov exponent can be defined in the L → ∞ limit, once n becomes sufficiently small, perhaps smaller that n c . However we have not been able explore this possibility systematically and leave it for future investigation. Before closing this section let us make one final comment. Here we have studied the ensemble obtained by randomly choosing some fraction of matrix elements to vanish. In appendix C we discuss another model where the nonvanishing matrix elements are chosen in a more systematic manner. We first choose an integer q, and then only take the matrix elements, H ij , for i − j = 0 mod q, to be non-zero. The resulting matrix then can be block diagonalised into q × q blocks. Each of these blocks behaves like a GUE. E.g., the level spacing statistics of each block follows the Wigner surmise, however when we combine all the blocks the resulting level statistics becomes Poisson. Local SYK model 4.1 Local SYK model and the reduction of random couplings In this section, we consider a variation of the SYK model where the number of random couplings is vastly reduced and only O(N ). The fermions in this model can be thought of as lying on a circle and the only couplings present are those between q nearest neighbours. The Hamiltonian is given by H local−SYK = i q/2 i 1 ,i 2 ,···iq j i 1 i 2 ···iq ψ i 1 ψ i 2 · · · ψ iq , j i 1 i 2 ···iq = 0 (except i 1 = i 2 + 1 = i 3 + 2 = · · · = i q + q − 1, (4.1) and i 1 = i 2 − 1 = i 3 − 2 = · · · = i q − (q − 1)), with the fermions satisfying periodic boundary conditions: ψ i=N +1 = ψ i=1 . (4.2) To distinguish between this model and the conventional SYK model, with all to all couplings, we will refer to the model above as the local SYK model, in contrast to the conventional SYK model which we have often referred to as the non-local SYK model above. The couplings j i 1 i 2 ...iq present are taken to be random with vanishing mean and variance j i 1 i 2 ···iq j j 1 j 2 ···jq = (q − 1)!Ĵ 2 δ i 1 ,j 1 δ i 2 ,j 2 · · · δ iq,jq (4.3) We are particularly interested in the behaviour as N → ∞, withĴ being held fixed. Here we will also keep q fixed. Notice that the varianceĴ 2 above has a different N dependence from the non-local SYK model, eq. (1.2). The choice above is appropriate for a local system and ensures that a good thermodynamic limit exists with energy, entropy, etc, scaling extensively with the number of sites, N , when N → ∞, at fixed T,Ĵ. Also notice that with the choice we made in eq. (4.3), we get the dispersion in energy TrH 2 = LN (q − 1)!Ĵ 2 (4.4) which has the same linear N dependence as in the non-local SYK model in the large N limit TrH 2 = LN J 2 q . (4.5) One more comment is worth making here. While the four-fermi interaction in eq. (4.1) does preserve a notion of locality, bilinear terms which would typically be present in a local theory are absent. For Majorana fermions, which we are considering here, the bilinear terms couple nearest neighbours and result in an additional "hopping" contribution to H H hop = i J 2,i ψ i ψ i+1 . (4.6) More generally one can consider bilinear terms which couple next to nearest neighbours, etc, as well. We will neglect all such terms below. Study of the local SYK model with these hopping contributions are done in [40]. We restrict ourselves to the particular case of q = 4 in this section. We will compare and contrast the behaviour of the local and non-local SYK model by studying various quantities such as density of states, specific heat, 2-pt Green's functions, spectral form factor and OTOCs. Before proceeding let us mention one potentially confusing point while reading the various plots of this section. The numerical results have been obtained for finite values of N . In all the plots, we take J = 1 for the non-local model and also takeĴ = 1 for the local model. The density of states (DoS) and specific heat The unit normalised density of states for the non-local and local models are presented in fig. 18 below for a few different values of N . We see qualitatively that the states are more clumped at the center (near E = 0 in the plot) in the local SYK model and the behaviour close to the two ends is more like the tails of a normal distribution. The reader will recall that when we discussed the non-local SYK and GUE models ( fig. 1) we saw that compared to the GUE the spectrum of the non-local model indicates that the repulsion between eigenvalues is weaker, leading to the eigenvalues being more concentrated near the center; we see now that this tendency is even more accentuated in the local case. We have scaled the axes in fig. 18 appropriately with N that ensures the width of the distribution is almost same for all values of N . The non-local SYK model has a specific heat which is linear in temperature T in the low energy region, eq. (2.17), eq. (2.15). The resulting behaviour of ln(Z) in the non-local case also has a constant and T linear term in the temperature regime, eq. (2.15). The plot for entropy S vs T in the non-local SYK and the GUE was presented in fig. 3. The corresponding plot for the local SYK model is in fig. 19. We see from fig. 19 that there are qualitative similarities between them, essentially due to the fact that in all cases the entropy vanishes at T → 0, and increases monotonically achieving its maximum value, S = N ln 2, as T → ∞. An important feature for the non-local SYK is the low-temperature regime eq. (2.15), where the entropy has the characteristic behaviour, eq. (2.17). The presence of the constant S 0 and the linear T behaviour are closely tied to the presence of the Schwarzian action. It would be very interesting to check whether the same kind of behaviour arises in the local case. Unfortunately, the numerics which have been carried out at finite N do not allow us to obtain a conclusive answer and we leave this question for the future. Let us make one comment before proceeding. We see from fig. 19 that the rate of change of S/N from its low-temperature value, where it is close to vanishing, to its asymptotic value as T → ∞, becomes more rapid as N increases, suggesting that this change should be quite rapid in the N → ∞ limit. Green's function The two point function of fermions is another important quantity and is given by G ψ (t) = 1 N N i=1 ψ i (t)ψ i (0) (4.7) In the non-local SYK model, the two point function can be computed analytically in the large N limit, in the low-energy regime. Due to the emergent conformal symmetry, it takes the scaling form, G N L (ω) = g 0 √ ω (4.8) where g 0 is a constant. The corresponding plot for the Green's function in the non-local model are shown in fig. 20. The left panel shows log(Im (G(ω))) against log(ω). There are three regions which can be fitted to a power law. For ω ≥ 1 we have a good fit, shown in green, to the free field result Im G(ω) = 1 ω . For 0.22 < ω < 0.65, the red line is a good fit to the conformal result, eq.(4.8). For sufficiently small ω we find the fit to eq.(4.8) breaks down, and at very low ω < 0.1 a new scaling regime arises shown in black, with Im (G) ∼ 1 ω 0.16 . This break down and the new scaling regime are finite N effects. The middle panel in fig. 20 is Im (G) vs ω showing the region ω > 1 in green, and the right panel shows that the region 0.22 < ω < 0.65 is well fitted by eq.(4.8). fig. 21. Once again there are three panels. The left panel is a plot of log(Im G(ω)) vs log(ω) showing that for ω > 0.1 there is a reasonably good fit with the free field result, Im G(ω) = 1 ω . The middle and right panel show that the regions 0.1 < ω < 1 and ω > 1 respectively, are well fitted by the free field form. There are deviations for ω < 0.1. We have not been able to reliably estimate the behaviour in this region. Here we rescale ω as ω R = ω × N 3/2 . (b) Im (G(ω R )) vs ω R for the region 0 < ω R < 100. (c) Im (G(ω R )) vs ω R for the region ω R > 100 . Spectral form factor (SFF) The SFF was discussed for the non-local SYK and GUE in subsection 2.2, fig. 4 and for the random sparseness model in section 3.2, fig. 12. In the figure below we plot the disconnected, connected and total SFF for the non-local and local models. The most immediate difference that catches the eye is in the total SFF ((c) panel) where we see that the ramp in the non-local model is almost flattened in the local SYK. For the case with random sparseness the reader will recall, see fig. 12 that we had seen a similar behaviour for the SFF, namely, the flattening of the ramp as we increase sparseness, in particular for 1 n n c . It therefore seems that the reduction of eigenvalue repulsion which occurs in the random sparseness case, is also present here with O(N ) randomness. We had noticed in the case of sparse random matrix analysis that as the sparseness is increased, when 1 n n c , the ramp in the SFF starts to disappear along with the level spacing becoming more Poisson-like, see fig.13. Recall that the SFF of the non-local SYK model is quite similar to that of the GUE where the ramp can be understood as a consequence of the sine kernel for the two point density correlator. The absence of a ramp in the local SYK model indicates that the two-point density correlator is quite different for this system. One more comment worth making about fig. 22 pertains to the disconnected part ((a) panel). For the non-local model, the disconnected piece falls like 1/t 3 (at t β). We see from the figure that in the local model, the disconnected term follows a similar fall-off. From fig. 23 we also see that the local maximum in the connected SFF for the local case occurs somewhat early, at a time when the disconnected part has still not decayed away and is comparable in magnitude to the connected one. As a result such a local maximum does not appear in the full SFF. Level Spacing The nearest neighbour level spacings statistics is also a useful diagnostic of chaos. After unfolding the spectrum, as described in subsection 3.2, and taking care to separate the eigenvalues for eigenvectors of different parity, the resulting statistics is plotted in fig. 28. The left panel is the probability P (s) vs the spacing s. The green, blue and brown points are the data for the GOE, non-local SYK and local SYK models respectively (the GOE is the relevant ensemble since N = 0 mod 8, [27]). The solid curves are the best fits to this data, with the green, blue and brown curves being the fit to the green, blue and brown points, respectively. We see that the non-local SYK model is in good agreement with the GOE RMT result, eq. (2.29), see subsection 2.3 and [29], while the local SYK model is in good agreement with Poisson statistics, P (s) ∝ e −s . (4.9) In the right hand panel the same nearest level statistics is presented now as a plot of the variable r, eq. (3.5), along x-axis, vs P (ln(r)), eq.(3.7). The GOE, non-local SYK and local SYK data is in green, blue and orange respectively, and the corresponding curves for the GOE RMT, eq. (2.29) and Poisson statistics, eq. (4.9), are in green and red respectively. We see that again the non-local and local SYK model are in good agreement with GOE and Poisson statistics respectively. The bottom line is that we find for the local SYK model that the nearest neighbour level statistics agrees with the Poisson distribution to very good approximation, showing, at least as far as this diagnostic is concerned, that the model is not chaotic. This is also in agreement with what we have observed in the analysis of SFF earlier in subsection 4.4. Out-of-time-ordered correlators Next we turn to OTOCs. As was discussed in [24,37] , one needs a small parameter to be present, analogous to 1 N in a large N vector model or gauge theory, or to , for a well defined notion of scrambling time and the associated Lyapunov exponent in OTOCs, to exist. The small parameter suppresses the connected component of the OTOC compared to the disconnected one. We do not have such a parameter in the local model and in its absence it is unclear if the OTOCs can serve as a useful diagnostic of chaos. The parameter N present in the local model, as was discussed at the beginning of this section, should be thought of as labelling a site with one fermion being present per site. The N → ∞ limit in the local model is therefore an infinite volume limit which is different from the vector model or gauge theory cases where in this limit the degrees of freedom per site diverge. Despite this observation it is worth examining the behaviour of the OTOCs in the local model and comparing it to the non-local case. An important limitation of our analysis below is that we will be working numerically at finite N and our conclusions will mostly be of a qualitative nature. Let us also note the following. Drawing on the discussion about quantum chaos in higher dimensional local systems [24,25], even in systems without a small parameter of the type mentioned above, one can often define a velocity dependent Lyapunov exponent λ(v) which vanishes at the butterfly velocity v B . For v = |x|/t where |x| is the spatial distance, with v > v B , the OTOC C(x, t), defined for an local operator O(t) at x = 0 and another local operator W x at x, takes the form C(x, t) = 1 2 [O 0 (t) , W x ] † [O 0 (t) , W x ] ∼ e λ(v)t (4.10) λ(v) = −c(v − v B ) α (4.11) with λ(v) being the velocity dependent Lyapunov exponent. When there is a large N (or small ) parameter which suppresses the connected component, α is unity and the behaviour of C(x, t) also continues to be of the same form when v < v B , resulting in exponential growth of the OTOC. However, in the absence of such a parameter suppressing the connected component, often α is bigger than unity and the growth of C(x, t) does not continue for v < v B . The local SYK model we are studying is not a conventional local system since there is no quadratic hopping term in the Hamiltonian, eq. (4.6). Even so, it would be interesting to ask whether a butterfly velocity exists in the system and the behaviour of the OTOC close to it. We will not be able to address this question in a definitive manner here and leave it for the future. For the non-local model the OTOCs were discussed in section 2.4. By averaging over flavours we defined the function G 4 , eq. (2.32) whose behaviour is plotted in figure 6, obtaining, at N = 24, good agreement for the Lyapunov exponent with the N → ∞ result. The plots for local SYK are shown in fig. 26, 27. In obtaining the plots for the OTOC, we have considered G ij = 1 −F ij (4.12) whereF ij is given in eq. (2.30), then averaged the value of G ij for all possible pairs (i, j) keeping the distance p = |j − i| fixed. This averaged correlation is denoted asḠ i,i+p below. Fig. 26 (a) is for p = 1, and varying β, fig. 26 (b) is for p = 2, and varying β and fig. 26 (b) is for various p and two different values of β. We see that the OTOCs in general grow rapidly before saturating to a maximum value, in a manner, qualitatively speaking, which is similar to the non-local SYK model and also the GUE RMT, section 2.4. The rate of growth is fastest forḠ i,i+1 , i.e., the nearest neighbour sites and then slows down as p increases. It is worth noting in this context that for p = 4, 5 there are no terms in the Hamiltonian which directly connect the two sites, and this might account for some of the slow down. Also for p = 1, 2 the rate of growth slows down as β increases, this is similar to what happens for the non-local SYK model too. In conclusion, qualitatively the OTOcs in the local case are similar to the non-local and GUE cases. A more systematic analysis of the OTOCs in the local model, especially in the large N limit, is left for the future. It will be especially interesting to study whether a butterfly velocity can be defined in the system, and whether there is evidence for exponential growth of the OTOCs inside the light cone, v < v B . Some further discussion on the OTOCs in this model can be found in appendix F. Operator Growth in SYK Models The growth of operators from simple to complex ones is also a way to characterise the scrambling properties of a system, [26,33]. In the non-local SYK model it was suggested that the number of fermionic terms which appear in an operator can be a measure of its complexity. Thus a single fermion ψ i is a simple operator whereas another having a product of many fermions would be complex. Staring with the single fermion ψ i at t = 0 its time evolution is given by ψ i (t) = e iHt ψ i (0)e −iHt (4.13) Products of fermions of the type ψ a 1 , ψ a 2 · · · ψ as for a complete basis in the space of operators and one can therefore expand ψ i (t) in this basis, ψ i (t) = s,a 1 <···<as 2 s−1 2 c {i} a 1 ···as (t)ψ a 1 ψ a 2 · · · ψ as (4. 14) The probability of having an operator of size s for a particular combination can then be computed by finding all the c and this leads to c {i} a 1 ···as (t) = 2 1+s 2 ψ a 1 · · · ψ as , ψ i (t) (4.16) The resulting probability distribution of having operators of size s is then P s (t) = a 1 <···<as |c {i} a 1 ···as (t)| 2 (4.17) P s (t) = O | O, ψ 1 (t) | 2 O, O ψ 1 , ψ 1 (4.18) In the equation above the operator O denotes all the N s operators of size s; ψ a 1 ψ a 2 · · · ψ as , with condition a 1 < a 2 < · · · < a s . Starting with the operator ψ 1 at t = 0 (setting i = 1 entails no loss of generality), we can find P s (t) numerically. Fig. 28 and 29 are plots obtained for the non-local and local SYK models for s = 1, 3, 5. We see that in both cases the operator growth is qualitatively similar. P 1 (t) decays with time. In the non-local case, P s (t) increases, reach a maximum and then decreases, with P 3 (t) peaking before P 5 (t). In the local case, the behaviour is similar, but there is a longer tail characterising the slow decrease of P 5 (t). Gaussian Ensembles with Varying Randomness As was discussed in the introduction, the SYK model, like the GUE, can be thought of as a random Hamiltonian with matrix elements being drawn from a suitable Gaussian distribution. We explored some other random systems above where the random Hamiltonians were obtained by choosing other Gaussian distributions, with interesting differences in their behaviour. Here, we will make some additional comment on Gaussian ensembles. Before proceeding let us note that the low-energy dynamics of the nonlocal SYK model, which is JT gravity, is known to also arise in Random Matrix theory by choosing a non-Gaussian potential V (H) and considering a suitable double scaling limit [9]. We restrict ourselves to commenting on Gaussian ensembles below, since this class of random systems is already quite rich. General Observations An Hermitian matrix H can be thought of as being a vector in an L 2 dimensional Hilbert space H of L × L Hermitian matrices. H can be expanded in any basis of H, H = L 2 a=1 c a T a ,(5.1) where T a are Hermitian matrices which are the basis elements, and c a are real coefficients. An inner product in this space is given by the trace, T a |T b = Tr(T a T b ). (5.2) By drawing the coefficients c a from various Gaussian distributions one can get different types of Gaussian ensembles. Under a change of basis the T a 's, and H, transform by conjugation, T a → U T a U † , H → U HU † ,(5.3) where U is the L × L unitary matrix specifying the change of basis. A few different bases for H -are : • A standard basis corresponding to the root vectors of U (L) [38]. • For L = 2 N : a basis made out of tensor products of the Pauli matrices. More precisely, the basis of matrices σ α 1 ⊗ σ α 2 · · · ⊗ σ α N where the indices α 1 , α 2 · · · α N each take 4 values, and σ α , α = 1, 2, 3, 4, are four 2 × 2 matrices, given by the Pauli matrices, σ i , i = 1, 2, 3, with σ 4 = I, where I is the identity. • When N is a multiple of 2 and L = 2 N 2 : a basis made out of products of N Majorana fermions, ψ i , i = 1, · · · N described in eq. (1.4), eq. (1.6) and eq. (1.7). Each of these bases is useful in their own way, since they allow us to use different types of physical intuition in the study of randomness. The first basis connects to Random matrix theory, the second to the study of spin systems, and the third to the study of the SYK model. An important property of the GUE is that the resulting probability distribution is U (L) symmetric. Many key properties of the GUE, including the fact that OTOCs and operators grow in an hyper-fast manner, as discussed in section 2, are tied to this U (L) invariance. For other random ensembles the U (L) symmetry will be broken in general. This is true for the non-local SYK model, and the breaking of the U (L) symmetry is in part responsible for interesting differences in behaviour with the GUE. It is also true for the random sparseness model studied in section 3, where we keep n < L 2 matrix elements, or the local SYK model studied in section 4, where we keep only O(N ) random variables. We can also consider other ways of breaking the U (L) symmetry, these could also yield interesting properties from the point of view of thermodynamics or chaos. For example we can take H to be block diagonal with |H IJ | 2 2σ 2 M + L α,β=M +1 |H αβ | 2 2σ 2 L + M I=1 L α=M +1 |H Iα | 2 σ 2 M L (5.4) If σ M L = 0, the off diagonal terms in the matrix integral go to zero and the distributions breaks up into two GUEs of rank M, L − M respectively. But when σ M L σ M , σ L , the system behaves like two GUEs with a weak coupling between them. E.g., the density of states is only perturbed slightly by the offdiagonal terms, which are "very massive", where mass m ∼ O (1/σ M L ). This perturbation can have interest effects for chaos related properties though. The off-diagonal terms could alter the nearest-neighbour spacings which are of O(1/L). And for operator growth, one expects that a simple operator acting in the M × M block will at first scramble in an hyperfast manner into a matrix filling out the M × M block, but its subsequent growth into a full L × L matrix would be much slower and governed by the off-diagonal terms 5 . By changing the strength of the off-diagonal term compared to the diagonal ones, once can in this way explore a range of behaviours. Other random ensembles are also worth studying. The GUE eq. (2.1), eq. (2.2), corresponds to choosing the coefficients c a (for unit normalised T a 's) to be independent Gaussian random variables with the probability distribution Taking σ a → 0, we will have a highly ordered matrix, where the dispersion about the meanc a is small. As σā ca increases the amount of disorder increases, and in this manner one can study the change in behaviour from the ordered to disordered situations. P (c a ) = L 2π e − Lc We will report on the behaviour of some of these systems in a subsequent study [36]. More on SYK Models Let us now turn to the SYK model in the context of the wider discussion above. As was noted in the introduction, eq. (1.6), eq. (1.9), the GUE can be obtained by summing over all ψ q terms with equal variance. The non-local SYK model corresponds to only keeping the terms of the ψ q kind, i.e., involving products of q number of ψ i fields. The large N saddle point arises when the coefficients of these terms are drawn from a Gaussian distribution with variance given in eq. (1.2). The fixed q model only preserves an SO(N ) subgroup of the U (L) symmetry mentioned above. In [14] a version of the fixed q model was studied where the number of couplings was reduced in a random manner. This model is reviewed in appendix A. In the large N limit the fixed q model, to begin, has O(N q ) couplings. As long as the number of random couplings after reduction scale like N α , α > 1, a saddle point analogous to the fixed q case exists in the large N limit, for a suitably chosen variance for the non-zero couplings. In fact, this is also true when q does not take a fixed value, as long as it grows with N as a power q ∼ N β , with β < 1 2 . This sparse SYK model, [14], is analogous to the sparse random matrix model studied in section 3. If we think of the GUE, as was mentioned above, by summing over all ψ q terms, then at large N the maximum number of terms are contributed by operators with q = N/2. In fact, the number of random variables for q = N/2 is N C N/2 ∼ 2 πN 2 N , (5.8) which is exponential to N . Thus, instead of starting with the GUE and making the matrix H sparse, we could at large N , only take terms with q = N 2 , and consider the effects of making the randomness sparser by setting some terms to vanish in a random fashion 6 . The difference with the [14] case is that the value of q = N 2 now scales linearly with N . As we saw in section 3 as the number of terms n is reduced an interesting change in behaviour occurs in this case when n ∼ n c , with nc L 2 , scaling with L as given in eq. (3.3). It would be interesting to try and understand this transition using saddle point methods but we have unfortunately not been able to do so 7 . The saddle point referred to above, obtained by starting with the fixed q model and keeping O(N α ) terms is analogous to the one obtained for the fixed q case (where all terms are kept). In particular, the saddle point continues to break time reparametrisation invariance giving rise to the Schwarzian mode and the low-energy behaviour is a CFT with the fermionic fields having the anaomalous dimension 1/q. In this example we then see that most of the essential features found in the non-local SYK model -tied to the realisation of the time reparametrisation symmetry and the Schwarzian action are quite robust and continue to occur even for much smaller number of random couplings and are independent of the underlying Hamiltonian. Some of the more detailed features though depend on the underlying model. For example, we can have two different models with the same number of couplings O(N α ), obtained by starting with two different values of q, q 1 , q 2 , and reducing the number of couplings to different extents. The anamolous dimensions of the ψ i fields in the low energy CFT will then be different, 1 q 1 and 1 q 2 respectively. Let us also comment on symmetries. When we keep only even q terms in the sum to obtain H there is a fermion symmetry, ψ → −ψ symmetry, which is preserved. This symmetry is broken once odd q terms are included. Also, among even q terms if we only keep q = 0 ( mod 4) terms then H has time reversal invariance -as can be seen from the fact that the couplings are real. Terms of the same symmetry type can be transformed into one another under U (L) transformations. Nevertheless and quite interestingly, as mentioned in the previous paragraph, if we take terms for two different powers q 1 , q 2 both of which preserve the same class of symmetries and also take the same number of couplings in both cases, the low energy limit has some differences in the resulting anomalous dimensions of the fermionic fields. We can now comment on the figures plotted below. Fig. 30 (a) plots the density of states for the GUE, the ψ q model where all even powers of q are included with the same variance, and the model where all ψ q terms with odd powers and the same variance are included. Fig. 30 (b) included the three models again, but with a rescaling carried out along the x (energy axis) so that the extent along the x axis of the three curves coincides. The y axis is also correspondingly rescaled to keep the area the same under each curve. We see that after this rescaling, the density of states for the models obtained by keeping only even or odd terms agree with the GUE. The density of states for the model obtained by keeping all q terms agrees with the GUE even before rescaling, as is to be expected. In fig. 31 working with the N = 20 case, we show the density of states when only some terms of the ψ q type, with q = 0 (mod 4), are kept in H. From our discussion of symmetries above we see that in this case all the terms we retain preserve the fermion and time reversal symmetries. The left panel of the figure shows the density of states for various values of nthe number of terms which have been kept, with the black curve showing density of states, ρ, for the SYK q = 4 model. The right hand panel is the same data after rescaling the energy E and ρ axis, so that all the curves coincide in their extent along the x axis, as discussed above. In addition, in the right panel we have included the RMT (all q) case and also the case where all q = 0 (mod 4) terms are included -these are shown as yellow and red curves respectively. We see after the rescaling, ρ for the GUE and the model where all q = 0 (mod 4) terms are kept, are in good agreement. As the total number of terms in H is decreased ρ begins to change. The black curve in the right panel is also for the q = 4 SYK model (where all q = 4 terms are included) -this model has a total of N C 4 = 4845 number of terms. Qualitatively, the right panel shows that for intermediate values of n, ρ near the center, around E = 0, agrees more with the q = 4 case. Fig. 32 shows a similar plot, with rescaled energy and ρ axes, for N = 24. The yellow curve is for the GUE and the black curve is for the q = 4 SYK. We see in this plot more clearly that at intermediate values of n, for E close to the edges, ρ differs from both the GUE and the SYK 4 theories, while for E close to the center, ρ lies in between the GUE and SYK 4 cases. It could be that some of these differences are due to finite N effects. In all the figures above the variance for all the non-zero couplings was taken to be the same. One can also examine what happens if we keep terms whose couplings are normal distributed with different values for the variance. An example of this is shown in Fig. 33 where we have kept all terms of ψ 4 and ψ 6 type, for N = 28. The ψ 4 , ψ 6 terms are drawn from distributions, eq. (1.2), with q = 4, 6 respectively, and we have set J = 1. The full Hamiltonian is taken to be H full = λH SYK 4 + (1 − λ)H SYK 6 (5.9) The resulting density of states is plotted in fig.33 for various values of λ, where λ is an interpolating parameter. Let us also note that a large N saddle point in terms of the G and Σ fields, [7], exists in this case. However solving for the values of the G, Σ fields analytically, at the saddle point, is difficult in general. When λ is small it is easy to see that the ψ 4 coupling is relevant in the conformal theory to which the ψ 6 model flows, thus at very low energies the behaviour will be governed by the ψ 4 fixed point, while at intermediate energies E J one will have the Schwarzian mode coupled to the ψ 4 perturbation. As λ is increased one expects that the model will flow from the UV more directly to the CFT governed by the ψ 4 coupling. The left panel of Fig. 33 is a plot of the density of states for varying values of λ and the right hand panel is a plot where the density of states is plotted after rescaling the E axis and also ρ, as discussed above. We see from both plots that the density of states interpolates smoothly from the SYK 4 to the SYK 6 case as λ goes from 1 to 0. It is also worth noting that in the right hand panel, once the rescaling is carried out, the density of states at the edges (i.e., at low and high energies), for all values of λ = 0, agrees with the SYK 4 model. This connects with our comment above that the low-energy behaviour should be of the SYK 4 type. The discussion in this section is indicative of the rich set of behaviours one can get for various Gaussian ensembles, with varying degrees of randomness. A more definitive analysis is clearly called for, especially in the large N limit, and we hope to return to this in subsequent work, as was also mentioned above. Summary and Discussion The conventional SYK model, with ψ q couplings, is referred to as the nonlocal SYK model in this paper, due to the all-to-all nature of its fermionic couplings. This model and the Gaussian Unitary Ensemble (GUE) are both random systems, with some similarities. Their spectral form factors are similar and the level statistics, of nearest neighbour energy levels, in both follow the same Wigner distribution. However, there are some important difference as well. In the large N limit, the non-local SYK model has interesting lowenergy behaviour characterised by the breaking of time reparametrisation invariance which gives rise to a Schwarzian action. This is not true in the GUE. The pattern of symmetry breaking for the SYK model determines much of its low energy behaviour. Its low-energy thermodynamics is governed by a saddle point resulting in a linear specific heat, and a big entropy, S 0 , scaling like N , eq. (2.17), eq. (2.18), close to ground state. And the OTOCs exhibit chaotic behaviour with a Lyapunov exponent saturating the chaos bound. These features are absent in the GUE. Many properties of the GUE, for L × L matrices, are tied to the U (L) invariance of the resulting probability distribution. This invariance results in hyperfast scrambling for operators and also for the OTOCs, which exhibit a saturation time of order the thermalisation time, see appendix B. The low-energy thermodynamics for the GUE is also quite different, see section 2. The behaviour of the non-local SYK model is of considerable interest from the point of view of gravity, agreeing in particular with JT gravity and being akin, quite universally, to near-extremal black holes with an AdS 2 factor in their near horizon geometry, [7,11]. The differences in behaviour with the GUE prompt one to ask why these differences arise and more generally when can behaviour of interest from the point of view of gravity arise, as the randomness is varied 8 . Such questions were some of the central motivations behind this paper. Both the GUE and the SYK model correspond to Gaussian Ensembles, see section 5. In the GUE, with L × L Hermitian matrices, H, all L 2 matrix elements are random variables drawn from independent Gaussian distributions with the same variance and no mean. Equivalently, one can expand a general matrix H in the basis of products of fermionic operators, eq. (1.6). The number of fermion flavours, N , needed is related to L by L = 2 N 2 . The total number of coupling constants in this fermionic basis is 2 N = L 2 , which is of course the same as the number of independent matrix elements in H, and for the GUE these couplings in the fermionic basis are also independent Gaussian random variables with equal variance and vanishing mean. In contrast, the non-local SYK model has fewer random variables. For the SYK q model, only operators with products of q fermions are present in the Hamiltonian, giving rise to N C q ∼ O(N q ) random variables, which is much smaller than L 2 . It is quite clear that some of the difference between the two models arise because of this reduction in randomness. To investigate this further, we studied two kinds of models in this paper. First, starting with the GUE we reduced the number of random variables by setting some number of matrix elements to vanish, at random, and choosing the remaining non-vanishing elements from the same kind of Gaussian distribution as in the GUE. This gave rise to the sparse random matrix model studied in section 3. As the number of non-vanishing matrix elements, n, is reduced, we move away from the GUE and one can study the effects of reduced randomness. Second, at the other end with much reduced randomness, we considered the case of the local SYK model where only 4 adjacent fermions were coupled to each other through ψ 4 type couplings, unlike the non-local SYK 4 model where all fermions are coupled to each other in ψ 4 type couplings, section 4. This model has only O(N ) random couplings, which is even smaller than the O(N 4 ) couplings in the corresponding non-local SYK 4 model. For the sparse random matrix model we found that as the number of random variables was reduced there was a marked change of behaviour when n reached a value n c , which was numerically found to scale like nc L 2 ∼ L −0.48 , for large L. This change of behaviour is tied to the reduction in eigenvalue repulsion, resulting in energy eigenvalues clumping in an increasingly pronounced manner near the origin (E = 0), fig. 8. At values of n < n c our analysis showed that the behaviour becomes quite different from the GUE and several properties begin to differ. The spectral form factor now showed evidence of attraction between eigenvalues; the level statistics began to depart from the Wigner statistics as n decreases and become more Poisson-like distributed; the ramp in the SFF is flattened and the OTOCs indicated a slower growth of the correlators. A key limitation of our study is that the numerical work could only be carried out at finite N and many of our conclusions could not be reliably extrapolated to the infinite L, N, limit. In particular, we could not investigate the change of behaviour near the transition at n c , in the L → ∞ limit. It would also be very interesting to know if for n < n c a Schwarzian mode arises in this limit. For the local SYK model, studied in section 4, we found that the behaviour was quite different from the non-local SYK case. These differences could be seen in the spectral form factor, which has some similarities in fact with the sparse random matrix model case (for 1 n n c ); in the level statistics, which was Poisson and not Wigner; and also, in a qualitative sense, in the OTOCs. It would be interesting to understand our results in greater depth, augmenting the numerical work presented here with additional analytical insights. As mentioned above, the GUE and the SYK models both belong to the class of Gaussian ensembles, where the Hamiltonian is determined by couplings drawn from Gaussian random distributions. Our investigation reveals that this class is in fact quite rich and such a detailed understanding, we think, will prove quite rewarding. Some analytical results along the lines of our investigation are already known. It was shown in [13,14] that starting with the non-local SYK q model, if we set some of the ψ q couplings to vanish at random, the behaviour would continue to be of the SYK q type, as long as the total number of non-vanishing couplings scaled like N α , α > 1, in the large N limit. The local model we have studied has O(N ) couplings, which is smaller and lies outside this range. In contrast, the sparse random matrix model of section 3 exhibits a change of behaviour when the number of couplings is n c ∼ L 1.52 -this is much bigger than any power of N . The results of [14] can be extended to the case where q is not fixed and also diverges when N → ∞. One finds, as discussed in appendix A, that a G − Σ type of saddle point which arises in the non-local model, continues to arise here, with the same breaking of time reparametrisation invariance etc, as long as q < √ N and the total number of non-vanishing terms scales like N α , with α > 1. If we think of the GUE in the fermionic basis, eq. (1.4), when N → ∞, its Hamiltonian will be determined by the ψ q terms with q = N 2 , since these terms out-number all others. Thus we see that the sparse random matrix model of section 3 is related to the fixed q model of [14], with q now scaling linearly with N . More generally, one could consider many other models in the class of Gaussian ensembles. For example, we can include a mean value as well as a variance in the normal distributions, see section 5 and [36]. Varying their ratio allows us to explore what happens as we go from ordered to disordered systems. We could also study such Gaussian models from the perspective of the breaking of the U (L) symmetry present in the GUE, see section 5. For example, one can have blocks in the random Hamiltonian of GUE type, and couple these blocks to each other with intermediate blocks that have reduced randomness, including of the SYK type. The resulting behaviour would then be quite rich. As far as the spread of entanglement goes for example, which could be measured through OTOCs, or the growth of operators, one would expect GUE like blocks to exhibit hyperfast scrambling, with t scrambling ∼ O(1), this would become somewhat slower t scrambling ∼ O(ln N ), in the non-local SYK blocks of the random matrix and then even slower, t scrambling ∼ O(N ), in the blocks where the couplings preserve spatial locality. Bottle neck for the spread of entanglement, may perhaps arise in highly ordered blocks. A detailed study along these lines is also left for the future. A Sparse SYK In this appendix, we discuss the sparse version of SYK following [14]. The sparse SYK model considered in [14] is a variant of the original SYK model where each of the q-fermi interaction are assigned a probability. The Hamiltonian for the model is given by H = Q j Q x Q ψ Q , Q = {i 1 , . . . i q }, i 1 < i 2 < . . . i q (A.1) where the variables j Q and x Q are independent random variables, with j Q being drawn from a Gaussian distribution and x Q is a boolean variable with a binary distribution. Note that in the notation above Q denotes a specific set of q flavour indices, {i 1 , i 2 , · · · i q }, and ψ Q denotes the product of the corresponding q fermions, ψ i 1 ψ i 2 · · · ψ iq . By the probability distribution we mean more precisely p(j Q ) = 1 √ 2πJ 2 σ e − j 2 Q 2J 2 σ =⇒ j Q j Q = J 2 σ δ Q,Q (A.2) and p(x Q = 1) = p, p(x Q = 0) = 1 − p (A.3) The case of p = 1, σ = (q−1)! N q−1 would correspond to the usual SYK model, where N is the total number of fermion flavours. The variable x Q being a binary variable will be unity for some bonds and zero for others and so only some of the bonds are turned on in any instantiation. The total number of terms in the Hamiltonian, n, is when p = 1 n = N C q . (A.4) For p < 1, the number of terms in any instantiation of the model is given by n = p N C q (A.5) The disorder averaged path integral for the model is given by Z = [Dψ]e i dt(ψ i i∂tψ i ) Q [dJ Q ][dx Q ]p(J Q )p(x Q )e −i dtj Q x Q ψ Q (A.6) Doing the integral over j Q first, we get Z ∝ [Dψ]e i dt(ψ i i∂tψ i ) Q [dx Q ]p(x Q )e − J 2 σ 2 dt dt x 2 Q ψ Q (t)ψ Q (t ) (A.7) Now doing the average over the sparseness variable x Q , we get Z ∝ [Dψ]e i dt(ψ i i∂tψ i ) Q pe − J 2 σ 2 dt dt ψ Q (t)ψ Q (t ) + 1 − p (A.8) Now, if p is such that the average number of bonds turned on is n = N α (A.9) for some α > 0, we have p = N α N C q (A.10) A.1 q = fixed case Let us first consider where q is fixed with large N limit. Then we get p ∼ q! N q−α . (A.11) Let us consider the case where both p and σ small, i.e., p → 0 , σ → 0 . (A.12) For p → 0 in the large N limit with q fixed, we require q > α . (A.13) Then we can exponentiate the term in the brackets in (A.8) and so we get Z ∼ [Dψ]e i 2 dt(ψ i i∂tψ i ) Q exp p{e − J 2 σ 2 dt dt ψ Q (t)ψ Q (t ) − 1} (A.14) ∼ [Dψ]e i 2 dt(ψ i i∂tψ i ) Q exp − pJ 2 σ 2 dt dt ψ Q (t)ψ Q (t ) (A.15) In the second line, we assume σ → 0 and also that ψ Q ψ Q ∼ O(1). We will see the self-consistency of these assumption soon. We now introduce the flavour dependent bi-local fields Σ Q (t, t ), G Q (t, t ) and follow the same manipulations as in the case of usual SYK. The path integral then becomes Z ∝ [Dψ][DΣ i ][DG i ] e i 2 dt(ψ i i∂tψ i )+ 1 2 dt dt Σ i (t,t )(G i (t,t )−ψ i (t)ψ i (t )) Q exp − pJ 2 σ 2 dt dt ψ Q (t)ψ Q (t ) = [Dψ][DΣ i ][DG i ]e 1 2 dt(ψ i (−∂t−Σ i )ψ i )+ 1 2 dt dt Σ i (t,t )G i (t,t )+ Q − pJ 2 σ 2 dt dt G Q (t,t ) = [DΣ i ][DG i ]e 1 2 log det(−δ(t−t )∂t−Σ i )+ 1 2 dt dt Σ i G i + Q − pJ 2 σ 2 dt dt G Q (t,t ) (A.16) where the symbol Q is the summation symbol with sum being taken over all sets Q of the form in eq. (A.1) and in the second line above we have used the notation that G Q (t, t ) = G i 1 (t, t ) . . . G iq (t, t ) (A.17) For a moment let us assume that there exists a solution where all flavour two point functions are same, i.e., The equation of motion obtained by varying with respect to Σ and G are then given by Σ i 1 (t, t ) = Σ i 2 (t, t ) ≡ Σ(t, t ) G i 1 (t, t ) = G i 2 (t, t ) ≡ G(t,G(t) = (−∂ t − Σ(t)) −1 Σ = q p σ N C q N J 2 G q−1 (A.20) First, the way it works in the conventional SYK model is as follows. The parameters for the conventional SYK model are p = 1 , σ = (q − 1)! N q−1 ⇒ q p σ N C q N = 1 (A.21) Then it is clear that eq. (A.20) becomes Σ = J 2 G q−1 , (A.22) which gives exactly the conventional SYK saddle point, where G ∼ O(1) in terms of N counting. However it must be clear now that the way to obtain conventional SYK saddle point, therefore G ∼ O(1), is not only (A.21), but more generically possible as long as q p σ N C q N = 1 , with p → 0 , σ → 0 (A.23) is satisfied. The condition of p → 0 and σ → 0 is due to the consistency with (A.12). With (A.11), this is equivalent to p ∼ q! N q−α , σ ∼ N 1−α q , with α > 1 (A.24) in the large N limit. In this case, the number of random variables, n, is n = N α with α > 1. Thus, we conclude that even in sparse case, as long as (A.24) is satisfied, we end up with the same saddle point equation and therefore solution as in the conventional SYK, (A.22). It is also worth mentioning, as we see from eq.(A.24), that as the sparseness increases, i.e., as α becomes smaller, the variance σ increases. On the other hand, for α = 1 case, the model has become sufficiently sparse that the effective number of terms is of O(N ). In this case, the approximation we used in (A.15) breaks down.The action then becomes Z ∝ [DΣ][DG]e N 2 log det(−δ(t−t )∂t−Σ)+ N 2 ΣG+ N Cq pe − J 2 σ 2 G q −p (A.25) It is straightforward to see the modified equation of motion obtained by varying G in this case Σ = −J 2 σq N C q p N e − J 2 σ 2 G q G q−1 (A.26) Another way to see the conclusion in eq. (A.24) is to compute tr(H 2 ) , which is given by tr(H 2 ) = N C q p J 2 σ = J 2 N α σ (A.27) For the saddle point solutions to exist we require tr(H 2 ) ∼ J 2 N and so we find J 2 N ∼ J 2 N α σ =⇒ σ ∼ O N 1−α (A.28) Thus, we reach the conclusion that α > 1 in this viewpoint. Note that all the manipulations above are done for a fixed q. Another interesting limit to consider is when q also becomes large. This can be done in two ways. First, we can take N → ∞ followed by q → ∞. Alternatively, we can consider q ∼ N a , 1 > a > 0, so that q 1 a /N is fixed as N → ∞. In both these cases, the above manipulations have to be redone carefully. A.2 Large q limit In the previous subsection, we focussed on the case of finite q and large N . Another interesting regime to consider is when both q and N are large. Let us first analyze the case of large q but with N taken to infinity first without introducing any sparseness. This part will be a review of [7]. The equations of motion for both G and Σ fields are G(t) = (−∂ t − Σ(t)) −1 Σ(t) = (J 2 σq) N C q N G q−1 (A.29) In momentum space this becomes G(ω) = (−iω − Σ(ω)) −1 (A.30) In the large q limit, we can take the ansatz for G to be given by G(t) = 1 2 sgn(t)e g(t) q−1 (A.31) so that we can do an expansion in 1 q as G(t) ∼ 1 2 sgn(t) 1 + g(t) q + . . . (A.32) The fourier transform of the above is given by G(ω) i ω + 1 q [sgn × g](ω) (A.33) The crucial point here is that, usually in the analysis at fixed q, low energies correspond to small ω and so the explicit ω dependence in eq. (A.30) is dropped in solving the saddle point equation. However, in the large q limit, even in the regime of low energies, we should not drop this term but instead do a perturbation around the free theory result as shown in eq. (A.32). Using eq. (A.33) in eq. (A.30), it is easy to see that Σ(ω) ∼ − ω 2 q [sgn × g](ω) . (A.34) We can self-consistently assume that in terms of q, N counting g(t ∂ 2 t (sgn(t)g(t)) = (J 2 σq 2 ) N C q 2 q−1 N sgn(t)e g(t) (A.36) We can choose σ to cancel all factors of N and q above, so that g(t) can be obtained to be of order unity. We take, σ = 2 q−1 N q 2 N C q (A.37) In this analysis, we have only assumed that both q and N are large but have not assumed anything about q/N . As we shall argue later, the saddle point analysis breaks when q ∼ O( √ N ). Before we discuss about the breakdown of saddle point, let us generalize the above discussion to the case where we introduce sparseness as mentioned in the previous section for the case of finite q by introducing sparseness variables x Q . The following discussion is also applicable to the case when q ∼ N a , 1 2 > a > 0 (A. 38) with the upper limit on ρ due to the breakdown of the saddle point estimate. In fact, both these generalizations can be carried out together which is what we shall present now. The steps leading to eq. (A. 19) are valid even when q is large as long as Σ(t) = (J 2 σq) N C q p 2 q−1 sgn(t)e g(t) (A.40) As before, equating eq. (A.40) and eq. (A.34) we get ∂ 2 t (sgn(t)g(t)) = (J 2 σq 2 ) N C q p 2 q−1 N sgn(t)e g(t) (A.41) Thus, for a good large N and large q limit to exist, we now have to take σ to be given by σ = 2 q−1 N q 2 N C q p = 2 q−1 N q 2 N α (A.42) where we used N C q p = N α . Naively it may appear that σ is large in the large q and large N if q ∼ N a , a > 0 due to the factor of 2 q−1 . However this factor is cancelled by a similar factor coming from G q in eq. (A.39). From eq. (A.42) and using eq. (A.38), we see that eq. (A.39) is satisfied as long as α > 1 − 2a (A.43) Thus, we see that eq. (A.39) is satisfied even if the total effective number of terms is linear in N , i.e., α = 1, if q is large. This is to be contrasted to the case of finite q where we required α > 1, see eq. (A.24). However, we can argue that we cannot make q arbitrarily large as the saddle point approximation will break down for q ∼ O( √ N ). One way to understand this is to compute the fluctuations around the saddle point and see when they become important. To see this consider the expansion of the fields around the saddle Σ = Σ 0 + Σ, G = G 0 + δG . (A.44) Expanding the action in eq. (A.19) by considering the fluctuations above, and noting the expansions (G 0 + δG) q G q 0 + G q−1 0 δG + q C 2 G q−2 0 δG 2 + q C 3 G q−3 0 δG 2 ln det(−∂ t − (Σ 0 + δΣ)) = ln det(G −1 0 ) + ln det(1 + G 0 δΣ) det(1 + A) 1 + Tr(A) + 2 (TrA 2 + (Tr(A)) 2 ) + 3 (TrA 3 + (Tr(A)) 3 + TrA 2 TrA + (Tr(A)) 2 TrA) (A.45) where A is any matrix in the last line above and is a small parameter in which the expansion is done. In our case Tr corresponds to the trace in time coordinates t, t , the arguments of the functions G, Σ. We find that the quadratic and cubic terms in fluctuations have the schematic form (ignoring the time integrals and traces) −S N (a 1 (G 0 δΣ) 2 + a 2 (G 0 δΣ) 3 ) + N δΣδG − N C q pJ 2 σ 2 q 2 2 G q−2 0 δG 2 + q 3 6 G q−3 0 δG 3 (A.46) where a 1 , a 2 are some O(1) coefficients. Noting that Σ 0 ∼ O(q −1 ), G 0 ∼ O(1) in terms of N and q counting given by eq. (A.31),(A.35) and that σ is given by eq. (A.42), we immediately see from the quadratic terms in the fluctuations that δΣ ∼ O 1 √ N , δG ∼ O 1 √ N . (A.47) Using these estimates in the cubic terms, we find that the cubic terms are suppressed compared to the quadratic terms if and only if q √ N 1. (A.48) So, precisely when q 2 ∼ O(N ), the saddle point estimates fail. In these cases there are other methods such as Chord diagram techniques [28] that can used to solve the model in such a limit. We shall not delve further into these in this appendix. B Four Point function in RMT In this section, we shall elaborate on the four point function calculations in GUE. To begin with, we illustrate the calculation in detail for the case of 2 point function which can then be straightforwardly generalized to four point function. Let us consider the averaged two point function of two operators A, B, with the averaging done over the Haar ensemble. This is given by A(t)B(0) GU E = dHTr(A(t)B(0)) = dHTr(e −iHt A(0)e iHt B(0)) = 1 Vol(U ) dU dHTr(U e −iHt U † A(0)U e iHt U † B(0)) (B.1) where in obtaining the third line we used the fact that the Hamiltonian H and its measure are invariant under H → U HU † . We now use the identity dU (U † ) a b U c d (U † ) e f U g h = 1 L 2 − 1 δ a d δ c b δ e h δ g f + δ a h δ g b δ e d δ c f − 1 L δ c b δ e d δ g f δ a h + δ a d δ c f δ e h δ g b (B.2) where L is the rank of the matrix. Using this in eq. (B.1) it is easy to see that we get Further, for simplicity, if we consider operators such that Tr(A(0)) = 0 = Tr(B(0)), then we get A(t)B(0) GU E = ZZ H L 2 − 1 Tr(A(0)B(0)) − 1 L Tr(A(0))A(t)B(0) GU E = ZZ H − 1 L 2 − 1 Tr(A(0)B(0)) (B.5) To leading order in large L, the quantity ZZ H goes like O(L 2 ) and thus gives the dominant contribution to the two point function above. So, in the large L limit, we have A(t)B(0) GU E ZZ H L 2 Tr(A(0)B(0)) (B.6) We now turn to the 4-pt function. Since we are mainly interested in the OTOCs, we consider a 4-pt function of the form A(t)B(0)C(t)D(0) GU E = dH Tr(A(t)B(0)C(t)D(0)) = dH Tr(e −iHt A(0)e iHt B(0)e −iHt C(0)e iHt D(0)) = 1 Vol(U ) dU dH Tr(U e −iHt U † A(0)U e iHt U † B(0)U e −iHt U † C(0)U e iHt U † D(0)) (B.7) The analog of the identity eq. (B.2) we now need invoves four U s and same number of U † s. This identity is very cumbersome and so we shall not write it out explicitly here. Again, assuming all one point functions are zero, we find that, in the large L limit, the leading term in the four point function is given by A(t)B(0)C(t)D(0) GU E ZZZZ H L 4 Tr(A(0)B(0)C(0)D(0)) J 1 (2t) t 4 Tr(A(0)B(0)C(0)D(0)) (B.8) The discussion so far pertained to the case of β = 0. We can generalize these results for the case of finite β as follows. The appropriate 4-pt function we for the finite temperature case is of the form yA(t)yB(0)yC(t)yD(0) , y = e − βH 4 (B.9) The analog of eq. (B.7) now becomes yA(t)yB(0)yC(t)yD(0) = 1 Vol(U ) dU dH Tr(U e −iHt U † A(0)U e iHt * U † B(0)U e −iHt U † C(0)U e iHt * U † D(0)) (B.10) wheret = t − i β 4 ,t * = t + i β 4 (B.11) Thus, in the limit of large L, the 4-pt function essentially is given by eq. (B.8) with the simple modification that now Z = Tr(e −iHt ),Z = Tr(e iHt * ) (B.12) witht,t * as defined in eq. (B.11). Thus, the 4-pt function in the limit of large L is given by A(t)B(0)C(t)D(0) GU E J 1 (2t) t 4 Tr(A(0)B(0)C(0)D(0)) (B.13) C Systematic Sparseness In this appendix, we consider a different way of sparsing a random matrix. In this version of constructing a sparse random matrix, we start by picking a positive integer p ∈ [1, L]. We then take M ij to have the following form, M ij = 0 when i − j = 0 (mod p) = 0 when i − j = 0 (mod p) (C.1) More precisely: the GUE, eq.(2.1), corresponds to the partition function Z = 1 Z 0 i dM ii i<j dR(M ij )dI(M ij ) e − L 2 i M 2 ii + i<j 2R(M ij ) 2 +2I(M ij ) 2 (C.2) where Z 0 is some normalization factor. This shows that the matrix elements are independent Gaussian random variables with the diagonal elements M ii having variance 1 L ; among the off-diagonal elements M ij , with i < j, can be taken to be the independent variables, with M ij for i > j being determined by the condition M † = M . R(M ij ), I(M ij ) denote the real and imaginary parts of M ij and these have variance 1 2L . Now in the systematic randomness case the matrix elements which are non-zero, M ij , with i − j = 0 mod p, eq. (C.1) are drawn in the same manner as independent Gaussian Random variables; i.e., the diagonal elements M ii are taken to have variance 1 L and among the off diagonal elements, M ij with i − j = 0 mod p that are non-zero, with M ij , i < j, being the independent ones and the variance for R(M ij ), I(M ij ) being 1 2L . Let us now consider a few special cases. For p = 1, the random matrix is fully random, i.e., all the elements of the matrix are non-zero random elements. For p = L, the only non-zero elements are the diagonal elements. More generally, let L = p q, then the matrix breaks up into p block diagonal sub-matrices 9 , each of rank q. The first block has the matrix elements M ij with i and j both being equal to 1 mod p. The second block has the matrix elements M ij , with i, j = 2 mod p, etc. The total number of (real) random elements is therefore equal to q 2 p = L 2 /p. It is also easy to see that each block is a GUE of q × q matrices with variance being L. Thus we see that the amount of sparseness can be controlled by varying the value of p with only a fraction 1/p of the L 2 total matrix elements being present. Due to the block diagonal nature, with each block being a GUE, most of the properties of this ensemble easily follow from those of the GUE as we will see below in Fig 34 and 35. The above is a plot of the rescaled density of states for various values of p. It is evident from the above plot that the except for highly sparse matrix, the case of p = 576 in the plot above when only L 2 p random elements are kept, the rest of the cases fit well with a Wigner distribution. For the case of p = 576, the size of each block is L p ∼ 7 and hence the wiggles around the Wigner distribution. The case of p = 2 is shown explicitly in the right panel where it can be seen that the two blocks have identical distribution, each contributing half to the total distribution, as expected. The above plot shows the level spacing distribution for the systematically sparsed random matrix. The left panel is the level spacing distribution for unfolded eigenvalues for the full random matrix and for varying values of p. As can be seen from the plots, for p > 1, when the full matrix splits in to blocks, the level spacing distribution is no longer follows Wigner surmise. As p increases, the distribution moves from Wigner Surmise to a Poisson distribution. This is just because of the fact that as the value of p increases, the full matrix splits into more number of blocks each of which are uncorrelated. So, increasing p corresponds to decreasing correlations between the eigenvalues of the full matrix. Each of the sub blocks in each matrix are still completely correlated as can be seen from the right panel. The mixing of the blocks leads to the distribution becoming more Poisson-like. We shall now calculate the 2-pt function for the case of sparse random matrix corresponding to systematic sparseness. As already mentioned above the hamiltonian for the systematically sparse random matrix becomes a block diagonal matrix. In particular, the unitary matrices which preserve this block diagonal form the symmetry group for such Hamiltonians. So, we shall compute the averaged 2-pt function in a manner similar to the previous section B but with the understanding that the corresponding unitary matrices are also block diagonal to preserve block diagonal nature of the Hamiltonian. Let the Hamiltonian be a block diagonal matrix with p blocks each of size q. Any matrix element of a block diagonal unitary matrix will be of the form U a a = 0 iff a = αq + i, a = αq + i , 1 ≤ α ≤ p, 1 ≤ i, i ≤ q. (C. 3) The expression for the two point function is still given by the expression in eq. (B.1). The operators A and B need not be block diagonal. The measure dHdU in eq. (B.1) means that dHdU = γ dU γ κ dH κ (C.4) where the quantities with subscripts ρ, κ indicate the corresponding blocks. Also, since we are assuming A, B are non block diagonal, the product U † AU will be of the form (U † AU ) a b = (U † ) a a A a b U b b = (U † α ) i i (A αβ ) i j (U β ) j j (C.5) where in the second line the quantities with the subscripts corresponds to the appropriate blocks. A similar expression can be written for operator involving B. The two point function eq. (B.1) written explicity in terms of blocks reads A(t)B(0) Sys = dU dH e −iHt d a e iHt b c U † A(0)U a b (U † B(0)U ) c d = dH e −iHt d a e iHt b c (A αβ (0)) i j (B ρσ (0)) k l × γ dU γ (U † α ) i i (U β ) j j (U † ρ ) k k (U σ ) l l = κ dH κ e −iHαt l i e iHρt j k δ ασ δ βρ (A αβ (0)) i j (B ρσ (0)) k l × γ dU γ (U † α ) i i (U β ) j j (U † ρ ) k k (U σ ) l l (C.6) where we used a = αq + i, b = βq + j, c = ρq + k, d = σq + l with 1 ≤ α, β, ρ, σ ≤ p, 1 ≤ i, j, k, l ≤ q (C.7) We now need the analog of the identity eq. (B.2). It is given by γ dU γ (U † α ) i i (U β ) j j (U † ρ ) k k (U σ ) l l = δ αβρσ q 2 − 1 δ i j δ j i δ k l δ l k + δ i l δ k j δ l i δ j k − 1 q (δ i j δ j k δ k l δ l i + δ j i δ k j δ l k δ i l ) + 1 − δ αβρδ q 2 δ αβ δ ρσ δ i j δ j i δ k l δ l k + δ ασ δ ρβ δ i l δ l i δ j k δ k j (C.8) The delta function with four indices means that it gives unity only if block diagonal indices are such that α = β = ρ = σ and zero otherwise. Using the identity eq. (C.8) in eq. (C.6) and contracting the indices appropriately, we get A(t)B(0) Sys = α 1 q 2 − 1 q − 1 q Z αZα Tr(A αα (0))Tr(B αα (0)) + (Z αZα − 1)Tr(A αα (0)B αα (0)) + α,β (1 − δ αβ ) q 2 Z αZβ Tr(A αβ (0)B βα (0)) (C.9) where we have explicitly written the summation to make it clear and where Z α = Tr(e −iHαt ),Z α = Tr(e iHαt ). The first term is the usual term that one get if the operators A, B are also block diagonal. The second term is an extra contribution that arises due to the off diagonal blocks in A, B. The four-point can also be generalized in a manner analogous to the two point function but involves many more terms. We do not explicitly write it here except to note that the leading term in the OTOC will also be given by eq. (B.8). D Wigner surmise In this appendix we outline the derivation of the distribution of the ratio of nearest neighbour eigenvalues, eq. (3.9). For Wigner Surmise p(r) is given by p(r) = K (r + r 2 ) β A quick derivation of the above result for the case of 3 × 3 matrix is shown below. D.1 3 × 3 matrix Consider 3 × 3 matrix H. We consider both GOE (β = 1) and GUE (β = 2) here. One can diagonalize this matrix by orthogonal matrix for β = 1 and unitary matrix for β = 2 and obtain the three eigenvalues, where λ 1 < λ 2 < λ 3 (D.2) Then the eigenvalue ratio r is given by r ≡ λ 3 − λ 2 λ 2 − λ 1 ≥ 0 (D.3) and the probability for finding eigenvalue ratio r is given by p(r) ∝ λ 2 −∞ dλ 1 +∞ −∞ dλ 2 +∞ λ 2 dλ 3 ∆(λ i ) δ r − λ 3 − λ 2 λ 2 − λ 1 e − β 4 Tr( i λ 2 i ) (D.4) Here we introduce Vandermonde determinant; ∆(λ i ) = Π 3 i<j (λ j − λ i ) β . (D.5) To simplify the calculation we change variables (λ 1 , λ 3 ) → (x, y) where x ≡ λ 2 − λ 1 , y ≡ λ 3 − λ 2 . where in the third line, we use δ r − y x = xδ(xr − y) (D. 10) and in the fourth line, integration over y is conducted. Therefore we obtain p(r) = K β r β (1 + r) β (1 + r + r 2 ) 1+ 3β 2 , (D.11) where K β is a normalization factor, which we can easily obtain from ∞ 0 dr p(r) = 1 as K β=1 = 27 8 for GOE (β = 1) , K β=2 = 81 √ 3 4π for GUE (β = 2) . (D.12) As is seen in this example, eq. (D.11) is correct expression only for 3 × 3 matrix. In higher L × L case, the expression deviates from eq. (D.11). See for example, [39]. E Error Analysis for non-local SYK OTOC fit In this appendix we present a detailed error analysis we have done for the case of OTOCs. We have taken non-local SYK OTOC as the set up to perform the error analysis since we already know the Lyapunov exponent is the limit of large N saturates the Chaos bound value of 2πT . The main point of doing the error analysis is to estimate the maximum time till which the growth of the OTOC can be approximated by an exponential. Starting with the smallest values of time, the exponential fit will be good till a certain time beyond which the exponential approximation to the OTOC will not be good and we are interested in estimating this critical time. We will denote this critical time as t c below. The error analysis is carried out using the Monte Carlo method. Given that our numerical data is a discrete set as a function of time, we first fit our data to the the function F th = A + Be λt , 0 < t < t max (E.1) for various values of t max . This is done using the best fit package of Python (scipy.optimize.curve f it). The python output will be the values of A, B, λ for the best fit case along with the associated covariance matrix σ αβ , of which we only show the diagonal elements corresponding to σ AA , σ BB , σ λλ denoted by σ A , σ B , σ λ in the table. This is the starting point for the analysis. We explain the methods for a single value of β. We start by choosing a value of t max , the maximum time till which we fit the exponential. Let y = {A, B, λ} and σ y = {σ A , σ B , σ λ }. E.1 Monte Carlo Simulation This method is described in [34] Chapter 15.6.1, 'Monte Carlo Simulation of Synthetic Data Sets'. We will also explain it, to some extent, below. From the quantities y = {A, B, λ} and σ y = {σ A , σ B , σ λ }, mentioned above, we can build three independent normal distribution of 5000 elements, around mean y i with variance σ 2 y i . So in total we have 5000 3 elements of type {A i , B j , λ k } where A i ∈ N (A, σ A ), B j ∈ N (B, σ B ), λ k ∈ N (λ, σ λ ). We retain only those data points which are within 1σ deviation from the mean value. Next we fix a time point (t n ) and generate the corresponding OTOC value using eq. (E.1) for F th (t n , A i , B j , λ k ). Around each t n we now have a total of ≈ 5000 3 such points. We take this distribution of ≈ 5000 3 points and compute the σ for that distribution. This σ can be denoted by σ tn , we use σ n . Now we use this to compute the value of χ 2 using the formula (6.14.38), & (15.1.6) in [34] χ 2 0 (t max ) = tn≤tmax (F numerical (t n ) − F th (t n , A, B, λ)) 2 σ n (E.2) then compute P (χ 2 > χ 2 0 ) given by P (χ 2 > χ 2 0 ) = χ 2 χ 2 0 1 2 ν 2 Γ ν 2 (χ 2 ) ν 2 −1 e − χ 2 2 dχ 2 (E.3) We repeat this for various values of t max . The one with lowest value of P will give the best fit and we set t c equal to this value of t max . Once we have fixed t c then our best estimate for the Lyapunov exponent is the corresponding fitted parameter(λ) from Python. In fact this more systematic way of obtaining t c and λ agrees with the more naive way of fitting "with the eye". This justify our choosing of t max for fitting the data naively with "the eye". The error in λ for the value of β under consideration corresponds to the standard deviation obtained from the best fit till t max . From the data shown below in fig.36, we see that for β = 20, 30, 50, correspondingly we get t max = 4, 5.5, 7. Thus for smaller β, the exponential fit is only good till small value of t. F Lyapunov Exponent for Local SYK In the local SYK section 4, we have discussed about OTOCs in Local SYK model forḠ i,i+1 andḠ i,i+2 , see fig. 26. We had argued there that there is no good large N limit that suppresses the connected part compared to the disconnected part. However, the plots for OTOCs are qualitatively similar to the case of Non-local SYK and so we can try to extract a Lyapunov exponent. This is shown in the plots below. In fig. 38, the blue curves are for fitting the Lyapunov exponents at different β with a functional form λ = A β + B. Black curves are for fitting with λ = A β . The green error bars correspond to 1σ error obtained from fitting the OTOCs with eq. (E.1). Within our numerical limitations we see that both fitting gives value of A to be lesser than 2π. Figure 1 : 1(a) The density of eigenvalues for 2 12 × 2 12 GUE matrix. (b) The density of states for q = 4, N = 24 SYK Hamiltonian. The number of ensemble elements used is 1000. Fig 2 shows the density of states ρ L (E) for L = 10, which can be contrasted with the case of L = 4096 in Fig 1. Figure 2 : 2The density of eigenvalues for 10 × 10 GUE random matrix ensemble. Fig 3. Figure 3 : 3Fig.(a) S(T ) = S −ln L vs T for GUE. Fig.(b) S N vs T for Non-local SYK model Figure 4 : 4Left: The spectral form factor for the size L = 2 12 random matrices. Right: The spectral form factor for the q = 4, N = 24 SYK model. The size of Hilbert space are both 2 12 = 4096. Figure 5 : 5Left: Nearest Neighbour Level Spacing for the size L = 2 12 Gaussian Orthogonal random matrices. Right: Nearest Neighbour Level Spacing for the q = 4, N = 24 SYK model. The size of Hilbert space in both cases is 2 12 = 4096. G 4 Figure 6 : 46(t) = 1 −F (t) (2.32) Non-Local SYK OTOC, q = 4, Non-local SYK model with N = 24, (a) Plot of G 4 = 1 −F as a function of time, fitted with A + Be λt , up to t ≤ t max , (b) Lyapunov coefficient as a function of β. Figure 7 : 7Plot of G 4 = 1 −F vs t. Solid curves are the theoretical result, eq.(2.35), in the L → ∞ limit. Points are numerical data. Figure 8 : 8The density of states for the random sparseness with L = 4096 and varying amount of randomness, n. Lower values of n implies a more sparse random matrix. For n ≥ 1.5 × 10 5 the spectrum is fitted by the Wigner semi-circle, after rescaling E, Fig(a)on left. Thereafter, for n ≤ 1.5 × 10 4 the behaviour changes and this is no longer true, Fig(b)on right. Figure 9 : 9Dependence of rescaling parameters B on n L 2 . Figure 10 : 10Plot of log n c vs log L. Error bars are at 1σ. Solid line is the best fit. Figure 13 : 13a) P (s) vs unfolded level spacing (b) P (ln r) vs ln(r) plot Level Spacing with changing randomness (n), for a matrix of rank 2 24/2 = 4096. n c ∼ 1.5 × 10 5 . Figure 14 : 14Fig(a)is the OTOC as a function of time t for different β. Solid curve is the best fit at early times. Fig(b)is the Lyapnov coefficient as a function of β. Figure 15 : 15Fig(a)is the OTOC as a function of time t for different β. Solid curve is the best fit at early times. Fig(b)is the Lyapnov coefficient as a function of β. Figure 16 : 16Fig(a)is the OTOC as a function of time t for different β. Solid curve is the best fit at early times. Fig(b)is the Lyapnov coefficient as a function of β. Figure 17 : 17Fig(a)is the OTOC as a function of time t for different β. Solid curve is the best fit at early times. Fig(b)is the Lyapnov coefficient as a function of β. Figure 18 : 18The density of states for non-local SYK model (left) and local SYK model (right) for different N . For the non-local SYK model, J = 1, and for N = 20, 22, 24, 26, 28, we take 10 4 , 4 × 10 3 , 10 3 , 150, 21 elements of the ensemble, respectively. For the Local SYK model,Ĵ = 1, and for N = 20, 22, 24, 26, 28, 30, we take 10 4 , 4 × 10 3 , 10 3 , 500, 100, 10 elements of the ensemble, respectively. Figure 19 : 19S/N vs T plot for Local SYK model. Here T R = T × N 3/2 is the rescaled temperature. Figure 20 : 20(a) log(Im G(ω)) vs log ω for N = 24, in non-local SYK Model. (b) Im (G(ω)) vs ω showing that the ω > 1 region is well fitted by the free field result. (c) Im (G(ω)) vs ω showing that the region 0.22 < ω < 0.65 is well fitted by the conformal result, eq. (4.8). . The corresponding plots for the Green's function in the local model are shown in Figure 21 : 21(a) log(Im G(ω R )) vs log ω R for N = 24, in local SYK Model. Figure 22 : 22(a) disconnected, (b) connected, (c) total spectral form factor respectively for the local (red) and non-local (blue) SYK models, at N = 24, β = 0.042, and 1000 ensembles. Figure 23 :Figure 24 : 2324Log-log plot of total (black), disconnected (blue), connected (red), SFF for the local SYK model.Finally, we have also analyzed the dependence of the SFF for varying values of the inverse temperature β, as shown in fig. 24. From the (b) panel (connected component) we see that the ramp and intermediate maximum get less and less pronounced in the local model as β increases, and finally disappear around β 70. This is connected to the fact that the starting value, at early times, for the connected component increases with increasing β while, in contrast, the height of the intermediate maximum roughly stays constant. The final value of the plateau also increases with increasing β, as can also be see in the right panel. (a) disconnected, (b) connected, (c) total SFF, respectively for local SYK model, at N = 24, 600 ensembles, and varying β. This β is in units ofĴ. And we rescale the time as t R = t × N −3/2 . Figure 25 : 25Nearest neighbour level spacing for the GOE, Non-local SYK Model and Local SYK Model. Figure 26 : 26SYK,Ḡ i,i+1 vs t (b) Local SYK,Ḡ i,i+2 vs t Local SYK modelḠ i,i+1 , &Ḡ i,i+2as a function of time, for varying values of β. β is inĴ = 1 unit, and we rescale t as t R = t × N −3/2 . Figure 27 : 27Local SYKḠ i,i+p as a function of time for different values of p. Left panel β = 0.34, Right panel β = 1.021. β is inĴ = 1 unit, and we rescale t as t R = t × N −3/2 . Figure 28 : 28Non-local SYK Model: operator growth as a function of time for different size of operators, s. Figure 29 : 29Local SYK Model: operator growth as a function of time for different sizes of operators, s. Here we rescale t as t R = t × N −3/2 .A quantitative understanding of this growth of operators, in the local case for N → ∞, is left for the future. two blocks of size M × M and (L − M ) × (L − M ) each, breaking U (L) → U (M ) × U (L − M ). Or more generally consider replacing the exponential term, eq. e., with zero mean and variance, σ = 1 √ L , independent of a. More generally we can consider distributions where there is a non-zero mean and both the mean and the variance vary for different generators. In this case eq. Figure 30 : 30(a) A comparison between models with different ψ q terms included. (b) After rescaling, all the different cases matches quite well. Figure 31 : 31Density of states for models with N = 20 and varying number of terms, n, in H. All terms included are of ψ q type with q = 0 (mod 4). Figure 32 : 32Density of states for models with N = 24 and differing number of terms, n, in H. All terms included are of ψ q type with q = 0 (mod 4). Figure 33 : 33(a) Density of states with varying λ, eq.(5.9), for N = 28 SYK model (b) Density of states with varying λ, after rescaling the axis appropiately. ported by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A "Extreme Universe" No. 21H05184. AM, SS and ST acknowledge support from the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4002, and acknowledge support from the Quantum Space-Time Endowment of the Infosys Science Foundation. AM, SS and ST acknowledge the use of the DTP, TIFR, computing cluster and especially thank Kapil Ghadiali and Ajay Salve for their help. AM thanks Nandini Trivedi and Ilya Gruzberg for valuable discussion. AM also thanks the Department of Physics, The Ohio State University for the University Fellowship. Z ∝ [DΣ][DG]e N 2 log det(−δ(t−t )∂t−Σ)+ N 2 ΣG− N Cq pJ 2 σ 2 G q (A.19) ) ∼ O(1) so that we get Σ(t) ∼ O(q −1 ). On the other hand, using eq. (A.31) in the second equation in eq. (A.29) gives Σ(t) = (J 2 σq) N C q 2 q−1 N sgn(t)e g(t) (A.35) Comparing eq. (A.34) and eq. (A.35), we get are the equations eq. (A.20). If σ is not small, this step is not valid and we should derive the appropriate equations from eq. (A.25). Let us for now consider the case where σ → 0. We use the ansatz eq. (A.31) for G(t). The first equation in eq. (A.20) then leads to eq. (A.34) as before where as the second equation now becomes traces on the rhs above involve only operators at time t = 0 and Z = Tr(e −iHt ),Z = Tr(e iHt ), ZZ H = dH Tr(e −iHt ) Tr(e iHt ) (B.4) Figure 34 : 34(a) Spectrum for the systematic sparseness with L = 4096 and various values of parameter p. (b) For p = 2 case both blocks follows Wigner semicircle. Figure 35 : 35(a) Nearest neighbour level spacing after unfolding the spectrum for different p values (b) Level spacing for p = 2 blocks. Each blocks satisfy Wigner surmise individually. (1 + r + r 2 ) 21+ β y β (x + y) β δ λ i ) = x β y β (x + y) β . β+1 y β (x + y) β δ (xr − y) e − β 4 (x 2 +y 2 β (1 + r) β (1 + r + r 2 ) 1+ 3β 2 (D.9) a) OTOC vs t plot different β (b) β = 20, 30, 50 Chi-square fitting Figure 36 : 36Non-local SYK model, G(t) = 1 −F (t) as a function of time and Chi-Square fit value for different β = 20, 30, 50. In (b) the values under [ ] denotes the corresponding P (χ 2 > χ 2 0 ) up to mentioned t max values in pervious line. Figure 37 : 37Non-local SYK model, G(t) = 1 −F (t) as a function of time and Chi-Square fit value for different β, with fitting parameters(A, B, λ), σ A , σ B , σ λ , degrees of freedom, χ 2 0 and P (χ 2 > χ 2 0 ) Figure 38 : 38Lyapunov exponent behavior at different termperature for Local SYK model. (a) λ vs β forḠ i,i+1 (b) λ vs β forḠ i,i+2 . We thank M. Tezuka for discussions of the sparse SYK models. The partition function eq. (2.1) and (2.3) are β → 0 limit of this thermal partition function. We thank Tezuka Masaki for helpful discussion on this point. For parrallelized matrix free OTOC computation we have used "dynamite" Python frontend, Krylov Subspace methods, PETSc and SLEPc Packages. Detailed discussion for this packages is discussed in[31]. To be more precise, in making the above statements we have in mind the limit where L → ∞, with M/L being held fixed. We thank Subir Sachdev for emphasising some of these points to us.7 It could be that a saddle point arises only when the variance of the non-zero couplings also scales with N suitably, we did not consider this possibility in section 3. Hyperfast scrambling might also be of interest from the point of view of gravity in dS space and more generally, cosmology, see[35] We thank Yoshinori Matsuo for pointing this out to us. 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[]
[ "Importance of electron-positron pairs on the maximum possible luminosity of the accretion columns in ULXs", "Importance of electron-positron pairs on the maximum possible luminosity of the accretion columns in ULXs" ]
[ "Valery F Suleimanov [email protected] \nInstitut für Astronomie und Astrophysik\nUniversitat Tuebingen\nSand 1D-72076TübingenGermany\n\nKazan Federal University\nKremlevskaya 18420008KazanRussia\n\nSpace Research Institute of the Russian Academy of Science\nProfsoyuznaya 84/32117997MoscowRussia\n", "Alexander Mushtukov \nSpace Research Institute of the Russian Academy of Science\nProfsoyuznaya 84/32117997MoscowRussia\n\nLeiden Observatory\nLeiden University\nNL-2300RALeidenThe Netherlands\n", "Igor Ognev \nP. G. Demidov\nYaroslavl State University\nSovietskaya 14150003YaroslavlRussia\n", "Victor A Doroshenko \nInstitut für Astronomie und Astrophysik\nUniversitat Tuebingen\nSand 1D-72076TübingenGermany\n\nSpace Research Institute of the Russian Academy of Science\nProfsoyuznaya 84/32117997MoscowRussia\n", "Klaus Werner \nInstitut für Astronomie und Astrophysik\nUniversitat Tuebingen\nSand 1D-72076TübingenGermany\n" ]
[ "Institut für Astronomie und Astrophysik\nUniversitat Tuebingen\nSand 1D-72076TübingenGermany", "Kazan Federal University\nKremlevskaya 18420008KazanRussia", "Space Research Institute of the Russian Academy of Science\nProfsoyuznaya 84/32117997MoscowRussia", "Space Research Institute of the Russian Academy of Science\nProfsoyuznaya 84/32117997MoscowRussia", "Leiden Observatory\nLeiden University\nNL-2300RALeidenThe Netherlands", "P. G. Demidov\nYaroslavl State University\nSovietskaya 14150003YaroslavlRussia", "Institut für Astronomie und Astrophysik\nUniversitat Tuebingen\nSand 1D-72076TübingenGermany", "Space Research Institute of the Russian Academy of Science\nProfsoyuznaya 84/32117997MoscowRussia", "Institut für Astronomie und Astrophysik\nUniversitat Tuebingen\nSand 1D-72076TübingenGermany" ]
[ "Neutron Star Astrophysics at the Crossroads: Magnetars and the Multimessenger Revolution, IAU Symposium N 363 Proceedings IAU Symposium" ]
One of the models explaining the high luminosity of pulsing ultra-luminous X-ray sources (pULXs) was suggested byMushtukov et al. (2015). They showed that the accretion columns on the surfaces of highly magnetized neutron stars can be very luminous due to opacity reduction in the high magnetic field. However, a strong magnetic field leads also to amplification of the electron-positron pairs creation. Therefore, increasing of the electron and positron number densities compensates the cross-section reduction, and the electron scattering opacity does not decrease with the magnetic field magnification. As a result, the maximum possible luminosity of the accretion column does not increase with the magnetic field. It ranges between 10 40 − 10 41 erg s −1 depending only slightly on the magnetic field strength.
10.1017/s1743921322001041
[ "https://export.arxiv.org/pdf/2208.14237v1.pdf" ]
251,928,864
2208.14237
37da5ba6cd4748b40b7442928966bd18991a6b1d
Importance of electron-positron pairs on the maximum possible luminosity of the accretion columns in ULXs 2022 Valery F Suleimanov [email protected] Institut für Astronomie und Astrophysik Universitat Tuebingen Sand 1D-72076TübingenGermany Kazan Federal University Kremlevskaya 18420008KazanRussia Space Research Institute of the Russian Academy of Science Profsoyuznaya 84/32117997MoscowRussia Alexander Mushtukov Space Research Institute of the Russian Academy of Science Profsoyuznaya 84/32117997MoscowRussia Leiden Observatory Leiden University NL-2300RALeidenThe Netherlands Igor Ognev P. G. Demidov Yaroslavl State University Sovietskaya 14150003YaroslavlRussia Victor A Doroshenko Institut für Astronomie und Astrophysik Universitat Tuebingen Sand 1D-72076TübingenGermany Space Research Institute of the Russian Academy of Science Profsoyuznaya 84/32117997MoscowRussia Klaus Werner Institut für Astronomie und Astrophysik Universitat Tuebingen Sand 1D-72076TübingenGermany Importance of electron-positron pairs on the maximum possible luminosity of the accretion columns in ULXs Neutron Star Astrophysics at the Crossroads: Magnetars and the Multimessenger Revolution, IAU Symposium N 363 Proceedings IAU Symposium 3632022accretionaccretion disksstars: neutronX-rays: binariesradiative transfer One of the models explaining the high luminosity of pulsing ultra-luminous X-ray sources (pULXs) was suggested byMushtukov et al. (2015). They showed that the accretion columns on the surfaces of highly magnetized neutron stars can be very luminous due to opacity reduction in the high magnetic field. However, a strong magnetic field leads also to amplification of the electron-positron pairs creation. Therefore, increasing of the electron and positron number densities compensates the cross-section reduction, and the electron scattering opacity does not decrease with the magnetic field magnification. As a result, the maximum possible luminosity of the accretion column does not increase with the magnetic field. It ranges between 10 40 − 10 41 erg s −1 depending only slightly on the magnetic field strength. Introduction At present a few pULXs are known (see e.g. Fabrika et al. 2021). One of the models describing their observed high luminosities is a strongly magnetized neutron star (NS) with high mass-accretion rate (Mushtukov et al. 2015) in a binary system. This model predicts increasing of the pULX luminosity with the strength of magnetic field, and, therefore, NSs with a magnetar-like magnetic fields are necessary for explaining the observed pULX luminosities (see also Eksi et al. 2015, Brice et al. 2021. Here we improve the model by changing of some geometry assumptions and considering the contribution of electron-positron pairs to the opacity. Model We made following improvements in the cross-section model of the accretion column. The magnetospheric radius R m in radiation-dominated discs as well as the propeller conditions were computed according to Chashkina et al. (2019). We fixed the relative thickness of the transition layer between the disc and the magnetosphere z d = ∆R/R m . We took into account the radiation friction between the radiation and the plasma outside the column. We changed the velocity law along the column height, V (h) = V 0 (h/H x ) ξ , where ξ > 0 is a parameter, and H x is the current column height, H x=0 = H 0 . The dependence of the horizontal radiation flux was taken as F (x) = F 0 (1 − τ x /τ 0 ) 1/β with a parameter β 1. e + -e − pairs were considered assuming thermodynamic equilibrium according to Kaminker & Yakovlev (1993) and Mushtukov et al. (2019). It was shown in these papers that a number density of the pairs strongly depends on the magnetic field strength and increases significantly if B > B cr = 4.414 × 10 13 G. We assume the solar 1 arXiv:2208.14237v1 [astro-ph.HE] 25 Aug 2022 2 V. F. Suleimanov, A. Mushtukov, I. Ognev, V. A. Doroshenko & K. Werner H/He mix as a chemical composition of the accreting plasma, and take into account all the opacity sources namely free-free transitions, electron scattering, cyclotron absorption, opacity due to two-photon annihilation and one-photon annihilation in a strong magnetic field. Electron number density vs. temperature for three magnetic field strengths and fixed plasma density. The thermodynamical equilibrium positron number density for the non-magnetic case is shown with the black dotted curve. Dependences of the Rosseland mean opacities on the plasma temperature for various magnetic field strengths and two plasma densities, ρ = 0.1 (dashed curves) and 10 g cm −3 (solid curves). The magnetic field strengths are marked at the curves. Continuum and cyclotron opacities in two modes for the given plasma parameters and fixed angle between direction of photon propagation and magnetic field. The two-photon and one-photon annihilation opacities are also shown. The latter opacity is given for three magnetic field strengths. Pairs prevent huge luminosity of the accretion columns in magnetars. Strong fields are not necessary for pulsing ULXs even at low beaming (see also Chashkina et al. 2019). * # = 0.3! 12 Positions of the pULXs on the luminosity -NS magnetic field strength plane. The upper limits on the magnetic field strength are found from the propeller conditions. The corresponding conditions for five spin periods are shown with the dotted lines. The approximate boundaries for the magnetospheres located in the radiation-and the advection-dominated parts of the discs are also plotted with the dashed and the dot-dashed lines respectively. The new recomputed maximum possible luminosities of the accretion columns (with H0=RNS) are plotted with the thick solid (β=0.1) and dashed (β=0.01) red curves. The corresponding old curve (Mushtukov et al. 2015) is drawn with the thin dashed red curve. * = 0.1 Λ = 0.5, 5 ! = 0.1, 0 = 0.5 2 Application to a long burst of 4U 1724 307, LMXB in a globular claster Terzan 2 • RXTE has observed three photospheric radius expanshion bursts (Fig.4), one long, and two short. • Differences in the observed properties can be explained by different accretion disk states (Fig.5). • Observed for a long burst dependense K 1/4 F can be fitted by theoretical f c L/L Edd dependences (Fig.6). Figure 4: Evolution of the observed blackbody fluxes, color temperatures and normalizations for a long (black circles) and short bursts from 4U 1724 307. Figure 5: Different spectral states of 4U 1724 307accretion disk before a short and a long bursts. Figure 6: Comparison of the X-ray burst data for 4U 1724 307 to the theoretical models of NS atmosphere. The crosses present the observed dependence of K 1/4 vs. F for the long burst, while diamonds represent two short bursts. Figure 7: Constraints on the mass and radius of the NS in 4U 1724 307 from the long burst spectra. These correspond to the three chemical compositions: green for pure hydrogen, blue for the solar ratio of H/He and subsolar metal abundance Z = 0.3Z appropriate for Terzan 2, and red for pure helium. Conclusions • The neutron star radius in 4U 1724 307is larger than 14 km for masses below 2.3 solar mass (Fig.7). Results Electron scattering on e + -e − pairs increases the Rosseland opacity at kT > 30 keV significantly (right panel in Fig. 1). We note that high luminosity accretion columns are optically thick at the considered parameters. Therefore, there are conditions for pair thermodynamical equilibrium and the pair numerical densities are significant along all the column height at high magnetic fields B > B cr . The opacity increase due to electron scattering on the pairs overcomes the opacity reduction due to magnetic field increase, and pairs prevent a huge luminosity of the accretion columns in magnetars at B > 10 14 G. (left panel in Fig. 1). The considered model provides the high enough accretion column luminosities at some parameters even at low magnetic fields, and strong NS fields are not necessary for pULXs even at low beaming due to the new propeller conditions derived by Chashkina et al. (2019). Appendix The poster itself presented as Fig. 2. Electron number density vs. temperature for three magnetic field strengths and fixed plasma density. The thermodynamical equilibrium positron number density for the non-magnetic case is shown with the black dotted curve. Dependences of the Rosseland mean opacities on the plasma temperature for various magnetic field strengths and two plasma densities, ρ = 0.1 (dashed curves) and 10 g cm −3 (solid curves). The magnetic field strengths are marked at the curves. Continuum and cyclotron opacities in two modes for the given plasma parameters and fixed angle between direction of photon propagation and magnetic field. The two-photon and one-photon annihilation opacities are also shown. The latter opacity is given for three magnetic field strengths. Pairs number density increases in strong magnetic fields (see e.g. Mushtukov, Ognev, Nagirner 2019). It leads to significant increasing of Rosseland opacity at kT > 30 keV and B > 10 13 G. Pairs prevent huge luminosity of the accretion columns in magnetars. Strong fields are not necessary for pulsing ULXs even at low beaming (see also Chashkina et al. 2019). # = 0.3 12 Positions of the pULXs on the luminosity -NS magnetic field strength plane. The upper limits on the magnetic field strength are found from the propeller conditions. The corresponding conditions for five spin periods are shown with the dotted lines. The approximate boundaries for the magnetospheres located in the radiation-and the advection-dominated parts of the discs are also plotted with the dashed and the dot-dashed lines respectively. The new recomputed maximum possible luminosities of the accretion columns (with H0=RNS) are plotted with the thick solid (β=0.1) and dashed (β=0.01) red curves. The corresponding old curve (Mushtukov et al. 2015) is drawn with the thin dashed red curve. Method • Observed spectra of X-ray bursting NSs are fitting by blackbody spectra with two parameters: color temperature T BB and normalization K = R 2 BB [km]/D 2 10 . • Real spectra of X-ray bursting NSs are close to diluted blackbody F E = f 4 c B E (f c T e↵ ) with color correction factrors f c in range 1.4 -1.8 (Fig.1), therefore K connected to real NS radius via f c and gravitational redshift z: K = (R(1 + z)) 2 /(f 2 c d 10 ) 2 . • In late phases of photospheric radius expantion bursts (when a photosphere radius equal to the neutron star radius) K depends on f c only. • We suggest to fit observed dependences K 1/4 F by theoretical dependences f c L/L Edd (Fig.2), where F is an observed bolometric flux. • Three parameters can be obtained from fitting: A = (R(1 + z)[km]/D 10 ) 1/2 , F Edd = GM c/(0.2(1 + X) (1 + z) D 2 ), where X is a hydrogen mass fraction, and T Edd,1 = 1.14 ⇥ 10 8 A F 1/4 Edd, 7 K, where F Edd, 7 = F Edd /10 7 erg cm 2 s 1 . • Some curve on a M R plane corresponds to each obtained parameter (Fig.3). Crossing points give necessary solutions for M and R. 2 Application to a long burst of 4U 1724 307, LMXB in a globular claster Terzan 2 • RXTE has observed three photospheric radius expanshion bursts (Fig.4), one long, and two short. • Differences in the observed properties can be explained by different accretion disk states (Fig.5). • Observed for a long burst dependense K 1/4 F can be fitted by theoretical f c L/L Edd dependences (Fig.6). : Different spectral states of 4U 1724 307accretion disk before a short and a long bursts. Figure 6: Comparison of the X-ray burst data for 4U 1724 307 to the theoretical models of NS atmosphere. The crosses present the observed dependence of K 1/4 vs. F for the long burst, while diamonds represent two short bursts. Figure 7: Constraints on the mass and radius of the NS in 4U 1724 307 from the long burst spectra. These correspond to the three chemical compositions: green for pure hydrogen, blue for the solar ratio of H/He and subsolar metal abundance Z = 0.3Z appropriate for Terzan 2, and red for pure helium. Conclusions • The neutron star radius in 4U 1724 307is larger than 14 km for masses below 2.3 solar mass (Fig.7). column shapes at different values of the parameter β=1, 0.1, and 0.01. The column shapes were computed with (solid curves) and without (dashed curves) taking radiation drag force into account. Other model parameters are fixed and shown in the plot. • Most possible the inner core of the neutron star has a stiff equation of state. Ditails see in papers: V. Suleimanov, J. Poutanen, K. Werner A&A, 527, A139, 2011a; V. Suleimanov, J. Poutanen, M. Revnivtsev, K. Werner ApJ (in press), 2011b (arXiv:1004.4871). Figure 1 . 1Left panel: Dependences of the Rosseland mean opacities on the plasma temperature for various magnetic field strengths and two plasma densities, ρ= 0.1 (dashed curves) and 10 g cm −3 (solid curves). The magnetic field strengths are marked at the curves. Right panel:Positions of the pULXs on the luminosity -NS magnetic field strength plane. The upper limits on the magnetic field strength are found from the propeller conditions. The corresponding conditions for five spin periods are shown with the dotted lines. The approximate boundaries for the magnetospheres located in the radiation-and the advection-dominated parts of the discs are also plotted with the dashed and the dot-dashed lines respectively. The new recomputed maximum possible luminosities of the accretion columns (with H0 = RNS) are plotted with the thick solid (β=0.1) and dashed (β=0.01) red curves. The corresponding old curve(Mushtukov et al. 2015) is drawn with the thin dashed red curve. column model from Mushtukov et al. 2015 (see also Lyubarsky & Sunyaev 1988) Improvements -magnetospheric radius in radiation-dominated disc after Chashkina et al. 2019; -Fixed relative thickness of the transition layer; -Radiation friction of the plasma outside the column; accretion column shapes at different values of the parameter β=1, 0.1, and 0.01. The column shapes were computed with (solid curves) and without (dashed curves) taking radiation drag force into account. Other model parameters are fixed and shown in the plot. Institute, Moscow, Russia, (4) Leiden University, Netherlands, (5) Yaroslavl State University, Russia e-mail: [email protected] !Figure 1 :Figure 2 : 12Color-correction factor as a function of the NS luminosity for different chemical composition (seeSuleimanov et al. 2010a). The surface gravity is taken to be g = 10 14.0 cm s 2 . The dashed curve shows the results for a hydrogen atmosphere at larger gravity of log g = 14.3. Illustration of the suggested new cooling tail method.Figure 3: Constraints on M and R from various observed values. If the assumed distance is too large, there are no solutions (the corresponding curves for F Edd =const and R1=const shown by thin lines do not cross). Figure 4 : 4Evolution of the observed blackbody fluxes, color temperatures and normalizations for a long (black circles) and short bursts from 4U 1724 307. Figure 5 5Figure 5: Different spectral states of 4U 1724 307accretion disk before a short and a long bursts. • Most possible the inner core of the neutron star has a stiff equation of state. Ditails see in papers: V. Suleimanov, J. Poutanen, K. Werner A&A, 527, A139, 2011a; V. Suleimanov, J. Poutanen, M. Revnivtsev, K. Werner ApJ (in press), 2011b (arXiv:1004.4871). Acknowledgements The work is supported by the DFG grant SFB / Transregio 7 "Gravitational Wave Astronomy" (V.S.), the Academy of Finland (grant 127512, J.P.), and DFG cluster of excellence "Origin and Structure of the Universe" (M.R.).Acknowledgements The work is supported by the DFG grant SFB / Transregio 7 "Gravitational Wave Astronomy" (V.S.), the Academy of Finland (grant 127512, J.P.), and DFG cluster of excellence "Origin and Structure of the Universe" (M.R.). cos . N Brice, S Zane, R Turolla, K Wu, MNRAS. 504701Brice N., Zane S., Turolla R., Wu K., 2021, MNRAS, 504, 701 . A Chashkina, G Lipunova, P Abolmasov, J Poutanen, A&A. 62618Chashkina A., Lipunova G., Abolmasov P., Poutanen J., 2019, A&A, 626, A18 . K Y Eksi, I C Andac, S Cikintoglu, MNRAS. 44840Eksi K. Y., Andac I. C., Cikintoglu S., et al., 2015, MNRAS, 448, L40 . S N Fabrika, K E Atapin, A S Vinokurov, O N Sholukhova, Astroph. Bull. 766Fabrika S. N., Atapin K. E., Vinokurov A. S., Sholukhova O. N., 2021, Astroph. Bull., 76, 6 . A D Kaminker, D G Yakovlev, Soviet JETPh. 76229Kaminker A. D., Yakovlev D. G., 1993, Soviet JETPh, 76, 229 . A A Mushtukov, I S Ognev, D I Nagirner, MNRAS. 485131Mushtukov A. A., Ognev I. S., Nagirner D. I., 2019, MNRAS, 485, L131 . A A Mushtukov, V F Suleimanov, S S Tsygankov, J Poutanen, MNRAS. 4542539Mushtukov A. A., Suleimanov V. F., Tsygankov S. S., Poutanen J., 2015, MNRAS, 454, 2539
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[ "Jacquet modules of principal series generated by the trivial K-type", "Jacquet modules of principal series generated by the trivial K-type" ]
[ "Noriyuki Abe [email protected] \nGraduate School of Mathematical Sciences\nthe University of Tokyo\n3-8-1 Komaba, Meguro-ku153-8914TokyoJapan\n" ]
[ "Graduate School of Mathematical Sciences\nthe University of Tokyo\n3-8-1 Komaba, Meguro-ku153-8914TokyoJapan" ]
[]
We propose a new approach for the study of the Jacquet module of a Harish-Chandra module of a real semisimple Lie group. Using this method, we investigate the structure of the Jacquet module of principal series representation generated by the trivial K-type. §1. IntroductionLet G be a real semisimple Lie group. By Casselman's subrepresentation theorem, any irreducible admissible representation U is realized as a subrepresentation of a certain non-unitary principal series representation. Such an embedding is a powerful tool to study an irreducible admissible representation but the subrepresentation theorem dose not tell us how it can be realized.Casselman[Cas80]introduced the Jacquet module J(U ) of U . This important object retains all information of embeddings given by the subrepresentation theorem. For example, Casselman's subrepresentation theorem is equivalent to J(U ) = 0. However the structure of J(U ) is very intricate and difficult to determine.In this paper we give generators of the Jacquet module of a principal series representation generated by the trivial K-type. Let Z be the ring of integers, g 0 the Lie algebra of G, θ a Cartan involution of g 0 , g 0 = k 0 ⊕ a 0 ⊕ n 0 the Iwasawa decomposition of g 0 , m 0 the centralizer of a 0 in k 0 , W the little Weyl group for (g 0 , a 0 ), e ∈ W the unit element of W , Σ the restricted root system for (g 0 , a 0 ), g 0,α the root space for α ∈ Σ, Σ + the positive system of Σ such that n 0 = α∈Σ + g 0,α , ρ = α∈Σ + (dim g 0,α /2)α, P = { α∈Σ + n α α | n α ∈ Z}, P + = { α∈Σ + n α α | n α ∈ Z, n α ≥ 0} and U (λ) the principal series representation with an infinitesimal character λ generated by the trivial K-type. In this paper we prove the following theorem.Theorem 1.1 (Theorem 3.9, Theorem 4.1). Assume that λ is regular. Set W(w) = {w ′ ∈ W | wλ − w ′ λ ∈ 2P + } for w ∈ W . Then there exist generators {v w | w ∈ W } of J(U (λ)) such that (H − (ρ + wλ))v w ∈ w ′ ∈W(w) U (g)v w ′ for all H ∈ a 0 , Xv w ∈ w ′ ∈W(w) U (g)v w ′ for all X ∈ m 0 ⊕ θ(n 0 ).
10.2977/prims/1231263783
[ "https://export.arxiv.org/pdf/math/0601462v2.pdf" ]
16,580,411
math/0601462
2c990e3d6c6ae3fcec728fb52a94dd4e84f7807b
Jacquet modules of principal series generated by the trivial K-type 20 Jan 2006 Noriyuki Abe [email protected] Graduate School of Mathematical Sciences the University of Tokyo 3-8-1 Komaba, Meguro-ku153-8914TokyoJapan Jacquet modules of principal series generated by the trivial K-type 20 Jan 2006arXiv:math/0601462v2 [math.RT] 1 Noriyuki Abe We propose a new approach for the study of the Jacquet module of a Harish-Chandra module of a real semisimple Lie group. Using this method, we investigate the structure of the Jacquet module of principal series representation generated by the trivial K-type. §1. IntroductionLet G be a real semisimple Lie group. By Casselman's subrepresentation theorem, any irreducible admissible representation U is realized as a subrepresentation of a certain non-unitary principal series representation. Such an embedding is a powerful tool to study an irreducible admissible representation but the subrepresentation theorem dose not tell us how it can be realized.Casselman[Cas80]introduced the Jacquet module J(U ) of U . This important object retains all information of embeddings given by the subrepresentation theorem. For example, Casselman's subrepresentation theorem is equivalent to J(U ) = 0. However the structure of J(U ) is very intricate and difficult to determine.In this paper we give generators of the Jacquet module of a principal series representation generated by the trivial K-type. Let Z be the ring of integers, g 0 the Lie algebra of G, θ a Cartan involution of g 0 , g 0 = k 0 ⊕ a 0 ⊕ n 0 the Iwasawa decomposition of g 0 , m 0 the centralizer of a 0 in k 0 , W the little Weyl group for (g 0 , a 0 ), e ∈ W the unit element of W , Σ the restricted root system for (g 0 , a 0 ), g 0,α the root space for α ∈ Σ, Σ + the positive system of Σ such that n 0 = α∈Σ + g 0,α , ρ = α∈Σ + (dim g 0,α /2)α, P = { α∈Σ + n α α | n α ∈ Z}, P + = { α∈Σ + n α α | n α ∈ Z, n α ≥ 0} and U (λ) the principal series representation with an infinitesimal character λ generated by the trivial K-type. In this paper we prove the following theorem.Theorem 1.1 (Theorem 3.9, Theorem 4.1). Assume that λ is regular. Set W(w) = {w ′ ∈ W | wλ − w ′ λ ∈ 2P + } for w ∈ W . Then there exist generators {v w | w ∈ W } of J(U (λ)) such that (H − (ρ + wλ))v w ∈ w ′ ∈W(w) U (g)v w ′ for all H ∈ a 0 , Xv w ∈ w ′ ∈W(w) U (g)v w ′ for all X ∈ m 0 ⊕ θ(n 0 ). j≥i U (g)v w j . Then by Theorem 1.1 we have the surjective map M (w i λ) → V i /V i+1 where M (w i λ) is a generalized Verma module (See Definition 4.4). This map is isomorphic. Namely we can prove the following theorem. Theorem 1.2 (Theorem 4.5). There exists a filtration J(U (λ)) = V 1 ⊃ V 2 ⊃ · · · ⊃ V r+1 = 0 of J(U (λ)) such that V i /V i+1 ≃ M (w i λ). Moreover if wλ − λ ∈ 2P for w ∈ W \ {e} then J(U (λ)) ≃ w∈W M (wλ). This theorem does not need the assumption that λ is regular. In the case of G is split and U (λ) is irreducible, Collingwood [Col91] proved Theorem 1.2. For example, we obtain the following in case of g 0 = sl(2, R): Choose a basis {H, E + , E − } of g 0 such that RH = a 0 , RE + = n 0 , [H, E ± ] = ±2E ± and E − = θ(E + ). Then Σ + = {2α} where α(H) = 1. Let λ = rα for r ∈ C. We may assume Re r ≥ 0. By Theorems 1.1 and 1.2, we have the exact sequence 0 / / M (−rα) / / J(U (rα)) / / M (rα) / / 0. Consider the case λ is integral, i.e., 2r ∈ Z. If r ∈ Z then this sequence splits by Theorem 1.2. On the other hand, if r ∈ Z then by the direct calculation using the method introduced in this paper we can show it does not split. Notice that U (rα) is irreducible if and only if r ∈ Z. Then we have the following; if λ is integral then J(U (λ)) is isomorphic to the direct sum of generalized Verma modules if and only if U (λ) is reducible. Our method is based on the paper of Kashiwara and Oshima [KO77]. In Section 2 we show fundamental properties of Jacquet modules and introduce a certain extension of the universal enveloping algebra. An analog of the theory of Kashiwara and Oshima is established in Section 3. In Section 4 we prove our main theorem in the case of a regular infinitesimal character using the result of Section 3. We complete the proof in Section 5 using the translation principle. Acknowledgments The author is grateful to his advisor Hisayosi Matumoto for his advice and support. He would also like to thank Professor Toshio Oshima for his comments. Notations Throughout this paper we use the following notations. As usual we denote the ring of integers, the set of non-negative integers, the set of positive integers, the real number field and the complex number field by Z, Z ≥0 , Z >0 , R and C respectively. Let g 0 be a real semisimple Lie algebra. Fix a Cartan involution θ of g 0 . Let g 0 = k 0 ⊕ s 0 be the decomposition of g 0 into the +1 and −1 eigenspaces for θ. Take a maximal abelian subspace a 0 of s 0 and let g 0 = k 0 ⊕ a 0 ⊕ n 0 be the corresponding Iwasawa decomposition of g 0 . Set m 0 = {X ∈ k 0 | [H, X] = 0 for all H ∈ a 0 }. Then p 0 = m 0 ⊕ a 0 ⊕ n 0 is a minimal parabolic subalgebra of g 0 . Write g for the complexification of g 0 and U (g) for the universal enveloping algebra of g. We apply analogous notations to other Lie algebras. Set a * = Hom C (a, C) and a * 0 = Hom R (a 0 , R). Let Σ ⊂ a * be the restricted root system for (g, a) and g α the root space for α ∈ Σ. Let Σ + be the positive root system determined by n, i.e., n = α∈Σ + g α . Σ + determines the set of simple roots Π = {α 1 , α 2 , . . . , α l }. We define the total order on a * 0 by the following; for c i , d i ∈ R we define i c i α i > i d i α i if and only if there exists an integer k such that c 1 = d 1 , . . . , c k = d k and c k+1 > d k+1 . Let {H 1 , H 2 , . . . , H l } be the dual basis of {α i }. Write W for the little Weyl group for (g 0 , a 0 ) and e for the unit element of W . Set P = { α∈Σ + n α α | n α ∈ Z}, P + = { α∈Σ + n α α | n α ∈ Z ≥0 } and P ++ = P + \ {0}. Let m be a dimension of n. Fix a basis E 1 , E 2 , . . . , E m of n such that each E i is a restricted root vector. Let β i be a restricted root vector such that E i ∈ g β i . For n = (n i ) ∈ Z m ≥0 we denote E n 1 1 E n 2 2 · · · E nm m by E n . For x = (x 1 , x 2 , . . . , x n ) ∈ Z n ≥0 , we write |x| = x 1 + x 2 + · · · + x n and x! = x 1 ! x 2 ! · · · x n !. For a C-algebra R, let M (r, r ′ , R) be the space of r×r ′ matrices with entries in R and M (r, R) = M (r, r, R). Write 1 r ∈ M (r, R) for the identity matrix. §2. Jacquet modules and fundamental properties Definition 2.1 (Jacquet module). Let U be a U (g)-module. Define modules J(U ) and J(U ) by J (U ) = lim ← − k U/n k U, J(U ) = J(U ) a-finite = {u ∈ J(U ) | dim U (a)u < ∞}. We call J(U ) the Jacquet module of U . Set E(n) = lim ← −k U (n)/n k U (n). Proposition 2.2. (1) The C-algebra E(n) is right and left Noetherian. (2) The C-algebra E(n) is flat over U (n). (3) If U is a finitely generated U (n)-module then lim ← −k U/n k U = E(n) ⊗ U (n) U . (4) Let S = (S k ) be an element of M (r, n E(n)) and (a n ) ∈ C Z ≥0 . Define ∞ n=0 a n S n = ( k n=0 a n S n k ) k . Then ∞ n=0 a n S n ∈ M (r, E (n)). Proof. Since Stafford and Wallach [SW82, Theorem 2.1] show that nU (n) ⊂ U (n) satisfies the Artin-Rees property, the usual argument of the proof for commutative rings can be applicable to prove (1), (2) and (3). (4) is obvious. Corollary 2.3. Let S be an element of M (r, E (n)) such that S − 1 r ∈ M (r, n E (n)). Then S is invertible. Proof. Set T = 1 r − S. By Proposition 2.2, R = ∞ n=0 T n ∈ M (r, E(n)). Then SR = RS = 1 r . We can prove the following proposition in a similar way to that of Goodman and Wallach [GW80, Lemma 2.2]. For the sake of completeness we give a proof. Proposition 2.4. Let U be a U (a ⊕ n)-module which is finitely generated as a U (n)-module. Assume that every element of U is a-finite. For µ ∈ a * set U µ = {u ∈ U | For all H ∈ a there exists a positive integer N such that (H − µ(H)) N u = 0}. Then J(U ) ≃ µ∈a * U µ . Proof. For k ∈ Z >0 put S k = {µ ∈ a * | U µ = 0, U µ ⊂ n k U }. Since U is finitely generated, dim U/n k U < ∞. Therefore S k is a finite set. Define a map ϕ : µ∈a * U µ → J(U ) by ϕ((x µ ) µ∈a * ) =   µ∈S k x µ (mod n k U )   k . First we show that ϕ is injective. Assume ϕ((x µ ) µ∈a * ) = 0. We have µ∈S k x µ ∈ n k U for all k ∈ Z >0 . Since n k U is a-stable and S k is a finite set, x µ ∈ n k U for all µ ∈ a * , thus we have x µ = 0. We have to show that ϕ is surjective. Let x = (x k (mod n k U )) k be an element of J (U ). Since every element of U is a-finite, we have U = µ∈a * U µ . Let p µ : U → U µ be the projection. The U (n)-module U is finitely generated and therefore for all µ ∈ a * there exists a positive integer k µ such that n kµ U ∩ U µ = 0. Notice that if i, i ′ > k µ then p µ (x i ) = p µ (x i ′ ). Hence we have ϕ((p µ (x kµ )) µ∈a * ) = x. We define an (a ⊕ n)-representation structure of U (n) by (H + X)(u) = Hu − uH + Xu for H ∈ a, X ∈ n, u ∈ U (n). Then U (n) is a U (a ⊕ n)-module. By Proposition 2.4 E(n) = µ∈a * U (n) µ . The following results are corollaries of Proposition 2.4. Corollary 2.5. A linear map C[[X 1 , X 2 , . . . , X m ]] −→ E(n) n∈Z m ≥0 a n X n −→ |n|≤k a n E n (mod n k U (n)) k is bijective, where X n = X n 1 1 X n 2 2 · · · X nm m for n = (n 1 , n 2 , . . . , n m ) ∈ Z m ≥0 . Proof. By the Poincaré-Birkhoff-Witt theorem {E n | i n i β i = µ} is a basis of U (n) µ . This implies the corollary since E(n) = µ∈a * U (n) µ . We denote the image of n∈Z m ≥0 a n X n under the map in Corollary 2.5 by n∈Z m ≥0 a n E n . Corollary 2.6. Let U be a U (g)-module which is finitely generated as a U (n)-module. Assume that all elements are a-finite. Then J(U ) = U . Proof. This follows from the following equation. J(U ) = J(U ) a-finite =   µ∈a * U µ   a-finite = µ∈a * U µ = U. Put E(g, n) = E(n) ⊗ U (n) U (g). We can define a C-algebra structure of E(g, n) by (f ⊗ 1)(1 ⊗ u) = f ⊗ u, (1 ⊗ u)(1 ⊗ u ′ ) = 1 ⊗ (uu ′ ), (f ⊗ 1)(f ′ ⊗ 1) = (f f ′ ) ⊗ 1, (1 ⊗ X)(f ⊗ 1) = n∈Z r ≥0 1 n! ∂ |n| ∂E n f ⊗ ((ad(E)) n ) ′ (X), where u, u ′ ∈ U (g), X ∈ g, f, f ′ ∈ E(n), ((ad(E)) n ) ′ = (− ad(E m )) nm · · · (− ad(E 1 )) n 1 and ∂ |n| ∂E n   m∈Z m ≥0 a m E m   = m∈Z m ≥0 a m m! (m − n)! E m−n . Notice that E(g, n) ⊗ U (g) U ≃ E(n) ⊗ U (n) U as an E(n)-module for a U (g)-module U . By Proposition 2.2, E(g, n) is flat over U (g). Notice that if b is a subalgebra of g which contains n then E(n) ⊗ U (n) U (b) is a subalgebra of E(g, n). Put E(b, n) = E(n) ⊗ U (n) U (b). Let U be a U (a ⊕ n)-module such that U = µ∈a * U µ . Set V =    (u µ ) µ ∈ µ∈a * U µ there exists an element ν ∈ a * 0 such that u µ = 0 for Re µ < ν    . Then we can define an a-module homomorphism ϕ : E(a ⊕ n, n) ⊗ U (a⊕n) U ≃   µ∈a * U (n) µ   ⊗ U (n)   µ ′ ∈a * U µ ′   → V by ϕ((f µ ) µ∈a * ⊗ (u µ ′ ) µ ′ ∈a * ) = ( µ+µ ′ =λ f µ u µ ′ ) λ∈a * . Notice that the composition U → U → V is equal to the inclusion map U ֒→ V . We consider the case U = U (g). Define an (a ⊕ n)-module structure of U (g) by (H + X)(u) = Hu − uH + Xu for H ∈ a, X ∈ n, u ∈ U (g). We have a map ϕ : E(g, n) →    (P µ ) µ∈a * ∈ µ∈a * U (g) µ there exists an element ν ∈ a * 0 such that P µ = 0 for Re µ < ν    . We write ϕ(P ) = (P (µ) ) µ∈a * . Put P (H,z) = µ(H)=z P (µ) for z ∈ C and H ∈ a such that Re α(H) > 0 for all α ∈ Σ + . By the definition we have the following proposition. Proposition 2.7. (1) Assume that U is finitely generated as a U (n)-module. Let ϕ : E(a⊕ n, n)⊗ U (n) U → µ∈a * U µ be an a-module homomorphism defined as above. Then ϕ is coincide with the map given in Proposition 2.4. In particular ϕ is isomorphic. (2) We have (P Q) (λ) = µ+µ ′ =λ P (µ) Q (µ ′ ) for P, Q ∈ E(g, n) and λ ∈ a * . (3) We have   n∈Z m ≥0 a n E n   (λ) = i n i β i =λ a n E n for λ ∈ a * . Proposition 2.8. Let U be a U (g)-module which is finitely generated as a U (n)-module. We take generators v 1 , v 2 , . . . , v n of an E(g, n)-module E(g, n) ⊗ U (g) U and set V = i U (g)v i ⊂ E(g, n) ⊗ U (g) U . Define the surjective map ψ : E(g, n) ⊗ U (g) V → E(g, n) ⊗ U (g) U by ψ(f ⊗ v) = f v. Assume that there exist weights λ i ∈ a * and a positive integer N such that (H − λ i (H)) N v i = 0 for all H ∈ a and 1 ≤ i ≤ n. Let ϕ : E(g, n) ⊗ U (g) V → µ∈a * V µ be the map defined as above. Then there exists a unique map E(g, n) ⊗ U (g) U → µ∈a * V µ such that the diagram E(g, n) ⊗ U (g) V ϕ / / ψ µ∈a * V µ E(g, n) ⊗ U (g) U 7 7 is commutative. Proof. Set U = E(g, n) ⊗ U (g) U and V = E(g, n) ⊗ U (g) V . Take f (i) ∈ E(g, n) and v (i) ∈ V such that ψ( i f (i) ⊗ v (i) ) = 0. We have to show ϕ( i f (i) ⊗ v (i) ) = 0. Since V = E(n) ⊗ U (n) V , we may assume f i ∈ E(n). We can write f (i) = (f (i) µ ) µ∈a * by the isomorphism E(n) ≃ µ∈a * U (n) µ . Since V = µ ′ ∈a * V µ ′ , we can write v i = µ ′ ∈a * v (i) µ ′ , v (i) µ ′ ∈ V µ ′ . We have to show i µ+µ ′ =λ f (i) µ v (i) µ ′ = 0 for all λ ∈ a * . Since U is a finitely generated U (n)-module we have U = lim ← −k U/n k U = lim ← −k U /n k U. It is sufficient to prove i µ+µ ′ =λ f (i) µ v (i) µ ′ ∈ n k U for all k ∈ Z >0 . Fix λ ∈ a * and k ∈ Z >0 . We can choose an element ν ∈ a * 0 such that Re µ≥ν U (n) µ ⊂ n k U (n). Then 0 = ϕ( i f (i) ⊗ v (i) ) ≡ i Re µ<ν f (i) µ v (i) µ ′ (mod n k U ) . Notice that following two sets are finite. {µ | Re(µ) < ν and there exists an integer i such that f (i) µ = 0}, {µ ′ | there exists an integer i such that v (i) µ ′ = 0}. This implies i µ+µ ′ =λ f (i) µ v (i) µ ′ ∈ n k U . The following result is a corollary of Proposition 2.8. Corollary 2.9. In the setting of Proposition 2.8, we have the following. Let P i (1 ≤ i ≤ n) be elements of E(g, n) such that n i=1 P i v i = 0. Then i P (λ−λ i ) i v i = 0 for all λ ∈ a * . §3. Construction of special elements Let Λ be a subset of P. Put Λ + = Λ ∩ P + and Λ ++ = Λ ∩ P ++ . We define vector spaces U (g) Λ , U (n) Λ , E(n) Λ and E(g, n) Λ by U (g) Λ = {P ∈ U (g) | P (µ) = 0 for all µ ∈ Λ}, U (n) Λ = {P ∈ U (n) | P (µ) = 0 for all µ ∈ Λ}, E(n) Λ = {P ∈ E(n) | P (µ) = 0 for all µ ∈ Λ}, E(g, n) Λ = {P ∈ E(g, n) | P (µ) = 0 for all µ ∈ Λ}. Put (nU (n)) Λ = nU (n) ∩ U (n) Λ and (n E(n)) Λ = n E(n) ∩ E(n) Λ . We assume that Λ is a subgroup of a * . Then U (g) Λ , U (n) Λ , E(n) Λ and E(g, n) Λ are C-algebras. Let U be a U (g) Λ -module which is finitely generated as a U (n) Λ -module. Let u 1 , u 2 , . . . , u N be generators of U as a U (n) Λ -module. Put u = t (u 1 , u 2 , . . . , u N ), U = U/(nU (n)) Λ U and u = u (mod (nU (n)) Λ ). The module U has an a-module structure induced from that of U . By the assumption we have dim U < ∞. Let λ 1 , λ 2 , . . . , λ r ∈ a * (Re λ 1 ≥ Re λ 2 ≥ · · · ≥ Re λ r ) be eigenvalues of a on U with multiplicities. We can choose a basis v 1 , v 2 , . . . , v r of U and a linear map Q : a → M (r, C) such that            Hv = Q(H)v for all H ∈ a, Q(H) ii = λ i (H) for all H ∈ a, if i > j then Q(H) ij = 0 for all H ∈ a, if λ i = λ j then Q(H) ij = 0 for all H ∈ a, where v = t (v 1 , v 2 , . . . , v r ). Take A ∈ M (N, r, C) and B ∈ M (r, N, C) such that v = Bu and u = Av. Set U = E(g, n) Λ ⊗ U (g) Λ U . Theorem 3.1. There exist matrices A ∈ M (N, r, E (n) Λ ) and B ∈ M (r, N, E (n) Λ ) such that the following conditions hold: • There exists a linear map Q : a → M (r, U (n) Λ ) such that            Hv = Q(H)v for all H ∈ a, Q(H) − Q(H) ∈ M (r, (nU (n)) Λ ) for all H ∈ a, if λ i − λ j ∈ Λ + then Q(H) ij = 0 for all H ∈ a, if λ i − λ j ∈ Λ + then [H ′ , Q(H) ij ] = (λ i − λ j )(H ′ )Q(H) ij for all H, H ′ ∈ a, where v = Bu. • We have u = ABu. • We have A − A ∈ M (N, r, (n E (n)) Λ ) and B − B ∈ M (r, N, (n E (n)) Λ ). For the proof we need some lemmas. Put w = Bu ∈ U r . Lemma 3.2. For H ∈ a there exists a matrix R ∈ M (r, (n E(n)) Λ ) such that Hw = (Q(H)+R)w in U r . Proof. Since w (mod ((nU (n)) Λ U ) r ) = v, we have Hw − Q(H)w ∈ ((nU (n)) Λ U ) r . Hence there exists a matrix R 1 ∈ M (N, r, (nU (n)) Λ ) such that Hw − Q(H)w = R 1 u. Similarly we can choose a matrix S ∈ M (N, (nU (n)) Λ ) which satisfies u = Aw + Su. Put R = R 1 (1 − S) −1 A. Then (H − Q(H) − R)w = R 1 u − R 1 (1 − S) −1 Aw = 0. Lemma 3.3. Let λ ∈ C and Q 0 , R ∈ M (r, C). Assume that Q 0 is an upper triangular matrix. Then there exist matrices L, T ∈ M (r, C) such that λL − [Q 0 , L] = T + R, if (Q 0 ) ii − (Q 0 ) jj = λ then T ij = 0. Proof. Since (Q 0 ) ij = 0 for i > j, we have (λL − [Q 0 , L]) ij = λL ij − r k=1 ((Q 0 ) ik L kj − L ik (Q 0 ) kj ) = λL ij − r k=i (Q 0 ) ik L kj + j k=1 L ik (Q 0 ) kj = (λ − ((Q 0 ) ii − (Q 0 ) jj ))L ij − r k=i+1 (Q 0 ) ik L kj + j−1 k=1 L ik (Q 0 ) kj . Hence we can choose L ij and T ij inductively on (j − i). α ∈ Σ + . Let Q 0 ∈ M (r, C), R ∈ M (r, (n E (n)) Λ ). Assume (Q 0 ) ij = 0 for i > j. Set L ++ = {λ(H) | λ ∈ Λ ++ }. Then there exist matrices L ∈ M (r, E (n) Λ ) and T ∈ M (r, (nU (n)) Λ ) such that            L ≡ 1 r (mod (nU (n)) Λ ), (H1 r − Q 0 − T )L = L(H1 r − Q 0 − R), if (Q 0 ) ii − (Q 0 ) jj ∈ L ++ then T ij = 0, if (Q 0 ) ii − (Q 0 ) jj ∈ L ++ then [H, T ij ] = ((Q 0 ) ii − (Q 0 ) jj )T ij . Proof. Set L = {λ(H) | λ ∈ Λ} and L + = {λ(H) | λ ∈ Λ + }. Put f (n) = i n i β i for n = (n i ) ∈ Z m . Set Λ = {n ∈ Z m ≥0 | f (n) ∈ Λ}. We define the order on Λ by n < n ′ if and only if f (n) < f (n ′ ). By Corollary 2.5, we can write R = n∈ Λ R n E n where R n ∈ M (r, C). We have R 0 = 0 where 0 = (0) i ∈ Λ since R ∈ M (r, (n E(n)) Λ ). We have to show the existence of L and T . Write L = 1 r + n∈ Λ L n E n and T = n∈ Λ T n E n . Then (H1 r − Q 0 − T )L = L(H1 r − Q 0 − R) is equivalent to f (n)(H)L n − [Q 0 , L n ] = T n + S n − R n , where S n is defined by n∈ Λ S n E n = T (L − 1 r ) − (L − 1 r )R. By Proposition 2.7 the above equation is equivalent to f (n)=µ S n E n = f (k)+f (l)=µ T k L l E k E l − f (k)+f (l)=µ L k R l E k E l for all µ ∈ a * . Notice that L 0 = T 0 = 0. S n can be defined from the data {T k | k < n}, {L k | k < n} and {R k | k < n}. Now we prove the existence of L and T . We choose the L n and T n which satisfy      L 0 = 0, T 0 = 0, f (n)(H)L n − [Q 0 , L n ] = T n + S n − R n , if (Q 0 ) ii − (Q 0 ) jj = f (n)(H) then (T n ) ij = 0. By Lemma 3.3, we can choose such L n and T n inductively. Put L = 1 r + n∈ Λ L n E n and T = n∈ Λ T n E n . Since [H, T ij ] = n∈ Λ (f (n)(H))(T n ) ij E n = ((Q 0 ) ii − (Q 0 ) jj )T ij , L and T satisfy the conditions of the lemma. Proof of Theorem 3.1. We can choose positive integers C = (C i ) ∈ Z l >0 such that {α ∈ Λ ++ | α( i C i H i ) = (λ i − λ j )( i C i H i )} = {λ i − λ j } (λ i − λ j ∈ Λ ++ ), ∅ (λ i − λ j ∈ Λ ++ ). The existence of such C is shown by Oshima [Osh84, Lemma 2.3]. Set X = i C i H i . Notice that (λ i − λ j )(X) ∈ {µ(X) | µ ∈ Λ ++ } if and only if λ i − λ j ∈ Λ ++ . By Lemma 3.4, there exist T ∈ M (r, (nU (n)) Λ ) and L ∈ M (r, E (n) Λ ) such that            L ≡ 1 r (mod (n E(n)) Λ ), (X1 r − Q(X) − T )L = L(X1 r − Q(X) − R), if λ i − λ j ∈ Λ ++ then T ij = 0, if λ i − λ j ∈ Λ ++ then [X, T ij ] = (λ i − λ j )(X)T ij . Let S ∈ M (N, (nU (n)) Λ ) such that u = Aw + Su. Put A = (1 − S) −1 AL −1 , B = LB and v = (v 1 , v 2 , . . . , v r ) = Bu = Lw then ABu = (1 − S) −1 AL −1 LBu = (1 − S) −1 Aw = u. Moreover, we have (X1 r − Q(X) − T )v = 0. Since [X, T ij ] = (λ i − λ j )(X)T ij , we have (X − λ i (X)) r v i = 0. Fix a positive integer k such that 1 ≤ k ≤ l. We can choose a matrix R k ∈ M (r, (n E (n)) Λ ) such that H k w = (Q(H k ) + R k )w by Lemma 3.2. Set T k = H k 1 r − Q(H k ) − L(H k 1 r − Q(H k ) − R k )L −1 . Then we have (H k 1 r − Q(H k ) − T k )v = 0, i.e., H k v i − r j=1 Q(H k ) ij v j − r j=1 (T k ) ij v j = 0 for each i = 1, 2, . . . , r. By Corollary 2.9, we have H k v i − r j=1 Q(H k ) ij v j − r j=1 (T k ) (X,(λ i −λ j )(X)) ij v j = 0. Define T ′ k ∈ M (r, (nU (n)) Λ ) by (T ′ k ) ij = (T k ) (X,(λ i −λ j )(X)) ij . Since (T k ) (µ) ij = 0 for µ ∈ Λ ++ , we have (T k ) (X,(λ i −λ j )(X)) ij = µ∈Λ ++ , µ(X)=(λ i −λ j )(X) (T k ) (µ) ij = (T k ) (λ i −λ j ) ij (λ i − λ j ∈ Λ ++ ), 0 (λ i − λ j ∈ Λ ++ ) by the condition of C. In particular [H, (T ′ k ) ij ] = (λ i − λ j )(H) for all H ∈ a. Put Q( x i H i ) = Q( x i H i ) + x i T ′ i for (x 1 , x 2 , . . . , x l ) ∈ C l then Q satisfies the condition of the theorem. Set ρ = α∈Σ + (dim g α /2)α. From the Iwasawa decomposition g = k ⊕ a ⊕ n we have the decomposition into the direct sum U (g) = nU (a ⊕ n) ⊕ U (a) ⊕ U (g)k. Let χ 1 be the projection of U (g) to U (a) with respect to this decomposition and χ 2 the algebra automorphism of U (a) defined by χ 2 (H) = H − ρ(H) for H ∈ a. We define χ : U (g) k → U (a) by χ = χ 2 • χ 1 where U (g) k = {u ∈ U (g) | Xu = uX for all X ∈ k}. It is known that an image of U (g) k under χ is contained in the set of W -invariant elements in U (a). Fix λ ∈ a * . We can define the algebra homomorphism U (a) → C by H → λ(H) for H ∈ a. We denote this map by the same letter λ. Put χ λ = λ • χ. Set U (λ) = U (g)/(U (g) Ker χ λ + U (g)k), u λ = 1 mod (U (g) Ker χ λ + U (g)k) ∈ U (λ) and U (λ) 0 = U (g) 2P u λ . Before applying Theorem 3.1 to U (λ) 0 , we give some lemmas. Lemma 3.5. Let u ∈ U (g) µ where µ ∈ a * . Then there exists an element x ∈ U (g)k such that u + x ∈ U (a ⊕ n) µ+2P . Proof. Set n = θ(n). We may assume u ∈ U (n) µ . Let {U n (n)} n∈Z ≥0 be the standard filtration of U (n) and n the smallest integer such that u ∈ U n (n). We will prove the existence of x by the induction on n. If n = 0 then the lemma is obvious. Assume n > 0. We may assume that there exist a restricted root α ∈ Σ + , an element u 0 ∈ U n−1 (n) µ+α and a vector E −α ∈ g −α such that u = u 0 E −α . Set E α = θ(E −α ), u 1 = u 0 E α , u 2 = E α u 0 and u 3 = u 1 − u 2 . Then u + u 2 + u 3 = u + u 1 ∈ U (g)k, u 1 , u 2 ∈ U (g) µ+2α and u 3 ∈ U n−1 (g) µ+2α . Using the Poincaré-Birkhoff-Witt theorem and the induction hypothesis we can choose an element u 5 ∈ U (a ⊕ n) µ+2P such that u 3 − u 5 ∈ U (g)k. Again by the induction hypothesis we can choose an element u 6 ∈ U (a ⊕ n) µ+α+2P such that u 0 − u 6 ∈ U (g)k. Then u + u 5 + E α u 6 ∈ U (g)k, u 5 ∈ U (a ⊕ n) µ+2P and E α u 6 ∈ U (a ⊕ n) µ+2α+2P . Lemma 3.6. The following equations hold. (1) Ker χ λ ⊂ U (g) 2P . (2) U (a ⊕ n) ∩ (Ker χ λ + U (g)k) ⊂ U (a ⊕ n) 2P ∩ (Ker χ λ + U (g)k). (3) U (a ⊕ n) ∩ (U (a ⊕ n) Ker χ λ + U (g)k) ⊂ U (a ⊕ n)(U (a ⊕ n) ∩ (Ker χ λ + U (g)k)). (4) U (a ⊕ n) ∩ (U (g) Ker χ λ + U (g)k) = U (a ⊕ n)((U (g) Ker χ λ + U (g)k) ∩ U (a ⊕ n) 2P ). Proof. (1) It is sufficient to prove U (g) k ⊂ U (g) 2P . Let G be a connected Lie group whose Lie algebra is g 0 and K its maximal compact subgroup such that Lie(K) = k 0 . Since K is connected, U (g) k = U (g) K = {u ∈ U (g) | Ad(k)u = u for all k ∈ K}. Assume that G has the complexification G C . Fix a maximal abelian subspace t 0 of m 0 . Let K split and A split be the analytic subgroups with Lie algebras given as the intersections of k 0 and a 0 with [Z g 0 (t 0 ), [Kna02,Thorem 7.55] and Lepowsky [Lep75, Proposition 6.1, Proposition 6.4]). Hence (1) follows. Z g 0 (t 0 )] where Z g 0 (t 0 ) is the centralizer of t 0 in g 0 . Let F be the centralizer of A split in K split . Since F ⊂ K, we have U (g) K ⊂ U (g) F . On the other hand, we have U (g) F ⊂ U (g) 2P (See Knapp (2) Let u ∈ Ker χ λ and x ∈ U (g)k such that u + x ∈ U (a ⊕ n). We can write u = µ u µ where u µ ∈ U (g) µ . By (1), we have u µ = 0 for µ ∈ 2P. Let µ ∈ 2P. By Lemma 3.5, there exists an element y µ ∈ U (g)k such that u µ + y µ ∈ U (a ⊕ n) µ+2P = U (a ⊕ n) 2P . Put y = µ y µ . Then u + y ∈ U (a ⊕ n) 2P . Since u + y ∈ U (a ⊕ n) and x, y ∈ U (g)k we have y = x by the Poincaré-Birkhoff-Witt theorem. Hence we have u + x = u + y ∈ U (a ⊕ n) 2P . ( 3) Let i x i u i ∈ U (a⊕n) Ker χ λ where x i ∈ U (a⊕n) and u i ∈ Ker χ λ . We write u i = j z (i) j v (i) j where z (i) j ∈ U (a ⊕ n) and v (i) j ∈ U (k). Let y ∈ U (g)k and assume i x i u i + y ∈ U (a ⊕ n). By the Poincaré-Birkhoff-Witt theorem, i x i u i + y = i,j x i z (i) j v (i) j,0 where v (i) j,0 is the constant term of v (i) j . Hence i x i u i + y = i x i (u i + j z (i) j (v (i) j,0 − v (i) j )) ∈ U (a ⊕ n)(U (a ⊕ n) ∩ (Ker χ λ + U (g)k)). (4) Since Ker χ λ ⊂ U (g) k , we have U (g) Ker χ λ + U (g)k = U (a ⊕ n)(Ker χ λ )U (k) + U (g)k = U (a ⊕ n) Ker χ λ + U (g)k. By (2) and (3), we have U (a ⊕ n) ∩ (U (g) Ker χ λ + U (g)k) = U (a ⊕ n) ∩ (U (a ⊕ n) Ker χ λ + U (g)k) ⊂ U (a ⊕ n)(U (a ⊕ n) 2P ∩ (Ker χ λ + U (g)k)) ⊂ U (a ⊕ n)((U (g) Ker χ λ + U (g)k) ∩ U (a ⊕ n) 2P ). This implies (4). Lemma 3.7. We have the following equations. (1) U (λ) 0 = U (a ⊕ n) 2P u λ . (2) U (n) ⊗ U (n) 2P U (λ) 0 ≃ U (λ) under the map p ⊗ u → pu. Proof. (1) This is obvious from Lemma 3.5. ( 2) Let I = U (a ⊕ n) ∩ (U (g) Ker χ λ + U (g)k). We have U (a ⊕ n) ⊗ U (a⊕n) 2P U = U (n) ⊗ U (n) 2P U for any U (a ⊕ n) 2P -module U since U (a ⊕ n) 2P = U (a) ⊗ U (n) 2P . By (1), we have U (λ) 0 = U (a ⊕ n) 2P /(I ∩ U (a ⊕ n) 2P ). Hence U (n) ⊗ U (n) 2P U (λ) 0 = U (a ⊕ n) ⊗ U (a⊕n) 2P U (λ) 0 = U (a ⊕ n) ⊗ U (a⊕n) 2P (U (a ⊕ n) 2P /(I ∩ U (a ⊕ n) 2P )) = U (a ⊕ n)/(U (a ⊕ n)(I ∩ U (a ⊕ n) 2P )) = U (a ⊕ n)/I = U (λ) by Lemma 3.6 (4). Lemma 3.8. Let {U n (n)} n∈Z ≥0 be the standard filtration of U (n) and U n (n) 2P = U n (n)∩U (n) 2P . Set U −1 (n) = U −1 (n) 2P = 0, R = Gr U (n) 2P = n U n (n) 2P /U n−1 (n) 2P and R ′ = Gr U (n) = n U n (n)/U n−1 (n). (1) R ′ is a finitely generated R-module. (2) U (n) is a finitely generated U (n) 2P -module. (3) U (n) 2P is right and left Noetherian. (4) U (λ) 0 is a finitely generated U (n) 2P -module. Proof. (1) Let Γ = {E ε | ε ∈ {0, 1} m }. We denote the principal symbol of u ∈ U (n) by σ(u). Notice that if u ∈ U (n) 2P then σ(u) is the principal symbol of u as an element of U (n) 2P . We will prove that {σ(E) | E ∈ Γ} generates R ′ as an R-module. Let x ∈ R ′ . We may assume that x is homogeneous, thus there exists an element u ∈ U (n) such that x = σ(u). Moreover we may assume that there exist non-negative integers p = (p 1 , p 2 , . . . , p m ) such that u = E p . Choose ε i ∈ {0, 1} such that ε i ≡ p i (mod 2). Set q i = (p i − ε i )/2 ∈ Z ≥0 , ε = (ε 1 , ε 2 , . . . , ε m ) and q = (q 1 , q 2 , . . . , q m ). Then we have x = σ(E p ) = σ(E 2q )σ(E ε ). Since σ(E 2q ) ∈ R, this implies that {σ(E) | E ∈ Γ} generates R ′ as an R-module. (2) This is a direct consequence of (1). (3) By the Poincaré-Birkhoff-Witt theorem, R ′ is isomorphic to a polynomial ring. In particular R ′ is Noetherian. By the theorem of Eakin-Nagata and (1), we have R is Noetherian. This implies (3). (4) Since U (λ) is a finitely generated U (n)-module and U (n) is a finitely generated U (n) 2Pmodule, U (λ) is a finitely generated U (n) 2P -module. Hence U (λ) 0 is a finitely generated U (n) 2Pmodule by (3). We enumerate W = {w 1 , w 2 , . . . , w r } such that Re w 1 λ ≥ Re w 2 λ ≥ · · · ≥ Re w r λ. Theorem 3.9. There exist matrices A ∈ M (1, r, E (n) 2P ) and B ∈ M (r, 1, E (a ⊕ n, n) 2P ) such that v λ = Bu λ ∈ ( E(g, n) ⊗ U (g) U (λ)) r satisfies the following conditions: • There exists a linear map Q : a → M (r, U (n) 2P ) such that            Hv λ = Q(H)v λ for all H ∈ a, Q(H) ii = (ρ + w i λ)(H) for all H ∈ a, if w i λ − w j λ ∈ 2P + then Q(H) ij = 0 for all H ∈ a, if w i λ − w j λ ∈ 2P + then [H ′ , Q(H) ij ] = (w i λ − w j λ)(H ′ )Q(H) ij for all H, H ′ ∈ a. • We have u λ = Av λ . • Let (v 1 , v 2 , . . . , v r ) = v λ . Then {v i (mod nU (λ))} is a basis of U (λ)/nU (λ). Proof. Let u 1 , u 2 , . . . , u N be generators of U (λ) 0 as a U (n) 2P -module. These are also generators of U (λ) as a U (n)-module by Lemma 3.7. We choose matrices C = t (C 1 , C 2 , . . . , C N ) ∈ M (N, 1, U (a ⊕ n) 2P ) and D = (D 1 , D 2 , . . . , D N ) ∈ M (1, N, U (n) 2P ) such that t (u 1 , u 2 , . . . , u N ) = Cu λ and u λ = D t (u 1 , u 2 , . . . , u N ). Notice that U (n) 2P + nU (n) = U (n). By Lemma 3.7, U (λ)/nU (λ) = (U (n)/nU (n)) ⊗ U (n) U (λ) = (U (n)/nU (n)) ⊗ U (n) U (n) ⊗ U (n) 2P U (λ) 0 = (U (n)/nU (n)) ⊗ U (n) 2P U (λ) 0 = ((U (n) 2P + nU (n))/nU (n)) ⊗ U (n) 2P U (λ) 0 = (U (n) 2P /(nU (n) ∩ U (n) 2P )) ⊗ U (n) 2P U (λ) 0 = (U (n) 2P /(nU (n)) 2P ) ⊗ U (n) 2P U (λ) 0 = U (λ) 0 /(nU (n)) 2P U (λ) 0 . On the other hand, U (λ)/nU (λ) = U (g)/(nU (g) + U (g) Ker χ λ + U (g)k) = (nU (g) + U (a) + U (g)k)/(nU (g) + U (g) Ker χ λ + U (g)k) = U (a)/((nU (g) + U (g) Ker χ λ + U (g)k) ∩ U (a)). By the definition of χ λ , we have (nU (g) + U (g) Ker χ λ + U (g)k) ∩ U (a) = p∈U (a) W U (a)(χ −1 2 (p) − λ(p)) where U (a) W is a C-algebra of W -invariant elements of U (a). By the result of Oshima [Osh88, Proposition 2.8], the set of eigenvalues of H ∈ a on U (a)/( p∈U (a) W U (a)(χ −1 2 (p) − λ(p))) is {(ρ + wλ)(H) | w ∈ W } with multiplicities. We take matrices A ′ ∈ M (N, r, E (n) 2P ) and B ′ ∈ M (r, N, E (n) 2P ) such that the conditions of Theorem 3.1 hold. Put A = DA ′ , B = B ′ C then A, B satisfy the conditions of the theorem. §4. Structure of Jacquet modules (regular case) In this section we assume that λ is regular, i.e., wλ = λ for all w ∈ W \ {e}. Let r = #W and v λ = (v 1 , v 2 , . . . , v r ) ∈ ( E(g, n) ⊗ U (g) U (λ)) r as in Theorem 3.9. Set W(i) = {j | w i λ − w j λ ∈ 2P + } for each i = 1, 2, . . . , r. Theorem 4.1. We have Xv i ∈ j∈W(i) U (g)v j for all X ∈ θ(n) ⊕ m. Let A = t (A (1) , A (2) , . . . , A (r) ) be as in Theorem 3.9 and A = t (A (1) , A (2) , . . . , A (r) ) an element of M (r, 1, C) such that A (i) − A (i) ∈ n E(n). Lemma 4.2. We have A (i) = 0 for each i = 1, 2, . . . , r. Proof. Put U (λ) = U (λ)/nU (λ), u λ = u λ (mod nU (λ)) and v i = v i (mod nU (λ)). Let B = (B (1) , B (2) , . . . , B (r) ) be a matrix in M (1, r, U (a)) such that v j = B (j) u λ . Then we have v j = i A (i) B (j) v i . By the regularity of λ, we have Hv j = (w j λ)(H)v j and HB (j) v i = (w i λ)(H)B (j) v i for all H ∈ a. This implies A (j) = 0 since λ is regular. Proof of Theorem 4.1. Put f (n) = i n i β i for n = (n i ) ∈ Z m . Set Λ = {n ∈ Z m ≥0 | f (n) ∈ 2P}. We write A (j) = n∈ Λ A (j) n E n . Let α ∈ Σ + and E α ∈ g α . Since ku λ = 0, we have (θ(E α ) + E α )u λ = 0. Hence (θ(E α ) + E α ) j n A (j) n E n v j = 0. By applying Corollary 2.9 we have r j=1   n∈ Λ A (j) n (θ(E α ) + E α )E n   (w i λ−w j λ−α) v j = 0 for i = 1, 2, . . . , r. On one hand if w i λ − w j λ ∈ 2P + then   n∈ Λ A (j) n (θ(E α ) + E α )E n   (w i λ−w j λ−α) = 0. On the other hand   n∈ Λ A (i) n (θ(E α ) + E α )E n   (−α) = A (i) 0 θ(E α ). Hence we have A (i) 0 θ(E α )v i ∈ j∈W(i) U (g)v j . Since A (i) 0 = A (i) = 0, we have θ(E α )v i ∈ j∈W(i) U (g)v j . Next let X be an element of m. By Corollary 2.9, we have r j=1   n∈ Λ A (j) n XE n   (w i λ−w j λ) v j = 0. We can prove Xv i ∈ j∈W(i) U (g)v j by the same argument. Put V (λ) = i U (g)v i ⊂ E(g, n) ⊗ U (g) U (λ). Corollary 4.3. V (λ) = J(U (λ)). Proof. By Theorem 4.1, V (λ) is finitely generated as a U (n)-module. By applying Proposition 2.4, we see that the map E(g, n) ⊗ U (g) V (λ) → µ∈a * V (λ) µ is bijective. Hence E(g, n) ⊗ U (g) V (λ) → E(g, n) ⊗ U (g) U (λ) is injective by Proposition 2.8. This map is also surjective since v 1 , v 2 , . . . , v r are generators of E(g, n) ⊗ U (g) U (λ). We have E(g, n) ⊗ U (g) V (λ) = E(g, n) ⊗ U (g) U (λ). Since U (λ) and V (λ) are finitely generated as U (n)-modules, we have E(g, n) ⊗ U (g) U (λ) = J(U (λ)), E(g, n) ⊗ U (g) V (λ) = J(V (λ)), by Proposition 2.2. Hence we have J(U (λ)) = J(V (λ)) = V (λ) by Corollary 2.6. Recall the definition of a generalized Verma module. Set p = θ(p), n = θ(n) and ρ = α∈Σ + (dim g α /2)α. Definition 4.4 (Generalized Verma module). Let µ ∈ a * . Define the one-dimensional representation C ρ+µ of p by (X + Y + Z)v = (ρ + µ)(Y )v for X ∈ m, Y ∈ a, Z ∈ n, v ∈ C ρ+µ . We define a U (g)-module M (µ) by M (µ) = U (g) ⊗ U (p) C ρ+µ . This is called a generalized Verma module. Set V i = j≥i U (g)v j . By the universality of tensor products, any U (p)-module homomorphism C ρ+µ → U is uniquely extended to the U (g)-module homomorphism M (µ) → U for a U (g)-module U . In particular we have the surjective U (g)-module homomorphism M (w i λ) → V i /V i+1 . We shall show that V i /V i+1 is isomorphic to a generalized Verma module using the character theory. Let G be a connected Lie group such that Lie(G) = g 0 , K its maximal compact subgroup with its Lie algebra k 0 , P the parabolic subgroup whose Lie algebra is p 0 and P = M AN the Langlands decomposition of P where Lie algebra of M (resp. A, N ) is m 0 (resp. a 0 , n 0 ). Since U (wλ) = U (λ) for w ∈ W , we may assume that Re λ is dominant, i.e., Re λ(H i ) ≤ 0 for each i = 1, 2, . . . , l. By the result of Kostant [Kos75, Theorem 2.10.3], U (λ) is isomorphic to the space of K-finite vectors of the non-unitary principal series representation Ind G P (1 ⊗ λ) K . The character of this representation is calculated by Harish-Chandra (See Knapp [Kna01, Proposition 10.18]). Before we state it, we prepare some notations. Let H = T A be the maximally split Cartan subgroup, h 0 its Lie algebra, T = H ∩ M , ∆ the root system of H, ∆ + the positive system compatible with Σ + , ∆ I the set of imaginary roots, ∆ + I = ∆ + ∩ ∆ I and ξ α the one-dimensional representation of H whose derivation is α for α ∈ h * . Under these notations, the distribution character Θ G (U (λ)) of U (λ) is as follows; Θ G (U (λ))(ta) = w∈W ξ ρ+wλ (a) α∈∆ + \∆ + I |1 − ξ α (ta)| (t ∈ T, a ∈ A). We will use the Osborne conjecture, which was proved by Hecht and Schmid [HS83a, Theorem 3.6]. To state it, we must define a character of J(U ) for a Harish-Chandra module U . Recall that J(U ) is an object of the category O ′ P , i.e., (1) the actions of M ∩ K and g are compatible, (2) J(U ) splits under a into a direct sum of generalized weight spaces, each of them being a Harish-Chandra modules for M A, Θ P (V ) of V by Θ P (V ) = µ∈a * Θ M A (V µ ), where V µ is a generalized µ-weight space of V . Let G ′ be the set of regular elements of G. Set A − = {a ∈ A | α(log a) < 0 for all α ∈ Σ + }, (M A) − = interior of      g ∈ M A α∈∆ + \∆ + I (1 − ξ α (ga)) ≥ 0 for all a ∈ A −      in M A. Then the Osborn conjecture says that Θ G (U ) and Θ P (J(U )) coincide on (M A) − ∩ G ′ (See Hecht and Schmid [HS83b, (42)Lemma]). It is easy to calculate the character of a generalized Verma module. We have Θ P (M (µ))(ta) = ξ ρ+µ (a) α∈∆ + \∆ + I (1 − ξ α (ta)) (t ∈ T, a ∈ A). Consequently we have Θ P (J(U (λ))) = w∈W Θ P (M (wλ)). This implies the following theorem when λ is regular. Theorem 4.5. There exists a filtration 0 = V r+1 ⊂ V r ⊂ · · · ⊂ V 1 = J(U (λ)) of J(U (λ)) such that V i /V i+1 is isomorphic to M (w i λ) for an arbitrary λ ∈ a * . Moreover if wλ − λ ∈ 2P for all w ∈ W \ {e} then J(U (λ)) ≃ w∈W M (wλ). §5. Structure of Jacquet modules (singular case) In this section, we shall prove Theorem 4.5 in the singular case using the translation principle. We retain notations in Section 4. Let λ ′ be an element of a * such that following conditions hold: • The weight λ ′ is regular. • The weight (λ − λ ′ )/2 is integral. • The real part of λ ′ belongs to the same Weyl chamber which real part of λ belongs to. First we define the translation functor T λ λ ′ . Let U be a U (g)-module which has an infinitesimal character λ ′ . (We regard a * ⊂ h * .) We define T λ λ ′ (U ) by T λ λ ′ (U ) = P λ (U ⊗ E λ−λ ′ ) where: • E λ−λ ′ is the finite-dimensional irreducible representation of g with an extreme weight λ − λ ′ . • P λ (V ) = {v ∈ V | for some n > 0 and all z ∈ Z(g), (z − λ( χ(z))) n v = 0} where Z(g) is the center of U (g) and χ : Z(g) → U (h) is the Harish-Chandra homomorphism. Notice that P λ and T λ λ ′ are exact functors. Theorem 4.5 in the singular case follows from following two equations. (1) T λ λ ′ (U (λ ′ )) = U (λ). (2) T λ λ ′ (M (wλ ′ )) = M (wλ). The following lemma is important to prove these equations. Lemma 5.1. Let ν be a weight of E λ−λ ′ and w ∈ W . Assume ν = wλ − λ ′ . Then ν = λ − λ ′ . Proof. See Vogan [Vog81, Lemma 7.2.18]. Proof of T λ λ ′ (U (λ ′ )) = U (λ). We may assume that λ ′ is dominant. Notice that we have U (λ ′ ) ≃ Ind G P (1 ⊗ λ ′ ) K . Let 0 = E 0 ⊂ E 1 ⊂ E 2 ⊂ · · · ⊂ E n = E λ−λ ′ be a P -stable filtration with the trivial induced action of N on E i /E i−1 . We may assume that E i /E i−1 is irreducible. Let ν i be the highest weight of E i /E i−1 . Notice that Ind G P (1⊗λ ′ )⊗E λ−λ ′ = Ind G P ((1⊗λ ′ )⊗E λ−λ ′ ). Then Ind G P (1 ⊗ λ ′ ) ⊗ E λ−λ ′ has a filtration {M i } such that M i /M i−1 ≃ Ind G P ((1 ⊗ λ ′ ) ⊗ (E i /E i−1 )). Since Ind G P ((1 ⊗ λ ′ ) ⊗ (E i /E i−1 ) ) has an infinitesimal character λ + ν i , P λ (M i /M i−1 ) = 0 if ν i = wλ − λ ′ for all w ∈ W . By Lemma 5.1 we have T λ λ ′ (Ind G P (1 ⊗ λ ′ ) = Ind G P ((1 ⊗ λ ′ ) ⊗ (E i /E i−1 )) where ν i = λ − λ ′ . By the conditions of λ ′ , the action of M on (λ − λ ′ )-weight space of E λ−λ ′ is trivial. Consequently T λ λ ′ (Ind G P (1 ⊗ λ ′ )) = Ind G P ((1 ⊗ λ ′ ) ⊗ (λ − λ ′ )) = Ind G P (1 ⊗ λ). Proof of T λ λ ′ (M (wλ ′ )) = M (wλ). We may assume w = e ∈ W . Since M (λ ′ )⊗E λ−λ ′ = U (g)⊗ (C λ ′ ⊗ E λ−λ ′ ), the equation follows by the same argument of the proof of T λ λ ′ (U (λ ′ )) = U (λ). Lemma 3 . 4 . 34Let H be an element of a such that α(H) > 0 for all ( 3 ) 3J(U ) is U (n)-and Z(g)-finite (See Hecht and Schmid [HS83b, (34)Lemma]). For an object V of O ′ P , we define the character Jacquet modules for real reductive groups. W Casselman, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) (Helsinki), Acad. Sci. Fennica. the International Congress of Mathematicians (Helsinki, 1978) (Helsinki), Acad. Sci. FennicaW. Casselman, Jacquet modules for real reductive groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) (Helsinki), Acad. Sci. Fennica, 1980, pp. 557- 563. Jacquet modules for semisimple Lie groups having Verma module filtrations. H David, Collingwood, J. Algebra. 1362David H. Collingwood, Jacquet modules for semisimple Lie groups having Verma module filtrations, J. Algebra 136 (1991), no. 2, 353-375. Whittaker vectors and conical vectors. Roe Goodman, Nolan R Wallach, J. Funct. Anal. 392Roe Goodman and Nolan R. Wallach, Whittaker vectors and conical vectors, J. Funct. Anal. 39 (1980), no. 2, 199-279. Characters, asymptotics and n-homology of Harish-Chandra modules. Henryk Hecht, Wilfried Schmid, Acta Math. 1511-2Henryk Hecht and Wilfried Schmid, Characters, asymptotics and n-homology of Harish- Chandra modules, Acta Math. 151 (1983), no. 1-2, 49-151. On the asymptotics of Harish-Chandra modules. Henryk Hecht, Wilfried Schmid, J. Reine Angew. Math. 343Henryk Hecht and Wilfried Schmid, On the asymptotics of Harish-Chandra modules, J. Reine Angew. Math. 343 (1983), 169-183. Representation theory of semisimple groups. Anthony W Knapp, Princeton Landmarks in Mathematics. Princeton University PressAn overview based on examples. Reprint of the 1986 originalAnthony W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001, An overview based on examples, Reprint of the 1986 original. Anthony W Knapp, Lie groups beyond an introduction. Boston, MABirkhäuser Boston Inc140Progress in MathematicsAnthony W. Knapp, Lie groups beyond an introduction, second ed., Progress in Mathe- matics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002. Systems of differential equations with regular singularities and their boundary value problems. Masaki Kashiwara, Toshio Oshima, Ann. of Math. 2Masaki Kashiwara and Toshio Oshima, Systems of differential equations with regular sin- gularities and their boundary value problems, Ann. of Math. (2) 106 (1977), no. 1, 145-200. On the existence and irreducibility of certain series of representations. Bertram Kostant, Lie groups and their representations (Proc. Summer School. Budapest; Halsted, New YorkBertram Kostant, On the existence and irreducibility of certain series of representations, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 231-329. Existence of conical vectors in induced modules. J Lepowsky, Ann. of Math. 2J. Lepowsky, Existence of conical vectors in induced modules, Ann. of Math. (2) 102 (1975), no. 1, 17-40. Boundary value problems for systems of linear partial differential equations with regular singularities. Toshio Oshima, Group representations and systems of differential equations. Tokyo; North-Holland, Amsterdam4Toshio Oshima, Boundary value problems for systems of linear partial differential equa- tions with regular singularities, Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 391- 432. A realization of semisimple symmetric spaces and construction of boundary value maps, Representations of Lie groups. Toshio Oshima, Adv. Stud. Pure Math. 14Academic PressToshio Oshima, A realization of semisimple symmetric spaces and construction of boundary value maps, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 603-650. The restriction of admissible modules to parabolic subalgebras. J T Stafford, N R Wallach, Trans. Amer. Math. Soc. 2721J. T. Stafford and N. R. Wallach, The restriction of admissible modules to parabolic sub- algebras, Trans. Amer. Math. Soc. 272 (1982), no. 1, 333-350. Representations of real reductive Lie groups. David A VoganJr, Progress in Mathematics. 15BirkhäuserDavid A. Vogan, Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser Boston, Mass., 1981.
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[ "Robust Multitask Diffusion Normalized M-estimate Subband Adaptive Filtering Algorithm Over Adaptive Networks", "Robust Multitask Diffusion Normalized M-estimate Subband Adaptive Filtering Algorithm Over Adaptive Networks" ]
[ "Wenjing Xu ", "Haiquan Zhao ", "Shaohui * Lv " ]
[]
[]
In recent years, the multitask diffusion least mean square (MD-LMS) algorithm has been extensively applied in the distributed parameter estimation and target tracking of multitask network. However, its performance is mainly limited by two aspects, i.e, the correlated input signal and impulsive noise interference. To overcome these two limitations simultaneously, this paper firstly introduces the subband adaptive filter (SAF) into the multitask network. Then, a robust multitask diffusion normalized M-estimate subband adaptive filtering (MD-NMSAF) algorithm is proposed by solving the modified Huber function based global network optimization problem in a distributed manner, which endows the multitask network strong decorrelation ability for correlated inputs and robustness to impulsive noise interference, and accelerates the convergence of the algorithm significantly. Compared with the robust multitask diffusion affine projection M-estimate (MD-APM) algorithm, the computational complexity of the proposed MD-NMSAF is greatly reduced. In addition, the stability condition, the analytical expressions of the theoretical transient and steady-state network mean square deviation (MSD) of the MD-NMSAF are also provided and verified through computer simulations. Simulation results under different input signals and impulsive noise environment fully demonstrate the performance advantages of the MD-NMSAF algorithm over some other competitors in terms of steady-state accuracy and tracking speed. ;In recent years, the multitask diffusion LMS (MD-LMS) algorithm has been well applied to the parameter estimation and target tracking of the multitask networks[6]. However, its performance is mainly limited by two aspects: On the one hand, its convergence speed largely depends on the characteristics of the input signal. Specifically, the large eigenvalue spread of the correlation matrix of the correlated input signal will lead to slow convergence of the algorithm. On the other hand, the optimality of the algorithm based on the MMSE criterion depends on the Gaussian noise assumption to a large extent. When the measurement noise does not conform to the Gaussian distribution, the performance of the MD-LMS algorithm will deteriorate significantly.In view of the former defect, reference [8] proposes the multitask diffusion affine projection algorithm (MD-APA) to improve the convergence speed of the algorithm under the correlated inputs. In fact, the MD-APA is a generalization of the normalized version of MD-LMS in the time domain. For the latter defect, on the basis of MD-APA, the multitask diffusion affine projection M-estimate (MD-APM) algorithm is proposed by applying the M-estimate function to remove outliers [9], [10]-[13]. The multitask diffusion affine projection maximum correntropy criterion (MD-APMCC) algorithm is also proposed by introducing the maximum correntropy criterion (MCC) into the MD-APA [14], [15], which is insensitive to large outliers. These algorithms are all robust to impulsive noise interference. Unfortunately, the above robust multitask algorithms are all based on the AP strategy, which not only improve the convergence speed of the algorithm, but also greatly increase the computational complexity. Inspired by the subband adaptive filter (SAF) [16]-[22], we extend the normalized version of the MD-LMS algorithm to the subband domain, and introduce the M-estimate function with the property of removing outliers, and propose a robust multitask diffusion normalized M-estimate subband adaptive filtering (MD-NMSAF) algorithm. The proposed MD-NMSAF algorithm inherits the strong decorrelation ability of SAF, improves the convergence speed of the algorithm under correlated inputs and endows the distributed network robustness to impulsive noise. Furthermore, compared with the MD-APA, the structure of SAF can greatly reduces the computational complexity of the MD-NMSAF algorithm. The main contributions of this paper are summarized as follows:1) The cluster multitask diffusion algorithm is extended to the subband domain, and the M-estimate function is introduced to propose a robust multitask diffusion SAF algorithm, i.e, MD-NMSAF algorithm, which has the strong decorrelation ability for correlated inputs and robustness to impulsive noise interference.2) The mean and mean square stability of the MD-NMSAF algorithm are analyzed in impulsive noise environment, and the stability conditions are also given. Then, the analytical expressions of the theoretical transient and steady-state network network mean square deviation (MSD) are provided.
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[ "https://export.arxiv.org/pdf/2210.11284v1.pdf" ]
253,018,692
2210.11284
d74f3e8d561907b38658f8c5f7535d8e8dd1a6e3
Robust Multitask Diffusion Normalized M-estimate Subband Adaptive Filtering Algorithm Over Adaptive Networks Wenjing Xu Haiquan Zhao Shaohui * Lv Robust Multitask Diffusion Normalized M-estimate Subband Adaptive Filtering Algorithm Over Adaptive Networks Distributed EstimationMultitask NetworkSubband Adaptive FilterImpulsive NoiseM-estimate Function In recent years, the multitask diffusion least mean square (MD-LMS) algorithm has been extensively applied in the distributed parameter estimation and target tracking of multitask network. However, its performance is mainly limited by two aspects, i.e, the correlated input signal and impulsive noise interference. To overcome these two limitations simultaneously, this paper firstly introduces the subband adaptive filter (SAF) into the multitask network. Then, a robust multitask diffusion normalized M-estimate subband adaptive filtering (MD-NMSAF) algorithm is proposed by solving the modified Huber function based global network optimization problem in a distributed manner, which endows the multitask network strong decorrelation ability for correlated inputs and robustness to impulsive noise interference, and accelerates the convergence of the algorithm significantly. Compared with the robust multitask diffusion affine projection M-estimate (MD-APM) algorithm, the computational complexity of the proposed MD-NMSAF is greatly reduced. In addition, the stability condition, the analytical expressions of the theoretical transient and steady-state network mean square deviation (MSD) of the MD-NMSAF are also provided and verified through computer simulations. Simulation results under different input signals and impulsive noise environment fully demonstrate the performance advantages of the MD-NMSAF algorithm over some other competitors in terms of steady-state accuracy and tracking speed. ;In recent years, the multitask diffusion LMS (MD-LMS) algorithm has been well applied to the parameter estimation and target tracking of the multitask networks[6]. However, its performance is mainly limited by two aspects: On the one hand, its convergence speed largely depends on the characteristics of the input signal. Specifically, the large eigenvalue spread of the correlation matrix of the correlated input signal will lead to slow convergence of the algorithm. On the other hand, the optimality of the algorithm based on the MMSE criterion depends on the Gaussian noise assumption to a large extent. When the measurement noise does not conform to the Gaussian distribution, the performance of the MD-LMS algorithm will deteriorate significantly.In view of the former defect, reference [8] proposes the multitask diffusion affine projection algorithm (MD-APA) to improve the convergence speed of the algorithm under the correlated inputs. In fact, the MD-APA is a generalization of the normalized version of MD-LMS in the time domain. For the latter defect, on the basis of MD-APA, the multitask diffusion affine projection M-estimate (MD-APM) algorithm is proposed by applying the M-estimate function to remove outliers [9], [10]-[13]. The multitask diffusion affine projection maximum correntropy criterion (MD-APMCC) algorithm is also proposed by introducing the maximum correntropy criterion (MCC) into the MD-APA [14], [15], which is insensitive to large outliers. These algorithms are all robust to impulsive noise interference. Unfortunately, the above robust multitask algorithms are all based on the AP strategy, which not only improve the convergence speed of the algorithm, but also greatly increase the computational complexity. Inspired by the subband adaptive filter (SAF) [16]-[22], we extend the normalized version of the MD-LMS algorithm to the subband domain, and introduce the M-estimate function with the property of removing outliers, and propose a robust multitask diffusion normalized M-estimate subband adaptive filtering (MD-NMSAF) algorithm. The proposed MD-NMSAF algorithm inherits the strong decorrelation ability of SAF, improves the convergence speed of the algorithm under correlated inputs and endows the distributed network robustness to impulsive noise. Furthermore, compared with the MD-APA, the structure of SAF can greatly reduces the computational complexity of the MD-NMSAF algorithm. The main contributions of this paper are summarized as follows:1) The cluster multitask diffusion algorithm is extended to the subband domain, and the M-estimate function is introduced to propose a robust multitask diffusion SAF algorithm, i.e, MD-NMSAF algorithm, which has the strong decorrelation ability for correlated inputs and robustness to impulsive noise interference.2) The mean and mean square stability of the MD-NMSAF algorithm are analyzed in impulsive noise environment, and the stability conditions are also given. Then, the analytical expressions of the theoretical transient and steady-state network network mean square deviation (MSD) are provided. Introduction Distributed adaptive estimation is an important technique for signal processing within a network, which performs specific tasks through continuous learning and adaptation of interconnected nodes in network. It has attracted extensive research due to its reliability, scalability, and resource efficiency [1]- [3]. Previous research on distributed estimation mainly focused on the distributed single-task network, the whole network only needs to estimate a single parameter vector. In this work, the distributed estimation of the cluster multitask network is considered, which is different from the single-task network in that there are multiple unknown parameter vectors need to be estimated. All nodes in the multitask network are divided into different clusters, each cluster has its own task (parameter vector to be estimated) [4]- [7]. 3) The computational complexity of the proposed MD-NMSAF algorithm with some other multitask diffusion algorithms are summarized in the form of table, and the computational advantages of the MD-NMSAF algorithm over the AP based multitask diffusion algorithm are explained in detail. 4) The influences of the step size and subband number on the performance of MD-NMSAF are studied, and the validity and accuracy of the mean square stable step size upper bound, theoretical transient and steady-state analysis results are verified under different input signals. 5) The convergence and tracking performance between the proposed MD-NMSAF algorithm and some other multitask diffusion algorithms are compared in different input signals, the superiority of the MD-NMSAF algorithm is verified. The remainder of this paper is organized as follows. In section 2, the MD-NMSAF algorithm is derived in detail. In section 3, the mean and mean-square stability, theoretical transient and steady-state performance and computational complexity of the proposed MD-NMSAF algorithm are analyzed. Computer simulations are provided in Section 4. Finally, the conclusion is given in Section 5. The proposed algorithm Consider a clustered multitask network consists of N nodes, the nodes on this network are divided into L clusters (L tasks). Node k has access to the reference signal ( ) k d t and input signal vector ( ) k t u , both of them are related by the following linear model: * ( ) ( ) ( )   u w T k k k k d t t v t(1) where the 1 M  vector * w k is the target parameter vector of node k, and ( ) k v n denotes the measurement noise at node k. The target parameter vectors of nodes in the same cluster , 1, 2,   l l L C are the same, i.e., * * ,  l k w w C for  l k C . In addition, it is assumed that there are some similarities between the target parameter vectors of neighboring clusters [4]. In order to accelerate the convergence of the MD-LMS algorithm under highly correlated input, we introduce the SAF into the multitask network and reduces the correlation of input signal by subband dividing and extracting [16]. Fig.1 ( ) ( ) ( ) k iD k iD k iD T k iD k i k e n d n y n d n n n     u w (2) where , ,, , ( ) ( ) ( )), ( 1),..., ( 1) T k i k i D k i D k i D n nN nN nN M         u u u u ., , ( ) ( Similar to the MD-LMS algorithm [6], to solve a multitask network global optimization problem in a distributed manner, the cost function must rely on the local interaction of measurement data within neighborhoods and the similarity relationship between adjacent tasks. At the same time, after introducing the M-estimate function, we can construct the following local cost function loc loc ( ) ( ) 1 2 , , 2 0 , 2 2 2 ( ) 0 \ ( ) ( ) , ( ) ( ) ( ) ( ) ( ) 1 2 2 ( )                             k D k k k k k k k l l l k T N r iD r i k k,r k,l k l m k k m r k i l k m k r i J J J d n n c b n    w w w u w w w w w u C N C N C N C N C(3) where k N represents the set of neighbor nodes of node k (including itself),  k N is the set that k N excluding node k, the symbol  denotes the set intersection, \ is the set difference, and ( ) k C denotes the cluster where node k is located. k,r c and k,l  are non-negative intra-task and inter-task combination coefficients, respectively. is a non-negative coefficient, please refer to [2] for details.  is called the regularization strength [6]. ( ) ( ), ,( 1)    k iD k iD k iD w e A n e n e n N is the sliding data window of length w N , the value of w N must be adjusted with the change of impulsive noise environment. In our simulations, we set 5 w  N . Taking the derivative of the local cost function (3) with respect to k w , we can get     ( ) ( ) 1 , , , 2 ( ) 0 , 0 , \ ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) 2 ( ) 2                            D k k k k k k k k T N r iD r i k k,r r i r k i r i k,l k l m k k m l k m k J J d n n c n n b    w w w u w u u w w w w C C N C N C N C (5) where [ ] [ ] [ ]         . By applying the stochastic gradient descent method, at instant n, the weight vector of the MD-NMSAF algorithm at node k is updated as     ( ) 1 , ,, 2 ( ) 0 ,0 , \ ( ) ( ) ( 1) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( )                            D k k k k k k k T N r iD r i k k k, where  is the step size, the step size of all nodes can be different, but for simplicity, the same value is chosen. By replacing the first three terms on the right-hand side of (6) with the intermediate estimate vector ( 1)  k n  , then updating formula (6) can be performed in the following two steps:     1 , ,, 2 ( ) 0 ,\ ( ) 0 , ( ) ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( 1) ( 1) 2 ( )                            D k k k T N r iD r i k k k k,r r i r k i r i k,l l k l k k k m k m k m k d n n n n n c n n n n n n b n      u w w u u w w w w w N C N C N C  (7) Actually, 0 m w is unavailable for node m and can be replaced by the intermediate estimate ( 1)  m n  . Moreover, we can use ( 1)  k n  to replace ( ) k n w since the former contains more information than latter. Therefore, (7) can be rewritten as   1 , , , 2 ( ) 0 , \ ( ) , ( ) ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( 1) ( 1)                                  D k k k T N r iD r i k k k k,r r i r k i r i k,l l k l k k m k m m k d n n n n n c n n n n n n      u w w u u w w w N C N C N C   (8) where , , 2 ( )   m k m k b m k   and ,, ( ) 1 2      k k k m k m k b   N C . It can be seen from (8) , i.e., node k only uses its own data for adaptive update, see [23] for the same operation. Thus, (8) can be further simplified as   1 , , 2 0 ( ) , , ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( 1) ( 1) D k k N k iD k k k i k,l l k i l k k i k m k m m k e n n n n n n n n n                                  w u w w u w N C N C   (9) where , , , ( ) ( ) ( ) ( ) T k iD k iD k i k e n d n n n   u w , and , , , , ( ), ( ) ( ) [ ( )] 0, other k iD k iD k iD k iD e n e n n e n         (10) Remark: In fact, the NSAF and APA are the extension of the NLMS algorithm in subband domain and time domain, respectively. From this point of view, the proposed MD-NMSAF algorithm can also be easily deduced from the update formula of MD-APM. The update formula of the MD-APM algorithm is given by [9]:       1 (a) ( ) , ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1) ( 1) k k T k k k k k k kl l k l k k m k m m k t t t t t t t t t t                                  ψ w U I U U e w w w ψ N C N C When the projection order P of MD-APA is reduced to 1, and the input signal and error signal in term (a) are replaced by the band-dependent input and error signals, respectively, i.e., the term (a) becomes 1 , , 2 0 , ( ) ( ) ( ) D N k iD k i i k i e n n n         u u , the proposed MD-NMSAF algorithm can be obtained, which also verifies the correctness of the derivation algorithm process in this section. Performance analysis To promote the performance analysis of the algorithm, the following commonly used assumptions are given: Assumption 1: The input signal , ( ) k i n u is temporally and spatially independent with zero-mean and positive definite auto-correlation matrix , , , [5]. E{ ( ) ( )}  T k i k i k i n n R u uAssumption 2: ( ) k n  w is statistically independent of , ( ) k i n u for all node k and 0, , 1 D i N    . Assumption 3: The Contaminated Gaussian (CG) model , , ( ) ( ) ( ) k g k k v t v t v t   + is used as the measurement noise [12], , ( ) g k v t denotes the background noise, which is white Gaussian noise (WGN) with zero-mean and variance 2 k g  , , ( ) ( ) ( ) k k v t b t c t    is a Bernoulli Gaussian process, ( ) b t is a Bernoulli process with probability mass function (PMF) ( ( ) 1) r P b t p   and ( ( ) 0) 1 r P b t p    , ( ) k c t is WGN with zero-mean and variance 2 k c  ( 2 2 k k c g    ). Using the update probability [12], the update formula (9) can be equivalently expressed   , , , ( ) ( ) ( ) k iD k iD k iD P n P e n n    [5],as   1 , , , , 2 ( ) 0 , \ ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( )                      D k m N m iD m iD k m k m m i m k i m i m,l l m l m P n e n n n n n n n     w w u u w w N C N C(11) The decimated subband reference signals can be rewritten as * , ,, ( ) ( )( ) T k iD k i k k iD d n n v n   u w , where , ( )( ) ( ) ( ) ( ) T k iD k i k g k iD e n n n v n    u w , where , , ( ) g k iD v n denote the decimated subband background noises of , ( ) g k v t . Furthermore, (11) can be simplified as According to the above global quantities, the global form of (12) is given by    , ( ) \ ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( )                k m k m k m m m m k k m,l l m l m n n n n n n n      w w B w T w w N C N C (12) where 1 , , 0 ( ) ( ) ( ) D N k k iD k i i n P n n     B A , ,, , 2 , ( ) () ( ) ( ) T k i k i k i k i n n n n  u u A u ,1 , , , , 0 ( )( 1) ( ) ( ) ( ) ( ) ( )       n n n n n n    w Gw GB w GT GQw (13) where T M   G C I , , , [ ] m k m k   C , NM M    Q I P I , , , [ ] k m k m   P and * ( ) ( ) n n    w w w is the network weight error vector. Combining the network weight error vector ( )  n w and (13), we can obtain   ( 1) ( ) ( ) ( ) NM n n n n          w G I Z w GT (14) where ( ) ( ) n n    Z B Q and * GQw   . Mean stability In this subsection, the mean stability of the proposed MD-NMSAF algorithm is analyzed, and the stable step size range is also given. Taking the expected values on both sides of (14), we have (16) According to the properties of the matrix spectral radius and norm [4], yields:   E{ ( 1)} E{ ( )} E{ ( )} NM n n n        w G I Z w (15)      , , , E{ ( )} E{ ( )} E{ ( )} NM NM b NM b b n n n             G I Z G I Z G I Z (17) where , b   is the block-maximum-norm [4]. Since the combination weight matrix satisfies T  1 1 C , where 1 is the column vector whose elements are all 1, , 1 T b     G C holds, and the following relation is established         , ,, , , (b) where (b) is obtained due to , 1 M b    P I [4]. Then, the mean stability condition of the MD-NMSAF algorithm becomes     E ( ) 1 NM NM n         I B I(19) Finally, the range of step size is given by max 2 0 max{ (E{ })} 2 k k       B(20) where max ( )   represents the largest eigenvalue of the matrix. Mean square stability By using Σ-weighted Euclidean norm on both sides of (14) and taking the expected values, we obtain              and  denote the vectorization operation and Kronecker product operator, respectively. Using the vectorization operation on both sides of (22) and the property vec( ) ( ) vec( ) T X Y Y X     , yields 1 vec( ) ( )vec( ) T T     K G G (23) where 2 2 2 N M      K I Γ  ,     =E ( ) E ( ) T T n n  Γ Z Z and         E ( ) E ( ) T T NM NM n n     Z I I Z  . It is known from [24] that E{ ( ) ( )} T n n T T and   E ( ) n B are finite values, so the terms       E ( ) ( ) =Tr E ( ) ( )   T T T T n n n n T G GT G G T T and   E ( ) T n Z in the right side of (21) are finite values. When the MD-NMSAF algorithm satisfy the mean stability,   E ( ) T n  w will converge to a finite value, so the last three terms on the right side of (21) are all finite values. To ensure the mean square stability of the MD-NMSAF, the spectral radius of   T T  K G G must satisfy   ( ) 1 T T    K G G . According to the properties of the spectral radius and matrix norm, we have the following relation     , ,, ( )T T T T T T b b b          K G G K G G K G G (24) Since , 1 T T b    G G , and K is a block diagonal Hermite matrix whose block maximum norm is equal to its spectral radius. Inequation (24) can be rewritten as     ( ) max 1    T T   K G G K(25) Following the same argument used in [25], the mean square stability condition of the MD-NMSAF algorithm is given by   1 max 1 1 0 min , ( ) max ( )                    Γ   (26) where 2 2 1 1 = 2 2            N M Γ I 0   . Transient and Steady-state network MSD Defining the auto-correlation matrix ( ) E{ ( ) ( )}    T n n n w w W of the network weight error vector ( )  n w . And according to (14), we have         2 ( 1) E [ ( )] ( ) ( )[ ( )] E [ ( )] ( ) E ( ) ( ) E ( )[ ( )] The network MSD at time instant n is defined as   1 MSD( ) Tr ( )  n n N W(28) Vectoring both sides of (27) yields                       2 vec ( 1) ( )vec ( ) ( )vec E{ ( )} ( )vec E ( ) ( ) ( )vec E ( ) ( ) ( )vec E{ ( )} ( )vec E ( ) ( ) vec                  TI G w I G Z w G G T T G I w G I w Z W F W      (29) where       2 2 2 ( ) ( ) E{ ( )} E{ ( )} E ( ) ( )             When the MD-NMSAF algorithm converges to the steady-state, (29) can be rewritten as                     2 2 1 2 vec( ( )) ( ) ( )vec E{ ( )} ( )vec E ( ) ( ) ( )vec E ( ) ( ) ( )vec E{ ( )} ( )vec E ( ) ( ) vec                                 N M T T T T T T T    I I G w I G Z w G G T T G I w G I w Z W F      (32) The steady-state network MSD can be expressed as input signal. unfortunately, the computational complexity has also increased significantly. Compared with the AP-type multitask diffusion algorithms, the MD-NMSAF algorithm proposed in this paper reduces the computational complexity a lot, and is close to the MD-LMS algorithm when the subband number is small.                     2 2 1 2 1 MSD( ) vec( ) ( ) ( )vec E{ ( )} ( )vec E ( ) ( ) ( )vec E ( ) ( ) ( )vec E{ ( )} ( )vec E ( ) ( ) vec                                 T NM N M T T T T T T T N    I I I G w I G Z w G G T T G I w G I w Z F      (33) where     1 E{ ( )} E{ ( )}        NM NM  w I G I Z  is obtained from (15) when   n , and       2 2 2 ( ) ( ) E{ ( )} E{ ( )} E ( ) ( )                  NM N M NM    G G I I Z Z I Z Z F (34) Computational Complexity   1 ( 1) 1 N k k k m n M        1 ( ) N k k k m n M    0 MD-APA 2 3 2 1 ( 2 ) N k k k P P m n M P P            2 3 1 ( 2 2 1) N k k k P P m n M P            3 ( ) O P MD-APM 2 3 2 1 ( 2 ) N k k k P P m n M P P            2 3 1 ( 2 2 1) N k k k P P m n M P            3 ( ) O P MD-APMCC 2 3 2 1 ( 2 ) 6 N k k k P P m n M P P P             2 3 1 ( 2 2 1) N k k k P P m n M P            3 ( ) O P MD-NMSAF   1 ( 2) 2 N k k D k m n N M        1 ( 2 1) ] N k k D D k m n N M N       0 Simulation results In this section, two multitask networks with different topologies as shown in Fig.2 are used to verify the performance of the proposed MD-NMSAF algorithm. The multitask network with node N=7 is used to verify the validity of the theoretical analysis of the proposed MD-NMSAF algorithm. The multitask network with node N=15 is used to demonstrate the performance advantages of the MD-NMSAF algorithm over some existing multitask algorithms. As shown in Fig.2, the nodes in different color regions (clusters) in the topology correspond to different estimation tasks (target parameter vectors). The target parameter vectors for the three clusters can be generated by * * 1 2 3 { , , } l l C C l C C C C      , w w w , where 1 0.025 C   , 2 0.05 C    , 1 0.075 C   , and * rand( ,1)  M w . The combination weight coefficients k,l  in the multitask algorithms are selected as 1 \ ( )    k,l k k  N C for \ ( )  k l k N C , where    denotes the cardinality of a set, and , m k  follow the Uniform rule in [26]. The network MSD is used as the performance measure of algorithms, which is defined as follows   2 2 1 1 MSD( ) E ( ) N k k k n n N      w w(35) Three different input signals are used for our simulations, and described as follows: (1) Gaussian input signal: The input signal of each node k is a white Gaussian process with zero-mean and variance 2 k   ; (2) AR (1) input signal: The input signal is produced by the first-order auto-regressive system 1 ( )= ( 1)+ ( )  k k k u t u t t   , where 1  is the correlation coefficient, ( ) k t  is white Gaussian process with zero-mean and variance 2 k   . (3) AR (2) input signal: The input signal is produced by the second-order auto-regressive system 2 3 ( )= ( 1)+ ( 2)+ ( ) k k k k u t u t u t t      , where 2  and 3  are the correlation coefficients. Verification of the theoretical analysis results Steady-state performance In this part, the correctness of the steady-state analysis result of the proposed MD-NMSAF algorithm is verified under the multitask network with 7 N  nodes shown in Fig. 2(a) Transient performance In this subsection, our purpose is to verify the validity of the transient performance analysis result, and study the influence of the number of subbands and step size on the proposed MD-NMSAF algorithm. Unless otherwise specified, the multitask network and parameters selection are the same as those in Section 4.1.1. Fig.5 shows are also provided in Fig.6, where the step size is set to 0.01, 0.02, 0.03, and the number of subbands D N is fixed to 8. As displayed in Fig.5 and Fig.6, the theoretical transient network MSD curves computed by (31) are in good agreement with the simulation curves under different input signals, subband numbers and step sizes. Fig.5 demonstrates that the smaller the number of subbands, the lower the steady-state network MSD of MD-NMSAF, but the slower the convergence speed also. From Fig.6, one can see that the convergence speed of MD-NMSAF is gradually increased as the step size increases, but the steady-state network MSD is also increased. Stable step size range In this simulation, the validity of the upper bound of the stable step size range calculated in Section 3 is verified under different input signals. The number of subbands D N of the MD-NMSAF algorithm is set to 4. The green dotted lines in Fig.7 depict the upper bound of the stable step size range (26), and the blue and red curves are the simulated and theoretical steady-state network MSD values, respectively. Simulation results show that when the step size of MD-NMSAF is near the upper bound of the stable step size range, the algorithm appears divergent, which verifies the accuracy of the stable step size bound. Performance comparison In this subsection, a multitask network with nodes 15 N  as shown in Fig.2 Fig.3 (b). The parameters of CG noise are as follows: 0.01 r p  , nodes are shown in Fig.3(b). To maintain fairness, the parameters of all algorithms are carefully selected to achieve the same initial convergence rate, as provided in Table II. It can be seen from Fig.8 that the convergence performances of the MD-APM and MD-NMSAF algorithms are almost the same under the white Gaussian inputs, while the MDAPA performs the worst since its not robust to impulsive noise. For AR(1) input signal, the MD-NMSAF algorithm has the lowest steady-state network MSD. And for AR(2) input signal, the convergence speed of MD-NMSAF is much faster than the MD-APM algorithm, which is attributed to the outstanding decorrelation ability of the SAF. The tracking performance of algorithm is particularly important for time-varying systems, so we will checkout the tracking performance of the proposed MD-NMSAF algorithm through computer simulations. In this experiment, the unknown parameter vectors of all clusters change from l C  w to l C  w at the middle iteration. The parameter settings of all algorithms are the same as those in Fig.8. As shown in Fig.9 (a), the tracking speeds of all algorithms are the same for white Gaussian inputs, but the steady-state network MSD of the MD-APM and MD-NMSAF algorithms are the lowest. On the contrary, the proposed MD-NMSAF algorithm can simultaneously provide the fastest tracking speed and the smallest steady-state error under correlated inputs and impulsive noise, which is due to the strong decorrelation ability of the SAF and the robustness of M-estimate function. Conclusions In this paper, a novel robust multitask adaptive algorithm called MD-NMSAF is proposed, which inherits the strong decorrelation ability of the SAF for correlated inputs and the robustness of the M-estimate function. Compared with the AP based multitask adaptive algorithms, the MD-NMSAF algorithm has greatly improved the convergence speed and steady-state accuracy. More importantly, the computational complexity of the MD-NMSAF algorithm is much less than those of multitask AP algorithms. In addition, the stability condition, analytical expressions of the transient and steady-state network MSD of the MD-NMSAF algorithm in impulsive noise environment are provided and verified by computer simulations under different input signals, number of subbands and step sizes. Finally, the performance comparison between the MD-NMSAF algorithm and some other multitask algorithms fully confirms the performance advantages of the proposed algorithm. Fig. 1 . 1Multiband structure of subband adaptive filter. of node k at time instant n on the i-th subband can be expressed as , , , ,  S S S S , the network MSD at time instant n is given by Ⅰ summarizes the computational complexity of the proposed MD-NMSAF algorithm, MD-LMS algorithm and some AP-based multitask diffusion algorithms at each iteration in terms of addition/minus, multiplication, and direct matrix inversion (DMI) operations. N is the number of node in the network; k m and k n denote the number of node in the sets , respectively; M is the length of filter; D N is the number of subband; P is the projection order. Although the AP based multitask diffusion adaptive algorithms significantly improve the convergence speed of the MD-LMS algorithm under correlated Fig. 2 Multitask network topology. (a) 7 N  . (b) 15 N  . . The filter order is =8 M . The parameter settings of three different input signals are as follows: all nodes are shown inFig.3 (a). The parameters of CG noise all node are shown inFig.3(a). The simulated steady-state MSD values are obtained by taking the average of the last 100 iterations of the total 50000 iterations.InFig.4, the steady-state network MSD of the MD-NMSAF algorithm versus step size and subband number are presented, and we can see from it that the simulated values are matched well with the theoretical analysis results, which confirms the validity of the steady-state analysis result (33) derived in Section 3.3. In addition, the conclusions under three different inputs are similar, that is, the smaller the number of subbands, the lower the steady-state network MSD of the MD-NMSAF. And the steady-state network MSD gradually increases with the increase of step size. the theoretical and simulated transient network MSD curves of the MD-NMSAF algorithm under different input signals, where the number of subbands D N is set to 2, 4, 8, and the step size is fixed to 0.005   . Moreover, the theoretical and simulated transient network MSD curves under different step sizes Fig. 4 Fig. 5 Fig. 6 456Theoretical and simulated steady-state network MSD versus step size and subband number. (a) Gaussian input signal. (b) AR (1) input signal. (c) AR (2) input signal. Theoretical and simulated transient network MSD curves under different subband numbers. The step size is fixed to 0.005   . (a) Gaussian input signal. (b) AR (1) input signal. (c) AR (2) input signal. Theoretical and simulated transient network MSD curves under different step sizes. The number of subband is fixed to 8 D N  . (a) Gaussian input signal. (b) AR (1) input signal. (c) AR (2) input signal. Fig. 7 Fig. 8 Fig. 9 789Theoretical and simulated steady-state network MSD versus step size and the stability threshold given in(26).(a) Gaussian input signal. (b) AR (1) input signal. (c) AR (2) input signal. Convergence performances of the multitask diffusion algorithms. (a) Gaussian input signal. (b) AR (1) input signal. (c) AR (2) input signal. Tracking performances of the multitask diffusion algorithms. (a) Gaussian input signal. (b) AR (1) input signal. (c) AR (2) input signal. The variables t and n represent the index of the original and decimated sequences, respectively.shows the multiband-structured SAF with D N subbands. For each node k, the reference signal ( ) k d t and input signal vector ( ) k t u are partitioned into D N band-dependent signals , ( ) k i d t and , ( ) k i t u by the analyzing filter bank { ( ), 0,1, , 1} i D H z i N    , respectively. The subband output signal , ( ) k i y t is obtained by inputting the subband signal , ( ) k i t u into the SAF with tap-weight vector ,1 ,2 , ( ) [ ( ), ( ), , ( )] T k k k k M n w n w n w n   w . Then, , ( ) k i d t and , ( ) k i y t are critically decimated to , , ( ) ( ) k iD k i D d n d nN  and , , ( ) ( ) k iD k i D y n y nN  , respectively. 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[ "Deep Neural Review Text Interaction for Recommendation Systems", "Deep Neural Review Text Interaction for Recommendation Systems" ]
[ "Parisa Abolfath \nComputer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran\n", "Beygi Dezfouli \nComputer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran\n", "Saeedeh Momtazi [email protected] \nComputer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran\n", "Mehdi Dehghan \nComputer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran\n" ]
[ "Computer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran", "Computer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran", "Computer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran", "Computer Engineering and Information Technology Department\nAmirkabir University of Technology\nTehranIran" ]
[]
Users' reviews contain valuable information which are not taken into account in most recommender systems. According to the latest studies in this field, using review texts could not only improve the performance of recommendation, but it can also alleviate the impact of data sparsity and help to tackle the cold start problem. In this paper, we present a neural recommender model which recommends items by leveraging user reviews. In order to predict user rating for each item, our proposed model, named M atchP yramid Recommender System (MPRS), represents each user and item with their corresponding review texts.Thus, the problem of recommendation is viewed as a text matching problem such that the matching score obtained from matching user and item texts could be considered as a good representative of their joint extent of similarity. To solve the text matching problem, inspired by MatchPyramid (Pang et al., 2016), we employed an interaction-based approach according to which a matching matrix is constructed given a pair of input texts. The matching matrix, which has the property of hierarchical matching patterns, is then fed into a Convolutional Neural Network (CNN) to compute the matching score for the given user-item pair. Our experiments on the small data categories of Amazon review dataset show that our proposed model gains from 1.76% to 21.72% relative improvement compared to DeepCoNN model (Zheng et al., 2017), and from * Principal corresponding author 0.83% to 3.15% relative improvement compared to TransNets model(Catherine and Cohen, 2017). Also, on two large categories, namely AZ-CSJ and AZ-Mov, our model achieves relative improvements of 8.08% and 7.56% compared to the DeepCoNN model, and relative improvements of 1.74% and 0.86% compared to the TransNets model, respectively.
10.1016/j.asoc.2020.106985
[ "https://arxiv.org/pdf/2003.07051v1.pdf" ]
212,725,084
2003.07051
ac73373ce67e8d5388d08f9cc2236683cd662412
Deep Neural Review Text Interaction for Recommendation Systems March 17, 2020 16 Mar 2020 Parisa Abolfath Computer Engineering and Information Technology Department Amirkabir University of Technology TehranIran Beygi Dezfouli Computer Engineering and Information Technology Department Amirkabir University of Technology TehranIran Saeedeh Momtazi [email protected] Computer Engineering and Information Technology Department Amirkabir University of Technology TehranIran Mehdi Dehghan Computer Engineering and Information Technology Department Amirkabir University of Technology TehranIran Deep Neural Review Text Interaction for Recommendation Systems March 17, 2020 16 Mar 2020Preprint submitted to .Recommender SystemsRating PredictionNeural Text SimilarityReview ProcessingConvolutional Neural Networks Users' reviews contain valuable information which are not taken into account in most recommender systems. According to the latest studies in this field, using review texts could not only improve the performance of recommendation, but it can also alleviate the impact of data sparsity and help to tackle the cold start problem. In this paper, we present a neural recommender model which recommends items by leveraging user reviews. In order to predict user rating for each item, our proposed model, named M atchP yramid Recommender System (MPRS), represents each user and item with their corresponding review texts.Thus, the problem of recommendation is viewed as a text matching problem such that the matching score obtained from matching user and item texts could be considered as a good representative of their joint extent of similarity. To solve the text matching problem, inspired by MatchPyramid (Pang et al., 2016), we employed an interaction-based approach according to which a matching matrix is constructed given a pair of input texts. The matching matrix, which has the property of hierarchical matching patterns, is then fed into a Convolutional Neural Network (CNN) to compute the matching score for the given user-item pair. Our experiments on the small data categories of Amazon review dataset show that our proposed model gains from 1.76% to 21.72% relative improvement compared to DeepCoNN model (Zheng et al., 2017), and from * Principal corresponding author 0.83% to 3.15% relative improvement compared to TransNets model(Catherine and Cohen, 2017). Also, on two large categories, namely AZ-CSJ and AZ-Mov, our model achieves relative improvements of 8.08% and 7.56% compared to the DeepCoNN model, and relative improvements of 1.74% and 0.86% compared to the TransNets model, respectively. Introduction Recommender systems autonomously gather information on the preference of users for a set of items (e.g., hotels, books, movies, songs and etc.) upon which they proactively predict the preference each user would give to an item. Thus, the problem that recommender systems seek to solve could be viewed as a rating prediction problem. Today, with the prevalence of online services and due to the abundance of choice in quite all such services, recommender systems play an increasingly significant role. As deep learning approaches have recently obtained considerably high performance across many different machine learning applications, recent researches on recommender systems have also focused on using deep neural networks which have shown impressive results. In this approach, neural networks are mostly used to construct latent representations of users and items using the content associated with them. Content of a user or an item could include any property of each; e.g., demographic information and product preferences are potential content for users, whereas the content linked to items could include their price or any other attribute with respect to their application in general. To make rating prediction, latent representations are then used as input to a regression or collaborative filtering model. However, unlike the content information that describes only the user or only the item, review text could be considered a valuable common context information, featuring joint user-item interaction. Most neural recommender models such as those proposed by Bansal et al. (2016), 2 Elkahky et al. (2015), Kim et al. (2016), Li et al. (2015) and Wang et al. (2015) have focused on the content information of either users or items. On the other hand, only a limited number of neural recommender models such as proposed models by Almahairi et al. (2015), Seo et al. (2017), Zheng et al. (2017), Catherine and Cohen (2017), Li et al. (2017) and Chen et al. (2018) use user-provided reviews. These review-based models have shown that utilizing reviews could considerably improve the performance of recommendation compared to traditional approaches such as collaborative filtering techniques. Using the valuable information in reviews employed in such models have particularly proved to alleviate the well-known rating sparsity and cold-start problems. Most review-based neural recommender models work upon text matching approaches; i.e., they work similar to neural information retrieval models with the text corresponding to user as the query and the text corresponding to item as the document. The task of text matching is the central part of many natural language processing applications, such as question answering (Xue et al., 2008) and paraphrase identification (Socher et al., 2011). In general, neural text matching models can be divided into two families: representation-based models and interaction-based models. The state-of-the-art neural recommender models which are based on text matching (Seo et al., 2017;Zheng et al., 2017;Catherine and Cohen, 2017), use the representation-based learning approach. Although interaction-based models have been successfully used for different applications such as question answering, paraphrase identification, and document retrieval, none of the previous proposed neural recommender systems have used this approach. According to Nie et al. (2018), in general, interaction-based approach performs better than representation-based approach in convolutional models for information retrieval. Because with representation-based models, it is very hard to learn global, informative representation of a user document or an item document which is usually very long; it is not surprising that representing every aspect of a long review document with a single vector is almost impossible. On the other hand, interaction-based models determine local matching signals between user document and item document by considering the interaction of every 3 term of user document with every term of item document. It is important to note that traditional information retrieval models essentially work on the basis of similar local matching signals. In this paper, we propose to model the problem of rating prediction as a text matching problem using an interaction-based approach. To the best of our knowledge, this is the first research to use an interaction-based text matching approach for recommendation. In this formulation, each user is represented by all reviews she/he has written for different items, denoted as user document, and, similarly, each item is represented by all reviews written for it by different users, denoted as item document. Given the representations of all users and all items, our goal is to find the matching score for each pair of user-item for which we intend to determine whether to recommend the item to the user. Based on this idea, we believe that the matching score would be a good representative of the similarity between user and item which is essentially the main guideline for recommendation. Inspired by MatchPyramid (Pang et al., 2016), we employed a CNN architecture fed by the matching matrix of corresponding reviews for a pair of user-item. Our model, MPRS, achieves a better performance with respect to rating prediction compared to the state-of-the-art deep recommendation systems which benefit from users' reviews too. The rest of this paper is organized as follows: Section 2 describes related works. The proposed model and architecture are introduced in detail in Section 3. The experiments and results are discussed in Section 4, and we conclude the paper in Section 5. Related Works In this section, we provide an overview of the literature that benefits from user reviews for recommendation. We classify this line of research into two categories: non-neural models and neural models. Non-Neural Models The Hidden Factors as Topics (HFT) model by McAuley and Leskovec (2013) works by regularizing the latent user and item parameters obtained from ratings with hidden topics in reviews. To this end, LDA topic modeling on reviews is combined with a matrix factorization model to be used as an objec- In their proposed RMR model ratings are sampled from a Gaussian mixture. Almahairi et al. (2015) used matrix factorization to learn the latent factors of users and items. This model benefits from review texts to overcome the data sparsity problem in matrix factorization technique. Following the multitask learning framework by Caruana (1997), the model jointly predicts rating given by the user u to the item i and models the review written by the user u for the item i. The model consists of two components: (1) rating prediction, and (2) review modeling which shares some of the parameters from the former component. Almahairi et al. (2015) then proposed two representations for modeling the likelihood of the review texts, namely bag-of-words and LSTM embedding. Seo et al. (2017) proposed attention-based CNNs to extract latent representations of users and items according to which the model makes rating prediction. One of the recent neural models which efficiently predicts rating is the Deep Cooperative N eural N etworks (DeepCoNN) model (Zheng et al., 2017). This model consists of two deep neural networks to obtain the latent representations of users and items from their corresponding reviews. User and item representations are then input to a layer to estimate their corresponding rating. A more recent study which successfully outperformed DeepCoNN on rating prediction is the T ransf ormational N eural N etworks (TransNets) model by Catherine and Cohen (2017). This model is an extension to Deep-CoNN model by adding a latent layer to obtain an approximate representation of the review corresponding to the user-item pair in the input. While training, this layer is regularized to be similar to the latent representation of the actual review of the target user-item pair which is computed through training a sentiment analysis network. The main idea is that the joint review of a user-item pair gives an insight into the user's experience with the item. At test time, the joint review is not given; therefore, an approximation of the user-item joint review could improve rating prediction. Li et al. (2017) proposed a neural recommendation system, named N eural Rating and T ips Generation (NRT), which jointly predicts ratings and generates abstractive tips using a multi-task learning approach. Chen et al. (2018) introduced a neural attention mechanism to simultaneously learn the usefulness of each review and predict ratings. In this setting, highly-useful reviews are obtained to provide review-level explanations which help users to make better and faster decisions. Neural Models As mentioned, all the models in the literature have built their architecture on the top of representation-based text matching. On the contrary, the current research is pioneer in utilizing an interaction-based text matching model for neural recommendation to benefit from more local text matching signals. Model Architecture The main idea of the proposed model comes from modeling user-item similarity as text matching, since each user or item could be represented by the text of their corresponding reviews. We perform the text matching task by building a matching/interaction matrix as the joint representation of the user-item pair. To compute the matching score between a user text and an item text, we then use an interaction-based matching model on the top of the matching matrix. In this model, we represent the input of text matching as a matching matrix with entries standing as the similarity between words. We consider each data entry as a tuple (j, k, r jk , rev jk ) which denotes a 6 review written by user j for item k with rating r jk and the review text rev jk . To this aim, first all the reviews written by user j are concatenated together into a single document, denoted as d j 1:n , consisting of n words. The same process is applied to get each item's corresponding document, denoted as d k 1:m for the item k, where m is the length of the document. The input of our proposed model for the pair of user j and item k is a matching matrix, M j,k , with each element m j,k p,q as the similarity between the words w p and v q denoting the pth and qth words in d j 1:n and d k 1:m respectively. The similarity between two words can be computed by calculating the cosine similarity of their word embeddings to capture the semantic matching between words. Cosine similarity is calculated as follows: m j,k p,q = α j p · β k q α j p β k q(1) where α j p and β k q are the word embeddings corresponding to w p and v q respectively. The matching matrix constructed in this way represents the joint representation of the user-item pair. The architecture of our proposed model for rating prediction is shown in Figure 1. In the first layer, the matching matrix for the given user-item pair is constructed, as it is described above. The matrix, which captures the joint semantic information in the review texts, is then fed into a CNN architecture followed by a regression network to predict the matching score for the corresponding user-item pair. The CNN part consists of common layers of CNN-based models, including convolution layer, batch normalization layer, and max-pooling layer to learn the underlying interaction patterns. At the end, a fully-connected layer followed by a regression layer is added to the top of the last max-pooling layer to make a prediction on the ratingr jk that the user j would give to the item k. The CNN model is used to extract effective features in a high-dimensional feature space which has disturbance and distortion. When the effective feature space is obtained, we can use fully-connected layers to perform the desired task Each convolution layer uses filter K i ∈ c×t to produce the feature map z i which is defined as: z i = f (M j,k * K i + b i )(2) where M j,k ∈ n×m is the input matching matrix, and symbol * is the convolution operator, b i is a bias term and f is an activation function. z i ∈ (n−c+1)×(m−t+1) , if the stride is 1. For our model, we use Rectified Linear Units (ReLUs) as the activation function which is defined as: f (x) = max(0, x)(3) It should be added that we perform a convolution operation regarding each kernel K i in the convolution layer. Each convolution layer removes infrequent sub-patterns (disturbances) and extracts frequent sub-patterns. We then apply a max-pooling operation over each feature map obtained from the convolution layer, and take the maximum value in each pooling window as one of the features for the corresponding kernel. In fact, each pooling layer reduces the dimension of the feature maps by sub-sampling. After each pooling operation, the extracted features are more invariant to distortion and scale. By repeating the convolution and pooling layers, we allow filtering disturbances and concurrently extracting each feature with a certain scale. The result from the final max-pooling layer is passed to a fully connected layer, followed by a regression layer which is a single neuron layer with a linear activation function. Accordingly, the matching scorer jk is calculated as: r jk = W × O + g(4) where O is the results from the fully-connected layer, W is the regression layer's weights and g is the bias. In essence, the interaction structures in the process of text matching are compositional hierarchies of signals which might appear in the matching matrix, i.e. when matching two texts, word-level matching signals together form the phrase-level signals, and phrase-level signals assemble into sentences-level signals. Thus, the hierarchical convolutions performed in our model would capture the matching patterns between two texts. The approach is designed based on a deep architecture for text matching, MatchPyramid (Pang et al., 2016 to predict if two texts are similar or not. In our model, however, instead of classifying two texts as relevant/similar or non-relevant/dissimilar, we aim at estimating the relevance degree of two texts. Therefore, our architecture is adapted to make regression prediction for the user-item similarity instead of classification. Experiments In order to evaluate the effectiveness of our proposed model, we performed different experiments. In this section, the dataset, the setup of the experiments as well as their results are presented. Dataset We evaluate the performance of our proposed model by using the most recent release More statistics about the length of reviews in these datasets are presented in Table 2. Evaluation Metric In our experiments, we employed the Mean Squared Error (MSE) to evaluate the performance of our proposed model. Let N be the total number of datapoints in the test set. MSE can be computed as follows: M SE = 1 N N i=1 (r i −r i ) 2(5) where r i is the ith observed value andr i is the ith predicted value. Setup of Experiments For each category, we divided the dataset into three sets of training, validation and test which are 80%, 10% and 10% of the whole dataset, respectively. Given training, validation and test sets, the document associated with each user or item is then constructed. Let rev jk denote the review written by user j for item k. Let d j and d k be the documents to be constructed corresponding to user j and item k respectively. For all (user j , item k ) pairs in the test set, rev jk is omitted from both d j and d k , since at test time, as the simulation of a real world situation, the joint review of a user-item pair is not obviously given. We build the train, validation and test sets accordingly for all our experiments for all the models including the baselines. Baselines In order to evaluate the performance of our model, we selected DeepCoNN (Zheng et al., 2017) and TransNets (Catherine and Cohen, 2017) to compare our results with. They are two of the most recent neural recommender models known as state-of-the-art models in this field. The performance of the Deep-CoNN model reported in work by Zheng et al. (2017) corresponds to an obsolete version of the dataset to which we had no access, therefore, we implemented the DeepCoNN model and tested the model on the latest version of the Amazon dataset. For the TransNets model and its variant, TransNet-Ext, we used the provided code by the authors that is available on github 2 . Results The MSE values of MPRS, DeepCoNN, TransNets and TransNet-Ext models on 7 different categories of Amazon datasets are given in Table 3. As it is shown in the models. As can be seen, our model gains the maximum relative improvement (3.15%) on AZ-MI category which is the smallest data category in terms of the total number of reviews. It indicates that our model can achieve superior results on small number of instances. Overall, the relative improvements of our model on two large datasets are 0.86% and 1.74%, and the relative improvement on small datasets is from 0.83% to 3.15% . In the next step of our experiments, we evaluate the robustness of our proposed model in case of changing the order of reviews in user/item documents. According to the structure of the problem and the way we represent users and items in this work, it is expected that if we change the order of reviews in the user document or item document, the model's prediction of rating does not undergo considerable changes. On the other hand, by changing the order of reviews in a document the matching matrix changes as well. As a result, the input to our model would be a different matrix and the following question is raised: "given our model trained with a specific order of reviews in each user/item document, if we change the order of reviews in user document and item document in test set, does the model's performance in rating prediction undergo considerable 13 changes?". Table 3 and the other with shuffled document of each user and item. As can be seen in Figure 2, for all three categories the amount of changes in the MSE value as a result of changing the order of reviews in documents is insignificant. In the next part of our experiments, we compare the performance of our proposed model with the baseline models in case of data sparsity. Some users might rate only a limited number of items. Data sparsity problem arises when there is not enough rating data available for some users and items, therefore recommender systems are not able to learn preferences regarding such users and items. Cold start problem, specifically refers to the difficulty of recommendation to new users and items due to lack of data. Data sparsity problem is one of the main challenges in designing recommender systems. Recent studies have shown that using reviews can help to considerably resolve the data sparsity problem. The diagrams in Figure is set to five. For each data category, MSE values of the models are plotted for both users and items groups. As it is shown in Figure 3, our proposed model performs better than the baseline models on all three data categories for almost any number of training reviews (1-5). Especially, when having only one review available, which is the worst case of data sparsity in this formulation, our model significantly outperforms the baselines. Conclusion In this paper, we propose a novel neural recommender system which recommends items by leveraging user reviews. We represent each user and item with their corresponding reviews and view the problem of recommendation as a text matching problem. We used an interaction-based model to adress the text matching problem. To this aim, a CNN architecture followed by a regression network was employed to predict user-item pair matching score as a representative of user's rating for item. Experimental results on various data categories of Amazon show that our proposed model outperforms the state-of-the-art neural recommender systems which are based on reviews. There are multiple possible directions to extend the model proposed in this paper. One way to extend the model is to use an attention-based CNN to locate the attention to most representative parts of the user-item matching matrix which would probably improve the performance of the base model. Another approach to further improve the performance of the model is to use a target network as well as the main CNN which should be followed by a Transform layer as inspired by the TransNets model (Catherine and Cohen, 2017). The Transform layer transforms user-item joint feature map obtained from the CNN into an approximation of their joint review which would give an insight into the users experience with the item during train as well as evaluation. The target network as used in the TransNets model, provides the latent representaion of target joint review to train the main CNN and the Transform layer. tive function. A modified version of HFT is the T opicM F model byBao et al. (2014), where latent user and item factors learned using matrix factorization are jointly optimized with the topic modeling of their joint review. To this aim, they utilized the non-negative matrix factorization technique.Ling et al. (2014) proposed the Rating Meets Reviews (RMR) model which extends the HFT model. 7 Figure 1 : 71The architecture of the proposed MPRS model of classification or regression. The key operations to extract effective features, or what is called dimension reduction, are filtering and scaling which are performed by convolution and pooling layers respectively. Figure 2 : 2The impact of shuffling reviews in user and item documents on MSE Figure 2 shows the MSE values of MPRS model on three different categories of Amazon dataset. For each category, there are two MSE values reported, one corresponds to the MSE of MPRS on test set with the same order reported in Figure 3 : 33 depict the MSE values of MPRS, DeepCoNN , TransNets and TransNet-Ext models on three categories of the Amazon dataset. To test the models' performances in this condition, for each dataset category, users and items are grouped based on the number of reviews in training set. In this experiment, the maximum number of training reviews MSE values of MPRS compared to DeepCoNN, TransNets and TransNet-Ext for users and items groups with different number of training reviews on three data categories Table 1 : 1Dataset statistics separated by category of Amazon dataset 1 (McAuley et al., 2015a,b) which includes reviews and ratings given by users for products purchased on amazon.com. The datasetcontains Amazon product reviews and metadata from May 1996 to July 2014, separated by category. In the following, we present the results of experiments Dataset User Review Average Length Item Review Average Length AZ-CSJ 78.77 214.4 AZ-Mov 197.87 2047.3 AZ-OP 1298.4 2630.65 AZ-IV 545.69 1659.32 AZ-Auto 486.69 775.99 AZ-PLG 1015.79 1779.53 AZ-MI 526.52 835.4 Table 2 : 2Review statistics separated by category on 7 different categories of this dataset. The statistics of these categories are given inTable 1.The categories presented in the table are from both large and small Amazon datasets. As can be seen, the average number of review and rating pairs providedby each user on the large datasets is less than three, which shows that these categories are extremely sparse. This issue may considerably deteriorate the performance of recommender systems. 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[ "Measuring NIR Atmospheric Extinction Using a Global Positioning System Receiver", "Measuring NIR Atmospheric Extinction Using a Global Positioning System Receiver" ]
[ "Cullen H Blake \nDepartment of Astrophysical Sciences\nPrinceton University\nPeyton Hall, Ivy Lane08544PrincetonNJ\n", "Margaret M Shaw \nDepartment of Astrophysical Sciences\nPrinceton University\nPeyton Hall, Ivy Lane08544PrincetonNJ\n" ]
[ "Department of Astrophysical Sciences\nPrinceton University\nPeyton Hall, Ivy Lane08544PrincetonNJ", "Department of Astrophysical Sciences\nPrinceton University\nPeyton Hall, Ivy Lane08544PrincetonNJ" ]
[]
Modeling molecular absorption by Earth's atmosphere is important for a wide range of astronomical observations, including broadband NIR photometry and high-resolution NIR spectroscopy. Using a line-by-line radiative transfer approach, we calculate theoretical transmission spectra in the deep red optical (700 < λ < 1050 nm) for Apache Point Observatory. In this region the spectrum is dominated by H 2 O, which is known to be highly variable in concentration on short timescales. We fit our telluric models to high-resolution observations of A stars and estimate the relative optical depth of H 2 O absorption under a wide range of observing conditions. We compare these optical depth estimates to simultaneous measurements of Precipitable Water Vapor (PWV) based on data from a Global Positioning System (GPS) receiver located at Apache Point. We find that measured PWV correlates strongly with the scaling of H 2 O absorption lines in our spectra, indicating that GPS-based PWV measurements combined with atmospheric models may be a powerful tool for the real-time estimation of total molecular absorption in broad NIR bands. Using photometric measurements from the Sloan Digital Sky Survey (SDSS) DR8 database we demonstrate that PWV biases the calibrated r-z colors and z-band fluxes of mid-M stars but not mid-G stars. While this effect is small compared to other sources of noise in the SDSS z-band observations, future surveys like the Large Synoptic Survey Telescope aim for higher precision and will need to take time-variable molecular transmission into account for the global calibration of NIR measurements of objects having strong spectral features at these wavelengths. Empirical calibrations based on PWV may be immediately applicable to ongoing efforts to make mmag differential measurements of M stars to detect transiting exoplanets.Subject headings: Astronomical Techniques 1 \protecthttp://www.lsst.org/ 2 http://www.darkenergysurvey.org/ 3 http://pan-starrs.ifa.hawaii.edu/public 4 http://www.naoj.org/Projects/HSC/index.html Point Observatory 3.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium. We thank UNAVCO for installing and operating the GPS station at Apache Point and the SuomiNet project at UCAR for producing real-time PWV estimates for this station.
10.1086/662980
[ "https://arxiv.org/pdf/1109.6703v1.pdf" ]
118,504,158
1109.6703
4ef913d51511b610a8f3985771cc73daeafcf172
Measuring NIR Atmospheric Extinction Using a Global Positioning System Receiver 30 Sep 2011 Cullen H Blake Department of Astrophysical Sciences Princeton University Peyton Hall, Ivy Lane08544PrincetonNJ Margaret M Shaw Department of Astrophysical Sciences Princeton University Peyton Hall, Ivy Lane08544PrincetonNJ Measuring NIR Atmospheric Extinction Using a Global Positioning System Receiver 30 Sep 2011Subject headings: Astronomical Techniques -2 - Modeling molecular absorption by Earth's atmosphere is important for a wide range of astronomical observations, including broadband NIR photometry and high-resolution NIR spectroscopy. Using a line-by-line radiative transfer approach, we calculate theoretical transmission spectra in the deep red optical (700 < λ < 1050 nm) for Apache Point Observatory. In this region the spectrum is dominated by H 2 O, which is known to be highly variable in concentration on short timescales. We fit our telluric models to high-resolution observations of A stars and estimate the relative optical depth of H 2 O absorption under a wide range of observing conditions. We compare these optical depth estimates to simultaneous measurements of Precipitable Water Vapor (PWV) based on data from a Global Positioning System (GPS) receiver located at Apache Point. We find that measured PWV correlates strongly with the scaling of H 2 O absorption lines in our spectra, indicating that GPS-based PWV measurements combined with atmospheric models may be a powerful tool for the real-time estimation of total molecular absorption in broad NIR bands. Using photometric measurements from the Sloan Digital Sky Survey (SDSS) DR8 database we demonstrate that PWV biases the calibrated r-z colors and z-band fluxes of mid-M stars but not mid-G stars. While this effect is small compared to other sources of noise in the SDSS z-band observations, future surveys like the Large Synoptic Survey Telescope aim for higher precision and will need to take time-variable molecular transmission into account for the global calibration of NIR measurements of objects having strong spectral features at these wavelengths. Empirical calibrations based on PWV may be immediately applicable to ongoing efforts to make mmag differential measurements of M stars to detect transiting exoplanets.Subject headings: Astronomical Techniques 1 \protecthttp://www.lsst.org/ 2 http://www.darkenergysurvey.org/ 3 http://pan-starrs.ifa.hawaii.edu/public 4 http://www.naoj.org/Projects/HSC/index.html Point Observatory 3.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium. We thank UNAVCO for installing and operating the GPS station at Apache Point and the SuomiNet project at UCAR for producing real-time PWV estimates for this station. Introduction For many astronomical measurements it is necessary to understand and precisely measure the attenuation of light by Earth's atmosphere. While attempts have been made to do this in an absolute sense (Hayes & Latham 1975), in general astronomical measurements at optical and NIR wavelengths are fundamentally differential, measuring flux relative to either a standard star or comparison stars within a given field of view. As shown in Figure 1, across the optical and infrared Rayleigh scattering by molecules, Mie scattering by aerosols, and line absorption by molecules leads to atmospheric transmission that is a rapidly varying function of wavelength and can also vary rapidly in time (Stubbs et al. 2007). Massive imaging surveys like the Large Synoptic Survey Telescope 1 (LSST), Dark Energy Survey 2 (DES), Pan-STARRS 3 , and Hyper Suprime-Cam 4 seek to produce broadband photometric measurements across the UV and near infrared (NIR) that are stable and uniform to better than 1% over periods of years. Even if the response of the telescope to monochromatic light of a given intensity is known very precisely (e.g. Stubbs & Tonry 2006, global calibration of photometric measurements requires correcting flux for atmospheric variations. Similarly, precise radial velocity measurements at NIR wavelengths require a detailed understanding of atmospheric transmission so that subtle Doppler shifts in stellar spectral features can be separated from changes in telluric absorption features. The flux in ADU measured by a single-element detector from an astronomical source, ̥, can be expressed as a function of the wavelength-dependent properties of the telescope, atmosphere, and source as ̥ = A∆T ∞ 0 F λ T (λ)S(λ)dλ(1) where A is the effective collecting area of the telescope, ∆T is the exposure time, T (λ) is the transmission of the atmosphere as a function of wavelength, S(λ) is the response function of the telescope in units of ADU/photon (including the effects of optics, filters, and detectors), and F λ is the flux density of the source in units of photons s −1 cm −2 cm −1 . Historically, for broadband photometry the band-integrated value of the product S(λ)T (λ) is measured for a given night using a combination of observations of a uniformly illuminated screen (the "flat field") and observations of photometric "standard" stars. Using this methodology it is possible, under excellent observing conditions, to measure the absolute brightnesses of objects with a precision of 1%, though these measurements are only as accurate as the measurements of the standard stars themselves. Temporal changes in T (λ) and S(λ) become important when the target and standard are not observed simultaneously. For many applications, such as precise photometry of transiting extrasolar planet hosts, only the relative brightness of an object, as compared to other nearby stars measured simultaneously, is required. This differential photometric approach results in such high photometric precision, often 0.1% (e.g. Chan et al. 2011), in part because the temporal variations in T (λ) and S(λ) are intrinsically removed. Given multiple observations of a sufficiently large number of objects, it is possible to generalize this differential approach and apply it to data from large-area surveys to internally calibrate the brightnesses (and colors) of objects. Padmanabhan et al. (2008) applied a technique of this type calledÜbercal to the Sloan Digital Sky Survey (SDSS) photometric measurements and demonstrated that 1% global photometric precision can be achieved for a survey of millions of objects spanning years (see also Ivezić et al. 2007). Differential calibration strategies may not account for the fact that F λ varies strongly with stellar spectral type (or quasar redshift), resulting in higher-order effects that can arise when measuring relative fluxes between objects having very different Spectral Energy Distributions (SEDs). This is particularly important at NIR wavelengths, where strong molecular bands due to H 2 O, CH 4 , and CO 2 dominate T (λ) and F λ may have strong absorption or emission features (Bailer-Jones & Lamm 2003;Blake et al. 2008;Mann et al. 2011). In these cases, temporal variations in T (λ) are not effectively removed through differential measurement, and residuals at the level of 1 mmg are expected in differential photometry of cool stars. Ultimately, the highest precision will be achieved for a survey like LSST with the help of detailed models of T (λ) that are contemporaneous with broadband photometric observations and localized to particular sight lines. Burke et al. (2010) demonstrate an ambitious strategy whereby simultaneous spectrophotometry of calibration stars in LSST fields can be combined with radiative transfer calculations for the atmosphere above the observatory to measure all important components of T (λ). From the observational point of view, the challenges of precise radial velocities at NIR wavelengths and the global calibration of NIR photometry in large surveys are closely related since both require contemporaneous modeling of transmission through Earth's atmosphere. Precise stellar radial velocity measurements at wavelengths longer than 600 nm require a detailed understanding of the impact of absorption due to molecules in Earth's atmosphere (Wang & Ge 2011). These telluric absorption features themselves can also be used as a simultaneous wavelength reference, a measurement technique conceptually similar to the I 2 cell described by Butler et al. (1996), though first proposed by Griffin & Griffin (1973). The use of telluric lines as a wavelength reference has a long history in the literature (e.g. Smith 1982, Deming et al. 1987, Hatzes & Cochran 1993, and recently several authors have demonstrated that telluric lines are a viable velocity reference for planet searches, including Seifahrt & Käufl (2008), Blake et al. (2010), Figueira et al. (2010a), andFigueira et al. (2010b). This technique requires an excellent model of T (λ), which can be derived empirically from high-resolution observations of the Sun (Blake et al. 2010) or calculated directly based on assumptions about the composition and structure of Earth's atmosphere (Seifahrt et al. 2010). Atmospheric constituents like CH 4 , CO 2 , and O 2 are well-mixed from Earth's surface through the mesosphere and have seasonal variations in total column density. For wavelengths 900 < λ < 2500 nm, absorption by H 2 O becomes very important, having large optical depth at some wavelengths. Modeling these absorption features poses a significant challenge since the total water column can have large and rapid variations. We demonstrate that Precipitable Water Vapor (PWV) estimates derived from a Global Positioning System (GPS) receiver can be used to calculate theoretical transmission spectra having no free parameters and resulting in excellent fits to astronomical spectra in the wavelength range 900 < λ < 1000 nm. In a region of non-saturated H 2 O lines we find fit residuals < 2%, likely dominated by residual fringing signals in our spectra. We find that GPS-based PWV estimates are also useful for estimating total transmission in the SDSS z band and demonstrate that changes in PWV bias photometric measurements of mid-M stars and determine an empirical correction factor for the z-band photometric measurements in the SDSS database. ARC Echelle Observations As part of an ongoing program to measure the radial velocities of M stars, we observed 74 A star telluric standards with the ARCES echelle spectrograph (Wang et al. 2003) on the 3.5 m telescope at Apache Point Observatory (APO). A single ARCES observation contains data in more than 100 spectral orders spanning 360 nm to 1010 nm at a resolution of R∼30,000. These A star observations were collected during 2010 under a wide range of observing conditions and at airmass (AM) up to AM=2.0. Exposure times were tailored so that S/N per pixel in the extracted spectra would be ∼ 500 at λ ∼ 800 nm. The ARCES spectral orders are closely spaced and marginally sampled in the spatial direction, requiring special care in the spectral extraction process so as not to induce aliasing. We collected a large number of bias and quartz lamp frames on each night and used the quartz lamp frames to define the positions of the centers of each of 75 spectral orders, which run approximately parallel to the detector rows. After subtracting an average nightly bias, we estimated scattered light in both the stellar spectra and quartz lamp spectra by fitting a loworder polynomial surface to the 74 inter-order minima of each column. Next, we extracted the stellar and quartz lamp spectra using an aperture extraction algorithm with fractional pixel weighting at the edges of the aperture. Since the orders are so narrow, it was necessary to flat field the ARCES data using the 1D extracted quartz lamp spectra, so we made a master flat for each spectral order for each observing night by averaging the individually normalized 1D quartz spectra. For the red orders of interest here there is large amplitude fringing in the ARCES spectra, which is partially removed from the stellar spectra by dividing by the normalized master flats. There are slight (∼1 pixel) offsets between the positions of the spectral orders in the quartz and stellar frames, resulting in incomplete fringe removal. Atmospheric Models Modeling telluric H 2 O absorption in our A star spectra requires high-resolution telluric templates. In Blake et al. (2010) an observed atmospheric transmission spectrum, derived from solar observations by Livingston & Wallace (1991), was scaled to fit telluric absorption features due to CH 4 around 2300 nm. We found this approach to be problematic for absorption by H 2 O between 800 nm and 1000 nm, so instead we created theoretical transmission spectra. Closely following Gordley et al. (1994) we developed a custom line-by-line radiative transfer code to calculate template spectra that contain only molecular absorption (no scattering, no line mixing, no H 2 O continuum absorption) given an atmospheric model and molecular transition data from the HITRAN 2008 database (Rothman et al. 2009). Following the Beer-Lambert law for extinction through a medium by a single transition of a single absorber, the transmission, T ν , of the Earth's atmosphere at frequency ν is given by T ν = e −uσ(ν) = e −τν(2) where σ(ν) is the frequency-dependent cross section and u is the mass path u = qP x kT (3) where x is the path length through the atmosphere, P is pressure, T is temperature, k is Boltzman's constant, and q is the Volume Mixing Ration (VMR), which is defined as the fraction of the particles in a volume that are the given constituent. Earth's atmosphere contains many gasses and the properties of the atmosphere, including P and T , change with altitude and time. The transmission through the atmosphere can be calculated by breaking the atmosphere up into a number of levels in the vertical direction, z, within which the bulk properties are assumed to be constant. The total transmission due to a number of different absorbing gasses, g, each with a number of absorption lines, i, can then be calculated T ν = e − z g ug,z i σ g,i,z (ν)(4) The shapes and strengths of the absorption lines depend on the cross section, σ(ν). Absorption lines in Earth's atmosphere have intrinsic shapes both due to the bulk properties of the gas mixture and the quantum mechanical properties of the transition. The approximate shape of an absorption line is given by the Voigt profile, V (ν), which is the convolution of line broadening kernels due to Doppler and pressure (Lorentzian) broadening mechanisms. At low pressure the motions of the absorbing molecules dominate the broadening and at higher pressure collisions between molecules come to dominate. Both regimes occur at different levels in Earth's atmosphere. The Voigt profile is defined V (ν) = 1 α D π 3/2 α L α D e −t 2 [(ν − ν c ) /α D − t] 2 + [α L /α D ] 2 dt (5) where α D is the Doppler half width of the line, α L is the Lorentzian half width of the line, and ν c is the line center, each of which can depend on atmospheric temperature and/or pressure. The cross section σ(ν) is a function of the line broadening as well as the line intensity, S = S 0 Q, where Q is a scaling factor based on the partition function of the molecule and the temperature of the gas. The total transmission can now be expressed as T ν = e −τν where τ ν is τ ν = z g u g,z i V g,i,z (ν)S g,i,z(6) We retrieved line parameters including half-widths, strengths, line centers, and pressure shifts of those line centers, for transitions of H 2 O in the wavelength range 600 to 11000 nm from HITRAN 2008 and calculated atmospheric transmission at zenith in wavelength steps of 0.01Å across a plane parallel atmosphere with 30 layers logarithmically spaced between 0 km and 80 km. This radiative transfer calculation requires a model of the structure and composition of Earth's atmosphere, including temperature, pressure, and chemical composition, as a function of height. We used the NASA-MSFC Earth Global Reference Atmospheric Model (Earth-GRAM 2010; Justus et al. 2004 and references therein) to approximate the average properties of the atmosphere above APO, located in southeastern New Mexico at an altitude of 2,780 m. Earth-GRAM combines a large number of historic atmospheric measurements with modern meteorological models to predict atmospheric properties, and the statistical distributions of those properties, as a function of altitude, location on the Earth, and month. While Earth-GRAM is often used for engineering applications, such as estimating the effect of Earth's atmosphere on trajectories, it is also a powerful resource for quantifying the impact of the atmosphere on astronomical observations. Spectral Fitting Following Blake et al. (2010) we adopt a forward modeling approach to fitting our ARCES A star observations. We focus on order 3, spanning the wavelength range 970 < λ < 986 nm, where there are numerous unsaturated H 2 O lines and the A stars are intrinsically devoid of spectral features. The model, B(i), begins with the atmospheric transmission spectrum based on the average Earth-GRAM model, T 0 , scaled for water optical depth, τ , which is a function of zenith angle (or airmass) and water content of the atmosphere. This scaled spectrum T τ 0 is convolved with an estimate of the spectrograph line spread function (LSF), interpolated onto a lower resolution pixel grid that is defined through a polynomial relation between pixel, i, and wavelength, W (i), and finally normalized by a continuum function, N(i), that is a polynomial in i B(i) = (T τ 0 (λ) ⊗ LSF ) | W (i) × N(i)(7) where ⊗ indicates convolution and | indicates "evaluated at". This model has 10 free parameters: four for N(i), three for W (i), two for the LSF, which is a 1D Moffat function (Moffat 1969), and τ . We fit this model to order 3 of each A star spectrum using an implementation of the AMOEBA downhill simplex method (Nelder & Meade 1965, Press et al. 1992) to minimize χ 2 . Examples of fits are shown in Figures 2 and 3. The agreement between our theoretical telluric template and the observed spectra is excellent overall, with typical residuals of 1 − 2% and no evidence for degradation of the fits in the centers of prominent absorption lines for regions where the line density is modest and the average line optical depth is < 1. As shown in Figure 3, in regions having numerous lines with large optical depths the overall quality of the fits deteriorates and the models exhibit significant discrepancies in the cores of some absorption features. In all cases the fit residuals do exhibit slowly varying features that we attribute to imperfect correction for CCD fringing, which has a large amplitude at these red wavelengths. Uncertainty in the HITRAN H 2 O line parameters may also lead to discrepancies between our models and observations and we note that incorporating empirical line parameters, such as those from Alekseeva et al. (2010b), may help to improve the overall quality of the fits. PWV Measurements We found that overall the best-fit model parameters had very small scatter, demonstrating that ARCES is physically stable, with small shifts in wavelength solutions over the period of a year. However, we expect that τ , which is presumably related to zenith angle and the amount of H 2 O in the atmosphere, should vary widely since H 2 O, unlike CH 4 (Blake et al. 2010), is in-homogeneously distributed in the lower atmosphere and its concentration can change by more than 10% in under an hour. As a result, our best fit values of τ spanned the ranged 0.1 < τ < 0.9. The amount of H 2 O in the atmosphere is related to Precipitable Water Vapor (PWV), a measurable quantity that should be proportional to the optical depth to absorption by this molecule. Historically, PWV, which is critically important for mm and sub-micron observations, has been measured with balloon-borne radiosondes or differential radiometers operating in the mid-IR. In the 1990s it was demonstrated that data from a multi-band geodetic quality Global Positioning System (GPS) receiver and a high accuracy barometer can be combined to derive PWV (Bevis et al. 1992, Bevis et al. 1994, Rocken et al. 1995, making these devices extremely important for studies of the time and spatial dependence of water vapor for atmospheric science applications. The index of refraction of the atmosphere, and therefore the speed of propagation for the microwaves used by GPS, is a strong function of pressure, temperature, and water content. Both the ionosphere and neutral atmosphere induce delays in signals from GPS satellites that are measured precisely by the receiver stations. The ionospheric delays are frequency dependent, so GPS signals at multiple frequencies (typically 1.2 and 1.6 GHz) can be used to correct for this effect. The total delay at zenith (TD) induced by the neutral atmosphere can be broken into two components: "zenith hydrostatic delay" (ZHD), which is due primarily to dry air, and "zenith wet delay" (ZWD), which is due to H 2 O vapor. The ZHD is a slowly varying function of time and can be calculated given a barometric pressure measurement to a precision better than 0.01%. A measurement of the TD and an estimate of the ZHD therefore provide an estimate of the ZWD (ZWD=TD-ZHD), which is proportional to PWV with a constant of proportionality that is a function of surface temperature. While it is possible to calculate this constant of proportionality to make absolute measurements of PWV, relative measurements of PWV can be made with quoted precision better than ∼ 0.2 mm given contemporaneous surface temperature and barometric pressure measurements. A large number of GPS-based PWV monitors are maintained around the world. Data from several receiver networks in North America are maintained by UCAR through SuomiNet 5 , including a station operated by UNAVCO located in close proximity to the APO 3.5 m telescope. Using the SuomiNet web interface we downloaded PWV estimates at 30-minute intervals from the APO station for the 2009 and 2010 seasons and interpolated the PWV estimates to the times of our A star observations, excluding cases where there was not a PWV estimate within 1 hour of the time when an A star spectrum was obtained. Using multiple linear regression, we fit for our measured values of τ , the scale factor specific to our telluric template, as a function of PWV and (AM-1) for PWV< 8 mm. We found that measured PWV and AM are strongly correlated with τ and determined a best-fit linear relation τ = 0.036 + 0.098 × P W V (mm) + 0.36 × (AM − 1) ± 0.055(8) The results of this fit are shown in Figure 4. We note that this result is qualitatively similar to those reported by Querel et al. (2008) and Thomas-Osip et al. (2007) who found that PWV estimates derived from radiometer observations strongly correlated with the measured depths of H 2 O features in optical spectra. As shown in Figure 5, our A star observations spanned a relatively narrow range of PWV compared to the overall distribution of PWV at APO, but we are biased toward lower PWV since periods of large PWV are likely to be unusable for astronomical observations. In practice, the coefficients of a relation like the one given in Equation 8, which is based on telluric models calculated using an "average" atmospheric profile for APO, will need to be calibrated once for a given observing site. Once this is done with a set of A star observations, the technique presented here should be broadly applicable to NIR observations at sites other than APO. SDSS Observations Our fits to the A star spectra demonstrate that given an estimate of PWV we can generate a theoretical model for atmospheric transmission in z band that has no free parameters and matches the observed telluric absorption around 980 nm to better than 2% on a perpoint basis. Integrated across a wide observing band, the discrepancy in total transmission between our models and the A star observations is significantly smaller, less than 0.2%. This opens up the possibility that GPS-based PWV estimates can be used as input to theoretical radiative transfer calculations to enable the precise estimation of molecular absorption integrated across NIR bands in real time. When combined with information about the spectra of the objects being observed, this technique could potentially lead to significant improvements in global NIR photometric precision for surveys like LSST and augment an overall calibration strategy like the ones outlined by Burke et al. (2010) and Jones et al. (2010). We test this hypothesis by considering z-band photometric measurements of M stars in the SDSS photometric database. The SDSS (York et al. 2000) made an imaging survey of the sky using a large-format CCD camera (Gunn et al. 1998) mounted on the Sloan Foundation 2.5 m telescope (Gunn et al. 2006) at APO to image the sky in five optical bands -u, g, r, i, and z (Fukugita et al. 1996). The imaging data were reduced by a set of software pipelines that produced a catalog of objects with calibrated magnitudes and positions (Lupton et al. 2002, Hogg et al. 2001, Padmanabhan et al. 2008. SDSS Data Release 8 (DR8; Aihara et al. 2011) includes photometric measurements in five bands of over 260 million stars over an area of more than 15,000 square degrees. The GPS-based PWV monitor at APO (where both the SDSS and 3.5 m telescopes are located) was installed in 2007 and began regular operations in 2008. Unfortunately, nearly all the SDSS imaging was acquired prior to this. A similar GPS-based PWV monitor operated by NOAA 6 is located at White Sands (WS), New Mexico, 50 km southwest of APO at an elevation of 1200 m (1580 m below APO). We interpolated the PWV estimates from WS during 2009 to the times of those from APO having PWV< 10 mm, only considering APO measurements within 30 minutes of a WS measurement. We found that the APO and WS PWV measurements are highly correlated (Pearson's r=0.85), though with significant deviations from linearity at PWV< 1 mm. We adopt an empirical relation between WS and APO by interpolating the average PWV data given in Table 1. While the scatter in this relation is large, particularly at low PWV, it is sufficient for considering the relationship between the overall statistical properties of SDSS z-band photometry and PWV. The photometric measurements in the SDSS DR8 database have undergone the calibration procedure described in Padmanabhan et al. (2008), which has been shown to achieve global photometric precision of ∼ 2% in z band, and ∼ 1% in g, r, and i bands. In z band, in addition to the components of atmospheric extinction that vary smoothly with wavelength, such as Rayleigh scattering by molecules, Mie scattering by aerosols, and "gray" absorption by thin clouds, line absorption by H 2 O becomes important. This means that the extinction for a given object depends on the SED of the object, which is not the case for the other bands.Übercal is fundamentally a differential approach, where flux is calibrated relative to a large ensemble of objects having a wide range of SEDs that, when averaged, at NIR wavelengths have a smooth SED like a solar type star. Given this, we expect systematic residuals in the SDSS z-band measurements for objects with complex NIR SEDs, such as M stars, that will correlate with PWV. We selected two samples of M stars from the SDSS DR8 database 7 to investigate this effect. The first sample is based on the spectroscopically-classified catalog of M stars produced by West et al. (2011). We selected 7,300 objects having spectral types from M7 to M4 and brighter than i=17.0 and retrieved the CLEAN photometric measurements of these objects from the STAR view in the DR8 context. We estimated PWV at the time of each observation based on the WS data and rejected epochs where no WS data was available. We also rejected outliers in (r-i) vs.(r-z) color space by excluding objects with (r-z) colors falling more than 0.2 mag from a linear fit to the stellar locus and calculated ∆(r − z), the color of each object relative to the stellar locus for a given (r-i). We also calculated the mean, with outlier rejection, of ∆(r − z) in each of the six camcols and found significant offsets which we subtracted off. The relation between ∆(r − z) and PWV is shown in Figure 6. The best fit linear relation between change in (r-z) color and PWV is ∆(r − z) = (−0.0019 ± 0.0002) × (P W V − 3.3 mm) mag. The second sample contains objects in Stripe 82, a 300 square degree area along DEC=0 that was repeatedly observed by SDSS (Ivezić et al. 2007;Bramich et al. 2008;Sako et al. 2008). Based on the stellar locus defined in Covey et al. (2007) we selected a random 3,000 G5-F5 stars and 2,500 M7-M4 stars from the Stripe 82 standard star catalog of Ivezić et al. (2007), restricting our samples to objects brighter than z=17.0. While parts of Stripe 82 have been observed up to 80 times, only a portion of those visits are included in DR8 (Aihara et al. 2011). While all the visits are included in DR7, these data were not globally calibrated usingÜbercal. We selected photometric observations of our Stripe 82 targets from the DR8 PHOTOOBJ view requiring all the photometric flags required for CLEAN status except for PRIMARY. We culled observations at epochs where no WS PWV data was available, as well as observations with colors offset more than 0.1 mag from the (g-i) stellar locus defined in Covey et al. (2007). In cases where observations of the same object spanned multiple camcols, we chose a single camcol for that object and rejected observations from other camcols. We also excluded objects with fewer than five remaining observations at PWV< 4 mm. The final samples contain 14,597 observations of 983 G5-F5 stars and 25,005 observations of 1,553 M7-M4 stars. For each object in each sample we calculate z 4 , the average of the z-band magnitudes only for PWV< 4 mm, and then ∆z = z − z 4 . The relation between ∆z and PWV for both samples is shown in Figure 7. As expected, the effect of differential extinction due to H 2 O is small for G and F stars, which have SEDs similar to the SED of the average point source used in the photometric calibration, but the effect is large for M stars, with ∆z H 2 O = (0.0012 ± 0.00008) × (P W V − 2.6 mm) mag. The slope of this relation is consistent with that of the ∆(r − z)-PWV relation, indicating that our results obtained using two different sets of observations are very similar. Considering that the distribution of PWV for the Stripe 82 observations is roughly uniform from 1 mm to 10 mm, this is potentially an important component of the global calibration of SDSS M star observations. However, taking the standard deviation of the empirical corrections for all the SDSS measurements indicates that this is a small effect overall: ∆z H 2 O = 0.003 mag. For the M stars we recalculated ∆z after applying this empirical PWV correction. Using the F-statistic (Press et al. 1992) we compared the variances of ∆z before and after the PWV correction and found no statistically significant reduction in the variance of ∆z (P (σ 2 1 = σ 2 2 ) = 0.44). Using the Stripe 82 sample we can also investigate changes in overall z-band transmission with PWV by considering uncalibrated flux measurements. Using the PSFFLUX and NMGYPERCOUNT entries in the PHTOTOOBJ view we can convert the calibrated fluxes retrieved from the DR8 database back into uncalibrated, raw fluxes. Using the same procedure as for the calibrated Stripe 82 data, for each object in the G5-F5 sample we calculate ∆z raw = z raw / z raw 4 and ∆r raw = r raw / r raw 4 . As a result of cloud cover and variations in the optical depth of molecular scattering, there are variations at the level of ∼ 10% in the uncalibrated r and z fluxes, but ∆r raw and ∆z raw are highly correlated. We fit a linear relation, F (∆r raw ), for ∆z raw as a function of ∆r raw and then calculate ∆ ′ z raw = ∆z raw − F (∆z raw ). Since the SDSS r band is free from strong H 2 O line absorption and probes instead the transmission variations due to scattering, we use ∆ ′ z raw as a proxy for the molecular component of absorption in z band. The change in raw z flux as a function of PWV is shown in Figure 8. The best fit linear relation between molecular absorption and PWV is ∆ ′ z raw (%) = (−0.0096 ± 0.0001) × (P W V − 4.3 mm). We compared these empirical relations between z raw flux, z magnitude, and the (rz) colors of M stars to theoretical estimates of the impact of changing PWV on SDSS z-band transmission. We numerically integrated the synthetic stellar spectra described by Brott & Hauschildt (2005) across the SDSS passbands 8 including our H 2 O absorption template for APO scaled following our derived τ (P W V ) relation given in Equation 8 and assuming AM=1.2. We focused on stellar templates representative of a mid-M dwarf (T eff =3000 K, log(g)=5.0) and a mid-G dwarf (T eff =6000 K, log(g)=4.5). For values of PWV in the range 0.5 to 12 we calculated the expected z-band and r-band fluxes for each template as a function of PWV, normalizing to the expected flux at PWV=3 mm. While the expected z-band fluxes change by more than 8% when PWV increases from 1 mm to 10 mm, to first approximation these variations are removed by the SDSS photometric calibration. If the appropriately weighted average of the SEDs of all SDSS stars is well-approximated by a G-star spectrum, then the differential fluxes we calculate in r and z bands should match the variations we measure in SDSS z-band photometry as a function of PWV. These theoretical expectations are shown as dashed lines in Figures 6 and 7 and are in very good agreement with our measurements. We experimented with including different Aerosol and Rayleigh scattering models in these calculations, but as functions that vary smoothly with wavelength they had a negligible effect on the predicted differential M dwarf colors, fluxes, and differential magnitudes. Discussion Empirical corrections for extinction due to H 2 O absorption may be important for a wide range of spectroscopic and photometric astronomical observations at deep red and infrared wavelengths. In many cases, contemporaneous PWV measurements derived from GPS may provide an important external parameter that can be used to correct for these effects. Precise differential measurements of stellar brightness can be used to find rocky planets transiting low-mass stars (Blake et al. 2008;Nutzman & Charbonneau 2008;Charbonneau et al. 2009) as well as measure the properties of the atmospheres of these planets in transmission (Bean et al. 2011;Crossfield et al. 2011;Croll et al. 2011). For these observations, precision at the level of 1 part in 10 3 or better is the objective. At this level, second-order extinction effects stemming from variations in molecular absorption may be a dominant source of noise, particularly for wide NIR photometric bands. Given a mid-M target and an ensemble of G-type reference stars, the theoretical (dashed) curve in the bottom panel of Figure 7 indicates the dependence of this effect on PWV. We simulated the impact of PWV variations on a long-term M dwarf survey by calculating the differential flux for a mid-M star and a mid-G comparison star given the overall distribution of PWV at APO corresponding to the times of 6,200 SDSS M star observations. The results of this calculation, shown in Figure 9 indicate that for the SDSS z band and a wide i+z band (Nutzman & Charbonneau 2008), the expected size of this effect is 0.003 to 0.004 mag, comparable to the signal from a super-Earth transiting a mid-M star. Large changes in PWV can occur on rapid timescales, comparable to the duration of a transit event, further complicating matters. A multiple linear regression of the differential flux of an M star target relative to the ensemble reference against PWV and AM may reveal significant correlation. The coefficients of this correlation are uniquely determined by the SEDs of the target and the reference stars and could be used as an empirical relation to improve photometric measurements by correcting for the impact of variations in H 2 O absorption (see Burke 2011). Future wide-area photometric surveys like the Large Synoptic Survey Telescope (LSST; Ivezić et al. 2008) will use sophisticated strategies for calibrating the absolute transmission of Earth's atmosphere in real time. Burke et al. (2010) demonstrate that spectrophotometric observations of a grid of standard stars can be used to calibrate band-integrated atmospheric transmission to much better than 1% by modeling both scattering and absorption. While absorption by H 2 O is just one component of the complex and variable atmospheric transmission function, our results indicate that GPS-based PWV measurements may augment spectrophotometric observations of standard stars, narrow-band photometric observations at key wavelengths (Alekseeva et al. 2010a), or an overall internal calibration procedure (Jones et al. 2010) and help improve photometric precision for objects with strong spectral features in the NIR. Another approach to mitigating the effect of variable H 2 O absorption is to design filters that avoid prominent absorption bands, such as those between 900 and 1000 nm, though at the cost of the total number of stellar photons detected. Some possible designs for the LSST y band filter incorporate a blue edge that avoids H 2 O absorption (High et al. 2010). We simulated the impact of variable PWV on differential measurements of a mid-M star and a mid-G comparison star using the y 3 filter 9 shown in Figure 1. The results of this simulation, shown in Figure 9, indicate that careful selection of NIR filters can help suppress the impact of variable PWV on M star photometry to below 1 mmag. Careful modeling of telluric lines is critically important for deriving precise stellar radial velocities from high-resolution NIR spectra. Wang & Ge (2011) showed that to approach the theoretical photon limit in Doppler measurements of low-mass stars, telluric lines need to be modeled with residual depths < 1%. While several methods exist for empirical estimation of telluric models using observations of A stars (e.g. Vacca et al. 2003), our results indicate that theoretical telluric templates combined with GPS-based PWV data can also be used for this purpose. Telluric templates that match observed line depths to better than 1% may be an important component of the analysis of data from a new generation of high-resolution NIR Doppler instruments (e.g. Mahadevan et al. 2010). Conclusion We calculate theoretical telluric transmission templates in the deep red appropriate for Apache Point Observatory (APO) and demonstrate that these models, which contain almost exclusively absorption by H 2 O, are excellent fits to high-resolution observations of A stars. These results are qualitatively similar to those from Thomas-Osip et al. (2007), Querel et al. (2008 and Seifahrt et al. (2010) who fit theoretical models to observations of a wide range of molecular absorption features in the optical and IR. We compare the scaling parameter for the depths of the H 2 O absorption features around 980 nm, τ , to contemporaneous estimates of Precipitable Water Vapor (PWV) derived from a Global Positioning System (GPS) receiver located at APO and find that they are strongly correlated. A multiple linear regression of PWV and airmass (AM) against τ indicates that the relative optical depth of H 2 O can be predicted to better than 10%, leading to band-integrated estimates of total water absorption that are better than 0.2%. For regions where absorption features have modest optical depth and are well-separated, the agreement between the models and telluric absorption features in the A star spectra is better than 2% even in the cores of lines. For regions with dense forests of lines, leading to saturation, residuals exist at the level of 5 − 10% within the cores of some lines. The real-time characterization of atmospheric molecular absorption is important for the calibration of broadband NIR photometry. Even using sophisticated differential global calibration techniques, changes in PWV can bias measurements of the fluxes of objects having strong spectral features in the NIR given that the average SED of the reference ensemble of stellar objects is likely to be much smoother. We show that globally calibrated SDSS measurements of the z-band fluxes and r-z colors of mid-M stars are correlated with PWV. This is not the case for G and F stars, since their NIR SEDs are similar to that of the "average" star included in theÜbercal photometric calibration. We compare the observed relation between PWV and the M dwarf photometric measurements to theoretical predictions based on transmission of the SDSS filters, synthetic stellar templates, and our telluric models and find that they are in excellent agreement. We measure the size of the impact of changing PWV on SDSS z-band measurements of mid-M stars to be ∆z H 2 O = (0.0012 ± 0.00008) × (P W V − 2.6 mm) mag. We applied this empirical correction to repeat observations from Stripe 82, but found no statistically significant reduction in the variance of 25,000 relative z-band measurements of mid-M stars. Given the range of PWV during the Stripe 82 observations, the expected size of the correction is ∆z H 2 O = 0.003 mag, significantly smaller than the photon errors for the majority of stars in our sample. GPS-based estimates of H 2 O absorption may be an important component of the calibration strategy for future surveys like LSST and, when combined with atmospheric monitoring (Burke et al. 2010) or internal calibration (Jones et al. 2010) and a strategy for precisely measuring the spectral response of the instrument (Stubbs & Tonry 2006, may help to improve differential photometric measurements for objects with complex NIR SEDs. In the near term, modeling H 2 O absorption through PWV measurements may help to improve the precision of differential NIR photometry of bright M stars, enabling transiting planet surveys to improve their sensitivity to small planets. At the same time, precise models of telluric absorption may enable Doppler measurements in new wavelength regimes, enhancing ongoing efforts to find Earth-like planets orbiting low-mass stars. We thank an anonymous referee for thoughtful comments that helped to focus and improve this work. We would also like to thank Robert Lupton, Chris Stubbs, and Ji Wang for helpful conversations that contributed to this work. We would like to thank the members of the Natural Environments Branch at Marshall Space Flight Center for developing and supporting the Earth-GRAM model and for making it available to us. CHB thanks the NSF for support through an Astronomy and Astrophysics Postdoctoral Fellowship (AST-0901918). CHB would also like to thank George Rybicki for teaching an excellent class about radiative processes. This work is based in part on observations obtained with the Apache Here we see evidence for significant residuals at the cores of some lines, as well as slowly varying residuals that we attribute in part to incomplete correction for fringing in the CCD detector. Here, ∆(r − z) is the difference in the measured color of a star relative to the stellar locus in (r-i) vs. (r-z) space.The dashed line is the result of a theoretical prediction that takes into account telluric H 2 O absorption scaled following Equation 8, the SDSS filter passbands, and synthetic stellar spectra from Brott & Hauschildt (2005). The theoretical (r-z) color difference is calculated between a mid-M star and a mid-G star reference, which should be representative of the average SED of all the bright stars in SDSS. Error bars are the formal statistical error on the mean of each bin and the null hypothesis of ∆(r − z) = 0 is ruled out at > 99% confidence. Bottom panel shows results of a consistency check were an empirical ∆(r − z) vs. PWV correction is calculated from a random half of the mid-M stars and then this correction is applied to the other half of the observations. After correction, the data are consistent with ∆(r − z) = 0 (P (χ 2 ) = 0.73) As discussed in Section 6, ∆ ′ z raw is an estimate of the change in raw z-band flux after correction for extinction as measured simultaneously in r band. The dashed line is based on the same theoretical calculations shown in Figure 6. The changes in total absorption in z-band with PWV are significant, up to several percent, but these are largely removed via relative calibration techniques, with residuals that depend on SED. Statistical errors on the means are smaller than the points. Fig. 1 . 1-Components of atmospheric absorption for a site like Apache Point Observatory (see alsoBurke et al. 2010). The transmission curves for NIR bands are also shown, each normalized to a peak transmission of 50%. The y band filter shown here is based on the proposed LSST y 3 filter(High et al. 2010). Fig. 2 . 2-Two examples of theoretical telluric H 2 O templates (solid lines) fit to ARCES A star observations (small points). These spectra highlight the strong impact changes in the total H 2 O column have on the telluric absorption. In both cases, the residuals of the fits are excellent, having scatter < 2%. Fig. 3 . 3-Two examples of theoretical telluric H 2 O templates (solid lines) fit to ARCES A star observations (small points) in a spectral region with strong H 2 O absorption, including saturated lines. Fig. 4 .Fig. 5 . 45-Results of a multiple linear regression of line depth scale factor, τ , against Precipitable Water Vapor and Airmass. -Histogram of PWV for ARCES A star observations (shaded region) compared to the overall distribution of PWV measurements at Apache Point Observatory. Astronomical observations are typically made from APO when PWV< 10 mm. Fig. 6 . 6-Top panel shows mean colors of 6,177 mid-M stars binned as a function of PWV. Fig. 7 .Fig. 8 . 78-Mean differential z-band magnitude, ∆z, binned as a function of PWV for objects in Stripe 82 that were repeatedly observed by SDSS. The dashed lines are based on the same theoretical calculations shown in Figure 6, again assuming a mid-G star as the reference SED. Formal statistical errors on the means are smaller than the points. 14,597 observations of 983 stars are included in the top panel and 25,005 observations of 1,553 stars in the bottom panel. -Mean values of 14,597 uncalibrated z-band flux measurements for the Stripe 82 G5-F5 star sample binned as a function of PWV. Fig. 9 . 9-Histograms of the expected variation in differential magnitude for a mid-M target and mid-G comparison star in different NIR bands given the PWV values at the times of all of the SDSS observations in our M star sample. The standard deviations of these distributions are: σ ∆z = 0.003, σ ∆i+z = 0.004, and σ ∆y = 0.0003 mag. 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[]
[ "0 A survey on measuring indirect discrimination in machine learning", "0 A survey on measuring indirect discrimination in machine learning", "0 A survey on measuring indirect discrimination in machine learning", "0 A survey on measuring indirect discrimination in machine learning" ]
[ "Indrėžliobaitė \nAalto University and Helsinki Institute for Information Technology HIIT\n\n", "Indrėžliobaitė \nAalto University and Helsinki Institute for Information Technology HIIT\n\n" ]
[ "Aalto University and Helsinki Institute for Information Technology HIIT\n", "Aalto University and Helsinki Institute for Information Technology HIIT\n" ]
[ "ACM Journal Name", "ACM Journal Name" ]
Nowadays, many decisions are made using predictive models built on historical data. Predictive models may systematically discriminate groups of people even if the computing process is fair and well-intentioned. Discrimination-aware data mining studies how to make predictive models free from discrimination, when historical data, on which they are built, may be biased, incomplete, or even contain past discriminatory decisions. Discrimination refers to disadvantageous treatment of a person based on belonging to a category rather than on individual merit. In this survey we review and organize various discrimination measures that have been used for measuring discrimination in data, as well as in evaluating performance of discriminationaware predictive models. We also discuss related measures from other disciplines, which have not been used for measuring discrimination, but potentially could be suitable for this purpose. We computationally analyze properties of selected measures. We also review and discuss measuring procedures, and present recommendations for practitioners. The primary target audience is data mining, machine learning, pattern recognition, statistical modeling researchers developing new methods for non-discriminatory predictive modeling. In addition, practitioners and policy makers would use the survey for diagnosing potential discrimination by predictive models.General Terms: fairness in machine learning, predictive modeling, non-discrimination, discriminationaware data mining
null
[ "https://arxiv.org/pdf/1511.00148v1.pdf" ]
8,372,620
1511.00148
44fc97a629103d67601fcdd518b289e9ea05467c
0 A survey on measuring indirect discrimination in machine learning 31 Oct 2015 Indrėžliobaitė Aalto University and Helsinki Institute for Information Technology HIIT 0 A survey on measuring indirect discrimination in machine learning ACM Journal Name 0031 Oct 20150, Publication date: October 2015.0:2 I.Žliobaitė Contents Nowadays, many decisions are made using predictive models built on historical data. Predictive models may systematically discriminate groups of people even if the computing process is fair and well-intentioned. Discrimination-aware data mining studies how to make predictive models free from discrimination, when historical data, on which they are built, may be biased, incomplete, or even contain past discriminatory decisions. Discrimination refers to disadvantageous treatment of a person based on belonging to a category rather than on individual merit. In this survey we review and organize various discrimination measures that have been used for measuring discrimination in data, as well as in evaluating performance of discriminationaware predictive models. We also discuss related measures from other disciplines, which have not been used for measuring discrimination, but potentially could be suitable for this purpose. We computationally analyze properties of selected measures. We also review and discuss measuring procedures, and present recommendations for practitioners. The primary target audience is data mining, machine learning, pattern recognition, statistical modeling researchers developing new methods for non-discriminatory predictive modeling. In addition, practitioners and policy makers would use the survey for diagnosing potential discrimination by predictive models.General Terms: fairness in machine learning, predictive modeling, non-discrimination, discriminationaware data mining INTRODUCTION Nowadays, many decisions are made using predictive models built on historical data, for instance, personalized pricing and recommendations, credit scoring, automated CV screening of job applicants, profiling of potential suspects by the police, and many more. Penetration of machine learning technologies, and decisions informed by big data has raised public awareness that automated decision making may lead to discrimination [House 2014;Miller 2015;Burn-Murdoch 2013]. Predictive models may discriminate people, even if the computing process is fair and well-intentioned [Barocas and Selbst 2016;Citron and III 2014;. This is because most machine learning methods are based upon assumptions that the historical data is correct, and represents the population well, which is often far from reality. Discrimination-aware machine learning and data mining is an emerging discipline, which studies how to prevent discrimination in predictive modeling. It is assumed that non-discrimination regulations, such as which characteristics, or which groups of people are considered as protected, are externally defined by national and international legislation. The goal is to mathematically formulate non-discrimination constraints, and develop machine learning algorithms that would be able to take into account those constraints, and still be as accurate as possible. In the last few years researchers have developed a number of discrimination-aware machine learning algorithms, using a variety of performance measures. Nevertheless, there is a lack of consensus how to define fairness of predictive models, and how to measure the performance in terms of discrimination. Quite often research papers propose a new way to quantify discrimination, and a new algorithm that would optimize that measure. The variety of approaches to evaluation makes it difficult to compare the results and assess the progress in the discipline, and even more importantly, it makes it difficult to recommend computational strategies for practitioners and policy makers. The goal of this survey is to present a unifying view towards discrimination measures in machine learning, and understand the implications of choosing to optimize one or another measure, because measuring is central in formulating optimization criteria for algorithmic discrimination discovery and prevention. Hence, it is important to have a structured survey at an early stage of development of this research field, in order to present task settings in a systematic way for follow up research, and to enable systematic comparison of approaches. Thus, we review and categorize measures that have been used in machine learning and data mining, and also discuss existing measures from other fields, such as feature selection, which in principle could be used for measuring discrimination. There are several related surveys that can be viewed as complementary to this survey. A recent review [Romei and Ruggieri 2014] presents a multi-disciplinary context for discrimination-aware data mining. This survey contains a brief overview of discrimination measures with does not go into analysis and comparison of the measures, since the focus is on approaches to solutions across different disciplines (law, economics, statistics, computer science). Another recent review [Barocas and Selbst 2016] discusses legal aspects of potential discrimination by machine learning, mainly focusing on American anti-discrimination law. A matured handbook on measuring racial discrimination [Blank et al. 2004] focuses on surveying and collecting evidence for discrimination discovery. The book is not considering discrimination by algorithms, only by human decision makers. The remainder of the article is organized as follows. Section 2 presents legal context, terminology, and provides an overview of research in developing nondiscriminatory predictive modeling approaches. Our intention is to keep this section brief. An interested reader is referred to focused surveys [Romei and Ruggieri 2014; Barocas and Selbst 2016] for more information. Section 4 reviews and organizes discrimination measures used in discrimination-aware machine learning and data mining, as well as potentially useful measures from other fields. Section 5 analyzes and compares a set of most popular measures, and discusses implications of using one or the other. Finally, Section 6 presents recommendations for researchers, and concludes the survey. BACKGROUND Discrimination and law Discrimination translates from latin as a distinguishing. While distinguishing is not wrong as such, discrimination has a negative connotation referring to adversary treatment of people based on belonging to some group rather than individual merits. Public attention to discrimination prevention has been increasing in the last few years. National and international anti-discrimination legislation are extending the scope of protection against discrimination, and expanding discrimination grounds. Adversary discrimination is undesired from the perspective of basic human rights, and in many areas of life non-discrimination is enforced by international and national legislation, to allow all individuals an equal prospect to access opportunities available in a society [for Fundamental Rights 2011]. Enforcing non-discrimination is not only for benefiting individuals. Considering individual merits rather than group characteristics is expected to benefit decision makers as well leading to more more informed, and thus likely more accurate decisions. Discrimination can be characterized by three main concepts: (1) what actions (2) in which situations (3) towards whom are considered discriminatory. Actions are forms of discrimination, situations are areas of discrimination, and grounds of discrimination describe characteristics of towards whom discrimination may occur. For example, the main grounds for discrimination defined in European Council directives [Commission 2011] (2000/43/EC, 2000/78/EC) are: race and ethnic origin, disability, age, religion or belief, sexual orientation, gender, nationality. Multiple discrimination occurs when a person is discriminated on a combination of several grounds. The main areas of discrimination are: access to employment, access to education, employment and working conditions, social protection, access to supply of goods and services. Discriminatory actions may take different forms, the two main of which are known as direct discrimination and indirect discrimination. A direct discrimination occurs when a person is treated less favorably than another is, has been or would be treated in a comparable situation on protected grounds. For example, property owners are not renting to a minority racial tenant. An indirect discrimination (also known as structural discrimination) occurs where an apparently neutral provision, criterion or practice would put persons of a protected ground at a particular disadvantage compared with other persons. For example, a requirement to produce an ID in a form of driver's license for entering a club may discriminate visually impaired people, who cannot have a driver's license. A related term statistical discrimination [Arrow 1973] is often used in economic modelling. It refers to inequality between demographic groups occurring even when economic agents are rational and non-prejudiced. Indirect discrimination applies to machine learning and data mining, since algorithms produce decision rules or decision models. While human decision makers may make biased decisions on case by case basis, rules produced by algorithms are applied consistently, and may discriminate more systematically and at a larger scale. Discrimination due to algorithms is sometimes referred to as digital discrimination (e.g. [Wihbey 2015]) . General population, and even many data scientists may think that algorithms are based on data, and, therefore, models produced by algorithms are always objective. However, models are as objective as the data on which they are applied, and as long as the assumptions behind the models perfectly match the reality. In practice, this is rarely the case. Historical data may be biased, incomplete, or record past discriminatory decisions that can easily be transferred to predictive models, and reinforced in new decision making . Lately, awareness of policy makers and public attention to potential discrimination has been increasing [House 2014;Miller 2015;Burn-Murdoch 2013], but there is a long way ahead before we can fully understand how such discrimination happens and how to prevent it. Discrimination-aware machine learning and data mining Non-discriminatory machine learning and data mining, a discipline at an intersection of computer science, law and social sciences, focuses on two main research directions: discrimination discovery, and discrimination prevention. Discrimination discovery aims at finding discriminatory patterns in data using data mining methods. Data mining approach for discrimination discovery typically mines association and classification rules from the data, and then assesses those rules in terms of potential discrimination [Ruggieri et al. 2010;Romei et al. 2012;Hajian and Domingo-Ferrer 2013;Pedreschi et al. 2012;Luong et al. 2011;Mancuhan and Clifton 2014]. A more traditional statistical approach to discrimination discovery typically fits a regression model to the data including the protected features (such as race, gender), and then analyzes the magnitude and statistical significance of the regression coefficients at the protected attributes (e.g. [Edelman and Luca 2014]). If those coefficients appear to be significant, then discrimination is flagged. Discrimination prevention develops machine learning algorithms that would produce predictive models, ensuring that those models are free from discrimination, while, standard predictive models, induced by machine learning and data mining algorithms, may discriminate groups of people due to training data being biased, incomplete, or recording past discriminatory decisions. The goal is to have a model (decision rules) that would obey non-discrimination constraints, typically the constraints directly relate to the selected discrimination measure. Solutions for discrimination prevention in predictive models fall into three categories: data preprocessing, model postprocessing, and model regularization. Data preprocessing modifies the historical data such that the data no longer contains discrimination, and then uses regular machine learning algorithms for model induction. Data preprocessing may modify the target variable [Kamiran and Calders 2009;Mancuhan and Clifton 2014;Kamiran et al. 2013a], or modify input data [Feldman et al. 2015;Zemel et al. 2013]. Model postprocessing produces a regular model and then modifies it (e.g. by changing the labels of some leaves in a decision tree) [Kamiran et al. 2010;Calders and Verwer 2010]. Model regularisation adds optimization constraints in the model learning phase (e.g. by modifying the splitting criteria in decision tree learning) [Kamiran et al. 2010;Kamishima et al. 2012]. An interested reader is invited to consult an edited book [Custers et al. 2013], a special issue in a journal [Mascetti et al. 2014], and proceedings of three workshops in discrimination-aware data mining and machine learning [Calders and Zliobaite 2012;Barocas and Hardt 2014;Barocas et al. 2015] for more details. Defining coherent discrimination measures is central for both lines of research: discrimination discovery and discrimination prevention. Discrimination discovery needs a measure in order to judge whether there is discrimination in data. Discrimination prevention needs a measure as an optimization criteria in order to sanitize predictive models. Hence, our main focus in this survey is to review discrimination measures, and analyze their properties, and understand implications of using one or another measure. MACHINE LEARNING SETTINGS, DEFINITIONS AND SCENARIOS Definition of fairness for machine learning In the context of machine learning non-discrimination can be defined as follows: (1) people that are similar in terms non-protected characteristics should receive similar predictions, and (2) differences in predictions across groups of people can only be as large as justified by non-protected characteristics. The first condition relates to direct discrimination, and can be illustrated by so called twin test: if gender is the protected attribute and we have two identical twins that share all characteristics, but gender, they should receive identical predictions. The first part is necessary but not sufficient condition to make sure that there is no discrimination in decision making. The second condition ensures that there is no indirect discrimination, also referred to as redlining. For example, banks used to deny loans for residents of selected neighborhoods. Even though race was not formally used as a decision criterion, it appeared that the excluded neighborhoods had much higher population of non-white people than average. Even though people from the same neighborhood ("twins") are treated the same way no matter what the race is, artificial lowering of positive decision rates in the non-white-dominated neighborhoods would harm the non-white population more than white. Therefore, different decision rates across neighborhoods can only be as large as justified by non-protected characteristics, and this is what the second part of the definition controls. More formally, let X be a set of variables describing non-protected characteristics of a person, S be a set of variables describing the protected characteristics, andŷ be the model output. A predictive model can be considered fair if: (1) the expected value for model output does not depend on the protected characteristics E(ŷ|X, S) = E(ŷ|X) for all X and S, that is, there is no direct discrimination; and (2) if non-protected characteristics and protected characteristics are not independent, then the expected value for model output dependence on those non-protected characteristics should be justified, that is if E(X|S) = E(X), then E(ŷ|X) = e ⋆ (ŷ|X), where e ⋆ is a constraint. Finding and justifying e ⋆ is non-trivial and very challenging, and that is where a lot of ongoing effort in discrimination-aware machine learning concentrate. Machine learning task settings Machine learning settings for decision support, where discrimination may potentially occur, can take many different forms. The variable that is to be predicted -target variable -may be binary, ordinal, or numeric, corresponding to binary classification, multiclass classification or regression tasks. As an example of a binary classification task in the banking domain could be deciding whether to accept or decline loan application of a person. Multiclass classification task could be to determine to which customer benefit program a person should be assigned (e.g. "golden clients", "silver clients", "bronze clients"). Regression task could be to determine the interest rate for a particular loan for a particular person. Discrimination can occur only when target variable is polar. That is, each task setting some outcomes should be considered superior to others. For example, getting a loan is better than not getting a loan, or the "golden client" package is better than the "silver", and "silver" is better than "bronze", or assigned interest rate 3% is better than 5%. If the target variable is not polar, there is no discrimination, because no treatment is superior or inferior to other treatment. The protected characteristic, in machine learning settings referred to as the protected variable or sensitive attribute, may as well be binary, categorical or numeric, and it does not need to be polar. For example, gender can be encoded with a binary protected variable, ethnicity can be encoded with a categorical variable, and age can be encoded with a numerical variable. In principle, any combination one or more personal characteristics may be required to be protected. Discrimination on more than one ground is known as multiple discrimination, and it may be required to ensure prevention of multiple discrimination in predictive models. Thus, ideally, machine learning methods and discrimination measures should be able to handle any type or a combination of protected variables. For instance, the authorities may want to enforce nondiscrimination with respect to ethnicity in determining interest rate, or non discrimination with respect to gender and age in deciding whether to accept loan applications. In discrimination prevention it is assumed that the protected ground is externally given, for example, by law. Principles for making machine learning non-discriminatory A typical machine learning process is illustrated in Figure 1. A machine learning algorithm is a procedure used for producing a predictive model from historical data. A model is the resulting decision rule (or a collection of rules). The resulting model is used for decision making for new incoming data. The model would take personal characteristics as inputs (for example, income, credit history, employment status), and output a prediction (for example, credit risk level). Algorithms themselves do not discriminate, because they are not used for decision making. Models (decision rules) that are used for decision making may potentially discriminate people with respect to certain characteristics. Algorithms, on the other hand, may be discrimination-aware by employing specific procedures during model construction to enforce non-discriminatory constraints into the models. Hence, one of the main goals of discrimination-aware machine learning and data mining is to develop discrimination-aware algorithms, that would guarantee that non-discriminatory models are produced. There is an ongoing debate in the discrimination-aware data mining and machine learning community whether models should or should not use protected characteristics as inputs. For example, a credit risk assessment model may use gender as input, or may leave the gender variable out. Our position on the matter is as follows. Using the protected characteristic as model input may help to ensure that there is no indirect discrimination (for example, as demonstrated in the experimental section of []). However, if a model uses the protected characteristic as input, the model is not treating two persons that share identical characteristics except for the protected characteristic the same way, a direct discrimination would be propagated. Therefore, such a model would be discriminatory discriminatory due to violation of condition #1 in the definition in Section ??. Hence, the model should not use the protected characteristic for decision making. However, we see no problem in using the protected characteristic in the model learning process, which often may help to enforce non-discrimination constraints. Thus, machine learning algorithms can use the protected characteristic in the learning phase, as long as the resulting predictive model does not require the protected characteristic when used for decision making. Ensuring that there is no indirect discrimination is much more tricky. In order to verify to what extent non-discriminatory constraints are obeyed, and enforce fair allocation of predictions across groups of people, machine learning algorithms must have access to the protected characteristics in the historical data. We argue that if protected information (e.g. gender or race) is not available during the model learning building process, the learning algorithm cannot be discrimination-aware, because it cannot actively control non-discrimination. The resulting models produces without access to sensitive information may be discriminatory, may be not, but that is by chance rather than discrimination-awareness property of the algorithm. Non-discrimination can potentially be measured on data (historical data), on predictions made by models, or on models themselves. Different task settings and application goals may require different measurement techniques. In order to select appropriate measures, which also typically serve as optimisation constraints in the nondiscriminatory model learning process, it is important to understand underlying assumptions and basic principles behind different discrimination measures. The next section presents a categorized survey of measures used in the discrimination-aware data mining and machine learning literature, and discusses other existing measures that could in principle be used for measuring fairness of algorithms. The goal is to present arguments for selecting relevant measures for different learning settings. DISCRIMINATION MEASURES Discrimination measures can be categorized into (1) statistical tests, (2) absolute measures, (3) conditional measures, and (4) structural measures. We survey measures in this order due to historical reasons, which is more or less how they came into use. First statistics tests were used which would answer yes or no, then absolute measures came into play that allow quantifying the extent of discrimination, then conditional measures appeared that take into account possible legitimate explanations of differences between different groups of people. Statistical tests, absolute measures and conditional measures are designed for indicating indirect discrimination. Structural measures have been introduced mainly in accord to mining classification rules, aiming at discovering direct discrimination, but in principle they can also address indirect discrimination. All these types are not intended as alternatives, but rather reflect different aspects of the problem, as summarized in Table I. Statistical tests indicate presence or absence of discrimination at a dataset level, they do not measure the magnitude of discrimination, neither the spread of discrimination within the dataset. Absolute measures capture the magnitude of discrimination over a dataset taking into account the protected characteristic, and the prediction deci- protected variable s i a value of a discreet/binary protected variable, s ∈ {s 1 , . . . , s m } typically index 1 denotes a protected group, e.g. s 1 -black, s 0 -white race X a set of input variables (predictors), X = {x (1) , . . . , x (l) } z explanatory variable or stratum z i a value of explanatory variable z ∈ {z 1 , . . . , z k } N number of individuals in the dataset n i number of individuals in group s i sion; no other characteristics of individuals are considered. It is assumed that all individuals are alike, and there should be no differences in decisions for the protected and the general group of people, disregarding any possible explanation. Absolute measures generally are not for using stand alone on a dataset, but rather provide core principles for conditional measures, or statistical tests. Conditional measures capture the magnitude of discrimination, which cannot be explained by any non-protected characteristics of individuals. Statistical tests, absolute and conditional measures are designed to capture indirect discrimination at a dataset level. Structural measures do not measure the magnitude of discrimination, but the spread of discrimination, that is, a share of people in the dataset that are affected by direct discrimination. Our survey of measures will use mathematical notation as summarized in Table II. For simplicity we will use the following short probability notation: p(s = 1) will be encoded as p(s 1 ), and p(y = +) will be encoded as p(y + ). Let s 1 denote the protected community, and y + denote the desired decision (e.g. positive decision to grant a loan). Upper indices will denote values, lower indices will denote counters of variables. Statistical tests Statistical tests are the earliest measures for indirect discrimination discovery in data. Statistical tests are formal procedures to accept or reject statistical hypotheses, which check how likely the result is to have occurred by chance. In discrimination analysis typically the null hypothesis, or the default position, is that there is no difference between the treatment of the general group and the protected group. The test checks, how likely the observed difference between groups has occurred by chance. If chance is unlikely then the null hypothesis is rejected and discrimination is declared. Two limitations of statistical tests need to be kept in mind when using them for measuring discrimination. (1) Statistical significance does not mean practical significance; statistical tests do not show the magnitude of the the differences between the groups, which can be huge, or can be minor. (2) If the null hypothesis is rejected then discrimination is present, but if null hypothesis cannot be rejected, this does not prove that there is no discrimination. It maybe that the data sample is too small to declare discrimination. Standard statistical tests are typically applied for measuring discrimination. The same tests are used in clinical trials, marketing, and scientific research. Regression slope test. The test fits an ordinary least squares (OLS) regression to the data including the protected variable, and tests whether the regression coefficient of the protected variable is significantly different from zero. A basic version for discrimination discovery considers only the protected characteristic s and the target variable y [Yinger 1986]. In principle s and y can be binary or numeric, but typically in discrimination testing s is binary. The regression may include only the protected variable s as a predictor, but it may also include variables from X that may explain some of the observed difference in decisions. The test statistic is t = b/σ, where b is the estimated regression coefficient of s, and σ is the standard error, computed as σ = √ n i=1 (yi−f (yi)) 2 √ (n−2) √ n i=1 (si−s) 2 , where n is the number of observations, f (.) is the regression model,. indicates the mean. The t-test with n − 2 degrees of freedom is applied. Difference of means test. The null hypothesis is that the means of the two groups are equal. The test statistic is t = E(y|s 0 )−E(y|s 1 ) σ √ 1/n0+1/n1 , where n 0 is the number of individuals in the regular group, n 1 is the number of individuals in the protected group, σ = ((n 0 − 1)δ 2 0 + (n 1 − 1)δ 2 1 )/(n 0 + n 1 − 2), where δ 2 0 and δ 2 1 are the sample target variances in the respective groups. The t-test with n 0 − n 1 − 2 degrees of freedom is applied. The test assumes independent samples, normality and equal variances. Difference in proportions for two groups. The null hypothesis is that the rates of positive outcomes within the two groups are equal. The test statistic is z = p(y + |s 0 )−p(y + |s 0 ) σ , where σ = p(y + |s 0 )p(y − |s 0 ) n0 + p(y + |s 1 )p(y − |s 1 ) n1 . The z-test is used. Difference in proportions for many groups. The null hypothesis is that the probabilities or proportions are equal for all the groups. This can be used for testing many groups at once. For example, equality of decisions for different ethnic groups, or age groups. If the null hypothesis is rejected that means at least one of the groups has statistically significantly different proportion. The text statistic is χ 2 = k i=1 (ni−np(y + |s i )) 2 p(y + |s i ) , where k is the number of groups. The Chi-Square test is used with k − 1 degrees of freedom. Other tests and related fields. Relation to clinical trials where protected attribute is the treatment, and outcome is recovery. Prove that there is an effect (there is a discrimination). Does not prove that there is no discrimination. Neither say anything about the magnitude. For example, reduce the flue recovery by 10 min. (practically irrelevant). It may be still relevant for discrimination. Also marketing (measuring the effects of intervention). Rank test MannWhitney U test is applied for comparing two groups when the normality and equal variances assumptions are not satisfied. The null hypothesis is that the distributions of the two populations are identical. The procedure is to rank all the observations from the largest y to the smallest. The test statistic is the sum of ranks of the protected group. Absolute measures Absolute measures are designed to capture the magnitude of the differences between (typically two) groups of people. The groups are determined by the protected characteristic (e.g. one group is males, another group is females). If more than one protected group is analyzed (e.g. different nationalities), typically each group is compared separately to the most favored group. Mean difference. Mean difference measures the difference between the means of the targets of the protected group and the general group, d = E(y + |s 0 ) − E(y + |s 1 ). If there is not difference then it is considered that there is no discrimination. The measure relates to the difference of means, and difference in proportions test statistics, except that there is no correction for the standard deviation. The mean difference for binary classification with binary protected feature, d = p(y + |s 0 ) − p(y + |s 1 ), is also known as the discrimination score [Calders and Verwer 2010], or sliftd [Pedreschi et al. 2009]. Mean difference has been the most popular measure in early work on non-discriminatory machine learning and data mining [Pedreschi et al. 2009;Calders and Verwer 2010;Kamiran and Calders 2009;Kamiran et al. 2010;Zemel et al. 2013]. Normalized difference. Normalized difference [Zliobaite 2015 ] is the mean difference for binary classification normalized by the rate of positive outcomes, δ = p(y + |s 0 )−p(y + |s 1 ) dmax , where d max = min p(y + ) p(s 0 ) , p(y − ) p(s 1 ) . This measure takes into account maximum possible discrimination at a given positive outcome rate, such that with maximum possible discrimination at this rate δ = 1, while δ = 0 indicates no discrimination. Area under curve (AUC). This measure relates to rank tests. It has been used in for measuring discrimination between two groups when the target variable is numeric (regression task), AU C = (s i ,y i )∈D 0 (s j ,y j )∈D 1 I(yi>yj ) n0n1 , where I(true) = 1 and 0 otherwise. For large datasets computation becomes time and memory intensive, since a quadratic number of comparisons to the number of observations is required. The authors did not mention, but there is an alternative way to compute based on ranking, which, depending on the speed ranking algorithm, may be faster. Assign numeric ranks to all the observations, beginning with 1 for the smallest value. Let R 0 be the sum of the ranks for the favored group. Then AU C = R 0 − n0(n0+1) 2 . We observe that if the target variable is binary, and in case of equality half of a point is added to the sum, then AUC linearly relates to mean difference as AU C = p(y + |s 0 )p(y − |s 1 ) + 0.5p(y + |s 0 )p(y + |s 1 ) + 0.5p(y − |s 0 )p(y − |s 0 ) = 0.5d + 0.5, where d denotes discrimination measured by the mean difference measure. Impact ratio. Impact ratio, also known as slift [Pedreschi et al. 2009], is the ratio of positive outcomes for the protected group over the general group, r = p(y + |s 1 )/p(y + |s 0 ). This measure is used in the US courts for quantifying discrimination, the decisions are deemed to be discriminatory if the ratio of positive outcomes for the protected group is below 80% of that of the general group. Also this is the form stated in the Sex Discrimination Act of U.K. r = 1 indicates that there is no discrimination. [Pedreschi et al. 2008] is similar to impact ratio, but instead of dividing by the general group, the denominator is the overall rate of positive outcomes r = p(y + |s 0 )/p(y + ). The same measure, expressed as p p(y,s) p(y)p(s) < 1 + η for all values of y and s, is later referred to as η-neutrality [Fukuchi et al. 2013]. Elift ratio. Elift ratio 4.2.6. Odds ratio. Odds ratio of two proportions is often used in natural, social and biomedical sciences to measure the association between exposure and outcome. The popularity is due to convenient relation with the logistic regression. The exponential function of the logistic regression coefficient translates one unit increase in the odds ratio. Odds ratio has been used for measuring discrimination [Pedreschi et al. 2009] as r = p(y + |s 0 )p(y − |s 1 ) p(y + |s 1 )p(y − |s 0 ) . Mutual information. Mutual information (MI) is popular in information theory for measuring mutual dependence between variables. In discrimination literature this measure has been referred to as normalized prejudice index [Fukuchi et al. 2013], and used for measuring the magnitude of discrimination. Mutual information is measured in bits, but it can be normalized such that the result falls into the range between 0 and 1. , where I(s, y) = (s,y) p(s, y) log p(s,y) p(s)p(y) , and H(y) = − y p(y) log p(y). For numerical variables the summation is replaces by integral. Balanced residuals. While other measures work on datasets, balanced residuals is for machine learning model outputs. This measure characterizes the difference between the actual outcomes recorded in the dataset, and the model outputs. The requirement is that underpredictions and overpredictions should be balanced within the protected and regular groups. proposed balanced residuals as a criteria, not a measure. That is, the average residuals should be equal, but in principle the difference could be used as a measure of discrimination d = i∈D 1 yi−ŷi n1 − j∈D 0 yj−ŷj n0 , where y is the true target value,ŷ is the prediction. Positive values of d would indicate discrimination towards the protected group. One should; however, use and interpret this measure with caution. If the learning dataset is discriminatory, but the predictive model makes ideal predictions such that all the residuals are zero, this measure would show no discrimination, even though the predictions would be discriminatory, since the original data is discriminatory. Suppose, another predictive model makes a constant prediction for everybody, and the constant prediction is equal to the mean of the regular group. If the learning dataset contains discrimination, then the residuals for the regular group would be smaller than for the protected group, and the measure would indicate discrimination, however, a constant prediction to everybody means tat everybody is treated equally, and there should be no discrimination detected. Other possible measures. There are many established measures in feature selection literature [Guyon and Elisseeff 2003] for measuring the relation between two variables, which, in principle, can be used as absolute discrimination measures. The stronger the relation between the protected variable s and the target variable y, the larger the absolute discrimination. There are three main groups of measures for relation between variables: correlation based, information theoretic, and one-class classifiers. Correlation based measures, such as the Person correlation coefficient, are typically used for numeric variables. Information theoretic measures, such as mutual information mentioned earlier, are typically used for categorical variables. One-class classifiers present an interesting option. In discrimination the setting would be to predict the target y solely on the protected variable s, and measure the prediction accuracy. We are not aware of such attempts in the non-discriminatory machine learning literature, but it would be a valid option to explore. 4.2.10. Measuring for more than two groups. Most of the absolute discrimination measures are for two groups (protected group vs. regular group). Ideas, how to apply those for more than two groups, can be borrowed from multi-class classification [Bishop 2006], multi-label classification [Tsoumakas and Katakis 2007], and one-class classification [Tax 2001] literature. Basically, there are three options how to obtain sub-measures: measure pairwise for each pair of groups (k(k−1)/2 comparisons), measure one against the rest for each group (k comparisons), measure each group against the regular group (k − 1 comparisons). The remaining question is how to aggregate the sub-measures. Based on personal conversations with legal experts, we advocate for reporting the maximum from all the comparisons as the final discrimination score. Alternatively, all the scores could be summed weighing by the group sizes to obtain an overall discrimination score. Even though absolute measures do not take into account any explanations of possible differences of decisions across groups, they can be considered as core building blocks for developing conditional measures. Conditional measures do take into account explanations in differences, and measure only discrimination that cannot be explained by non-protected characteristics. Table III summarizes applicability of absolute measures in different machine learning settings. Conditional measures Absolute measures take into account only the target variable y and the protected variable s. Absolute measures consider all the differences in treatment between the protected group and the regular group to be discriminatory. Conditional measure, on the other hand, try to capture how much of the difference between the groups is explainable by other characteristics of individuals, recorded in X, and only the remaining differences are deemed to be discriminatory. For example, part of the difference in acceptance rates for natives and immigrants may be explained by the difference in education level. Only the remaining unexplained difference should be considered as discrimination. Let z = f (X) be an explanatory variable. For example, if z i denotes a certain education level. Then all the individuals with the same level of education will form a strata i. Within each strata the acceptance rates are required to be equal. Unexplained difference. Unexplained difference [Kamiran et al. 2013b] is measured, as the name suggests, as the overall mean difference minus the differences that can be explained by other legitimate variable. Recall that mean difference is d = p(y + |s 0 ) − p(y + |s 1 ). Then the unexplained difference d u = d − d e , where d e = m i=1 p ⋆ (y + |z i )(p(z i |s 0 ) − p(z i |s 1 )), where p ⋆ (y + |z i ) is the desired acceptance rate within the strata i. The authors recommend using p ⋆ (y + |z i ) = p(y + |s 0 ,z i )+p(y + |s 1 ,z i ) 2 . In the simplest case z bay be equal one of the variables in X. The authors also use clustering on X to take into account more than one explanatory variable at the same time. Then z denotes a cluster, one strata is one cluster. Propensity measure. Propensity models [Rosenbaum and Rubin 1983] are typically used in clinical trials or marketing for estimating the probability that an indi-vidual would receive a treatment. Given the estimated probabilities, individuals can be stratified according to similar probabilities of receiving a treatment, and the effects of treatment can be measured within each strata separately. Propensity models have been used for measuring discrimination , in this case a function was learned to model the protected characteristic based on input variables X, that is s 1 = f (X). A logistic regression was used for modeling f (.). Then the estimated propensity scoresŝ 1 were split into five ranges, where each range formed one strata. Discrimination was measured within each strata, treating each strata as a separate dataset, and using absolute discrimination measures discussed in the previous section. The authors did not aggregate the resulting discrimination into one measure, but in principle the results can be aggregated into one measure, for instance, using the unexplained difference formulas, reported above. In such a case each strata would correspond to one value of an explanatory variable z. Belift ratio. Belift ratio [Mancuhan and Clifton 2014] is similar to Elift ratio in absolute measures, but here the probabilities of positive outcome are also conditioned on input attributes, belif t = p(y + |s 1 ,X r ,X a ) p(y + |X a ) , where X = X r ∪ X r is a set of input variables, X r denotes so caller redlining attributes, the variables which are correlated with the protected variable s. The authors proposed estimating the probabilities via bayesian networks. A possible difficulty for applying this measure in practice may be that not everybody, especially non-machine learning users, are familiar enough with the Bayesian networks to an extent needed for estimating the probabilities. Moreover, construction of a Bayesian network may be different even for the same problem depending on assumptions made about interactions between the variables. Thus, different users may get different discrimination scores for the same application case. A simplified approximation of belift could be to treat all the attributes as redlining attributes, and instead of conditioning on all the input variables, condition on a summary of input variables z, where z = f (X). Then the measure for strata i would be p(y + |s 1 ,z i ) p(y + ) . The measure has a limitation that neither the original version, nor the simplified version allow differences to be explained by variables that are correlated with the protected variable. That is, if a university has two programmes, say medicine and computer science, and the protected group, e.g. females, are more likely to apply for a more competitive programme, then the programmes cannot have different acceptance rates. That is, if the acceptance rates are different, all the difference is considered to discriminatory. Structural measures Structural measures are targeted at quantifying direct discrimination. The main idea behind structural measures is for each individual in the dataset to identify whether s/he is discriminated, and then analyze how many individuals in the dataset are affected. Currently The idea is to compare each individual to the opposite group and see if the decision would be different. In that sense, the measure relates to propensity scoring (Section 4.3), used for identifying groups of people similar according to the non-protected characteristics, and requiring for decisions within those groups to be balanced. The main difference is that propensity measures would signal indirect discrimination within a group, and situation testing aims at signalling direct discrimination for each individual in question. 4.4.2. Consistency. Consistency measure [Zemel et al. 2013] compares the predictions for each individual with his/her nearest neighbors. C = 1 − 1 κN N i=1 yj∈D κ−nearest −neighbours |y i − y j |. Consistency measure is closely related to situation testing, but considers nearest neighbors from any group (not from the opposite group). Due to this choice, consistency measure should be used with caution in situations where there is a high correlation between the protected variable and the legitimate input variables. For example, suppose we have only one predictor variable -location of an apartment, and the target variable is to grant a loan or not. Suppose all non-white people live in one neighborhood (as in the redlining example), and all the white people in the other neighborhood. Unless the number of nearest neighbors to consider is very large, this measure will show no discrimination, since all the neighbors will get the same decision, even though all black residents will be rejected, and all white will be accepted (maximum discrimination). Perfect consistency, but maximum discrimination. In their experimental evaluation the authors have used this measure in combination with the mean difference measure. ANALYSIS OF CORE MEASURES Even though absolute measures are naive in a sense that they do not take any possible explanations of different treatment into account, and due to that may show more discrimination that there actually is, these measures provide core mechanisms and a basis for measuring indirect discrimination. Conditional measures are typically built upon absolute measures. In addition, statistical tests often directly relate to absolute measures. Thus, to provide a better understanding of properties and implications of choosing one measure over another, in this section we computationally analyze a set of absolute measures, and discuss their properties. We analyze the following measures, introduced in Section 4.2: mean difference, normalized difference, mutual information, impact ratio, elift and odds ratio. From the measures analyzed in this section, mean difference and area under curve can be directly used in regression tasks. We focus on the classification scenario, since this scenario has been studied more extensively in the discrimination-aware data mining and machine learning literature, and there are more measures available for classification than for regression; the regression setting, except for a recent work , remains a subject of future research, and therefore is out of the scope of a survey paper. Table IV summarizes boundary conditions of the selected measures. In the difference based measures 0 indicates no discrimination, in the ratio based measures 1 indicates no discrimination, in AUC 0.5 means no discrimination. The boundary conditions are reached when one group gets all the positive decisions, and the other group gets all the negative decisions. Next we experimentally analyze the performance of the selected measures. We leave out AUC from the experiments, since in classification it is equivalent to the mean difference measure. The goal of the experiments is to demonstrate how the performance depends on variations in the overall rate of positive decisions, balance between classes and balance between the regular and protected groups of people in data. For this analysis we use synthetically generated data which allows to represent different task settings and control the levels of underlying discrimination. Given four parameters: the proportion of individuals in the protected group p(s 1 ), the proportion of positive outputs p(y + ), the underlying discrimination d ∈ [−100%, 100%], and the number of data points n, data is generated as follows. First n data points are generated assigning a score in [0, 1] uniformly at random, and assigning group membership at random according to the probability p(s 1 ). This data contains no discrimination, because the scores are assigned at random. If would contain full discrimination if we ranked the observations according to the assigned scores and all the members of the regular group would appear before all the members of the protected group. Following this reasoning, half-discrimination would be if in a half of the data the members of the regular group appear before all the members of the protected group in the ranking, and the other half of the data would show a random mix of both groups in the ranking. For the experimental analysis purposes we define this as 50% discrimination. It is difficult to measure discrimination in data this way, but it is easy to generate such a data. For a given level of desired discrimination d we select dn observations at random, sort them according to their scores, and then permute group assignments within this subsample in such a way that the highest scores get assigned to the regular group, and the lowest scores get assigned to the protected group. Finally, since the experiment is about classification, we round the scores to zero-one in such a way that the proportion of ones is as desired by p(y + ). Then we apply different measures of discrimination to data generated this way, and investigate, how these measures can reconstruct the underlying discrimination. For each parameter setting we generate n = 10000 data points, and average the results over 100 such runs 1 Figure 2 depicts the performance of mean difference, normalized difference and mutual information. Ideally, the performance should be invariant to balance of the groups (p(s 1 0)) and the proportion of positive outputs (p(y + )), and thus run along the diagonal line in as many plots, as possible. We can see that the normalized difference captures that. The mean difference captures the trends, but the indicated discrimination highly depends on the balance of the classes and balance of the groups, therefore, this measure to be interpreted with care when data is highly imbalanced. The same holds for mutual information. For instance, at p(s 1 ) = 90% and p(y + ) = 90% the true discrimination in data may be near 100%, i.e. nearly the worst possible, but both measures would indicate that discrimination is nearly zero. The normalized difference would capture the situation as desired. In addition to that, we see that the mean difference and normalized difference are linear measures, while mutual information is non-linear, and would show less discrimination that actually in the medium ranges. Moreover, mutual mean difference normalized difference mutual information information dos not indicate the sign of discrimination, that is, the outcome does not indicate whether discrimination is reversed or not. For these reasons, we do not recommend using mutual information for the purpose of quantifying discrimination. Therefore, from the difference based measures we advocate normalized difference, which was designed to be robust to imbalances in data. The normalized difference is somewhat more complex to compute than the mean difference, which may be a limitation for practical applications outside research. Therefore, if data is closed to balanced in terms of groups and positive-negative outputs, then the mean difference can be used. Figure 3 presents similar analysis of the measures based on ratios: impact ratio, elift and odds ratio. We can see that the odds ratio, and the impact ratio are very sensitive to imbalances in groups and positive outputs. The elift is more stable in that respect, but still has some variations, particularly at high imbalance of positive outputs (p(y + ) = 90% or 10%), when discrimination may be highly exaggerated (far from the diagonal line). In addition, measured discrimination by all ratios grows very fast at low rates of positive outcome (e.g. see the plot p(y + ) = 10% and p(s 1 ) = 90%), while there is almost no discrimination in the data, measures indicate high discrimination. We also can see that all the ratios are asymmetric in terms of reverse discrimination. One unit of measured discrimination is not the same as one unit of reverse discrimination. This makes ratios a bit more difficult to interpret than differences, analyzed earlier, especially at large scale explorations and comparisons of, for instance, different computational methods for prevention. Due to these reasons, we do not recommend using ratio based discrimination measures, since they are much more difficult to interpret correctly, and may easily be misleading. Instead recommend using and building upon difference based measures, discussed in Figure 2. −1 0 1 −1 0 1 measured discrim. p(y + ) = 10% −1 0 1 −1 0 1 30% −1 0 1 −1 0 1 50% −1 0 1 −1 0 1 70% −1 0 1 −1 0 1 p(s 1 ) = 10% 90% −1 0 1 −1 0 1 measured discrim. −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 30% −1 0 1 −1 0 1 measured discrim. −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 50% −1 0 1 −1 0 1 measured discrim. −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 70% −1 0 1 −1 0 1 discrim. in data measured discrim. −1 0 1 −1 0 1 discrim. in data −1 0 1 −1 0 1 discrim. in data −1 0 1 −1 0 1 discrim. in data −1 0 1 −1 0 1 90% discrim. in data The core measures that we have analyzed form a basis for assessing fairness of predictive models, but it is not enough to use them directly, since they do not take into account possible legitimate explanations of differences between the groups, and instead consider any differences between the groups of people undesirable. The basic principle is to try to stratify the population in such a way that in each stratum contains people that are similar in terms of their legitimate characteristics, for instance, have similar qualifications if the task is candidate selection for job interviews. propensity score matching, reported in Section 4.3, is one possible way to stratification, but it is not the only one, and outcomes may vary depending on internal parameter choices. Thus, the principle to measuring is available, but there are still open challenges ahead to make the approach more robust to different users, and more uniform across different task setting, such that one could diagnose potential discrimination or declare fairness with more confidence. RECOMMENDATIONS FOR RESEARCHERS AND PRACTITIONERS As attention of researchers, media and general public to potential discrimination is growing, it is important to be able to measure fairness of predictive models in a systematic and accountable way. We have surveyed measures used (and potentially usable) for measuring indirect discrimination in machine learning, and experimentally analyzed the performance of the core measures in classification tasks. Based on our analysis we generally recommend using the normalized difference, and in case the classes and groups of people in the data are well balanced, it may be sufficient to use the simple (unnormalized) mean difference. We do not recommend using ratio based measures challenges associated with their interpretation in different situation. The core measures stand alone are not enough for measuring fairness correctly. These measures can only be applied to uniform populations considering that everybody within the population is equally qualified to get a positive decision. In reality this is rarely the case, for example, different salary levels may be explained by different education levels. Therefore, the main principle of applying the core measures should be by first segmenting the population into more or less uniform segments according to their qualifications, and then applying core measures within each segment. Some of such measuring techniques have been surveyed in Section 4.3 (Conditional measures), but generally there is no one easy way to approach it, and presenting sound arguments to justify the methods of allocating people into segments is very important in research and practice. We hope that this survey can establish a basis for further research developments in this important topic. So far most of the research has concentrated on binary classification with binary protected characteristic. While this is a base scenario, relatively easy to deal with in research, many technical challenges for future research lie in addressing more complex learning scenarios with different types and multiple protected characteristics, in multi-class, multi-target classification and regression settings, with different types of legitimate variables, noisy input data, potentially missing protected characteristics, and many more. Fig. 1 . 1A typical machine learning setting. 4.4. 1 . 1Situation testing. Situation testing [Luong et al. 2011] measures which fraction of individuals in the protected group are considered discriminated, as f = y i ∈D(y 0 |s 1 ) I(diff (yi)≥t) |D(y 0 |s 1 )| , where t is a user defined threshold, I is the indicator function that takes 1 if true, 0 otherwise. The situation testing for an individual i is computed as diff (y i ) = y j ∈D 0 κ−nearest −neighbours κ − y j ∈D 1 κ−nearest −neighbours κ . Positive and negative discrimination is handled separately. Fig. 2 . 2Analysis of the measures based on differences: discrimination in data vs. measured discrimination. Fig. 3 . 3Analysis of the measures based on ratios: discrimination in data vs. measured discrimination. Table I . IDiscrimination measure typesMeasures Indicate what? Type of discrimination Statistical tests presence/absence of discrimination indirect Absolute measures magnitude of discrimination indirect Conditional measures magnitude of discrimination indirect Structural measures spread of discrimination direct or indirect Table II . IISolutionsSymbolExplanation y target variable, y i denotes the i th observation y i a value of a binary target variable, y ∈ {y + , y − } s Table III . IIISummary of absolute measures. Checkmark ( ) indicates that it is directly applicable in a given machine learning setting. 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[ "https://github.com/zliobaite/paper-fairml-survey.ACM" ]
[ "Neutrino Mixing and Leptogenesis in a L e − L µ − L τ model", "Neutrino Mixing and Leptogenesis in a L e − L µ − L τ model", "Neutrino Mixing and Leptogenesis in a L e − L µ − L τ model", "Neutrino Mixing and Leptogenesis in a L e − L µ − L τ model" ]
[ "Giorgio Arcadi [email protected]†[email protected]‡[email protected] \nDipartimento di Scienze Matematiche e Informatiche\nScienze Fisiche e Scienze della Terra\nUniversita degli Studi di Messina\nVia Ferdinando Stagno d'Alcontres 31I-98166MessinaItaly\n", "Simone Marciano \nDipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly\n\nINFN Sezione di Roma Tre\nVia della Vasca Navale 8400146RomaItaly\n", "Davide Meloni \nDipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly\n", "Giorgio Arcadi [email protected]†[email protected]‡[email protected] \nDipartimento di Scienze Matematiche e Informatiche\nScienze Fisiche e Scienze della Terra\nUniversita degli Studi di Messina\nVia Ferdinando Stagno d'Alcontres 31I-98166MessinaItaly\n", "Simone Marciano \nDipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly\n\nINFN Sezione di Roma Tre\nVia della Vasca Navale 8400146RomaItaly\n", "Davide Meloni \nDipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly\n" ]
[ "Dipartimento di Scienze Matematiche e Informatiche\nScienze Fisiche e Scienze della Terra\nUniversita degli Studi di Messina\nVia Ferdinando Stagno d'Alcontres 31I-98166MessinaItaly", "Dipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly", "INFN Sezione di Roma Tre\nVia della Vasca Navale 8400146RomaItaly", "Dipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly", "Dipartimento di Scienze Matematiche e Informatiche\nScienze Fisiche e Scienze della Terra\nUniversita degli Studi di Messina\nVia Ferdinando Stagno d'Alcontres 31I-98166MessinaItaly", "Dipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly", "INFN Sezione di Roma Tre\nVia della Vasca Navale 8400146RomaItaly", "Dipartimento di Matematica e Fisica\nUniversità di Roma Tre\nVia della Vasca Navale 8400146RomaItaly" ]
[]
We present a simple extension of the Standard Model with three right-handed neutrinos in a SUSY framework, with an additional U(1)F abelian flavor symmetry with a non standard leptonic charge Le − Lµ − Lτ for lepton doublets and arbitrary right-handed charges. We show that the model is able to reproduce the experimental values of the mixing angles of the PMNS matrix and of the r = ∆m 2 sun /∆m 2 atm ratio, with only a moderate fine tuning of the Lagrangian free parameters. The baryon asymmetry of the Universe is generated via thermal leptogenesis through CP-violating decays of the heavy right-handed neutrinos. We present a detailed numerical solution of the relevant Boltzmann equation accounting for the impact of the distribution of the asymmetry in the various lepton flavors. *
10.1140/epjc/s10052-023-11268-6
[ "https://export.arxiv.org/pdf/2205.02565v1.pdf" ]
248,524,702
2205.02565
4e7f1a3547a7bb0c3b0edb7c0ec5066982867b56
Neutrino Mixing and Leptogenesis in a L e − L µ − L τ model 5 May 2022 Giorgio Arcadi [email protected]†[email protected]‡[email protected] Dipartimento di Scienze Matematiche e Informatiche Scienze Fisiche e Scienze della Terra Universita degli Studi di Messina Via Ferdinando Stagno d'Alcontres 31I-98166MessinaItaly Simone Marciano Dipartimento di Matematica e Fisica Università di Roma Tre Via della Vasca Navale 8400146RomaItaly INFN Sezione di Roma Tre Via della Vasca Navale 8400146RomaItaly Davide Meloni Dipartimento di Matematica e Fisica Università di Roma Tre Via della Vasca Navale 8400146RomaItaly Neutrino Mixing and Leptogenesis in a L e − L µ − L τ model 5 May 20222 CONTENTS We present a simple extension of the Standard Model with three right-handed neutrinos in a SUSY framework, with an additional U(1)F abelian flavor symmetry with a non standard leptonic charge Le − Lµ − Lτ for lepton doublets and arbitrary right-handed charges. We show that the model is able to reproduce the experimental values of the mixing angles of the PMNS matrix and of the r = ∆m 2 sun /∆m 2 atm ratio, with only a moderate fine tuning of the Lagrangian free parameters. The baryon asymmetry of the Universe is generated via thermal leptogenesis through CP-violating decays of the heavy right-handed neutrinos. We present a detailed numerical solution of the relevant Boltzmann equation accounting for the impact of the distribution of the asymmetry in the various lepton flavors. * I. INTRODUCTION The Standard Model (SM) of particle physics has proven to be one of the most accurate theories to explain microscopic interactions at an unprecedented level. In spite of its many successes, it fails to account for relevant low energy data, such as the structure of fermion masses and mixings (in particular, the non-vanishing neutrino masses) and the value of the baryon asymmetry of the Universe (BAU), which is commonly expressed by the parameter: η B ≡ n B − nB n γ 0 ,(1) where n B , nB and n γ are the number densities of the baryons, antibaryons and photons, while the subscript "0" stands for "at present time". Latest observations provide a numerical value of η B ≈ 6.1 · 10 −10 [1]. In recent times, an enormous experimental progress has been made in our knowledge of the neutrino properties and it has been clearly shown that the lepton mixing matrix contains two large and one small mixing angle, and that the two independent mass-squared differences are both different from zero [2][3][4]. Although several abelian and non-abelian symmetries acting on flavour space have been proposed to explain such a pattern, not an unique framework emerged as the optimal one [5]. Thus, one is still motivated to explore scenarios where different symmetries and/or field (charge) assignments to the group representations are studied in details. In this context, a less explored possibility (compared to the most famous discrete non-abelian symmetries) is given by the U (1) F flavor symmetry with non-standard leptonic charge L e − L µ − L τ for lepton doublets [6] and arbitrary right-handed charges [7]. As it is well known, in the limit of exact symmetry, the neutrino mass matrix assumes the following structure: m ν = m 0   0 1 x 1 0 0 x 0 0   ,(2) le lµ lτ l c e l c µ l c τ F1 F2 Hu H d N1 N2 N3 U(1)F +1 -1 -1 −13 7 3 2 1/2 0 0 -1 1 0 which leads to a spectrum of inverted type and to θ 12 = π/4, tan θ 23 = x (i.e. large atmospheric mixing for x ∼ O(1)) and θ 13 = 0. While the previous texture can be considered a good Leading Order (LO) result, it evidently fails to reproduce two independent mass differences (two eigenvalues have the same absolute values) and, with the exception of the atmospheric angle, also fails in the correct description of the solar and reactor angles. Models based on the see-saw mechanism [8,9] have been proven to be sufficiently realistic as to accommodate solar and atmospheric splittings but either the solar angle was too large or the reactor angle was (almost) vanishing. With the increasing precision in the measurement of oscillation parameters, it turned out that both θ 12 and θ 13 were substantially different from their LO results; the observation that corrections of O(λ) (λ being the Cabibbo angle) are needed to bring both mixing angles to their experimental values, encouraged to explore the contributions to the neutrino mixing matrix from the charged lepton sector [10,11]; in this context, a natural value of r = ∆m 2 sol /∆m 2 arm ∼ O(λ 2 ) was also obtained [12], thus showing that models based on L e −L µ −L τ are capable to successfully describe low energy neutrino data. An important missing piece of the previous constructions is the possibility to explain the value of the BAU through leptogenesis. In [9] it was clearly shown that the baryon-to-photon ratio of the Universe η B is proportional to the neutrino mass m 1 and, for vanishing lightest mass m 3 , m 1 ∼ ∆m 2 atm , thus producing an hopelessly small η B . Providing a quantitative leptogenesis analysis has become sophisticated in recent years, due to the addition of many ingredients, such as various washout effects [13,14] or thermal corrections to the particle masses [14,15]. Also, the flavor effects can have a significant impact on the final value of the baryon asymmetry, as widely shown in [16][17][18]. With the present paper we aim to go beyond the existing literature, assessing whether see-saw models based on the L e − L µ − L τ quantum number can simultaneously account for neutrino masses and mixing and explain the BAU through thermal leptogenesis. The paper is structured as follows. In Sect.II we describe our model and derive the analytic expressions for the mass ratio r and the mixing angles, showing that appropriate choices of the Lagrangian parameters lead to a satisfactory description of low energy data; in Sect.III we face the problem of reproducing the value of η B , analyzing the resonant and hierarchical scenarios and solving the related Boltzmann equations. Our conclusions are drawn in Sect.IV. II. THE MODEL In the following, we summarize the relevant features of our see-saw flavor model based on a broken U (1) F symmetry. In the proposed scenario, the left handed lepton doubles have charge L e − L µ − L τ [11] under the U (1) F , while the right-handed SU (2) singlets l c e,µ,τ have the charges reported in Tab.I. The spectrum of the theory also contains three heavy sterile neutrinos N i=1,2,3 , needed for the generation of the light neutrino masses as well as for the implementation of the leptogenesis process. The flavor symmetry is broken by vacuum expectation values (vev's) of SU (2) singlet scalar fields (flavons) suitably charged under the U (1) F symmetry. Non-vanishing vevs are determined by the D-term potential [19]: V D = 1 2 (M 2 FI − g F |F 1 | 2 − g F |F 2 | 2 ) ,(3) where g F denotes the gauge coupling constant of the U (1) F symmetry while M FI is the Fayet-Iliopulos term. Non-zero vevs are obtained by imposing the SUSY minimum V D = 0. Without loss of generality, we can assume equal vevs for the two flavons and define λ = F 1 /M F = F 2 /M F the common ratio between the vevs of the flavons and the scale M F at which the flavour symmetry is broken. A. Charged lepton sector In the charged lepton sector, many operators of different dimensions enter the Lagrangian; to avoid cumbersome expressions, we quote here the lowest dimensional operators contributing to each entry of the mass matrix: L =a 11 l e l c e F 1 M F 6 H d + a 12 l e l c µ F † 1 M F 4 H d + a 13 l e l c τ F † 1 M F 2 H d + a 21 l µ l c e F 1 M F 7 H d + a 22 l µ l c µ F 1 M F 3 H d + a 23 l µ l c τ F † 1 M F H d + a 31 l τ l c e F 1 M F 7 H d + a 32 l τ l c µ F 1 M F 3 H d + a 33 l µ l c τ F † 1 M F H d + h.c. ,(4) where all a ij coefficients are generic O(1) free parameters. After flavor and electroweak symmetry breakings, the previous Lagrangian generates a mass matrix whose elements have the general structure: (m l ) ij ∼ a ij l i l c i F 1 M F αij F 2 M F βij H d ,(5) where α ij and β ij denote the appropriate powers of the flavon fields needed to generate a singlet under U (1) F . Factorizing out the τ mass, the charged lepton mass matrix assumes the following form: m l ∼ m τ    a 11 λ 5 a 12 λ 3 a 13 λ a 21 λ 6 a 22 λ 2 e iφ22 a 23 e iφ23 a 31 λ 6 a 32 λ 2 e iφ32 1    .(6) For λ < 1, the following mass ratios m e : m µ : m τ = λ 5 : λ 2 : 1 is found, which naturally reproduces the observed pattern if λ ∼ 0.22. It is not difficult to derive the left-handed rotation U l , which contributes to the total neutrino mixing matrix U P M N S [20,21], diagonalizing the hermitean m l m † l combination: U l =          −         + O(λ 2 ) .(7) As expected, the diagonal elements of U l are unsuppressed; also, the (23) and (33) entries are of O(1), which indicates that the charged lepton contribution to the atmospheric angle will be large. Notice also that, being the (12) and (13) elements of O(λ), we expect a similar corrections to the solar and reactor angles. B. Neutrino sector In the neutrino sector, masses are generated through the standard type-I see-saw mechanism; at the renormalizable level, the see-saw Lagrangian reads: L LO = 1 2 MW N c 1 N 2 + 1 2 MZN c 3 N 3 − aN 1 H u l µ + − bN 1 H u l τ − cN 2 H u l e + h.c. ,(8) where M is an overall Majorana mass scale while W, Z, a, b, c are dimensionless coefficients which will be regarded as free parameters. When H u acquires a vev v u , Majorana and Dirac mass matrices are generated: M R = M    0 W 0 W 0 0 0 0 Z    , m D = v u    0 a b c 0 0 0 0 0    .(9) Next-to-leading order (NLO) contributions are given by higher dimensional operators suppressed by the large scale M F ; up to two-flavon insertions, we get 1 : L N LO = 1 2 Mm 11 N c 1 N 1 F 1 M F + 1 2 Mm 13 N c 1 N 3 F 2 M F 2 + 1 2 Mm 22 N c 2 N 2 F † 1 M F + + 1 2 Mm 23 N c 2 N 3 F † 2 M F 2 − d 11 N 1 H u l e F † 1 M F − d 22 N 2 H u l µ F 1 M F + − d 23 N 2 H u l τ F 1 M F − d 31 N 3 H u l e F † 2 M F 2 − d 32 N 3 H u l µ F † 2 M F 2 + − d 33 N 3 H u l τ F † 2 M F 2 + h.c. .(10) 1 Notice that we use the same overall scale M in the Majorana mass terms. Their main effects is to fill the vanishing entries in eq.(9); however, as we have numerically verified, some of the free parameters in eq.(10) needed to be slightly adjusted to fit the low energy data. In particular, only a moderate fine-tuning is necessary on m 11 , m 22 , d 11 , d 22 and d 23 , which we rescale according to: (m 11 , m 22 , d 11 , d 22 , d 23 ) → λ (m 11 , m 22 , d 11 , d 22 , d 23 ) , (m 11 , m 22 , d 11 , d 22 , d 23 ) ∼ O(1) . With the previous position (and reminding that λ = F 1 /M F = F 2 /M F ), the following Dirac and Majorana mass matrices are obtained: Y = m D v u ∼ N l ∼    λ 2 d 11 ae iΣ be iΩ ce iΦ λ 2 d 22 λ 2 d 23 e iΘ λ 2 d 31 λ 2 d 32 λ 2 d 33    ,(11) and M R ∼ N N ∼ M    λ 2 m 11 W λ 2 m 13 W λ 2 m 22 λ 2 m 23 λ 2 m 13 λ 2 m 23 Z    .(12) Notice that the Dirac mass matrix contains un-suppressed entries because of the choice Q N1 = −Q N2 for two of the right-handed neutrinos. The four physical phases Σ, Ω, Φ, Θ in Y , obtained after a suitable redefintion of the fermion fields, are the only source of CP violation of our model and are not fixed by the symmetries of the Lagrangians. For the sake of simplicity and without any loss of generality, we can assume the parameters m ij ∼ m and consider m as a real quantity. From the type-I seesaw master formula, m ν −v 2 u Y T M −1 R Y , we get the following matrix for the light SM neutrinos, up to O(λ 2 ): m ν v 2 u M × ×     ce iΦ (ce iΦ m−2d11W )λ 2 W 2 • • − ace i(Σ+Φ) W ae iΣ (ae iΣ m−2d22W )λ 2 W 2 • − bce i(Φ+Ω) W (abe i(Σ+Ω) m−ad23e i(Θ+Σ) W −bd22e iΩ W )λ 2 W 2 be iΩ (be iΩ m−2d23e iΘ W )λ 2 W 2     .(13) This mass matrix provides, as usual for models based on the L e − L µ − L τ symmetry, an inverted mass spectrum. From now on, we will distinguish two different scenarios, that will be further elaborated when studying the BAU generated in our model, and identified by different assumptions on the parameter M. In order to better understand this distinction, let us assume that all parameters in eq.(13) are of O(1); thus, the light neutrino mass matrix can be recast in the following form: m ν = m 0    λ 2 x 1 1 x 1 x 2 λ 2 x 3 λ 2 x x 3 λ 2 x 4 λ 2    ,(14) where m 0 = v 2 u /M × O(1) coefficients and (x, x i ) are suitable combinations of the coefficients present in Dirac and Majorana matrices in eqs. (11) and (12). At the leading order in λ, m ν has two degenerate eigenvalues m 1 = −m 2 = √ 1 + x 2 and a vanishing one, m 3 = 0; therefore, we can only construct the atmospheric mass difference ∆m 2 atm = |m 1 | 2 − |m 3 | 2 , which results in: x 2 = ∆m 2 atm m 2 0 − 1 .(15) To maintain x ∼ O(1), we can choose the overall mass scale to m 0 ∼ O(10 −2 ) eV, which corresponds to the choice M ∼ 10 15 GeV. Notice also that, taking into account the corrections of O(λ 2 ), the eigenvalue degeneracy is broken and the solar mass difference can be accounted for, which results in: ∆m 2 sol = λ 2 x 1 (1 + x 2 ) + x 2 + 2xx 3 + x 4 x 2 √ 1 + x 2 .(16) Since the masses of the three heavy right-handed neutrinos are simply given by M i = M M i with: M 1 W + mλ 2 + O(λ 3 ), M 2 W − mλ 2 + O(λ 3 ), M 3 Z + O(λ 3 ) ,(17) and, in particular, the relation W Z holds, we dubbed such a situation as the resonant scenario. As expected, two mass eigenstates are degenerate, up to corrections of order λ 2 . The second possibility arises when M ∼ 10 13 GeV and, consequently, W should be around 10 2 to maintain m 0 ∼ O(10 −2 ) eV. With all other parameters again of O(1), the light neutrino mass matrix is now as follows: m ν v 2 u MW    −2ce iΦ d 11 λ 2 −ace i(Σ+Φ) −bce i(Φ+Ω) −ace i(Σ+Φ) −2ae iΣ d 22 λ 2 − ad 23 e i(Θ+Σ) + bd 22 e iΩ λ 2 −bce i(Φ+Ω) − ad 23 e i(Θ+Σ) + bd 22 e iΩ λ 2 −2bd 23 e i(Θ+Ω) λ 2    . (18) The right-handed neutrino spectrum will feature two, almost degenerate states with mass M 1 M 2 ∼ 10 15 GeV, and a lighter one with mass M 3 ZM ∼ 10 13 GeV; we call this scenario as the hierarchical scenario. Even in this case, the left-handed neutrino mass matrix can be recast in the form (14), with the obvious redefinition of (x, x i ) in terms of the parameters appearing in eq. (18). Thus, in both resonant and hierarchical scenarios, we get the same structure of the diagonalizing matrix U ν : U ν =         − 1 √ 2 1 √ 2 0 1 2(1 + x 2 ) 1 2(1 + x 2 ) − x 2(1 + x 2 ) x 2(1 + x 2 ) x 2(1 + x 2 ) − 1 √ 1 + x 2         + O(λ 2 ).(19) The final expressions of the neutrino mixing angles (from the relation U P M N S = U † l U ν ) and the ratio r = ∆m 2 sol /∆m 2 atm in terms of the model parameters are quite cumbersome. Thus, we prefer [2]. to report here the order of magnitude in the expansion parameter λ, which are valid for both scenarios considered in this paper: r ∼ O(λ 2 ), tan θ 12 ∼ 1 + O(λ), sin θ 13 ∼ O(λ), tan θ 23 ∼ 1 .(20) Our order of magnitude estimates are in good agreement with their measured values, reported in Tab.II. This conclusion has been further strengthened by a succesful numerical scan [22] over the model free parameters, with moduli extracted flat in the intervals [λ, 5] and all the phases in [−π, π]. Notice that the model provides a CP conserving leptonic phase 2 (still compatible with the data at less than 3σ). III. LEPTOGENESIS As stated above, our study of leptogenesis will be performed within two reference scenarios, identified by different mass patterns for the heavy right-handed neutrinos: • the resonant scenario, with M 1 M 2 M 3 around 10 15 GeV, and the mass difference comparable to the decay width ∆M ij ∼ Γ i ; • the hierarchical scenario, with M 3 M 1,2 10 15 GeV . The three Majorana neutrinos decay in the early Universe creating a lepton asymmetry, which is consequently conversed in a baryon asymmetry through non perturbative processes, known as sphaleron processes [13,14]. As we will clearify in the following, a different Majorana neutrino mass spectrum can lead to different CP-violating parameters, affecting the final amount of baryon asymmetry in the Universe. Resonant Scenario We are interested in computing the BAU via thermal leptogenesis. To facilitate the understanding of the numerical results, we will first provide analytical (approximated) expressions of all relevant quantities entering our computations, which will be validated against a full numerical solution of an appropriate system of Boltzmann's equations. Being mass degenerate, we expect that all the three right-handed neutrinos contribute to the leptogenesis process. Following [23], we write the baryon asymmetry as: η B 7.04 · 10 −3 i ε i η i with i=1,2,3 ,(21) with ε i being the CP-asymmetries produced in the decay of the i-th neutrino and η i the corresponding efficiency factors. As will be clarified below, the generation of the lepton asymmetry in the resonant scenario occurs in an unflavored regime; the final asymmetry is consequently just the sum of the contributions associated to the individual neutrinos. The efficiency factors can be written in terms of decay parameters K i [18]: η i = 1 K i , K i ≡ m i m * ,(22) with: m i = Y * ii Y ii v 2 u M i , m * =1.1 × 10 −3 eV .(23) As it is well known, different values of K i define different washout regimes, namely strong (K i 1), intermediate (K i 1) and weak (K i 1). In terms of the parameters of the Dirac and Majorana neutrino mass matrices of eq.(11), their expressions up to O(λ 4 ) are as follows: K 1 27.7 W c 2 + d 2 11 + d 2 31 λ 4 + 2 (W − Z) c 2 m 2 (W − Z) + √ 2 c d 31 m cos(Φ) λ 4 , K 2 27.7 W a 2 + (d 2 22 + d 2 33 )λ 4 , K 3 27.7 Z b 2 + (d 2 23 + d 2 33 )λ 4 .(24) Being all the above parameters of O(1), we straightforwardly conclude K i ∼ O(10), implying a intermediate/strong washout regime. We are now in the position to discuss the CP asymmetry. Since the heavy neutrinos are close in mass, the CP-asymmetry can be resonantly enhanced [23][24][25]. To check whether such an enhancement occurs, and hence properly evaluate the CP asymmetry parameters, a good rule of thumb consists in computing the ratios ∆M ij /Γ i between the mass splittings and the decay widths of the righthanded neutrinos, verifying that the resonance condition ∆M ij ∼ Γ i is satisfied. In the scenario under scrutiny, with degenerate masses and not strongly hierarchical Yukawa couplings, we can assume Γ 1 ∼ Γ 2 ∼ Γ 3 , so that: ∆M 12 Γ 1 ∆M 12 Γ 2 32 mπ a 2 W λ 2 + O(λ 4 ) , ∆M 23 Γ 2 ∆M 23 Γ 3 16 (Z − W ) π b 2 + Z + O(λ 2 ) , ∆M 13 Γ 1 ∆M 13 Γ 3 16 (Z − W ) π b 2 + W + O(λ 2 ) ,(25) where we have used Γ i = M i Y † Y ii /(16π). As evident, for W−Z O(0.1), ∆M i3 Γ i . Consequently, leptogenesis occurs in the resonant regime. In such a case, the self energy contribution dominates the CP violation parameters. Furthermore, as shown in [26], the asymmetry parameters are time dependent. Following [26,27], we rewrite the latter as: ε i (z) j =i Im Y † Y 2 ij (Y † Y ) ii (Y † Y ) jj ∆M ij /Γ j 1 + (∆M ij /Γ j ) 2 f mix ij (z) + f osc ij (z) ,(26) where z = M/T , with M the sterile neutrino mass. The coefficient in front of the squared parenthesis is the (constant) usual CP-asymmetry and it is resonantly enhanced for ∆M ∼ Γ, while the second one is the sum of two z (and hence time) dependent functions: f mix ij (z) =2 sin 2 (∆M ij t) = +2 sin 2 K i z 2 ∆M ij 4Γ i , f osc ij (z) = − Γ j ∆M ij sin (∆M ij t) = − Γ j ∆M ij sin K i z 2 ∆M ij 2Γ i ,(27) In the case ∆M ij t ≡ Kiz 2 ∆Mij 4Γi 1, eq.(27) is a strongly oscillating function. Making the average over a generic time interval (or, equivalently, in z) t ∈ [0, τ ], we have that: 2 sin 2 ∆M ij t − Γ ij ∆M ij sin ∆M ij t = 1 − sin 2∆M ij τ 2∆M ij τ − Γ i ∆M ij 1 − cos ∆M ij τ ∆M ij τ(28) As evident, the average tends to 1 if ∆M ij τ 1. In such a limit we have that the CP asymmetry averages to the following constant value: (z) = j =i Im Y † Y 2 ij (Y † Y ) ii (Y † Y ) jj ∆M ij /Γ j 1 + (∆M ij /Γ j ) 2 .(29) In the case of a large decay parameter, the regime ∆M ij t 1 occurs for small values of z. In good approximation the whole leptogenesis process can be described by replacing the time dependent CP-asymmetry with its average. Being, in our case, K i O(10), we can adopt this approach and, consequently, propose an analytic estimate of the baryon asymmetry neglecting the time dependence of the ε i 's. The η B will be nevertheless compared against a complete numerical treatment. The time-independent piece of the CP-asymmetry parameters in eq. (26) fig. (1). The parameters are selected close to 1 as to avoid particular effects of enhancement or suppression in the ε i . of the small parameter λ as follows: ε 1 = + 16 π (W − Z) Z c 2 (256π(W − Z) 2 + b 4 Z 2 ) c 2 d 2 23 sin [2 (Θ − Φ)] + + bd 11 (bd 11 sin [2Ω] + 2cd 23 sin [Θ − Φ + Ω]) λ 4 + O(λ 5 ) ε 2 = − 16 b 2 π(W − Z)Z sin [2(Σ − Ω)] 256π 2 (W − Z) 2 + b 4 Z 2 + + 16b 2 mπZ 256π 2 (W − Z) 2 − b 4 Z 2 sin [2 (Σ − Ω)] λ 2 (256π 2 (W − Z) 2 + b 4 Z 2 ) 2 + O(λ 4 ) ε 3 = + 16 b 2 π(W − Z)Z sin [2(Σ − Ω)] 256π 2 (W − Z) 2 + b 4 Z 2 + − 16b 2 mπZ 256π 2 (W − Z) 2 − b 4 Z 2 sin [2 (Σ − Ω)] λ 2 (256π 2 (W − Z) 2 + b 4 Z 2 ) 2 + O(λ 4 ) .(30) By inspecting the analytical expressions above we see that, for W − Z O(0.1), the parameters ε 2,3 tend approximately to O(0.1), while ε 1 is of order 10 −4 . Therefore, the main contribution to the final baryon asymmetry of the Universe is carried by ε 2,3 , leading to an η B value which exceeds the experimentally favoured one η B 6.1 · 10 −10 by several orders of magnitude. It is nevertheless possible to suppress the values of the ε i parameters by an ad-hoc assignations of the phases to trigger a destructive interference among them. First of all, by taking Σ − Ω → 0, the leading order contributions to ε 2,3 go to zero and, as for ε 1 , all ε's are of O(λ 4 ): ε 2 = 32bd 22 d 23 π(W − Z)Z sin [Θ] λ 4 a (256π 2 (W − Z) 2 + b 4 Z 2 ) + O(λ 5 ) ε 3 = 16πW (Z − W ) b 2 2abd 22 d 23 sin [Θ] a 4 W + 256π 2 (W − Z) 2 + + b 2 d 11 sin [2Σ] + cd 23 (cd 23 sin [2(Θ − Φ)] + 2bd 11 sin [Θ + Σ − Φ]) c 4 W 2 + 256π 2 (W − Z) 2 λ 4 + O(λ 5 ) .(31) Upon numerical check, we have found that the λ 4 suppression was not enough to guarantee viable values of the ε i . Consequently, we need to further assume individually suppressed Θ, Φ and Σ phases. For the benchmark values reported in Tab.III, the efficiency factors turn to be: η 1 1.0 · 10 −2 , η 2 2.2 · 10 −2 , η 3 1.6 · 10 −2 ,(32) while the CP-asymmetry parameters are: ε 1 1.7 · 10 −5 , ε 2 3.2 · 10 −6 , ε 3 −1.1 · 10 −5 .(33) Using the approximated formula in eq.(21), we can estimate the final baryon asymmetry, obtaining η B ≈ 5.20 · 10 −10 , in good agreement with the observations. We conclude our study of the resonant scenario with a numerical validation of the analytical results presented above. We have hence solved the following set of coupled Boltzmann's equations [13]: dN i dz = − (D i + S i ) (N i − N eq i ) i = 1, 2, 3 dN B−L dz = 3 i ε i D i (N i − N eq i ) − W i N B−L ,(34) where N i stands for number density of the RH sterile neutrinos, while N B−L is the amount of B − L asymmetry, both normalized by comoving volume [13]. ε i = ε i (z) are the full time dependent asymmetry parameters as given in eqs. (26)(27). D i and S i indicate, respectively, inverse decay and scattering contributions to the production of the right-handed neutrinos while the W i represent the total rate of Wash-out processes including both inverse decay and ∆L = 0 scattering contributions (see Appendix A for further details). From the benchmark values of Tab.III, the following entries of the Yukawa matrix are obtained: Y =    +0.969 − 0.004i −0.343 − 0.002i −0.405 + 0.002i −1.102 + 0.004i −0.408 − 0.002i −0.491 + 0.002i +0.02851 + 0.00005i +1.168 − 0.008i +1.355 − 0.009i    ,(35) where the symbol hat refers to the Yukawa evaluated in the physical mass basis of the Majorana neutrinos, i.e.Ŷ = U Y , where U is the unitary matrix such that U T M −1 R U = M diag R −1 . Plugging the latter values in the interaction rates appearing in the Boltzmann's equations, we have solved the system assuming null initial abundance for the right-handed neutrinos in the primordial plasma. The B-L yield N B−L as a function of z is shown with a blue line in fig.1, while the abundance of the right-handed neutrinos is displayed with a green line. For reference, we have reported the corresponding equilibrium function as a dashed orange line. To better pinpoint the impact on our result of the time dependency of the CP-asymmetry, we have shown, as red line in fig.1, solution of the same system of equation but retaining a constant CP violating parameter as given by eq.(33). As evident, there is a nice agreement between the curves, justifying the assumption of neglecting the time dependence of the asymmetry parameter in our analytical treatment. Some more comments are in order. Starting from a negligible abundance, the yield of the right-handed neutrino is driven by inverse decays toward the equilibrium value which is reached for z eq < 1. For z > z eq , the decays dominate and the neutrino abundance decreases, until it becomes almost zero around z 10. This means that the leptogenesis processes is completed at temperatures above 10 13 GeV. This justifies our assumption of neglecting flavor effects since the latter are relevant only if leptogenesis occurs at temperatures below 10 12 GeV. The shape of the N B−L also clearly evidences the time dependency of the ε i parameters. The solution of the system N B−L (∞) can be related to η B through the relation η B = (a sph /f )N B-L (∞). Here a sph = 28/79 [13] is the fraction of B − L asymmetry converted into a baryon asymmetry by the sphaleron processes while f = N rec γ /N * γ = 2387/86 is the dilution factor calculated assuming standard photon production from the onset of leptogenesis till recombination. The values of the latter parameters, as obtained from the numerical solution of the system are: N B−L (∞) =2.11 × 10 −8 , η B =3.01 × 10 −10 .(36) We conclude that, after the phase suppression discussed above, it is possible to provide a viable leptogenesis in the resonant scenario. Hierarchical scenario This alternative scenario is obtained by lowering the mass scale M down to a value of the order of 10 13 GeV, which brings to a hierarchical mass spectrum for the sterile neutrinos, with two heavy, almost degenerate, states with M 1 M 2 ∼ 10 15 GeV, and a lighter one with M 3 10 13 GeV. In this setup, the baryon asymmetry of the Universe can be generated through the conventional thermal leptogenesis only via the out-of-equilibrium decay of the lightest heavy neutrino. Similarly to the previous case, no flavor effects need to be accounted for. As discussed in [23], the latter becomes relevant when the rate of processes mediated by τ , µ and e exceeds the Hubble expansion rate. This occurs when the temperature of the Universe drops below, respectively: T τ 5 · 10 11 GeV , T µ 2 · 10 9 GeV , T e 4 · 10 4 GeV . In the scenario under consideration we have M 3 T τ T µ,e . Thus, we can work in the so-called unflavored regime, in which the lightest neutrino decays via flavor blind processes. The baryon asymmetry can then be parametrized as: η B 7.04 · 10 −3 ε 3 η 3 ,(38) with η i as in eq. (22) and the unflavored CP-asymmetry as [28][29][30][31]: ε i = 1 8π (Y † Y ) ii j =i Im Y † Y 2 ij f M 2 j M 2 i ,(39) where x j = M 2 j /M 2 i and the loop function is: f [x] = √ x 1 − (1 + x) log 1 + 1 x + 1 1 − x ,(40) that can be approximated to f [x] −3/(2 √ x) in the limit x 1. Expressing the elements of the Yukawa matrices in terms of the model parameters, we have: ε 3 = f M 2 2 M 2 3 1 8b 4 π · sin [2(Σ − Ω)] a 2 b 2 − 2a 2 (d 2 23 + d 2 33 ) + − 2a b d 22 d 23 sin [Θ − Σ + Ω] + + 2 a b d 32 d 33 sin [Σ − Ω] + O(λ 4 ) .(41) Using the values in Tab.IV, and noticing that f M 2 2 /M 2 3 10 −2 , we obtain ε 3 3.24 · 10 −4 . The corresponding efficiency factor η 3 can be simply approximated to: η 3 3.6 · 10 −4 Z b 2 + O(λ 4 ) 4.3 · 10 −4 ,(42) for the assignations in Tab.IV. The efficiency factor can be converted to the baryon asymmetry parameter η B : form: η B 7.04 × 10 −3 [ε 3 η 3 ] ≈ 9.5 × 10 −10 ,(43)dN 3 dz = − (D 3 − S 3 ) (N 3 − N eq 3 ) , dN B−L dz = ε 3 D 3 (N 3 − N eq 3 ) − W 3 N B−L .(44) confirming what we expected from the simplified analytical analysis. It is interesting to notice that, contrary to the resonant regime, in the hierarchical scenario it is possible to find a parameter assignation leading to viable leptogenesis without imposing a fine tuning on the CP violating phases. IV. CONCLUSIONS In this paper we have provided a proof of existence about the possibility of contemporary achieving viable masses and mixing patterns for the SM neutrinos and a value of the BAU, via leptogenesis, compatible with the experimental determination, in models based on the abelian flavor symmetry L e − L µ − L τ . Given the large parameter space of the model, we have identified two reference scenarios. The first one, dubbed the resonant scenario, provides a viable light neutrino mass spectrum and assures the existence of three degenerate right-handed neutrinos at a mass scale of 10 15 GeV. In this scenario, the generation of the lepton asymmetry is resonantly enhanced so that a baryon asymmetry exceeding the experimentally favored value is generically predicted. This problem can be overcome by invoking an ad-hoc suppression of the CP-violating phases in the Yukawa matrix. In the second scenario, that we called the hierarchical scenario, one of the right-handed masses is lowered down to 10 13 GeV without destroying the good agreement with the lepton masses and mixing. The BAU is generated via the conventional thermal leptogenesis. We have verified that it is possible to find parameter assignations leading to the correct value of the BAU without invoking specific assignations for the CP violating phases. Appendix A: Full Boltzmann Equations In this Appendix we discuss in greater detail the Boltzmann's equations solved in the main text. For the sake of simplicity, we will write them as a single equation for the right-handed neutrino specie N 1 . In the case of the resonant regime, we just need to consider multiple copies of the same equation. The relevant processes in the thermal plasma are: • N 1 decays (D) and inverse decays (ID) into leptons and Higgs bosons N 1 → φl, and into anti-leptons and anti-Higgs bosons N 1 →φl; • ∆L = 2 scattering processes mediated by the heavy Majorana neutrinos, lφ ↔lφ (N) and ll ↔φφ,ll ↔ φφ (Nt); • ∆L = 1 scattering, with an intermediate Higgs boson field φ, involving the top quark. The s-channel N 1 l ↔tq, N 1l ↔ tq (s) and the t-channel N 1 t ↔lq, N 1t ↔ lq (t); In brackets we have indicated a short-hand notation for such processes, to be used later on. Considering z = M/T , where M is the mass of the decaying neutrino, the Boltzmann's equations read: dN 1 (z) dz = − D(z) + S(z) N 1 (z) − N eq 1 (z) , dN B−L (z) dz =ε(z)D(z) N 1 (z) − N eq 1 (z) − W (z)N B−L (z) ,(A1) where ε is the usual CP-violation parameter. Instead of the number density n X of the particle species, it is useful to consider their number N X in some portion of the comoving volume, in such a way to automatically take into account the expansion of the Universe. In the literature, the comoving volume R 3 * (t) is usually chosen such that it contains one photon at the time t * , before the onset of the leptogenesis [32]: N X (t) = n X (t)R 3 * (t) ,(A2) with R * (t * ) = n eq γ (t * ) −1/3 ,(A3) so that N γ (t * ) = 1. Differently, one could even choose to normalize the number density to the entropy density s considering then Y X = n x /s, as widely done in literature, e.g., [13]; however, if the entropy is conserved during the Universe evolution, the two normalizations are related by a constant. Introducing the thermally averaged dilation factor 1/γ as the ratio of the modified Bessel functions of the second type: 1 γ = K 1 K 2 ,(A4) we can write the decay term D(z) as [33]: D(z) = Kz 1 γ . (A5) of the reduced cross sectionσ x given in [32]. In particular, for the scattering processes mediated by the three Majorana neutrinos, i.e. the ∆L = 2 scatterings, the reduced cross section reads [36]: σ N,N t (x) = 1 2π   i Y † Y 2 ii f N,N t ii (x) + i<j Re Y † Y 2 ij f N,N t ij (x)   ,(A15)with f N ii (x) = 1 + a i D i (x) + xa i 2D 2 i (x) − a i x 1 + x + a i D i ln 1 + x a i ,(A16)f N ij (x) = √ a i a j 1 D i (x) + 1 D j (x) + x D i (x)D j (x) + + 1 + a i x 2 a j − a i − 1 D j (x) ln 1 + x a i + + 1 + a j x 2 a i − a j − 1 D i (x) ln 1 + x a j ,(A17)f N t ii (x) = x x + a i + a i x + 2a i ln 1 + x a i ,(A18) f N t ij (x) = √ a i a j (a i − a j )(x + a i + a j ) (2x + 3a i + a j ) ln 1 + x a j + −(2x + 3a j + a i ) ln 1 + x a i . (A19) Here a j ≡ M 2 j /M 2 1 and 1/D i (x) ≡ (x−a i )/[(x−a i ) 2 +a i c i ] is the off-shell part of the N i propagator with c i = a i (Y † Y ) 2 ii /(8π) 2 . On the other hand, for the ∆L = 1 scattering processes, it is convenient to write the rates S s,t (z) in term of the functions f s,t (z) defined as [32]: f s,t (z) = ∞ z 2 dΨχ s,t (Ψ/z 2 ) √ ΨK 1 ( √ Ψ) z 2 K 2 (z) ,(A20) with the functions χ s,t (x) as follows: χ s (x) = x − 1 x 2 ,(A21)χ t (x) = x − 1 x x − 2 + 2a h x − 1 + a h + 1 − 2a h x − 1 log x − 1 + a h a h ,(A22) where a h = m φ /M 1 is the infrared cut-off for the t-channel and m φ is the mass of the Higgs boson which receives contributions from the thermal bath and its value can be written as m φ 0.4 T [37]. In this a way, the ∆L = 1 scattering terms are: S s,t (z) = K s 9ζ(3) f s,t (z) ,(A23) with K s = m 1 /m s * and [13]: m s * = 4π 2 9 g N h 2 t m * ,(A24) where h t is the top quark Yukawa coupling evaluated at T M 1 . The total scattering term S(z), related to the production of the sterile neutrinos, is given by S(z) = 2S s (z) + 4S t (z), where the s-channel acquires a factor of 2 stemming from the diagram involving the antiparticles, and an additional factor of 2 in the t-channel comes from the u-channel diagram. Therefore, the parameter S(z) can finally be written as: S(z) = 2K s 9ζ(3) (f s (z) + 2f t (z)) .(A25) Thus, for the total washout term W (z), which includes contributions from the inverse decay processes and the ∆L = 1, 2 scatterings, we have [35], [32]: W (z) = W ID (z) 1 + 2 D(z) 2S t (z) + N 1 (z) N eq 1 (z) S s (z) + 2S N (z) + 2S N t (z) .(A26) FIG. 1 : 1B − L asymmetry and neutrino abundance evolution during the expansion of the Universe in case of zero initial neutrino abundance. The blue line refers to the full solution of the Boltzmann's equations, obtained retaining the time dependence of the CP-violation parameter. The red line refers to the solution of the analogous system but adopting a time-constant value of the asymmetry parameters. Finally, the green line represents the abundance of the right-handed neutrinos, as given by the solution of the system. For reference, the latter is compared with the function N eq N (dashed line) rwhich represents a thermal equilibrium abundance for right-handed neutrinos. FIG. 2 : 2very close to the experimentally favoured value. Also in this case, we have verified the goodness of our analytical approximations by solving a suitable set of Boltzmann equations, which take the Evolution of the B − L symmetry in the hierarchical scenario. The color code is the same as infig.1. From the parameter choices in Tab.IV, we get the following Yukawa matrix : the imaginary parts of the third row is suppressed by a factor proportional to Z/W . This choice corresponds to the CP-violation parameter ε 3 = 4.3 · 10 −4 . For the benchmark values under consideration, the numerical solution of the Boltzmann's equations is shown in fig.2 and gives: N B−L (∞) = 2.75 · 10 −8 , η B = 3.96 · 10 −10 , TABLE I : IU (1) F charges for leptons, Higgses and flavon fields. TABLE II : IIBest fits and 1σ ranges for the oscillation parameters, from can be expressed in power seriesa b c d 11 d 22 d 23 d 31 d 32 1.28 1.49 1.46 1.07 1.13 1.45 1.70 1.01 d 33 m Σ Ω Θ Φ Z W 1.50 1.50 0.007 0.007 -0.003 -0.004 1.38 1.27 TABLE III : IIISet of parameter assignments which lead to the final baryon asymmetry value shown in TABLE IV : IVSet of parameter assignments which lead to a final baryon asymmetry close to the cosmological observations. See text for further details. We preferred not to report here its analytic cumbersome expression. K is the decay parameter, which is introduced in the context of the GUT baryogenesis[33], to control whether the decays of the sterile neutrinos are in equilibrium or not. 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[ "arXiv:physics/9703010v1 [physics.chem-ph] A Monte Carlo Simulation", "arXiv:physics/9703010v1 [physics.chem-ph] A Monte Carlo Simulation" ]
[ "A Werner \nUniversität Mainz\nD-55099Mainz\n", "F Schmid \nUniversität Mainz\nD-55099Mainz\n", "K Binder \nUniversität Mainz\nD-55099Mainz\n", "Johannes Gutenberg \nUniversität Mainz\nD-55099Mainz\n", "M Müller \nDepartment of physics\nUniversity of Washington\nBox351560, 98195Seattle, Washington\n" ]
[ "Universität Mainz\nD-55099Mainz", "Universität Mainz\nD-55099Mainz", "Universität Mainz\nD-55099Mainz", "Universität Mainz\nD-55099Mainz", "Department of physics\nUniversity of Washington\nBox351560, 98195Seattle, Washington" ]
[]
The properties of diluted symmetric A-B diblock copolymers at the interface between A and B homopolymer phases are studied by means of Monte Carlo (MC) simulations of the bond fluctuation model. We calculate segment density profiles as well as orientational properties of segments, of A and B blocks, and of the whole chain. Our data support the picture of oriented "dumbbells", which consist of mildly perturbed A and B Gaussian coils. The results are compared to a self consistent field theory (SCFT) for single copolymer chains at a homopolymer interface. We also discuss the number of interaction contacts between monomers, which provide a measure for the "active surface" of copolymers or homopolymers close to the interface.
10.1021/ma960614h
[ "https://arxiv.org/pdf/physics/9703010v1.pdf" ]
24,160,852
physics/9703010
79ec8306a44bba5e4b4b27c5e42b0f13f392691e
arXiv:physics/9703010v1 [physics.chem-ph] A Monte Carlo Simulation 6 Mar 1997 August 28, 2018 A Werner Universität Mainz D-55099Mainz F Schmid Universität Mainz D-55099Mainz K Binder Universität Mainz D-55099Mainz Johannes Gutenberg Universität Mainz D-55099Mainz M Müller Department of physics University of Washington Box351560, 98195Seattle, Washington arXiv:physics/9703010v1 [physics.chem-ph] A Monte Carlo Simulation 6 Mar 1997 August 28, 2018Diblock Copolymers at a Homopolymer-Homopolymer-Interface: The properties of diluted symmetric A-B diblock copolymers at the interface between A and B homopolymer phases are studied by means of Monte Carlo (MC) simulations of the bond fluctuation model. We calculate segment density profiles as well as orientational properties of segments, of A and B blocks, and of the whole chain. Our data support the picture of oriented "dumbbells", which consist of mildly perturbed A and B Gaussian coils. The results are compared to a self consistent field theory (SCFT) for single copolymer chains at a homopolymer interface. We also discuss the number of interaction contacts between monomers, which provide a measure for the "active surface" of copolymers or homopolymers close to the interface. Introduction Blending of polymeric substances is a straightforward and inexpensive way of creating new materials with improved mechanical properties [1]. However, long polymers of different type A and B are often immiscible already at high temperatures, since the total energy of the (usually repulsive) relative interaction is proportional to the total number of monomers and cannot be balanced by the entropy of mixing, which is proportional to the number of polymers in the mixture [2,3]. In order to overcome this problem, block copolymers containing both types of monomers can be used as effective compatibilizers [4,5]. Being partly compatible with both the A and B rich phase, they tend to aggregate at interfaces, where they reduce the number of direct contacts between A and B homopolymers, thereby reducing the interfacial tension [6]. Consequently the total area of interfaces increases, and the immiscible components may get finely dispersed in the mixture. Furthermore, copolymers improve the mechanical properties of such interfaces: Due to entanglement between homopolymers and copolymers, they increase the adhesive attraction and the fracture toughness [7,8,9]. At high enough copolymer concentrations, additional copolymer rich phases emerge which may display a diversity of structures ordered on a mesoscopic scale [10,11]. The simplest possible copolymers are diblocks, which consist of a block of A monomers connected to a block of B monomers. From a thermodynamic point of view, their effect on interfaces can be described as follows: Copolymers act as amphiphiles in the homopolymer mixture [12]. Increasing the copolymer concentration causes the interfacial tension to decrease monotonically, until this process is terminated by the formation of a third, copolymer rich phase, e.g. an ordered lamellar phase or microemulsion. The amphiphilic strength of a copolymer, i.e., the maximum reduction of the interfacial tension it can achieve, increases with the length of the A and B blocks relative to the size of the homopolymers. Mean field theories predict that the interfacial tension can be driven to zero for long copolymers, which implies that the copolymer rich lamellar phase evolves into the two-phase region via a continuous unbinding transition [13,14]. Fluctuations push the transition to first order in real systems [15]. One can argue that the low interfacial tensions at the presence of copolymers result from the affinity of the system to such an unbinding transition [16]. This macroscopic discussion however does not shed light on the microscopic mechanisms, why and how copolymers work as compatibilizer or amphiphiles. In a simple microscopic picture, the A block of copolymers at interfaces preferably stick into the A-rich phase, the B block into the B-rich phase, and the molecules as a whole act as reinforcing rods [7]. The conformations of the copolymers determine the properties of the interface. A more detailed microscopic description of copolymer properties at interfaces has been developed by Leibler [17] and refined by Semenov [18]. The analysis is based on the assumption, that the junction points between A and B blocks are confined to a narrow region, which is much smaller than the width of the copolymer layer at the interface. The presence of copolymers at the interface gives rise to two free energy contributions: the entropy of mixing, divided into a translational term and a swelling term, and the elastic energy of stretching of the copolymer blocks. A scenario emerges which distinguishes between four different regimes: In the dilute regime, copolymers aggregate at the interface, but do not yet 3 overlap. The A and B blocks are described as weakly perturbed coils, and the free energy is dominated by the mixing energy. When the mean interchain distance gets of the order of the block radii of gyration, copolymers start to stretch and form a "wet brush". At even higher copolymer concentrations, the "dry brush" regime is entered, where homopolymers do not penetrate into the interfacial region; A and B blocks have the conformation of stretched coils and the free energy is dominated by the elastic energy of stretching. Finally, in the saturated regime, the density of copolymers at the interface is close to one and the interfacial segregation of copolymers competes with the formation of micelles in the bulk [18]. Experimental studies of copolymers at homopolymer interfaces have been carried out by numerous groups, mostly using neutron reflectivity or forward recoil spectroscopy [19]- [25]. By deuterating individual parts of the homopolymers or the copolymers, the excess of copolymers at the interface can be measured as well as distributions of homopolymer segments, copolymer segments or even selected copolymer segments, like the junction point between A and B or the end segment [25,26,27]. These studies have provided detailed insight into the microscopic structure of such interfaces. Simple Leibler type theories are already quite successful in reproducing many of the experimental results [18,24]. In order to reach quantitative agreement, however, the Flory Huggins parameter χ has to be treated as adjustable, molecular weight dependent parameter. Less transparent, but more accurate mean field approaches are the self consistent field theories [28,29] or the density functional theories [30,31]. The self consistent field theory has first been applied to copolymer-homopolymer interfaces by Noolandi and Hong [32] and 4 adjusted to the case of a blend without solvent by Shull and Kramer [33,34]. It has been shown to be quantitatively successful in predicting the correct copolymer excess at the interface, using a χ-parameter which is taken from independent bulk measurements [25]. The calculated width of the segment interface is somewhat too low, but the discrepancies can be understood quantitatively if broadening due to capillary waves is accounted for [22,25]. Such a remarkable success of a mean field theory is characteristic for polymeric substances, in which high molecular weight polymers interact with a high number of other polymers [2,35]. In addition to reproducing experimental data, the self consistent field theory also has the advantage of yielding further structural information, e.g. on chain conformations, monomer orientations etc. [36,45], which may be hard to access experimentally. Unfortunately, the theory also has some serious drawbacks. In particular, the usual treatment of polymers as Gaussian random walks is questionable for chains of the minority component, e.g. A in the B rich phase, and generally on length scales smaller than the screening length of the excluded volume [45]. Computer simulations provide another way of obtaining additional information on the microscopic structure of interfaces. Simulations of inhomogeneous polymeric systems are computationally extremely demanding, and therefore rare. Minchau et al [37] and Fried and Binder [38,39] have investigated the phase behavior of pure copolymer systems. Pan et al studied microphase structures in systems of short copolymers, which are swollen by a small amount (volume fraction 10 %) of longer homopolymers [40]. Wang and Mattice have studied the adsorption of self avoiding copolymers at a stationary, sharp interface, modelled by an external field with a sharp kink [41]. A similar study 5 has been performed for a random copolymer by Peng et al [42]. More detailed simulations have been presented by Cifra [43]. In this work, we present a study of copolymers at the interface between homopolymer phases, where both homopolymers and copolymers are treated in microscopic detail. Pure homopolymer interfaces in immiscible A/B blends have been analyzed previously [44] and compared to the predictions of a self consistent field theory [45]. Here, we consider the effect of adding a small number of symmetric diblock copolymers to such a "known" homopolymer interface. We restrict ourselves to the diluted case, where the copolymer coils do almost not overlap, and where the static structure of the interface is essentially that of a pure homopolymer interface. The results are compared to self consistent field theory calculations for a single copolymer in a homopolymer interface. Our paper is organized as follows: The next section describes the simulation model and method, and ends with a few comments on the self consistent field theory. The results are presented in section three. In particular, we discuss segment density profiles, chain and bond orientations, and profiles for the number of interacting contacts between monomers. We summarize and conclude in the last section. The Bond Fluctuation Model With the presently available computational resources, molecular modelling of the phase behavior in polymer blends in atomistic detail is far beyond feasibility. Fortunately, many important features of such systems are already apparent in coarse grained models [46], such as the bond fluctuation model on a cubic lattice [47]. The latter models polymers as chains of N effective monomers, 6 which occupy each a cube of 8 neighboring sites and are connected by bond vectors of length 2, √ 5, √ 6, 3, or √ 10 in units of the lattice spacing a 0 . One such cube represents a group of n ≈ 3 − 5 chemical monomers. Hence a total chain length of 32, as has been used here, corresponds to a degree of polymerization of approx. 100-160 in a real polymer. At volume fraction φ = 0.5a −3 0 or monomer density ρ = 1/16 a −3 0 , the model reproduces many important properties of dense polymer melts, e.g. single chain configurations show almost ideal Gaussian chain statistics, the single chain structure factor follows a Debye function, and the collective scattering function has the experimental form [48]. The relative repulsion between monomers of type A and B is modelled by introducing symmetric energy parameters ε AA = ε BB = −ε AB = −k B T ε, which describe the pairwise interaction between monomers at distances of less than √ 6 a 0 . At ε = 0.1, the interactions between monomers are dominated by the effect of excluded volume, and the mixture can be described as a weakly perturbed athermal melt; in particular, the equation of state and the compressibility are almost not affected by the presence of the interactions [49]. From extensive previous study of this model, the relation of these model parameters to commonly used parameters in polymer theories is well known. At chain length N = 32, the statistical segment length b is given by b = 3.05 a 0 [44], the radius of gyration is R g = N/6 b ≈ 7 a 0 , the compressibility is k B T κ = 3.9 a 3 0 [49], and the Flory Huggins parameter χ can be calculated using χ = 2zε, where z = 2.65 is the effective coordination number in the bulk, i.e., the average number of interchain contacts of a monomer [50]. The interfacial properties were studied in a L × D × L geometry at system size D = 64 and L = 512. The dimensions of the system were chosen such that the width D of the slab is much larger than the gyration radius R g , i.e., almost ten times as large. The boundary conditions are periodic in x and z direction and "antiperiodic" in the y direction, i.e., A-chain parts leaving the right part of the simulation box reenter it on the left side as B-chain parts and vice versa. Since no mechanism fixes the interface at a certain position, the interface position is subject to diffusion due to thermal fluctuations. We choose the coordinate system such that the origin of the y-axis is at the center y 0 of the interfacial profile, which we determine via [51] | y 0 +20 y 0 −20 m(y)| = min.(1) Here m = ρ A − ρ B is the order parameter of the demixing transition and the relative monomer densities are defined by ρ A,B = φ A,B /φ, where φ A and φ B are the volume fractions taken by A and B monomers. The simulation box contains 32768 polymers of equal chain length 32. As initial configuration, we choose a relaxed configuration of a pure homopolymer interface [44], randomly pick 1024 chains with the center of mass at distances of less than ±δ from the interface with δ = 3 or 9, and turn them into copolymers. No effect of the choice of δ on the results has been found. The simulation algorithm involves random hopping of randomly chosen monomers by one lattice unit with Metropolis probability, but no grandcanonical moves, i.e., the number of copolymers remains fixed. After an initial equilibration time of 2.5 · 10 5 attempted moves per monomer (AMM), the concentration of copolymers in the bulk at ε = 0.1 is 0.05% ± 0.01% -estimates from grand canonical bulk simulations suggest that it should be around 0.04% [52]. We average over 86 configurations in total, where the data for averaging are taken every 10 4 AMM. The area covered by one copolymer can be roughly estimated by πR 2 g,b , where R 2 g,b = b 2 N/12 is the gyration radius of one copolymer block. Hence 1024 copolymers cover approximately 30% of the total interface area 512 × 512, i.e., the system is well in the diluted regime. We close this section with a brief comment on the self consistent field calculations. In a previous paper, we have compared the properties of homopolymer interfaces with the predictions of self consistent field theories for completely flexible and semiflexible chains with different chain rigidities [45]. The qualitative agreement between theory and simulation was over all very good. The main quantitative discrepancy was found in the interfacial width -at chain length 32, the self consistent field theory underestimates the profile widths by at least a factor of 2. This effect could not be explained by capillary waves alone, but seemed to be a consequence of the short chain length. At temperatures sufficiently below the critical point, so that critical fluctuations do not affect the interface any more, the interfacial width gets already comparable to the screening length of the excluded volume, on which length scale chains can not be treated as pure random walks. This leads to wrong predictions of the size of the interface. However, theory and simulations still agree quantitatively for other quantities, e.g. the reduction of the total density at the interface. Our present calculations are based on this work. We consider single wormlike copolymer chains at an interface of wormlike homopolymers. Chains are represented by space curves r(s) with s varying from 0 to 1, and the single chain partition function of a copolymer is given by Z = D{ r(·)} exp − 1/2 0 dsW A ( r(s)) − 1 1/2 dsW B ( r(s)) ,(2) where W i ( r) is the self consistent field acting on a monomer of type i in a homopolymer interface. Each space curve is assigned a statistical weight in 9 the functional integral, D{ r(·)} = D{ r(·)}P W { r(·)} with[53] P W { r(·)} = N s δ( u 2 − 1) exp[− η 2N 1 0 ds| d u ds | 2 ],(3) where a is the fixed monomer length, η a dimensionless stiffness parameter, u = d r ds /(Na) the dimensionless tangent vector constrained to unity by the delta function, and N the normalization factor. Here we choose η = 0.5, the value which best reproduces bond orientations in pure homopolymer interfaces in the bond fluctuation model [45]. The self consistent fields W i for such an interface of semiflexible homopolymers have previously been determined numerically in Ref. [45], based on the Helfand type free energy functional βF = ρ d r{χΦ A Φ B + 1 2ρk B T κ (Φ A + Φ B − 1) 2 }(4) with the total bulk monomer density ρ, the relative monomer densities Φ i ( r) = ρ i ( r)/ρ (i = A or B) , the Flory Huggins parameter χ, and the compressibility κ. The distribution of copolymer segments is calculated by solving appropriate diffusion equations for the end segment distributions Q( r, u; s) and Q + ( r, u; s) for chain parts of length sN < N, which begin on the A side (Q) or the B side (Q + ) of the copolymer [54,55]. From those one can calculate the density of monomers at position s with orientation u via Φ( r, s) = Q( r, s)Q + ( r, 1 − s) or Φ( r, u) = Q( r, u, s)Q + ( r, − u, 1 − s). The numerical treatment is facilitated by expanding the functions Q, Q + in Legendre polynomials and including only the three lowest moments [45]. Results All results presented here were obtained at ε = 0.1 or χN = 17, i.e., the system is far from the critical point ((χN) c = 2.4 [48,50]) in the strong segregation regime. Simulations were also performed at ε = 0.05; the results are qualitatively the same, but the effects are less marked. In the following, lengths are given in units of w SSL = b/ √ 6χ, the interfacial width of a homopolymer interface in the mean field strong segregation limit. The radius of gyration in these units is R g = 4.2 w SSL , and the slab thickness width also leads to quantitative discrepancies in the distribution of copolymer segments. However, the qualitative agreement is very good, and in the wings of the profile one even reaches quantitative agreement. 12 D = 37.4 w SSL . Density profiles and segment distributions Chain and bond orientations Next we discuss the orientational properties of the polymers. It is instructive to consider separately the orientations of single bonds, of chain segments, and of whole chains. Chains with rather weakly oriented single bonds can still be strongly oriented as a whole, as found by Müller et al for homopolymers at a homopolymer interface [44,45]. The orientation of whole chains involves two different factors. First, chains may be oriented without volume changes, i.e., the total gyration radius or end-to-end vector remains unaffected by the orientation. A weak orienting field is sufficient to bring about orientation of this kind. Second, chains may get compressed or stretched in one direction. As Figure 4 illustrates, the latter effect dominates close to an interface. The mean squared components of the end-to-end vector in directions parallel (x,z) and perpendicular (y) to the interface are shown for homopolymers and copolymers. The components parallel to the interface hardly vary throughout the system. Perpendicular to the interface, the end-to-end vector of homopolymers is reduced in the interfacial region, as is already the case in pure homopolymer systems. Homopolymers are thus squeezed perpendicular to the interface, and get effectively oriented parallel to the interface. Copolymers show the inverse behavior, they stretch in the direction perpendicular to the interface. As obtained both from self consistent field calculations and from the simulations, the effect is very strong for copolymers centered at about one to two radii of gyration away from the interface, and much weaker for those located in the wings of profile or at the middle of the interface (Figure 4). One can picture the latter as consisting of two almost independent 13 homopolymer blocks, which hardly feel the effect of being linked together at one end. Indeed, the vector connecting the ends of the single A or B blocks is on average hardly oriented (Table 1). Only the blocks centered deep in their majority phase (y/w SSL ≈ 7) stretch perpendicular to the interface, since they are pulled towards the interface by the other copolymer end ( Figure 5). Note that Figure 5 (Table 1). In sum, single blocks are mostly not oriented at all; the perpendicular orientation of whole copolymers results from the arrangement of the two constituent blocks. Diblock copolymers can be pictured as dumbbells [56] consisting of two mildly perturbed homopolymer coils. Similar copolymer shapes have already been found by Binder and Fried in simulations of block copolymers in the disordered phase, far above the ordering transition [39]. Consequently, one would expect that the only region, where the local conformation of a diblock differs significantly from a homopolymer conformation, is the region close to the link which connects the two blocks. We study this in more detail by looking at the orientation of single bonds b. In order to do so, it is useful to define a bond orientation parameter q( r) = b 2 y − 1 2 ( b 2 x + b 2 z ) b 2 .(5) Negative q implies orientation parallel to the interface, positive q perpendicular orientation. The orientations of homopolymer bonds are shown in Figure 6. Like whole 14 chains, but to a much lesser extent, bonds tend to align themselves parallel to the interface. Unlike the end-to-end vector of whole chains, the average squared bond length b 2 varies by less than 0.15% throughout the system (not shown here), hence the bond length is quasi fixed, the bonds get oriented without compression. The effect is strongest in the middle of the chains and qualitatively the same, but weaker for end bonds. The bond orientation parameter can be compared to the average orientation of the tangent vector u predicted by the self consistent field theory q = 1 2 (3 u 2 y − 1). The results show the same trend as the simulation data for the middle bonds, but do not predict any effect for the end bonds ( Figure 6, inset), unlike what is seen in the MC simulations. This reflects the difference between the tangent vector at the end of a continuous space curve, as assumed by SCFT, and an end bond connecting two discrete statistical monomers of a relatively short chain, as it is the case in our MC simulation. In contrast to homopolymer chains, MC data reveal a rather strong dependence of the orientation of bonds in copolymers on their position within the chain (Figure 7a). The bond which links the two blocks is preferably oriented perpendicular to the interface -the stronger, the further away its location from the center of the interface. Already the bonds next to the link bond show a much smaller effect. The larger the distance along the chain from the link bond, the stronger the tendency of parallel alignment at the center of the interface becomes, and the further it reaches out into the wings of the profile. Hence bonds in the middle and at the end of an A or B block behave very much like homopolymer bonds. We note en passant that, compared to the other bonds, the link bond is stretched by ∼ 4% due to the relative repulsion 15 between the adjacent A and B monomers. The results obtained by SCFT show the same trends as the simulation data (Figure 7b), and agree qualitatively, but the same remarks as above apply. Thus, SCFT does not reproduce the slight tendency of parallel alignment which is found in the MC data at the ends of the blocks. The orientational parameter of the link bond at the center of the profile is q ≈ +0.025 according to both MC results and SCFT. However, SCFT underestimates the average orientation (averaged over link bonds in the region shown in Figure 7a) by ∼ 30%. At distances of several gyration radii R g from the interface, the bond orientation parameter q drops back to zero, indicating that copolymers that deep in the bulk are oblivious to the effect of the interface. Note that two different length scales are reflected in the profiles of q. One of them, the width of the homopolymer interface, fixes the width of the central dip in the profiles, the other one, the gyration radius, determines the overall width of the region with nonzero q. Self and Mutual Contacts Another instructive quantity in the bond fluctuation model is the number of monomer "contacts", i.e., the number of interacting monomer pairs. It is expedient to distinguish between "self contacts", i.e., contacts between monomers belonging to the same chain ("intrachain contacts"), and "interchain contacts" of monomers from two different chains. The self contacts provide additional information on the conformation of single chains, and the interchain contacts on the arrangement of chains relative to each other. Figure 8 compares the number of self contacts per monomer N i,self /ρ i in homopolymer (i = h) and copolymer (i = c) chains. The main contribution to N i,self /ρ i comes from the two direct neighbors of a monomer in the chain, thus N i,self /ρ i is mostly slightly larger than 2. Note however that direct neighbors do not always interact with each other, since the maximum bond length √ 10a 0 is larger than the range of the interactions √ 6 a 0 . The number of self contacts in homopolymers is almost constant throughout the system. It is slightly enhanced in the wings of the profile and decreases again at the center, where A and B meet -suggesting that homopolymer chains tend to loop away from the interface, such that there is no room for intrachain contacts at the center, but that the number of contacts increases right next to it. In copolymers, one finds the opposite behavior -the relative number of intrachain contacts is higher at the center of the interface than in homopolymers, but diminishes rapidly as one moves away from the interface. This trend is promoted by three factors. First, far from the interface copolymers stretch towards the interface, and the number of self contacts associated with backfolding goes down. Second, the stretching goes along with a higher occupation of the (0, 3, 0) bond, which is outside of the interaction region, hence monomers gradually lose contact to their direct neighbors. Third, the further away one moves from the interface, the greater becomes the contribution of chain end monomers, which only have one direct neighbor in the chain. Obviously, it would be highly desirable to study separately the contributions to N i,self /ρ i from monomers which are direct neighbors in a chain, and from monomers which are further apart from each other. Unfortunately, such a distinction was not possible due to limited memory space. Whereas the number of self contacts reflects the conformational properties of a chain, the demixing of A and B chains in the melt is essentially driven by the interchain contacts. The number of interchain contacts per monomer can be interpreted as an effective coordination number z ef f . In homogeneous systems, the identification χ = 2z ef f ε makes contact with the Flory-Huggins theory [50]. In inhomogeneous systems, the effective coordination number is position dependent. As long as the finite range of interactions is neglected, mean field theory simply asserts that it is proportional to the local density of monomers: z i,ef f ( r) = N i,inter ( r) ρ i ( r) ∝ ρ( r)(6) A more elaborated mean field approach predicts for a system which is inhomogeneous in one direction y [57], z ef f ∝ ρ(y) + 1 2 k 2 d 2 dy 2 ρ(y) with k 2 = d rγ( r)V ( r)y 2 d rγ( r)V ( r) ,(7) where V ( r) is the integrable part of the interaction potential (i.e., excluding the hard core part) and γ( r) the normalized pair correlation function. Since the interaction region extends no further than |y| ≤ 2a 0 , an upper bound for the factor k 2 is given by k 2 ≤ 4a 2 0 . Figure 1 shows that ρ ′′ (y)/ρ(y) ≤ 0.01/a 2 0 . Hence the expected deviation from the simple proportionality law (6) is of order ≤ 2%. Furthermore, one would expect that the effective coordination number does not depend on whether a monomer belongs to a copolymer or a homopolymer. In contrast to these considerations, the Monte Carlo data reveal a much lower dip of the reduced coordination number z ef f (y)/ρ(y) (Figure 9) at the center of the interface. The discrepancy between mean field assumption and simulation data becomes even more manifest when looking at the number 18 of contacts N AB between A and B monomers. From the usual mean field assumption, one would expect N AB ∝ ρ A · ρ B (Figure 10). The simulation data however show that N AB /(ρ A ρ B ) is increased by a factor of 1.5, compared to the bulk value, at the distance of approximately two radii of gyration from the interface, where the chains are stretched (cf. Figure 5). At the center of the interface, on the other hand, it is decreased by 25%. One can deduce that chains are compactified in the interfacial region. Interestingly, this holds particularly for copolymer chains. We recall that the number of intrachain contacts in copolymers is enhanced at the interface, whereas the interchain contacts are obviously suppressed. Thus copolymers offer relatively little "active surface" for interaction with external monomers in the interfacial region. In contrast, the relative number of external contacts increases further away from the interface. Hence the active surface per monomer will presumably increase with the copolymer chain length. This may be an additional reason why short copolymers are relatively poor amphiphiles. Summary We have studied the ternary system of A and B homopolymers and symmetric AB diblock copolymers close to an A/B interface in the dilute regime, where copolymer coils do almost not overlap with each other, by Monte Carlo simulations of the bond fluctuation model. In this regime, the structural properties of the interface are not altered with respect to the pure homopolymer interface -in particular, we find no significant broadening of the interfacial width, as is expected at higher copolymer concentrations. According to our simulation results, copolymers resemble oriented dumbells with the A block sticking into the A rich phase, and the B block sticking into the B rich phase. The conformations of single blocks are not very different from conformations of pure homopolymer coils. Single blocks tend to orient themselves parallel to the interface, like homopolymers, whereas copolymers as a whole are oriented perpendicular to the interface. When looking at the orientations of single bonds, we find that the orientational properties of the central bond, which links the two blocks, differ distinctly from those of a homopolymer bond, but that already the neighboring bonds behave very similar to homopolymer bonds. Hence the chain loses the memory of the link very rapidly. Our results are reproduced qualitatively, but not quantitatively, by a self consistent field calculation. Furthermore, we found that copolymers are unusually compact in the interfacial region, i.e., monomers have more contacts to monomers of the same chain, and fewer contacts to monomers of different chains than monomers in the bulk phase. As a result, the copolymer-copolymer interchain contacts are particularly suppressed at the interface. This should become important as the concentration of copolymers is increased. Future work will be concerned with the structure of interfaces with higher content of copolymers and with the transition between different regimes, from the dilute to the wet brush to the dry brush regime, until the two phase region breaks down and a lamellar phase emerges. Table Table 1 Results are shown for homopolymers and copolymers, and compared to the prediction for copolymers of the self consistent field theory. ρ 1/2 / ρ sb ρ 1/4 / ρ sb ρ / ρ b ρ e / ρ sb Figure 1 1shows profiles of A and B monomer densities in systems with and without copolymers, profiles of just the A and B blocks of the copolymers, and total density profiles. As expected in the diluted regime, the A and B monomer density profiles are hardly affected by the presence of the copolymers. The interface as a whole is still very similar to a pure homopolymer interface. Due to the finite compressibility of the blend, the total density is slightly reduced in the interfacial region[44]. The distributions of copolymer monomers A and B agree qualitatively with the experimental results and confirm the simple picturepresented in the introduction: Monomers of type A are more concentrated in the A-rich phase, Monomers of type B in the B rich-phase. The distribution is rather broad, a fair portion of the A monomers sticks into the B-rich phase and vice versa. Distributions of single chain segments are shown in Figure 2 and 3. For homopolymers, one finds a relative enrichment of chain ends ρ e at the inter-face, whereas the concentration ρ 1/2 of middle segments (the 16th and 17thmonomer) is comparatively low there. The total density profile ρ h is best reproduced by the density profiles ρ 1/4 of the segments at one and three fourth of the chain (the 8th and 25th monomer). Looking at the higher concentration of chain ends at the interface, one is lead to suspect that homopolymers tend to form loops with two ends at the interface. However, this is not the case, as can be seen from the distribution ρ e−e of midpoints between the two ends of homopolymers. It is strongly reduced at the center of the interface and enhanced at the distance of one gyration radius from there (Figure 2).Copolymer segment densities show the inverse trend: The middle segments concentrate at the interface, whereas the chain ends stretch out into their favorite bulk phase. Hence our results are in qualitative agreement with the experimental findings of Russell et al[25]. As in the case of homopolymers, the total density profiles of A and B monomers are almost identical with the distribution of the 8th and 25th monomer, respectively, i.e., the distribution of segments in the middle of the A or B block. Note that the peak of the distribution of middle segments ρ 1/2 is relatively broad, broader than the radius of gyration, hence they are not strongly confined to the interface as assumed by the Leibler theory (Figure 3). The results of the self consistent field theory are shown in the inset. The fact that the SCFT underestimates the interfacial also demonstrates, how minority blocks (A blocks in the B phase, B blocks in the A phase) slightly shrink by a factor of approximately 0.8 due to the hostile environment. The vector D AB connecting the centers of mass of the A and B blocks is on average oriented and strongly stretched in the negative y direction : Orientational properties of copolymer chains. Averages are taken over all 1024 copolymers and over 86 independent configurations. Compared are x, z components (together) with y components. Lengths are given in units of w SSL .i = x, z i = y end-to-end vector: A and B blocks ( Figure 1 : 1Monomer density profiles as a function of the distance from the center of the interface y, in units of w SSL = b/ √ 6χ = 1.71 a 0 . Profiles are shown for the density of all monomers (ρ), of A and B monomers separately (ρ A and ρ B ), of just homopolymer monomers (ρ h ), and of A and B monomers belonging to a copolymer block (ρ A,c and ρ B,c ). Also shown for comparison are A and B monomer profiles in a pure homopolymer system (ρ o A and ρ 0 B , taken from [44]). Figure 2 : 2Homopolymer segment density profiles vs. y/w SSL in units of w SSL = b/ √ 6χ. Profiles are shown for the density of monomers in the middle of the chain (ρ 1/2 ), at the end of the chain (ρ e ), at one and three fourth of the chain (ρ 1/4 ), and of all homopolymer monomers (ρ h ). Also shown is the distribution of midpoints between the two ends of homopolymers, ρ e−e . Units of densities are the total bulk concentration of monomers ρ b , or ρ sb = ρ b /16 (as indicated). Figure 3 : 3Copolymer segment density profiles vs. y/w SSL in units of w SSL = b/ √ 6χ = 1.71a 0 . Profiles are shown for the density of A and B monomers in the middle of the chain (ρ 1/2 , squares), at the end of the chain (ρ e , circles), at one and three fourth of the chain (ρ 1/4 ,diamonds), and of all copolymer monomers (ρ c , broken line). Inset shows results of the self consistent field theory for A monomers. Full lines show the predictions for segment density profiles ρ 1/2 , ρ e , ρ 1/4 , broken line shows total density profile ρ c , and symbols compare with MC results (symbols like above).Units of densities are ρ b /2, or ρ sb = ρ b /32 (as indicated). Figure 4 : 4Mean square end-to-end vector components R 2 i with i = x, y, z in units of the average bulk value b 2 N/3, plotted vs. the distance of the center of the end-to-end vector from the interface y in units of w SSL = b/ √ 6χ. Figure 5 : 5Mean square end-to-end vector components R 2 i (i = x, z, or y) of the copolymer blocks in their minority phase (A block in B phase, B block in A phase), and majority phase (A block in A phase, B in B phase), in units of the average bulk value b 2 N/6, plotted vs. the distance of the center of the end-to-end vector from the interface y in units of w SSL = b/ √ 6χ. Figure 6 : 6Orientational order parameter q for end bonds (dashed line) and middle bonds (the bonds connecting the 16th and 17th monomer, solid line) in homopolymers, vs. y/w SSL in units of w SSL = b/ √ 6χ. Inset shows the prediction of the self consistent field theory. Figure 7 : 7(a) Orientational order parameter q for end bonds, link bonds (linking 16th and 17th monomer), bonds next to link bonds (15th to 16th monomer and 17th to 18th monomer), and bonds in the middle of a block (8th to 9th monomer and 24th to 25th monomer) in copolymers, vs. y/w SSL in units of w SSL = b/ √ 6χ. (b) Prediction for q of the self consistent field theory for s = (0, 1) (end bonds), s = 0.5 (link bonds), s = (0.25, 0.75) (block middle bonds). Figure 8 : 8Number of intrachain contacts per monomerN i,self /ρ i (y) (i = h,c)vs. y/w SSL in units of w SSL = b/ √ 6χ for homopolymer and copolymer chains. Figure 9 : 9Normalized effective coordination number z ef f (y)/ρ(y) vs. y/w SSL in units of w SSL = b/ √ 6χ, for homopolymer and copolymer chains. Figure 10 : 10Normalized number of AB contacts N AB /(ρ A (y)ρ B (y)), and number of AB contacts per monomer N AB /ρ(y) (inset), vs. y/w SSL in units of w SSL = Figure 2 Werner et al Figure 3 Werner et al Figure 4 Werner et al Figure 5 Werner et al Figure 7a Werner et al Figure 7b Werner et al Figure 8 Werner et al Figure 9 Werner et al Polymer Compatibility and Incompatibility -Principles and Practices; Harwood: Chur. See, D R Paul, S Newman, D S Walsh, J S Higgins, A Maconnachie, Polymer Blends and Mixtures. 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[]
[ "IDENTIFYING IONIZED REGIONS IN NOISY REDSHIFTED 21 CM DATA SETS", "IDENTIFYING IONIZED REGIONS IN NOISY REDSHIFTED 21 CM DATA SETS" ]
[ "Matthew Malloy ", "Adam Lidz " ]
[]
[]
One of the most promising approaches for studying reionization is to use the redshifted 21 cm line. Early generations of redshifted 21 cm surveys will not, however, have the sensitivity to make detailed maps of the reionization process, and will instead focus on statistical measurements. Here we show that it may nonetheless be possible to directly identify ionized regions in upcoming data sets by applying suitable filters to the noisy data. The locations of prominent minima in the filtered data correspond well with the positions of ionized regions. In particular, we corrupt semi-numeric simulations of the redshifted 21 cm signal during reionization with thermal noise at the level expected for a 500 antenna tile version of the Murchison Widefield Array (MWA), and mimic the degrading effects of foreground cleaning. Using a matched filter technique, we find that the MWA should be able to directly identify ionized regions despite the large thermal noise. In a plausible fiducial model in which ∼ 20% of the volume of the Universe is neutral at z ∼ 7, we find that a 500-tile MWA may directly identify as many as ∼ 150 ionized regions in a 6 MHz portion of its survey volume and roughly determine the size of each of these regions. This may, in turn, allow interesting multi-wavelength follow-up observations, comparing galaxy properties inside and outside of ionized regions. We discuss how the optimal configuration of radio antenna tiles for detecting ionized regions with a matched filter technique differs from the optimal design for measuring power spectra. These considerations have potentially important implications for the design of future redshifted 21 cm surveys. Subject headings: cosmology: theory -intergalactic medium -large-scale structure of Universe
10.1088/0004-637x/767/1/68
[ "https://arxiv.org/pdf/1212.2656v3.pdf" ]
119,190,795
1212.2656
f7cb8673f1c8f32ac38d4e7858fcea04531fb0d7
IDENTIFYING IONIZED REGIONS IN NOISY REDSHIFTED 21 CM DATA SETS Matthew Malloy Adam Lidz IDENTIFYING IONIZED REGIONS IN NOISY REDSHIFTED 21 CM DATA SETS May 5, 2014. To be submitted to ApJ. May 5, 2014. To be submitted to ApJ.Preprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: cosmology: theory -intergalactic medium -large-scale structure of Universe One of the most promising approaches for studying reionization is to use the redshifted 21 cm line. Early generations of redshifted 21 cm surveys will not, however, have the sensitivity to make detailed maps of the reionization process, and will instead focus on statistical measurements. Here we show that it may nonetheless be possible to directly identify ionized regions in upcoming data sets by applying suitable filters to the noisy data. The locations of prominent minima in the filtered data correspond well with the positions of ionized regions. In particular, we corrupt semi-numeric simulations of the redshifted 21 cm signal during reionization with thermal noise at the level expected for a 500 antenna tile version of the Murchison Widefield Array (MWA), and mimic the degrading effects of foreground cleaning. Using a matched filter technique, we find that the MWA should be able to directly identify ionized regions despite the large thermal noise. In a plausible fiducial model in which ∼ 20% of the volume of the Universe is neutral at z ∼ 7, we find that a 500-tile MWA may directly identify as many as ∼ 150 ionized regions in a 6 MHz portion of its survey volume and roughly determine the size of each of these regions. This may, in turn, allow interesting multi-wavelength follow-up observations, comparing galaxy properties inside and outside of ionized regions. We discuss how the optimal configuration of radio antenna tiles for detecting ionized regions with a matched filter technique differs from the optimal design for measuring power spectra. These considerations have potentially important implications for the design of future redshifted 21 cm surveys. Subject headings: cosmology: theory -intergalactic medium -large-scale structure of Universe INTRODUCTION The Epoch of Reionization (EoR) is the time period when early generations of galaxies first turn on and gradually photoionize neutral hydrogen gas in the surrounding intergalactic medium (IGM). The IGM is expected to resemble a two phase medium during reionization. One phase consists of highly ionized regions, termed 'ionized bubbles', that form around clustered groups of ionizing sources, while the other phase is made up of intervening mostly neutral regions that shrink and eventually vanish as reionization progresses. A primary goal of reionization studies is to determine the size distribution and volume-filling factor of the ionized bubbles. This should, in turn, significantly improve our understanding of high redshift galaxy and structure formation. A wide variety of current observations have started to provide tantalizing hints regarding the timing and nature of the EoR (e.g., Fan et al. 2006, Totani et al. 2006, Dunkley et al. 2009, Ouchi et al. 2010, Bouwens et al. 2012, Mortlock et al. 2011, Zahn et al. 2012, Schenker et al. 2012), but we still await a more detailed understanding. A highly anticipated way of improving our knowledge of the EoR is to directly detect intergalactic neutral hydrogen from the EoR using the redshifted 21 cm transition (e.g., Madau et al. 1997, Zaldarriaga et al. 2004. Indeed, several radio telescopes have been constructed, or are presently under construction, in effort to detect this signal, including the Giant Metrewave Radio Telescope (GMRT) (Paciga et al. 2011), the Low Frequency Array (LOFAR) (Harker et al. 2010), the Murchison Widefield Array (MWA) (Lonsdale [email protected] (Parsons et al. 2010). This method provides the most direct, and potentially most powerful, way of studying reionization, but several challenges need first to be overcome. In particular, upcoming surveys will need to extract the faint cosmological signal in the presence of strong foreground emission from our own galaxy and extragalactic point sources, and to control systematic effects from the instrumental response, polarization leakage, calibration errors, and other sources of contamination (e.g., Liu et al. 2009, Datta et al. 2010, Harker et al. 2010, Petrovic & Oh 2011, Parsons et al. 2012). In addition, thermal noise will prevent early generations of 21 cm experiments from making detailed maps of the reionization process. Instead, detections will mostly be of a statistical nature (McQuinn et al. 2006). For example, a primary goal of these experiments is to measure the power spectrum of 21 cm brightness temperature fluctuations by binning together many individually noisy Fourier modes (Zaldarriaga et al. 2004, Morales & Hewitt 2004, Bowman et al. 2006, McQuinn et al. 2006. It is unclear, however, how best to analyze the upcoming redshifted 21 cm data. Most previous work has focused only on the power spectrum of 21 cm brightness temperature fluctuations (e.g., Furlanetto et al. 2004a, Lidz et al. 2008). This statistic does not provide a complete description of the 21 cm signal from the EoR, which will be highly non-Gaussian, with large ionized regions of essentially zero signal intermixed with surrounding neutral regions. The power spectrum, and especially its redshift evolution, do encode interesting information about the volume-averaged ionized fraction and the bubble size distribution (e.g., Lidz et al. 2008). However, these inferences are somewhat indirect and likely model dependent, and so it is natural to ask if there are more direct ways of determining the properties of the ionized regions. The approach we explore here is to check whether it may be possible to directly identify ionized regions in noisy redshifted 21 cm observations by applying suitable filters to the noisy data. Our aim here is to blindly identify ionized bubbles across an entire survey volume, rather than to consider targeted searches around special regions, such as those containing known quasars (e.g., Wyithe & Loeb 2004. Since the 21 cm signal from reionization is expected to have structure on rather large scales -30 h −1 Mpc co-moving (Furlanetto et al. 2004b, Iliev et al. 2006) -it may be possible to make crude images of the large scale features even in the regime where the signal to noise per resolution element is much less than unity. Even if it is only possible to identify a few unusually large ionized regions in upcoming data sets, this would still be quite valuable. Any such detection would be straightforward to interpret, and would open-up several interesting possibilities for follow-up investigations. Towards this end, we extend previous work by Datta et al. (2007) and Datta et al. (2008), who considered the prospects for detecting ionized regions using an optimal matched filter. A matched filter is constructed by correlating a known 'template' signal with a noisy data set in order to determine whether the template signal is present in the noisy data. Matched filters are used widely in astrophysics: to name just a few examples, matched filters are used to detect clusters in cosmic microwave background (CMB) data (Haehnelt & Tegmark 1996), to identify galaxy clusters from weak lensing shear fields (e.g., Hennawi & Spergel 2004, Marian et al. 2009), and are central to data analysis efforts aimed at detecting gravitational waves (e.g., Owen & Sathyaprakash 1999). The outline of this paper is as follows. In §2 we describe the mock 21 cm data sets used in our investigations. We use the mock data to first consider the ability of future surveys to make maps of the redshifted 21 cm signal ( §3). In §4, we then quantify the prospects for identifying individual ionized regions using a matched filter technique. In §5 and §6 we consider variations around our fiducial choice of reionization history and redshifted 21 cm survey parameters. We compare with previous related work in §7, and conclude in §8. Throughout we consider a ΛCDM cosmology parametrized by n s = 1, σ 8 = 0.8, Ω m = 0.27, Ω Λ = 0.73, Ω b = 0.046, and h = 0.7, (all symbols have their usual meanings), consistent with the latest WMAP constraints from Komatsu et al. (2011). METHOD Briefly, our approach is to construct mock redshifted 21 cm data sets and check whether we can successfully identify known 'input' ionized regions in the presence of realistic levels of instrumental noise and the degrading impact of foreground cleaning. Here we describe the ingredients of our mock data sets: our simulations of reionization and the 21 cm signal, our model for thermal noise, and our approach for incorporating the impact of foreground cleaning. 2.1. The 21 cm Signal First, let us describe the underlying 21 cm signal and our reionization simulations. The 21 cm signal will be measured through its contrast with the cosmic microwave background (CMB). The brightness temperature contrast between the CMB and the 21 cm line from a neutral hydrogen cloud with neutral fraction x HI and fractional baryon overdensity δ ρ is (Zaldarriaga et al. 2004): δT b = 28x HI (1 + δ ρ ) T S − T γ T S 1 + z 10 1/2 mK. (1) Here T γ denotes the CMB temperature and T S is the spin temperature of the 21 cm line. Here and throughout we neglect effects from peculiar velocities, which should be a good approximation at the redshifts and neutral fractions of interest (e.g., Mesinger & Furlanetto 2007. Furthermore, throughout we assume that the spin temperature is globally much larger than the CMB temperature, i.e., we assume that T S T γ . In this case the 21 cm signal appears in emission and the brightness temperature contrast is independent of T S . This is expected to be a good approximation for the volume-averaged ionized fractions of interest for our present study, although it will break down at earlier times (e.g., Ciardi & Madau 2003). With these approximations, δT b = T 0 x HI (1 + δ ρ ),(2) where T 0 = 28 [(1 + z)/10] 1/2 mK. Throughout this paper, we refer to the brightness temperature contrast in units of T 0 . Semi-Numeric Simulations In order to simulate reionization we use the 'seminumeric' scheme described in Zahn et al. (2007) (see also e.g., Mesinger et al. 2010, for related work and extensions to this technique). This scheme is essentially a Monte Carlo implementation of the analytic model of Furlanetto et al. (2004b), which is in turn based on the excursion set formalism. The Zahn et al. (2007) algorithm allows us to rapidly generate realizations of the ionization field over large simulation volumes at various stages of the reionization process. The results of these calculations agree well with more detailed simulations of reionization on large scales Zahn et al. 2010). We start by generating a realization of the linear density field in a simulation box with a co-moving side length of 1 h −1 Gpc and 512 3 grid cells. The ionization field, x i , is generated following the algorithm of Zahn et al. (2007), assuming a minimum host halo mass of M min = 10 8 M , comparable to the atomic cooling mass at these redshifts (Barkana & Loeb 2001). Each halo above M min is assumed to host an ionizing source, and the ionizing efficiency of each galaxy is taken to be independent of halo mass. In our fiducial model, we adjust the ionizing efficiency so that the volume-averaged ionization fraction is x i = 0.79 at z fid = 6.9. We focus most of our analysis on this redshift and on this particular model for the volume-averaged ionized fraction. However, we consider additional redshifts in §5, as well as variations around our fiducial ionization history in effort to bracket current uncertainties in the ionization history (see e.g., Kuhlen & Faucher-Giguère 2012, Zahn et al. 2012. From the linear density field and the ionization field we generate the 21 cm brightness temperature contrast following Equation 2. Using the linear density field hererather than the evolved non-linear density field -should be a good approximation for the large scales of interest for our study; we focus on length scales of R 20 h −1 Mpc and high redshift (z 6) in subsequent sections. Redshifted 21 cm Surveys and Thermal Noise We mostly consider two concrete examples of upcoming redshifted 21 cm surveys. The first is based on the current, 128-tile version of the MWA (Tingay et al. 2012) and the second is based on an expanded, 500-tile version of the MWA (as described in Lonsdale et al. 2009, and considered in previous work such as McQuinn et al. 2006;Lidz et al. 2008). These two examples are intended to indicate the general prospects for imaging and bubble identification with first and second generation 21 cm surveys, respectively. Similar considerations would apply for other experiments, but we choose these as a concrete set of examples. We mainly focus on the 500-tile configuration in this paper because of its greater sensitivity. In §5.4, we shift to considering 128-tile configurations and in §6, we consider a LOFAR-style interferometer for comparison. Hereafter, we refer to the 500-tile configuration as the MWA-500 and the 128-tile version as the MWA-128. Throughout this paper, we work in co-moving coordinates described by Cartesian labels (x-y-z), with Fourier counterparts (k x -k y -k z ). The Fourier modes can be connected directly with the u-v-ν coordinate system generally used to describe interferometric measurements. Here u and v describe the physical separation of a pair of antennae in units of the observed wavelength, while ν describes the corresponding observed frequency. The instrument makes measurements for every frequency, ν, in its bandwidth, and for every antenna tile separation, (u, v), sampled by the array. In order to shift to a Fourier space description, the interferometric measurements must first be Fourier-transformed along the frequency direction. With our Fourier convention, the relation between the two sets of coordinates is given by: k x = 2πu D k y = 2πv D k z = 2π ∆χ ,(3) where D is the co-moving distance to the survey center and ∆χ is the co-moving distance corresponding to a small difference in observed frequency of ∆ν (e.g., Liu et al. 2009). For small ∆ν/ν, we can express ∆χ as ∆χ ≈ c(1 + z fid ) H(z fid ) |∆ν| ν ,(4) where H(z fid ) is the Hubble parameter at the fiducial redshift, and |∆ν|/ν is the absolute value of the fractional difference between two nearby observed frequencies. In order to test the prospects for imaging and bubble identification with the MWA, we must corrupt the underlying 21 cm signal described in §2.2 with thermal noise. We do this by generating a Gaussian random noise field in the k-space coordinate system described above, using an appropriate power spectrum. We assume that the covariance matrix of the thermal noise power is diagonal in k-space. We add the resulting noise field to the underlying 21 cm signal (Equation 2). The power spectrum of the thermal noise is given by (McQuinn et al. 2006, Furlanetto & Lidz 2006: P N (k, µ) = T 2 sys Bt int D 2 ∆D n(k ⊥ ) λ 2 A e 2 .(5) Here µ is the cosine of the angle between wavevector k = |k| and the line of sight, so that k ⊥ = 1 − µ 2 k is the transverse component of the wavevector. We assume a system temperature of T sky = 280 [(1 + z)/7.5] 2.3 K (Wyithe & Morales 2007) and a total observing time of t int = 1000 hours, which is an optimistic estimate for the observing time in one year. At our fiducial redshift of z fid = 6.9, the co-moving distance to the center of the survey is D = 6.42 × 10 3 h −1 Mpc. In this equation, λ denotes the observed wavelength of the redshifted 21 cm line, λ = 0.211(1 + z) m, and A e is the effective area of each antenna tile. We determine A e by linearly extrapolating or interpolating from the values given in Table 2 of Bowman et al. (2007); the effective area at z fid = 6.9 is A e = 11.25 m 2 . We assume that the full survey bandwidth of 32 MHz is broken into individual blocks of bandwidth B = 6 MHz to protect against redshift evolution across the analysis bandwidth (McQuinn et al. 2006). The co-moving survey depth depth corresponding to a B = 6 MHz chunk is ∆D = 69 h −1 Mpc. The n(k ⊥ ) term describes the configuration of the antenna tiles. More specifically, it is the number density of baselines observing modes with transverse wavenumber k ⊥ (McQuinn et al. 2006). Following Bowman et al. (2006) and McQuinn et al. (2006), we assume the antenna tiles are initially packed as closely as possible in a dense compact core, and that the number density of antenna tiles subsequently falls off as r −2 out to a maximum baseline of 1.5 km. The radius of the dense antenna core is set by the requirement that the antenna density falls off as r −2 outside of the core, and that it integrates to the total number of antennae. For the MWA-500, this gives r c = 20 m, while for the MWA-128, the core radius is r c ≈ 8 m. Equation 5 gives the noise power spectrum in units of mK 2 , and so we divide by T 2 0 to combine with the simulated 21 cm signal expressed in units of T 0 . Note that the volume of the MWA survey differs somewhat from that of our reionization simulation. In particular, the transverse dimension of the simulation is smaller than that of the MWA by a factor of ∼ 3, while the simulation is deeper in the line-of-sight direction by about the same factor, as compared with the full MWA bandwidth. However, we remove the long wavelength modes along the line-of-sight direction to mimic foreground cleaning ( §2.4), and so we do not, in practice, use the longer lineof-sight scales in our simulation box. As we will see, the ionized regions in the simulation are substantially smaller than the transverse length of the box. Transverse slices should therefore be representative of what the actual MWA will observe from a fraction of its larger field of view. We have checked that the coarser transverse k-space sampling in the simulation compared to in the actual MWA survey does not impact our results. Foregrounds Next, we need to consider contamination from foreground emission at the frequencies of interest. The relevant foregrounds include diffuse Galactic synchrotron radiation, extragalactic point sources, and Galactic Bremsstrahlung radiation. While these foregrounds are many orders of magnitude brighter than the cosmological 21 cm signal, they are expected to individually follow smooth power laws in frequency. Over a sufficiently small frequency range, the summed contributions can also be approximated as following a smooth power law, while the 21 cm signal will vary rapidly. This allows the foregrounds to be removed from the data by, for example, fitting a low-order function along each line of sight and subtracting it. While this procedure is effective at removing foreground contamination, it also removes long wavelength modes along the line of sight from the signal itself, and hence prevents measuring these modes. Several related methods for foreground removal have been discussed in the literature (e.g., Wang et al. 2006, Harker et al. 2009, Petrovic & Oh 2011, Chapman et al. 2012. In this work, we approximately mimic the degrading effects from foreground removal by subtracting the running mean from the noisy signal along each line of sight, rather than including realizations of the foregrounds in our simulation and excising them with one of the above algorithms. We generally remove the running mean over a bandwidth of 16 MHz, which corresponds to a co-moving distance of L fg = 185 h −1 Mpc at redshift z fid = 6.9; we consider the impact of other choices of L fg in §5.3. We defer more detailed models of foreground contamination, and foreground removal algorithms, to future work. PROSPECTS FOR IMAGING Having described our mock 21 cm data sets, we now turn to consider the prospects for constructing direct 'images' of the redshifted 21 cm signal. Previous studies already suggest that the prospects for imaging with the MWA-500 are limited (McQuinn et al. 2006). Here we emphasize that even a crude, low-resolution image of the redshifted 21 cm signal may be quite interesting, especially given that the ionized regions during reionization may be rather large scale features. We hence seek to quantify the imaging capabilities further using our corrupted reionization simulations. Here our work complements recent work in a similar vein by Zaroubi et al. (2012), who considered the prospects for imaging with LOFAR. While the central idea in this section is similar to this previous work, we focus on the MWA while Zaroubi et al. (2012) considered LOFAR. In order to construct the best possible images from the noisy mock 21 cm data, we apply a Wiener filter. We assess the ability of the MWA to image the redshifted 21 cm sky by comparing the filtered (recovered) noisy signal with the underlying noise-free 21 cm input signal. The Wiener Filter The Wiener filter is the optimal filter for extracting an input signal of known power spectrum when it is corrupted by additive noise, also with known power spectrum. As described in Press et al. (2002), this filter is optimal in that it minimizes the expectation value of the integrated squared error between the estimated signal field and the true signal field. The estimate of the true signal is a convolution of the Wiener filter and the corrupted signal in real space, and so is a product of the two quantities in Fourier space, S(k) = C(k)W (k),(6) where C(k), W (k), andS(k) are the Fourier transforms of the corrupted signal, Wiener filter, and estimated signal, respectively. Requiring that the filter be optimal in the least-square sense results in W (k) taking the form W (k, µ) = P S (k) P S (k) + P N (k, µ) ,(7) where P S (k) and P N (k, µ) are the power spectra of the signal and noise, respectively. We note that, while the signal power spectrum is roughly isotropic 2 , the noise power spectrum depends on µ and consequently so does the filter. The filter keeps a unity weighting for k-modes where P S (k) P N (k) and significantly downweights kmodes where P S (k) P N (k). This can allow for partial recovery of the original signal, provided that the signal power dominates for some k-modes. The Wiener filter requires an estimate of the signal power spectrum, P S (k), and of the total (signal plus noise) power spectrum, P S (k) + P N (k, µ), as inputs. These may not be precisely known. However, since the filter is the outcome of a minimization problem -i.e., it minimizes the expected difference between the estimated and true fields -the accuracy of the filter should be insensitive to small changes about its optimal value. In other words, the accuracy of the filter is not expected to change greatly by using estimates of the signal and noise power spectra rather than the true spectra. Furthermore, we do expect to have an estimate of the underlying signal power spectrum; measuring this statistic is a major goal of redshifted 21 cm surveys. Specifically, the underlying signal power can be estimated by cross-correlating redshifted 21 cm measurements made over two different time intervals (after foregrounds have been removed). The statistical properties of the signal should be identical across the two different time periods, but the thermal noise contributions will be independent. The cross-correlation between two time chunks then provides an unbiased estimate of the signal power (e.g., Liu et al. 2009). Estimates of the noise power spectrum can then be made by subtracting the estimated signal power from the power measured over the entire integration time, which contains both the signal and noise contributions. The Wiener filter does not actually require the noise power spectrum to be known on its own. However, in §4 we consider the optimal matched filter, which does have this requirement. Throughout this study, we assume perfect knowledge of the underlying power spectra. Before applying the Wiener filter to our corrupted simulations, it is useful to estimate the expected signal-tonoise ratio of the filtered maps analytically, using simulated signal power spectra and the noise power spectrum of Equation 5. The expected signal-to-noise ratio of the Wiener-filtered field is S wf =σ S /σ N , whereσ 2 S(N) is the filtered signal (noise) variance. The signal and noise variance can in turn be calculated as integrals over their respective power spectra, σ 2 S(N) = d 3 k (2π) 3 |W (k, µ)| 2 P S(N) (k, µ).(8) Here we use P S(N) to denote the power spectrum of the signal (noise). One can also consider the impact of foreground cleaning here by downweighting modes where the foreground power is large compared to the signal power. In order to consider the dependence of the signal-to-noise ratio on the stage of reionization, we consider simulation outputs in which the volume-averaged ionization fraction is x i = 0.51, 0.68, 0.79 and 0.89. We consider each of these models at our fiducial redshift of z fid = 6.9. 3 Presently, we don't consider still earlier stages of reionization since the prospects for imaging with the MWA-500 are especially poor for lower ionized fractions. The resulting Wiener filters for the different values of x i are shown in Figure 1, after integrating over angle µ. In this figure, foreground cleaning has been accounted for by subtracting a running mean along the line of sight, as described in §2.4. It is helpful to note, from Equation 7, that the filter is equal to 1/2 for modes where the signal and noise power are equal. The figure suggests that a small range of k-modes with k 0.1h Mpc −1 will have signal-to-noise ratio larger than unity for all four ionized fractions considered, although imaging is less promising for the smaller ionized fractions. If Top-Left: Spatial slice of the unfiltered and noise-less 21 cm brightness temperature contrast field (normalized by T 0 ). Top-Right: Simulated signal-to-noise field after applying the Wiener filter to a pure noise field. Bottom-Left: Simulated signal-to-noise field after applying the Wiener filter to the noisy signal. This can be compared with the uncorrupted input signal shown in the top-left panel and the noise realization in the top-right panel. Bottom-Right: Simulated signal-to-noise field after applying the Wiener filter to the noiseless signal. (The filtered noiseless signal shown here is normalized by the standard deviation of the noise to facilitate comparison with the other panels.) All panels show a square section of the MWA field of view transverse to the line of sight with sidelength L = 1 h −1 Gpc . All slice thicknesses are ∼ 8 h −1 Mpc . Unless noted otherwise, the simulation slices in subsequent figures have these same dimensions. the ionized regions are larger than in our fiducial model -as expected if, for example, rarer yet more efficient and more clustered sources dominate reionization (e.g., Mc-Quinn et al. 2007, Lidz et al. 2008) -then the prospects for imaging may improve somewhat. Performing the integrals in Equation 8, while incorporating foreground cleaning, we find that the total signal-to-noise ratio expected for the MWA-500 is S wf = 0.52, 0.79, 1, and 1.2 for x i = 0.51, 0.68, 0.79, 0.89, respectively. Application to a Simulated 21 cm Signal With the analytic signal-to-noise ratio estimates as a guide, we apply the Wiener filter to our mock noisy redshifted 21 cm data. The results of these calculations, for a particular slice through the simulation volume, are shown in Figure 2. The side length (1 h −1 Gpc) of each slice is a factor of ∼ 3 smaller than the transverse dimension of the MWA. One can asses how well the original signal is 'recovered' by comparing the top-left panel of the figure which shows the input signal with the bottom-left panel which shows the filtered noisy signal, after mimicking foreground removal. The two panels do not bear a striking resemblance since the average signal-to-noise ratio is only of order unity. Nonetheless, it is encouraging that many of the minima in the filtered noisy signal do indeed correspond to ionized regions in the input signal. Furthermore, we can compare the filtered noisy signal in the bottom-left panel with the top-right panel, which shows filtered pure noise. While these two panels do not look drastically different, they are easily distinguishable from each other given the increased contrast in the filtered noisy signal. Note that the noisy signal -in the absence of any filtering (not shown) -looks like pure noise since the signal to noise per resolution element is extremely low, and so filtering helps significantly. In addition, we see that the filtered noisy signal obtains signal-to-noise values exceeding 6 − σ, while the statistical significance of the filtered noise does not exceed ∼ 5 − σ. Quantitatively, ∼ 3% (∼ 0.03%) of the volume in the filtered noisy signal is occupied by pixels with statistical significance greater (in magnitude) than 3−σ (5−σ). This is expected given that the filtered data cube has an average signal-to-noise ratio of σ S /σ N ≈ 1, as anticipated in the analytic calculation of §3.1. Comparing the filtered noisy signal and the filtered pure noise, one can see that ionized regions in the underlying signal are diminished if they happen to be coincident with upward fluctuations in the noise, as expected. For example, the ionized region in the bottom-right corner of the unfiltered signal lies very close to a ∼ 3 − σ upward fluctuation in the filtered noise and, as a result, appears with weak statistical significance in the filtered noisy signal. Conversely, some of the most statistically significant regions in the filtered noisy signal occur when ionized regions overlap downward noise fluctuations. We can further compare the filtered noisy signal with the filtered noise-less signal, shown in the bottom right panel of Figure 2. The filtered noise-less signal is normalized by the standard deviation of the filtered noise so that it can be compared with the signal-to-noise slices in the other panels. This comparison reveals that high significance regions ( 5 − σ) in the filtered noisy signal only line up well with the corresponding regions in the filtered noiseless signal if they are coincident with downward fluctuations in the noise. On its own, the filtered noiseless signal only attains statistical significances of 4σ N . Finally, Figure 3 illustrates the impact of foreground cleaning, performed here over a bandwidth of 16 MHz ( §2.4). Foreground cleaning removes the long wavelength modes along the line of sight -which is along the vertical axis in the figure -and thereby compresses structures along the line of sight. However, the cleaning process only impacts the long wavelength line-of-sight modes which still leaves room to image other modes robustly. Note that the slice thickness (8 h −1 Mpc) in Figure 2 and 3 is somewhat arbitrary. However, the Wiener filter smooths out structure on significantly larger scales than this (Figure 1), and so we expect similar results for other values of the slice thickness, provided the slice is thin compared to the cut-off scale of the filter. In practice, of course, one can make many independent maps similar to Figure 2 from the MWA-500 or similar surveys. Collectively, our results mostly confirm previous wisdom; the prospects for imaging with the MWA-500 are limited. Nonetheless, it appears that a signal-to-noise ratio of order unity is achievable, suggesting that the MWA-500 can make low resolution maps of the reionization process. PROSPECTS FOR IDENTIFYING IONIZED REGIONS We now shift our focus to discuss whether it may also be possible to identify interesting individual features in upcoming 21 cm data cubes. In particular, we aim to identify ionized regions in noisy 21 cm data sets and, furthermore, to estimate the spatial center and approximate size of each ionized bubble. For this purpose, we will use an optimal matched filter technique. As we discuss, individual ionized regions may be identifiable as prominent minima in the filtered field. The Optimal Matched Filter The optimal matched filter is suited for the case of a corrupted signal containing a known feature that one would like to extract. The filter acts in Fourier space by cross-correlating the corrupted signal with a template describing the known feature, while downweighting kmodes in the corrupted signal by the noise power. The resulting form of the filter in Fourier space, M (k, µ), is M (k, µ) = T (k) P N (k, µ) ,(9) where T (k) is the Fourier profile of the known feature. The filter is optimal in the sense that it maximizes the signal-to-noise ratio in the filtered data cube at the location of the feature being extracted. While the Wiener filter requires an estimate of the signal and total (signal plus noise) power spectra, the matched filter requires a good estimate of the template profile, T (k), and the noise power spectrum, P N (k, µ). For our present application, we would like templates describing the ionized regions. An appropriate choice is not obvious; theoretical models predict that the ionization state of the gas during reionization has a complex, and somewhat uncertain, morphology, with ionized regions of a range of sizes and shapes (Iliev et al. 2006, Mc-Quinn et al. 2007). However, we find that the simplest conceivable choice of template filters, corresponding to completely ionized spherical bubbles of varying size, are nonetheless effective at identifying ionized regions with a more realistic and complex morphology. In this case, T (k; R) is just the Fourier transform of a spherical tophat of radius R and is given by (10) with V denoting the volume of the spherical top-hat. T (k; R T ) = V k 3 R 3 T [−kR T cos kR T + sin kR T ] , Note that the precise normalization of the filter is unimportant since we are mainly interested in the signal-tonoise ratio here, in which case the overall normalization divides out. Application to Isolated Spherical Ionized Regions with Noise It is instructive to first consider an idealized test case that can be treated analytically before applying the matched filter to our full mock 21 cm data sets. In particular, we consider the case of an isolated, spherical, and highly ionized region placed at the origin and embedded in realistic noise. We assume that the neutral fraction exterior to the ionized region is uniform, with a mass-weighted neutral fraction of x HI (1 + δ ρ ) . Ignoring foreground contamination for the moment, the 21 cm signal may be written as: δT b (x) − δT b =B(x; R B ) +Ñ (x),(11) where B(x; R B ) denotes our isolated bubble of radius R B , and N (x) denotes the thermal noise contribution to the signal. We have subtracted off the overall mean brightness temperature, δT b , since this will not be measured in an interferometric observation. The tildes indicate that the spatial average has been removed from each of the underlying signal and noise so thatB(x; R B ) and N (x) each have zero mean. In this caseB(x; R B ) has an inverted spherical top-hat profile, B(x; R B ) = − x HI (1 + δ ρ ) |x| < R B , 0 otherwise.(12) The Fourier transform of the isolated bubble is hence related to the Fourier transform of our template bỹ B(k; R B ) = − x HI (1 + δ ρ ) T (k; R B ). Note that we express brightness temperatures in units of T 0 (see Equation 2), and so all quantities here are dimensionless. It is straightforward to derive the expected signal-tonoise ratio at the center of the isolated ionized region, and thereby gauge the prospects for bubble detection with a matched filter technique. Let us assume that the radius, R B , of our template filter is well matched to the true radius of the ionized region. This will maximize the expected signal-to-noise ratio. Neglecting foregrounds for the moment, and using the fact that the thermal noise has zero mean, we find that the signal-to-noise ratio at bubble center for the optimal matched filter is: S(R B ) = − x HI (1 + δ ρ ) d 3 k (2π) 3 T 2 (k; R B ) P N (k, µ) 1/2 . (13) For our sign convention, in which the template and ionized regions have opposite signs, this quantity is negative -ionized bubbles are regions of low 21 cm signal. The contribution of a Fourier mode to the signal to noise ratio depends on the relative size of T 2 (k; R B ) and P N (k, µ): modes for which the template is much larger than the noise power contribute appreciably to S(R B ) while modes dominated by the noise power are not useful. The signal-to-noise ratio depends on the neutral fraction: a larger exterior neutral fraction increases the contrast between an ionized bubble and the exterior, and hence boosts the detectability of the ionized region. We would like to calculate the expected signal-to-noise ratio for ionized regions of different sizes and for various volume-averaged ionization fractions. To do this, we need to connect the volume-averaged ionized fraction with the mass-averaged fraction, x HI (1 + δ ρ ) , which enters into Equation 13. Here we should incorporate that large scale overdense regions are generally ionized before typical regions during reionization, i.e., the neutral fraction and overdensity fields are anti-correlated. Defining δ x = (x HI − x HI )/ x HI , we approximate δ x δ ρ as fixed at δ x δ ρ = −0.25 throughout the reionization process . The results of the signal to noise calculation are shown in Figure 4 for the MWA-500 and a LOFAR-type experiment. Here we consider only our fiducial redshift, z fid = 6.9. The (absolute value of) S(R B ) is evidently a strongly increasing function of bubble size. This occurs because the thermal noise is a strong function of scale and only the rather large scale modes are measurable. It is encouraging that the expected signal-to-noise ratio exceeds five, S(R B ) 5, for a range of radii and neutral fractions. This corresponds to a 5 − σ detection: 'false' bubbles at this significance from downward fluctuations in the noise are highly unlikely, with a fraction of only ∼ 3 × 10 −7 of pixels in the filtered noise having such a large (negative) significance on their own. For simplicity we neglect the impact of foreground cleaning in this figure: this will degrade the expected signal-to-noise ratios somewhat, as we will consider subsequently (see §4.3, §5.3). In order to estimate the number of bubbles that can be detected from these curves, we need to consider how many bubbles there are of different sizes, i.e., we need to fold in an estimate of the bubble size distribution. In particular, while the contrast of an ionized region increases with the neutral fraction, large ionized bubbles become increasingly scarce for larger values of the neutral fraction. For instance, we can consider the model bubble size distributions in Figure 4 of Zahn et al. (2007). This figure indicates that bubbles of radius larger than 30 h −1 Mpc are exceedingly rare for neutral fractions larger than x HI > 0.5, with only the tail end of the distribution extending past 25 h −1 Mpc. However, bubbles this size are relatively common later in reionization. Since Figure 4 indicates that only bubbles with R 30 h −1 Mpc exceed S(R B ) 5, this suggests that bubble detection is feasible for the MWA-500 after the Universe is more than ∼ 50% ionized, but that it will be difficult to use this method at earlier stages of the reionization process. Also, note again that the calculation here neglects the effects of foreground cleaning. However, we find that incorporating foreground cleaning only has a small effect on bubbles of this size ( 30 h −1 Mpc , §5.3). Bubble detection will also be challenging once the Universe is less than ∼ 10 − 20% neutral, owing mostly to the reduced contrast between the bubbles and typical regions. If the ionized bubbles at a given stage of the EoR are larger than in the model of Zahn et al. (2007), then the prospects for bubble detection will be enhanced. We refer the reader to McQuinn et al. (2007) for a quantitative exploration of the bubble size distribution across plausible models for the ionizing sources. Finally, it is interesting to consider a LOFAR-style interferometer, as discussed further in §6. This is shown as the red dot-dashed curve in Figure 4. The expected S(R) exceeds that of the MWA-500 for small bubble radii, before flattening off at larger radii. This occurs because the LOFAR-style interferometer has more collecting area per baseline, but a larger minimum baseline. This makes it more sensitive to the smaller ionized regions, but less sensitive to larger ones. While the signal-to-noise curves in this toy case provide a useful guide, we should keep in mind their limitations. First, it considers only the case of a single isolated ionized region. Next, we consider here only the signal to noise at the bubble center, while an ionized region will typically have a strong (negative) signal to noise over much of its volume. Specifically, for bubbles with 10 Mpc/h R B 60 Mpc/h, we find that the signal-to-noise value at the edge of bubble is roughly half that at its center. This can help significantly with detection. Finally, we consider only the average signal-to-noise ratio here. In practice, the signal-to-noise ratio in a filtered map may fluctuate significantly around this average, as we will see. Application to a Simulated 21 cm Signal With the estimates of the previous section as a rough guide, we now apply the matched filter to our noisy mock redshifted 21 cm data. In order to illustrate the results of passing our mock data through a matched filter, we start by examining simulated signal-to-noise fields for a single template radius of R T = 35 h −1 Mpc. This template radius corresponds to the typical size of the ionized bubbles we believe we can detect (see Figure 4). A representative slice through the simulation is shown in Figure 5. The results look promising, with signal-to-noise ratios comparable to the values anticipated in the idealized calculation of Figure 4. Although the Wiener filter provides the best overall map, or data cube, one can still detect individual features at greater significance by applying a matched filter. Comparing with Figure 2, it is clear that the Wiener filter is passing more small scale structure than the matched filter shown here. This results in the signal-to-noise ratio being larger (in absolute value) for the matched filter than for the Wiener filter. In particular, we find values of the signal-to-noise ratio that are as low as ∼ −10 in the matched filter data cube, a significant improvement over the global minimum of ∼ −6 for the Wiener filter. Moreover, we can compare the filtered noisy signal in the bottom-left panel with the filtered pure noise field in the top-right panel. They differ by more than in the case of the Wiener filter. Indeed, the very low signal to noise ratio regions (shown in dark blue/purple in the bottom-left panel) line up fairly well with ionized regions in the top-left panel. This is especially apparent when comparing the filtered noisy signal to the filtered noiseless signal, shown in the bottom-right panel. For the slice shown, almost all of the significant features in the filtered noiseless signal are preserved in the noisy case. Figure 6 shows the impact of foreground cleaning: as in the case of the Wiener filter (Figure 3), this compresses structures along the line of sight and reduces the overall signal-to-noise ratio in the data cube. The signal-to-noise ratio is still significant enough, however, to robustly identify ionized regions. As with the Wiener filter, and in what follows subsequently, we show slices of 8 h −1 Mpc thickness. This choice is arbitrary, but we expect similar results provided the slice thickness is small compared to the radius of the template filter. It is important to keep in mind, however, the full data cube will consist of many separate slices of this thickness. Also note that the transverse dimension of the MWA-500 is larger than that of our simulation box by a factor of ∼ 3, and so these slices represent only ∼ 1/9 of the MWA field of view. These results are promising, but they are for a single filtering scale, and so we can do significantly better by considering a range of template radii, and looking for extrema in the resulting signal-to-noise fields. In particular, we proceed to apply a sequence of filters with template radii up to R T ≤ 75 h −1 Mpc -see §4.5 for a justification of this maximum -across the simulation volume. We assign the minimum (most negative) signal-to-noise value obtained over the range of template radii to each simulation pixel and use this to construct a new field. The position of local minima in this field are chosen to be the centers of candidate bubbles, and each such bubble is assigned a radius according to the scale of the template filter that minimizes its signal-to-noise. We focus on minimum values since ionized regions are expected to appear as regions of low 21 cm signal. All candidate bubbles whose central signal to noise is lower than −5 are considered to be detected ionized regions. We find it important to apply one additional criterion to robustly identify ionized regions. The criterion is that a low signal to noise region on scale R B must additionally be low in signal to noise at all smaller smoothing scales, R T < R B . This guards against the possibility that a detected bubble will be centered on neutral material that is nevertheless surrounded by ionized hydrogen. A region like this will have a high (least negative) signal-tonoise when filtered on small scales and then dive down (gaining statistical significance) when filtered on scales -Application of the matched filter to simulated data and noise ( x i = 0.79 at z fid = 6.9). The template radius of the filter is 35 h −1 Mpc , since this is a commonly detected bubble radius for our matched filter search. Top-Left: Spatial slice of the unfiltered and noise-less 21 cm brightness temperature contrast field. Top-Right: Simulated signal-to-noise field after applying the matched filter to a pure noise field. Bottom-Left: Simulated signal-to-noise field after applying the matched filter to the noisy signal. This can be compared directly to the top-left panel. Bottom-Right: Simulated signal-to-noise field after applying the matched filter to the noiseless signal. All panels are at the same spatial slice. See text for discussion on interpreting signal-to-noise values. containing the surrounding ionized material. We discard such spurious bubbles by requiring that the field is low on all smaller smoothing scales. The only downside to this procedure is that it occasionally discards true ionized regions whose center happens to coincide with a significant upward noise fluctuation. Overall, however, it improves the quality of detected bubbles ( §4.4). This cut also requires a threshold choice; we reject candidate regions if their signal-to-noise ratio crosses above a threshold S max at any smoothing scale less than R T . After trying several thresholds, we found the most effective choice to be S max = −1σ. In principle, one might use the full curve of signal-to-noise ratio versus template radius for each candidate bubble to help verify the detection and determine the properties of the bubble. In practice, we found that individual signal-to-noise curves are noisy and difficult to incorporate into our analysis and so we don't consider this possibility further in what follows. We apply this algorithm to the mock redshifted 21 cm data and identify 220 ionized regions across the simulation volume (which is different than the MWA survey volume, as we will discuss subsequently). A representative example of a detected bubble is shown in Figure 7. The circle in the figure identifies the detected bubble size and the location of its center in both the filtered noisy signal (top-left and top-right panels), as well as in the input signal (bottom-left and bottom-right panels). The algorithm has convincingly identified an ionized region. The detected bubble overlaps a small fraction (%10) of neutral material in the input signal. Although this particular ionized bubble is well identified, most of the ionized regions in the signal will escape detection. This is because the significance levels of the detected bubbles are not that high, and an ionized region generally needs to be coincident with a downward fluctuation in the noise to pass our significance threshold. For example, consider the larger ionized region below and to the left of the detected region in the bottom-left panel of Figure 7. This region, while larger and therefore more detectable on average than the identified bubble, happens to coincide with a large upward noise fluctuation and hence fails to cross the significance threshold. While we can not identify all of the large ionized regions in the noisy mock data, we can robustly identify some regions; this may still be quite valuable. It is also clear that the underlying ionized regions are manifestly non-spherical, creating an ambiguity as to what the appropriate 'radius' of the region is. Focusing on the bottom right panel in Figure 7, we could imagine the size being reasonably described by a radius ∼ %50 larger, so as to enclose more of the nearby ionized material. However, our method naturally favors radii causing little overlap with neutral material at these size scales. Therefore, an ionized region like the one shown in Figure 7 is more likely to be detected as several small ionized regions than one large one, although both characterizations seem reasonable. Figure 8 gives a further example of how the algorithm identifies bubbles, and some of the ambiguities that can result. This figure shows an example of an irregular, yet contiguous, ionized region that is detected as more than one ionized bubble. Here we show spatial slices through the center of the middle sphere, marked with a solid circle, which happens to intersect neighboring ionized bubbles, whose cross sections are shown as dashed circles. Hence, our algorithm generally represents large, irregularly shaped, yet contiguous, regions as multiple ionized bubbles. It is important to emphasize further the difference between the simulated results shown here and the idealized test case of the isolated bubble shown in the previous section. In particular, we consider here the application of matched filters to the 21 cm signal during a late phase of reionization in which many ionized regions, with a broad size distribution, fill the survey volume: the ionized regions are not isolated bubbles in a sea of partly neutral material. When applying a matched filter of template radius R T around a point, the values of the field at many neighboring pixels impact the filtered field at the point in question. It is hence possible that a filtered pixel is affected by several distinct neighboring ionized regions. Indeed, this can result in even neutral regions having low signal-to-noise ratios provided they are surrounded by many nearby ionized regions. For instance, in the low noise limit, any region with volume-averaged neutral fraction lower than the cosmic mean would pass our significance threshold. To guard against this type of false detection, we implemented the requirement that a candidate bubble has low signal-to-noise for all template radii smaller than the detected radius. Another possibility might be to treat small ionized regions as an additional noise term in the filter. However, in practice, our attempts along these lines introduced an additional level of model dependence without significantly increasing the quality of the detected bubbles. Ultimately, it is important to keep in mind that the signal-to-noise values quoted here reflect only the likelihood that a value arises purely from noise, and so they are not strictly indicative of the quality of the detected bubbles. Success of Detecting Ionized Regions We hence turn to describe the characteristics of the detected ionized regions, and to quantify the method's level of success in detecting ionized bubbles. To do this, we calculate the fractional overlap of each detected bubble with ionized material in the underlying signal. Additionally, we estimate how many ionized bubbles should be detectable across the entire MWA-500 survey volume. The matched filter technique finds 220 bubbles across our simulation volume. However, the algorithm for determining bubble positions and sizes allows for bubbles to occupy overlapping areas, as shown in Figure 8. We find that ∼ 55% of the detected bubbles have some overlap with another bubble, although only ∼ 15% of the total volume occupied by detected bubbles is occupied by more than one. Regardless, ∼ 96% of the detected ionized bubbles have an average ionized fraction larger than x i = 0.79, which is the volume-averaged ionization fraction of the simulation box at the redshift under consideration. Furthermore, ∼ 42% of the detected bubbles have an ionized fraction greater than x i = 0.9. The lowest ionization fraction of a detected bubble is x i,min = 0.77, just slightly below the volume-averaged ionization fraction of the simulation. In total, we detect 9 bubbles whose ion- ized fractions are less than the average ionization fraction of the box. Inspection reveals that these regions happen to be coincident with significant (≤ −3 − σ) downward noise fluctuations. In Figure 9, we plot the volume-averaged ionized fraction within each of our detected bubbles against the detected bubble radius for a significance threshold of 5σ. For comparison, we show the 1 − σ spread in the ionized fraction enclosed by randomly distributed spheres of the same size. 4 The spread in ionization of the randomly distributed spheres around the box average ionized fraction, x i = 0.79, decreases with increasing radius; this reflects the drop off in the power spectrum of the ionization field towards large scales. Most of the detected bubbles are significantly more ionized than random regions, as expected, indicating a significant success level. There are a few poor detections which result mostly from downward noise fluctuations. There is a small overall decrease in the ionized fraction of detected bubbles larger than R B 40 h −1 Mpc, suggesting that we may no longer be detecting individual ionized regions here. These regions may potentially be distinguished from isolated bubbles by examining the signal-to-noise ratio as a function of template radius closely, as we discuss in §4.5. We can estimate the number of ionized regions detectable in the MWA-500 by scaling from our simulation volume to the MWA survey volume. At z fid = 6.9, for an ionized fraction of x i = 0.79, we expect to find 140 bubbles in a B = 6 MHz chunk of the MWA, over its entire field of view of ∼ 770 deg 2 . About 135 (60) of these detected bubbles are expected to have ionized fractions larger than 79% (90%). This estimate comes from simply scaling our simulation volume (which is deeper than the MWA bandwidth) to a 6 MHz portion of the MWA survey volume. Analyzing the MWA data over a 6 MHz chunk is meant to guard against redshift evolution: the full bandwidth of the survey is B = 32 MHz and so the prospects for bubble detection across the full survey are even better than this estimate suggests. The precise gain will be dependent on how rapidly the bubble size distribution evolves across the full survey bandwidth. One caveat with our estimate, however, is that B = 6 MHz corresponds to only ∼ 70 h −1 Mpc . This is comparable to the size of our larger bubbles, and so analyzing chunks this small might weaken our ability to detect large bubbles. This effect is not incorporated in our scaling estimate, which simply takes the ratio of the MWA survey volume and our simulation volume. In practice, one can perform the bubble extraction for different analysis bandwidths to help ensure robust detections. By increasing the significance threshold for detection, one might hope to improve the quality of the detected bubbles. However, we find that this leads to the detected bubbles being larger on average rather than being more ionized. This results from the fact that, at a fixed ionized fraction, a larger bubble will have a greater (in magnitude) signal-to-noise value than a smaller bubble. By increasing the significance threshold, some smaller, highlyionized bubbles are no longer detected, while larger, lessionized bubbles are. Specifically, we find that by increasing the threshold to 6 − σ (7 − σ) the number of detected bubbles drops to ∼ 100 (∼ 50) with 99% (100%) being more ionized than the box, but only 34% (33%) being more than 90% ionized. Hence, for any of these thresholds we robustly detect many bubbles, although the detailed success rate and overall number varies somewhat with the precise choice of threshold. Note that lowering the significance threshold has the undesirable effect of in-creasing the number of false positives. Ultimately, with real data one should explore a range of threshold choices and examine the impact. Range of Template Radius Considered It is worth mentioning one further detail of our algorithm. In the previous section, we set the maximum template radius considered at R T,max = 75 h −1 Mpc, without justification. In fact, we have a sensible and automated way for arriving at this choice. We discuss this procedure briefly here. A good candidate ionized region should in fact obey three criteria. First, it should have a large (negative) signal-to-noise ratio, so that it is unlikely to result from a noise fluctuation. Second, the signal-to-noise ratio should be small for all trial radii smaller than the optimal template radius, as discussed in §4.3. Finally, the total signal must itself be small in an absolute sense. In the limit of low noise, anything less neutral than average would qualify as a bubble by the first criterion, and so this third criterion may then become important for robustly identifying bubbles. This low noise limit is relevant for the MWA-500 only on very large smoothing scales, where the noise averages down significantly. Since this third criterion becomes important only on very large smoothing scales here, we use it only to set the maximum template radius considered. Without this third consideration, our algorithm generally identifies a few excessively large ionized bubbles, but this can be easily understood and avoided as follows. Consider, for the moment, the 21 cm brightness temperature field in the absence of noise and foregrounds. Let's further work in units of T 0 (Equation 2), and remove the average brightness temperature contrast across the data cube. In this case, the signal inside a highly ionized bubble is expected to be − x HI (1 + δ ρ ) . If we now spherically average the field on scales smaller than the bubble, the value at bubble center will not change from this value, − x HI (1+δ ρ ) . Once the smoothing scale becomes larger than the bubble scale, however, surrounding neutral material will increase the value of the filtered field at bubble center. Hence, if the filtered field becomes everywhere larger than − x HI (1 + δ ρ ) on some smoothing scale, it is clear that no larger ionized bubbles exist within the data cube. This suggests that we can set the maximum template radius by requiring that the filtered noisy signal reaches sufficiently small values, at some locations across the data cube, for there to still plausibly be completely ionized regions. Since the presence of noise only increases the variance, this should provide a conservative estimate of the maximum size of the ionized regions. In practice, we need to chose a threshold criterion without assuming prior knowledge of the neutral fraction. Here we set the maximum template radius to be the smallest smoothing scale at which the filtered noisy field everywhere exceeds − x HI (1 + δ ρ ) ≥ −0.075. This corresponds to the expected contrast at x HI = 0.1, assuming δ x δ ρ = −0.25, and yields a maximum template radius of R T,max = 70 h −1 Mpc , 73 h −1 Mpc , 75 h −1 Mpc , and 75 h −1 Mpc for x i = 0.51, 0.68, 0.79, and 0.89, respectively. The precise threshold used here, −0.075, is somewhat arbitrary but this choice is only being used to set the maximum template radius considered. 5 VARIATIONS ON THE FIDUCIAL MODEL So far, we have considered the prospects for bubble detection only in our fiducial model with x i = 0.79 at z fid = 6.9 and only for the MWA-500. Here we consider first alternate models in which the Universe is more or less ionized at z fid = 6.9 than in our fiducial case, and then consider how the sensitivity declines towards higher redshifts at fixed ionized fraction. In addition, we consider variations around our fiducial assumptions regarding the impact of foreground cleaning. Then we turn to consider the sensitivity of the MWA-128; this is meant to illustrate the prospects for bubble detection with the very first generation of redshifted 21 cm surveys, while the MWA-500 represents a second generation survey. Ionized Fraction In order to consider bubble finding at earlier and later stages of the EoR, we run our matched filter on simulation outputs with volume-averaged ionized fractions of x i = 0.51, 0.68, and 0.89. As discussed in §3.1 these outputs are actually at slightly different redshifts, but we generate the 21 cm field as though they were at z fid = 6.9. As far as bubble detection is concerned, varying the ionized fraction leads to two, generally competing, effects. First, the bubbles grow as reionization proceeds. This tends to boost detection, since it is only the large scale modes that are detectable over the thermal noise. Second, however, the contrast between an ionized region and a typical volume of the Universe is reduced as reionization proceeds. This makes bubble detection more difficult. Both of these effects are quantified in the idealized isolated bubble case in Figure 4. It is also clear that the ideal ionized fraction for bubble detection will be somewhat survey dependent. As already illustrated in Figure 4 and discussed further in §6, a LOFAR-type interferometer will perform better when the ionized regions are smaller. We find that the matched filter is capable of detecting ionized regions for each of the ionized fractions studied. In Figure 10 we show histograms of the detected bubble size distributions for each ionized fraction. Since we preferentially detect large ionized regions, we don't expect these distributions to be representative of the true underlying bubble size distributions. For example, in Figure 10, the size distribution peaks around 40 h −1 Mpc for the case of x i = 0.79, despite volume-weighted size distribution peaking around ∼ 30 h −1 Mpc in Figure 4 of Zahn et al. (2007) at roughly the same ionized fraction. Nonetheless, the histograms illustrate a general shift from smaller to larger detected bubble radius as the ionized fraction increases. By applying the matched filter to several redshift bins, one can potentially observe precisely this evolution with the MWA-500. This would complement studies of the 21 cm power spectrum evolution over the same redshift range (e.g., Lidz et al. 2008). From the histograms, one can see that -of the models shown -the best ionized fraction for bubble detection is x i = 0.79. This is apparently near the sweet spot for the MWA-500 where the bubbles are large enough in the model for detection, but the contrast with typical regions is still sufficiently large. The average ionized fraction within detected bubbles varies significantly across the different ionized fractions considered. Specifically, the percentage of detected bubbles that are more than 90% ionized is 0%, 15%, 43%, and 91% for x i = 0.51, 0.68, 0.79, and 0.89, respectively. However, in each case the percentage of detected bubbles with ionized fraction larger than the (global) volume-averaged ionization fraction is fixed at 95%. At first glance, one aspect of these results may appear to be in tension with the calculations of §4.2, where we estimated that bubble detection would be unsuccessful for neutral fractions larger than x HI 0.5. However, this estimate considered the detection of isolated bubbles. Inspection reveals that the detected bubbles at x i = 0.51 each correspond to clusters of smaller ionized regions. Evidently, these appear as a single larger ionized region after convolving with the template filter and downweighting the noisy short-wavelength modes. In practice, it may be possible to distinguish this case from that of an isolated bubble by analyzing the trend of signal-to-noise ratio versus trial template radius. The signal-to-noise ratio is expected to grow more rapidly with radius (before reaching the bubble scale) for a truly isolated bubble. Timing of Reionization We now consider how the prospects for bubble detection diminish if reionization occurs earlier and the observations are focused on the corresponding redshifts. In particular, we examine the case that our model with an ionized fraction of x i = 0.79 is observed at a higher redshift. We focus on this case since this ionized fraction appears close to optimal for bubble detection. Aiming for only a rough estimate here, we consider the prospects for detecting a R B = 40 h −1 Mpc bubble. Although several different factors in the noise power spectrum of Equation 5 scale with redshift, the dominant scaling is with the sky temperature. The noise power scales as P N ∝ T 2 sky , and the sky temperature follows T sky ∝ ν −2.6 ∝ (1 + z) 2.6 . Therefore, we expect the signal-to-noise ratio of a detected bubble to fall off with increasing observation redshift roughly as S(z) = S(z fid ) 1 + z fid 1 + z 2.6 .(14) This indicates the signal-to-noise ratio for a bubble detected with a signal-to-noise of S(z fid ) at redshift z fid = 6.9, if the bubble were instead observed at redshift z. A relatively large bubble with R B ≈ 40 h −1 Mpc has a typical signal-to-noise ratio at bubble center of S ≈ 4 at our fiducial redshift. This value is found by incorporating foreground cleaning into the corresponding curve in Figure 4. According to Equation 14, the signal-tonoise value will be reduced to a significance of S ≈ 2 (1) at z = 9.3 (12.5). The bubble will, in fact, be more detectable than implied by this one number -the signalto-noise ratio at bubble center -since an ionized region should have low signal-to-noise over much of its volume. From this, we conclude that bubble detection should be feasible with the MWA-500 if our fiducial ionized fraction occurs later than z 9 or so, but that the prospects are rather limited in the case of significantly earlier reionization. A range of recent work in the literature, however, suggests that reionization is unlikely to complete so early. See, for example, Figure 9 from the recent study of Kuhlen & Faucher-Giguère (2012) which combines Ly-α forest data (Fan et al. 2006), measurements of the Thomson optical depth from WMAP (Komatsu et al. 2011), and measurements of the Lyman-break galaxy luminosity function (Bouwens et al. 2011). Hence, the prospects for bubble detection appear good for the MWA-500. 5.3. Effects of Foreground Cleaning Next, we consider the impact of variations around our standard foreground cleaning model. As discussed previously ( §2.4), our standard assumption is that the impact of foreground cleaning can be approximately mimicked by removing the running mean, over a bandwidth of B = 16 MHz, across each line of sight. The optimal foreground cleaning strategy avoids 'over-fitting' by removing the smoothest possible function over the largest possible bandwidth, in order to preserve the underlying signal as much as possible. It also avoids 'under-fitting' by ensuring that foreground residuals do not excessively contaminate the signal. Liu & Tegmark (2012), for example, find that 21 cm foregrounds can be removed to one part in 10 5 or 10 6 by subtracting roughly three modes over ∼ 32 MHz of bandwidth. This should have a fairly similar impact to our fiducial cleaning model, but we would expect a bit more degradation in this case. A detailed investigation would add foreground contamination into our mock data cubes, and explore the impact of various cleaning algorithms directly. Here, we instead check how our results change for slightly more and less aggressive foreground cleaning. In particular, we remove the running mean over each of B = 32 MHz, 8 MHz, and 6 MHz and rerun our bubble finding algorithm (at z fid = 6.9, x i = 0.79). This has little impact on our results for the case of B = 32 MHz and 8 MHz. In particular, the number of identified bubbles varies by less than 10−15% and the quality of detected bubbles decreases slightly for the more aggressive cleaning model, and improves slightly in the most optimistic case. Specifically, for B = 16 MHz, 96% (43%) of bubbles have ionized fractions exceeding x i = 0.79 (0.9); for B = 8 MHz the corresponding numbers are 90% (32%); and for B = 32 MHz the same numbers are 96% (62%). For the most aggressive case of B = 6 MHz, we find the number of detected bubbles drops significantly to ∼ 50% of the number detected in the fiducial case. The larger degradation in the B = 6 MHz results because the foreground cleaning scale has become comparable to the scale of many of the detected bubbles. Despite this drop, we find the quality of the detected bubbles to be about the same as with the B = 8 MHz model, with 92% (36%) exceeding x i = 0.79 (0.9). The distribution of detected bubble sizes also depends on the bandwidth used for foreground cleaning, with smaller bandwidths generally corresponding to smaller detected bubble sizes when B 16 MHz. This behavior is consistent with work by Petrovic & Oh (2011) who found that a bubble whose size is comparable to the scale for foreground cleaning should have its contrast reduced. Even in the most aggressive case considered, however, we still detect many ionized regions robustly. While these estimates are encouraging, a more detailed study is warranted. It may also be advantageous to estimate the power spectrum of the foregrounds, and incorporate this as an additional noise term in Equation 7 and Equation 9 for each of the Wiener filter and the matched filter, respectively. 128 Antenna Tile Configurations So far our analysis has focused on the MWA-500, which is meant to represent a second generation 21 cm survey. In the near term, it is timely to consider the prospects for a 128 tile version of the MWA (the MWA-128) which is ramping up to take data in the very near future. This should be significantly less sensitive, since the number of baselines scales as the number of antenna tiles squared. In order to generate thermal noise representative of the MWA-128, we start by considering a similar antenna distribution as for the MWA-500. In particular, we assume all of the antenna tiles are packed as closely as possible within a core of radius 8 m and that the antenna distribution subsequently falls off as r −2 out to a maximum baseline of 1.5 km. After comparing the thermal noise power spectrum in this configuration with that in Beardsley et al. (2012), we find that our noise power is larger by up to a factor of a few. This could possibly be due to our approximation of a smooth antenna distribution being less valid for the MWA-128, or to the fact that our analytic formula for the noise power spectrum does not incorporate a full treatment of rotation synthesis. In an effort to bracket somewhat the impact of the detailed antenna distribution, we further consider a configuration where all antenna tiles are packed as closely in a dense core of radius ∼ 25 m. This resembles the 'supercore' configuration considered in Lidz et al. (2008) for the power spectrum. The results of applying the optimal matched filter for a single template radius of 35h −1 Mpc are shown in Figure 11 for the r −2 tile distribution. This shows that the sensitivity is much lower than for the MWA-500, as expected. It is much more difficult to distinguish the filtered noisy signal (bottom-left panel) from the filtered pure noise (top-left) panel here than in Figure 5. Most of the significant, dark blue regions in the filtered noisy signal correspond simply to low noise regions. However, applying the detection algorithm we do nonetheless detect 7 bubbles across a volume equivalent to a 6 MHz chunk of the MWA-128 survey. The success is generally lower than in the case of the MWA-500: here 75% of detected bubbles exceed the average ionized fraction of the box, while ∼ 42% exceed x i = 0.9. In the supercore configuration, we find slightly higher significance levels (up to 6.9−σ) but the identified regions generally correspond to several large clustered ionized regions, rather than a single ionized bubble. Altogether, the algorithm identifies 10 ionized regions across the MWA survey in the supercore configuration, but the identified regions have a lower overall quality than in the r −2 configuration. Our conclusion is that bubble detection is only marginally possible with the MWA-128. While the results are unlikely to be very compelling, it is worth applying the matched filter to the first generation surveys as an initial test. Even a few weakly identified candidate bubbles would provide compelling targets for follow-up observations. Another possibility is to focus on targeted searches around known bright sources for the MWA-128 (e.g., Wyithe & Loeb 2004). FAVORABLE ANTENNA CONFIGURATIONS FOR BUBBLE DETECTION The possibility of imaging or identifying ionized regions from second generation redshifted 21 cm surveys invites the question: how do we optimize future surveys for this goal? It seems unlikely that the optimal configuration for bubble detection is identical to that for measuring the power spectrum, although power spectrum measurements have mostly driven survey design considerations thus far. In the case of the power spectrum, one aims to minimize the error bar on power spectrum estimates in particular bins in wavenumber. The power spectrum error bar for each k-mode contains a thermal noise term and a sample variance (sometimes called 'cosmic variance') term. Because of the sample variance contribution, the gain from reducing the thermal noise for a given k-mode is limited: once the thermal noise is reduced sufficiently far below the sample variance, it is advantageous to instead measure a different k-mode on the sky within the |k| bin of interest. As a result, grouping individual antennas into only small tiles to achieve a wide survey, generally reduces the statistical error bars on power spectrum measurements compared to antenna configurations with larger tiles that probe narrower fields of view. For imaging and bubble detection, one aims for the best possible signal to noise on particular regions of the sky. In other words, for good imaging one wants to reduce the thermal noise to well below the sample variance level. Grouping individual antennas into larger tiles, in order to devote more collecting area to a narrower field of view, may be better for this purpose. In order to get some sense for these trade-offs, we consider here a LOFAR-style interferometer with the specifications listed in McQuinn et al. (2007). Although the detailed specifications for LOFAR have evolved somewhat (e.g., Zaroubi et al. 2012), (as have the MWA specifications), this is nonetheless a helpful case to consider. In particular, our toy LOFAR-style interferometer has many fewer antenna tiles than the MWA-500 but a substantially larger collecting area per tile. Specifically, the interferometer considered has N a = 32 antenna tiles, A e = 596 m 2 at our fiducial redshift (compared to A e = 11.25 m 2 for the MWA-500), d min = 100 m, and d max = 2 km. We assume that antenna tiles are packed as closely as possible, consistent with d min = 100 m, inside a compact core and that the density subsequently falls off as r −2 , out to a maximum radius of r max = 1000 m. These parameters are meant to broadly represent an upgraded version of the existing LOFAR array, analogous to our MWA-500 survey, which is an upgrade to the ongoing MWA-128 instrument. With these parameters, the LOFAR-style interferometer has more total collecting area than the MWA-500 setup by a factor of a few. The results of applying a matched filter to a data cube with simulated LOFAR noise are shown in Figure 12. Here we zoom in to show a portion of our simulation box that matches the smaller field of view of this LOFARlike instrument. From the figure it is evident that the filter removes large scale structures, a result of the relatively large minimum baseline of this interferometer. In addition, the maximum signal-to-noise achieved here is smaller than with the MWA-500 (it drops from 10 to 6.9). Nonetheless, many small-scale ionized regions in the unfiltered noise-less signal are well preserved in the filtered noisy signal. This is consistent with the idealized calculation of Figure 4, which showed that LOFAR should have a higher signal-to-noise detection of small ionized regions, but a reduced signal-to-noise otherwise. Because of this, the LOFAR-style configuration is more successful during earlier stages of reionization when the bubbles are still relatively small. In general, we find that the LOFARstyle configuration detects slightly fewer bubbles overall but with more success for x i 0.79, while the MWA-500 has a greater level of success at later stages of the EoR. This example suggests that the ideal configuration for bubble detection is likely intermediate between the MWA-style and LOFAR-style antenna configurations. It appears helpful to have more collecting area on fewer baselines than the MWA, but a smaller minimum baseline than in the LOFAR-style instrument is necessary to detect large bubbles. This deserves further study, however: for example, we have neglected calibration requirements and systematic concerns. These considerations will also certainly drive the experimental design. As a further concrete example of how systematic concerns could impact the design of future arrays, suppose foreground cleaning requires removing more large scale modes than anticipated. In this case, it would make sense to focus efforts on smaller bubbles. This would shift the ideal configuration closer to a LOFAR-style instrument with a larger minimum baseline. Figure 5, except it is for the LOFAR configuration rather than the MWA-500. Additionally, all boxes in this figure have a side length of 426 h −1 Mpc , corresponding to the field-of-view of the LOFAR-style interferometer at z = 6.9. COMPARISONS TO PREVIOUS WORK Previous work by Datta et al. (2007) and Datta et al. (2008) also considered the possibility of detecting ionized regions in noisy redshifted 21 cm data sets using a matched filter technique. The main difference between our study and this earlier work is that these previous authors considered the prospects for detecting a specific spherical ionized region of varying size, i.e., they considered the detectability of a spherical bubble at the origin, or offset slightly from the origin. These authors also considered the case where the bubble of interest was embedded in a variety of different ionization environments; the bubble under consideration was not always isolated. Altogether, their study is mostly similar to a targeted search, where one has a good prior regarding the likely location of an ionized region. It also provides a feasibility estimate for a more ambitious blind search. The main advantage of a targeted search is that, if a region is known a priori to be highly ionized, one need not worry about an entirely false detection from a downward noise fluctuation. This then allows a lower significance threshold for robust bubble detection, and may therefore be the most feasible approach for the MWA-128 and other first generation surveys. Nonetheless, our work is a significant extension to the earlier work by Datta et al. (2007) in that we conduct a blind search across an entire mock survey volume. A detailed comparison with their work is not straightforward given the difference between our approaches, but both studies have a similar bottom-line conclusion: ionized regions are detectable with surveys similar to the MWA-500. CONCLUSION We considered the prospects for making low-resolution images of the 21 cm sky and for direct, blind detection of ionized regions using first and second generation 21 cm surveys. We find that a 500-tile version of the MWA, the MWA-500, is potentially capable of detecting ionized regions. In our fiducial model, in which 79% of the volume of the Universe is ionized at z fid = 6.9, the MWA-500 can find ∼ 150 ionized regions in a B = 6 MHz chunk after ∼ 1, 000 hours of observing time. First generation surveys, such as the MWA-128, are substantially less sensitive. We find that the MWA-128 may, nonetheless, be able to detect a handful of ionized regions across its survey volume, with 7 expected in our fiducial model. The MWA-128 may be more effective at identifying ionized bubbles using targeted searches towards, for example, bright quasars (e.g. . There are several possible future directions for this work. First, while we incorporate realistic levels of thermal noise and mimic the effect of foreground cleaning, it will be important to test the robustness of bubble detection with a more detailed model for foreground contamination, and to consider systematic effects from calibration errors and the MWA instrumental response. These considerations can also help in determining the optimal design for future surveys aimed at bubble detection. Our first efforts considering which configurations of antenna tiles are favorable for bubble detection, detailed in §6, suggest that an observing strategy intermediate to that of the MWA and LOFAR is favorable. It would also be interesting to consider the prospects for bubble identification across a larger range of reionization models than considered here. If the ionized regions at a given stage of reionization are, in fact, larger than in the models considered here, this should increase their detectability. On the other hand, if the ionized regions are smaller than in our present models, this would likely diminish detectability, at least for the MWA-500. If blind bubble identification is indeed feasible in future 21 cm surveys, we believe this will open up several interesting avenues of investigation. First, direct identification of ionized regions can help to build confidence in early redshifted 21 cm detections. Next, if the centers of ionized regions can be robustly identified, one may be able to use the brightness temperature contrast between the signal near the bubble's center and its surroundings to directly constrain the cosmic mean neutral fraction (e.g., Petrovic & Oh 2011). These authors also discuss how detected bubbles can be used to calibrate foreground cleaning (Petrovic & Oh 2011). Finally, identifying ionized regions in redshifted 21 cm surveys allows one to commence follow-up observations, comparing galaxy properties inside detected bubbles with those in more typical regions. Typical regions and likely neutral regions can be identified as locations in the data cube with average and maximal signal-to-noise ratios, respectively, after applying the matched filter. Furthermore, if the edge of an ionized region can be identified precisely enough, one might imagine targeted searches for galaxies at the edge of bubbles, close to neighboring neutral gas. Spectroscopic observations of these galaxies might then help to reveal the damping wing redward of the Ly-α line (e.g., Miralda-Escude 1998). This would provide yet another means for constraining the neutral fraction. Fig. 1 . 1-Fourier profile of the Wiener filter, W (k). The filter is averaged over line-of-sight angle and the results are shown at z fid = 6.9 for simulated models with x i = 0.51 (blue dotted), x i = 0.68 (cyan dot-dashed), x i = 0.79 (green dashed), and x i = 0.89 (red solid). Fig. 2 . 2-Application of the Wiener filter to simulated data. The results are for our fiducial model with x i = 0.79 at z fid = 6.9. Fig. 3 . 3-Impact of foreground cleaning on the Wiener-filtered field. The top slice is a perpendicular, zoomed-in view of the simulated, unfiltered, noise-less brightness temperature contrast. The bottom slice is the signal-to-noise of the same region after applying the Wiener filter to the noisy signal field. The vertical axis shows the line-of-sight direction, with its extent set to the distance scale for foreground removal, L fg = 185 h −1 Mpc . The horizontal axis shows a dimension transverse to the line of sight and extends 1 h −1 Gpc . Fig. 4 . 4-Expected signal-to-noise ratio at the center of isolated, spherical, ionized bubbles as a function of bubble radius after applying the optimal matched filter. The curves show the signalto-noise ratio at z fid = 6.9 for the MWA-500 at various neutral fractions: x HI = 0.4 (blue solid), 0.3 (cyan dashed), and 0.2 (green dot-dashed). For contrast, the red dotted curve indicates the expected signal-to-noise for an interferometer with a field of view and collecting area similar to a 32-tile LOFAR-like antenna array (at x HI = 0.4). Fig. 5 . 5Fig. 5.-Application of the matched filter to simulated data and noise ( x i = 0.79 at z fid = 6.9). The template radius of the filter is 35 h −1 Mpc , since this is a commonly detected bubble radius for our matched filter search. Top-Left: Spatial slice of the unfiltered and noise-less 21 cm brightness temperature contrast field. Top-Right: Simulated signal-to-noise field after applying the matched filter to a pure noise field. Bottom-Left: Simulated signal-to-noise field after applying the matched filter to the noisy signal. This can be compared directly to the top-left panel. Bottom-Right: Simulated signal-to-noise field after applying the matched filter to the noiseless signal. All panels are at the same spatial slice. See text for discussion on interpreting signal-to-noise values. Fig. 6 . 6-Impact of foreground cleaning on the matched-filtered field. This is similar toFigure 3, except that the results here are for a matched filter with a template radius of R T = 35 h −1 Mpc . Fig. 7 . 7-An example of a detected ionized region. Top-left: Signal-to-noise field after applying the matched filter to the noisy signal. The detected bubble is plotted on top of the corresponding region in the map. Top-Right: Zoomed-in view of the detected bubble in the matched-filtered map. Bottom-Left: Detected bubble superimposed on a zoomed-in view of the noise-less unfiltered 21 cm brightness temperature contrast map. Bottom-Right: A perpendicular zoomed-in view of the bubble depicted in the bottom-left panel. All matchedfiltered maps use the template radius that minimizes the signal-to-noise at the center of the detected bubble. In the top-left case, the boxlength is L = 1 h −1 Gpc , while in the zoomed-in slices it is L ≈ 500 h −1 Mpc . Fig. 8 . 8-An example of an ionized region that our algorithm detects as several neighboring bubbles. Top-left: Signal-to-noise field after applying the matched filter to the noisy signal. The main detected bubble is plotted on top of the corresponding region in the map. Top-Right: Zoomed-in view of the main detected bubble in the matched filtered map (solid curve) along with two other nearby detected bubbles (dashed curve). Bottom-Left: The detected bubble superimposed on the zoomed-in, noise-less, unfiltered 21 cm brightness temperature contrast map. Again, the additional nearby detected bubbles are shown (dashed curve). Bottom-Right: A perpendicular view of the bubble depicted in the bottom-left panel, with the nearby detected bubbles visible. All matchedfiltered maps use the template radius that maximizes the signal to noise at the center of the main detected bubble. The box length in the top-left figure is L = 1 h −1 Gpc , while in the zoomed-in panels, the box length is L = 550 h −1 Mpc . Fig. 9 . 9-A measure of the bubble detection success rate. The points (×) show the volume-averaged ionized fraction of detected bubbles versus their detected radius. For comparison, the cyan shaded region shows the 1-σ spread in the ionized fraction of randomly placed bubbles of the same radii. The bubble depicted inFig. 7is marked with a large red square, while the three bubbles shown inFig. 8are marked with large green circles. Fig. 10 . 10-Size distributions of detected bubbles for varying (volume-averaged) ionization fractions. The histograms show the size distribution of (identified) ionized regions for simulation snapshots with volume-averaged ionized fractions of x i = 0.51 (topleft), 0.68 (top-right), 0.79 (bottom-left), and 0.89 (bottom-right). These figures demonstrate how the total number and size distribution of detected bubbles varies with ionized fraction. Fig. 11 . 11-Bubble detection with the MWA-128. This figure is similar toFigure 5, except it is for the MWA-128 configuration rather than for the MWA-500. Fig. 12 . 12-Bubble detection with a LOFAR-style interferometer. This figure is similar to .edu 1 Department of Physics & Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA et al. 2009), and the Precision Array for Probing the Epoch of Reionization (PAPER) Redshift-space distortions and redshift evolution across the observed bandwidth break isotropy (e.g.). However, for the bandwidth considered here (B = 6 MHz) and the neutral fractions of interest, the signal should be approximately isotropic. In practice, the simulated ionization fields for ionized fractions lower (higher) than our fiducial value ( x i = 0.79 at z fid = 6.9) come from slightly higher (lower) redshift simulation outputs. We generate the 21 cm signal and noise as though each data cube were in fact at z fid = 6.9. This is appropriate to the extent that the statistical properties of the ionized regions are mainly determined by the volume-averaged ionized fraction, and are relatively insensitive to the precise redshift at which a given ionized fraction is reached (seeMcQuinn et al. 2007 and Furlanetto et al. 2004b.) The 1 − σ spread shown in the figure extends past x i = 1, but this is only because the distribution of ionized fractions is not symmetric about the mean, i.e., the probability distribution function of the ionized fraction is non-Gaussian. This choice might appear to preclude the possibility of detecting bubbles at the end of reionization when x HI (1 + δ) ≤ 0.075. However, the threshold choice is only used to set the maximum template radius, and so ionized regions may still in principle be detected at these late stages of reionization. The ionized regions identified at the end of reionization are, however, generally less robust given the reduced contrast between fully ionized and average regions at these times (see §5.1). ACKNOWLEDGEMENTSWe thank Judd Bowman and Piyanat Kittiwisit for related collaboration and feedback, and Matt McQuinn for providing several helpful suggestions on a draft manuscript and for useful discussions. We also acknowledge a helpful report from the anonymous referee. We also thank Suvendra Dutta, Steve Furlanetto, Lars Hernquist, Peng Oh, Jonathan Pritchard, Oliver Zahn, and Matias Zaldarriaga for discussions regarding imaging and bubble finding in noisy 21 cm data. 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[]
[ "Chiral Separation by Flows: The Role of Flow Symmetry and Dimensionality", "Chiral Separation by Flows: The Role of Flow Symmetry and Dimensionality" ]
[ "Sunghan Ro \nGraduate School of Nanoscience and Technology\nKorea Advanced Institute of Science and Technology\n34141DeajeonKorea\n", "Juyeon Yi \nDepartment of Physics\nPusan National University\n46241BusanKorea\n", "Yong Woon Kim \nGraduate School of Nanoscience and Technology\nKorea Advanced Institute of Science and Technology\n34141DeajeonKorea\n" ]
[ "Graduate School of Nanoscience and Technology\nKorea Advanced Institute of Science and Technology\n34141DeajeonKorea", "Department of Physics\nPusan National University\n46241BusanKorea", "Graduate School of Nanoscience and Technology\nKorea Advanced Institute of Science and Technology\n34141DeajeonKorea" ]
[]
Separation of enantiomers by flows is a promising chiral resolution method using cost-effective microfluidics. Notwithstanding a number of experimental and numerical studies, a fundamental understanding still remains elusive, and an important question as to whether it is possible to specify common physical properties of flows that induce separation has not been addressed. Here, we study the separation of rigid chiral objects of an arbitrary shape induced by a linear flow field at low Reynolds numbers. Based on a symmetry property under parity inversion, we show that the rate-of-strain field is essential to drift the objects in opposite directions according to chirality. From eigenmode analysis, we also derive an analytic expression for the separation conditions which shows that the flow field should be quasi-two-dimensional for the precise and efficient resolutions of microscopic enantiomers. We demonstrate this prediction by Langevin dynamics simulations with hydrodynamic interactions fully implemented. Finally, we discuss the practical feasibility of the linear flow analysis, considering separations by a vortex flow or an extensional flow under a confining potential.An enantiomer is a molecule that cannot be superposed on its own mirror image, and such property is called chirality 1 . Despite the structural similarity, a pair of enantiomers often exhibits very different biochemical activities due to the chiral nature of living systems 2 . It is, therefore, of great importance to separate a racemic mixture by chirality in the pharmaceutical, agricultural, and environmental industries where a number of molecules in use are enantiomers 3 . The conventional separation techniques such as chromatography and capillary electrophoresis require molecular specific and expensive chiral agents or media.Among alternative physical separation methods that do not rely on a chiral selector 4-6 , chiral resolution by flows has recently received considerable attention with rapid developments in microfluidics 7-19 . Since the original suggestions 7,8 , chirality-dependent drift has been demonstrated by several experiments using shear or vortex flows 9-13 . While a number of numerical studies[14][15][16][17][18][19]have been done to propose various sorting strategies by assuming particular flow fields, e.g., microfluidic vortices 14 or asymmetric flows with different slip lengths15,16, there are few efforts to develop a general theoretical framework for arbitrary linear flow patterns and object shapes. Hence, important questions, what are the common characteristics, if any, of flows that cause separation? or what is the role of each flow component in separation?, are still to be answered.In this work, we address this problem by introducing a theoretical framework to understand the motion of a rigid chiral object of any shape in an arbitrary linear Stokes flow. The essential flow component for separation is elucidated from a symmetry argument using parity inversion and mirror reflection. We also show that the velocity gradient tensor has to be nearly singular, i.e., quasi-two-dimensional, to induce the separation of high precision. In order to validate our analytic results, we perform Langevin dynamics simulations that explicitly incorporate hydrodynamic interactions among object elements. In simulations, the separation precision is quantified by the Jensen-Shannon divergence between the probability distributions of particles of opposite chirality.ResultsModel. Consider a rigid chiral object moving through a viscous incompressible fluid. Let  r be the position of any point O fixed to the particle, say, the center of mass, with respect to a reference frame. At low Reynolds number (i.e., in the inertialess regime), the equations of motion are obtained from the Stokes equation for translational velocity  v of that point and angular velocity ω of the object, in relative to the flow 20-22 :
10.1038/srep35144
null
4,582,513
1611.03223
295eb95ee50a08e4ef48a90221f02c8ab54f5744
Chiral Separation by Flows: The Role of Flow Symmetry and Dimensionality Published: 14 October 2016 Sunghan Ro Graduate School of Nanoscience and Technology Korea Advanced Institute of Science and Technology 34141DeajeonKorea Juyeon Yi Department of Physics Pusan National University 46241BusanKorea Yong Woon Kim Graduate School of Nanoscience and Technology Korea Advanced Institute of Science and Technology 34141DeajeonKorea Chiral Separation by Flows: The Role of Flow Symmetry and Dimensionality Published: 14 October 201610.1038/srep35144received: 23 May 2016 Accepted: 21 September 20161 Scientific RepoRts | 6:35144 | Correspondence and requests for materials should be addressed to Y.W.K. ( Separation of enantiomers by flows is a promising chiral resolution method using cost-effective microfluidics. Notwithstanding a number of experimental and numerical studies, a fundamental understanding still remains elusive, and an important question as to whether it is possible to specify common physical properties of flows that induce separation has not been addressed. Here, we study the separation of rigid chiral objects of an arbitrary shape induced by a linear flow field at low Reynolds numbers. Based on a symmetry property under parity inversion, we show that the rate-of-strain field is essential to drift the objects in opposite directions according to chirality. From eigenmode analysis, we also derive an analytic expression for the separation conditions which shows that the flow field should be quasi-two-dimensional for the precise and efficient resolutions of microscopic enantiomers. We demonstrate this prediction by Langevin dynamics simulations with hydrodynamic interactions fully implemented. Finally, we discuss the practical feasibility of the linear flow analysis, considering separations by a vortex flow or an extensional flow under a confining potential.An enantiomer is a molecule that cannot be superposed on its own mirror image, and such property is called chirality 1 . Despite the structural similarity, a pair of enantiomers often exhibits very different biochemical activities due to the chiral nature of living systems 2 . It is, therefore, of great importance to separate a racemic mixture by chirality in the pharmaceutical, agricultural, and environmental industries where a number of molecules in use are enantiomers 3 . The conventional separation techniques such as chromatography and capillary electrophoresis require molecular specific and expensive chiral agents or media.Among alternative physical separation methods that do not rely on a chiral selector 4-6 , chiral resolution by flows has recently received considerable attention with rapid developments in microfluidics 7-19 . Since the original suggestions 7,8 , chirality-dependent drift has been demonstrated by several experiments using shear or vortex flows 9-13 . While a number of numerical studies[14][15][16][17][18][19]have been done to propose various sorting strategies by assuming particular flow fields, e.g., microfluidic vortices 14 or asymmetric flows with different slip lengths15,16, there are few efforts to develop a general theoretical framework for arbitrary linear flow patterns and object shapes. Hence, important questions, what are the common characteristics, if any, of flows that cause separation? or what is the role of each flow component in separation?, are still to be answered.In this work, we address this problem by introducing a theoretical framework to understand the motion of a rigid chiral object of any shape in an arbitrary linear Stokes flow. The essential flow component for separation is elucidated from a symmetry argument using parity inversion and mirror reflection. We also show that the velocity gradient tensor has to be nearly singular, i.e., quasi-two-dimensional, to induce the separation of high precision. In order to validate our analytic results, we perform Langevin dynamics simulations that explicitly incorporate hydrodynamic interactions among object elements. In simulations, the separation precision is quantified by the Jensen-Shannon divergence between the probability distributions of particles of opposite chirality.ResultsModel. Consider a rigid chiral object moving through a viscous incompressible fluid. Let  r be the position of any point O fixed to the particle, say, the center of mass, with respect to a reference frame. At low Reynolds number (i.e., in the inertialess regime), the equations of motion are obtained from the Stokes equation for translational velocity  v of that point and angular velocity ω of the object, in relative to the flow 20-22 : Scientific RepoRts | 6:35144 | DOI: 10.1038/srep35144 ω µ µ µ µ ξ ξ ζ ζ      − − Ω       =                          +                v U E : (1) tt tr rt rr te re t r where ≡ U U r ( ) denotes the ambient fluid velocity at  r , and Ω = ∇ × U r ( ) 1 2 is half the fluid vorticity. Here the ambient flow is approximated as a linear Stokes flow = + U r U r J ( ) (0) with a constant velocity gradient tensor = ∇ U J 21 . The rate-of-strain field E is the symmetric part of J, given as E = (J + J T )/2 with the transpose operator T. The mobility tensors, μ's, are related to the strengths of thermal noises, ξ  's, through the fluctuation dissipation theorem ξ ξ µ 〈 〉= k T 2 i m j n B ij mn , with m, n = t, r and k B T being the thermal energy at temperature T. Here the superscripts, t and r, stand for 'translational' and 'rotational' , respectively; μ's couple two degrees of freedom corresponding to its superscripts. The third-rank resistance tensors, ζ's, determine hydrodynamic friction force and couple the translational and rotational motions of the object with the rate-of-strain field (for which we assign 'e' as a superscript symbol, and see also index notation of Carrasco et al. 23 ). The tensor product in Eq. (1) is defined as (ζ : E) i ≡ ∑ j, k ζ ijk E jk . In calculating μ and ζ, the no-slip condition on object surface is imposed, and they depend on the position vectors of the surface elements relative to O. The mobility and resistance tensors are thus independent of the choice of the origin, and they are functions only of the geometry of the particle such as object orientation φ and handedness α. Therefore, the chirality-dependent drift can be induced only by the rate-of-strain field E. We write it in shorthand notation, ϕ µ µ ζ ζ ≡ + α v E E ( ) ( : ) ( : ),(2) E tt te tr re with chirality index α = R or L and refer to it as drift velocity. Symmetry under parity inversion. Using a symmetry property under parity-inversion operation, we first show that the chiral separation (objects move in opposite directions according to handedness indeed) occurs by the parity-even rate-of-strain field. In particular, this is proven even with considering the rotational motions of object orientations. Using parity-inversion () operation about the point O, a linear flow is decomposed into two parts; one is a parity-odd flow whose direction is changed by  operation, and the other is a parity-even flow that remains invariant under  operation. In Fig. 1, translation motions of a chiral molecule and its chiral partner under parity-odd and parity-even flows are illustrated, respectively. Shown on the left panel of Fig. 1(a) is a left-handed (L) object in a parity-odd flow, which is supposed to drift to the left by the flow. Applying  operation on this system, the flow direction is inverted, and the object handedness as well as the drift direction are reversed (middle panel). If the flow field becomes again inverted (right panel), the original flow field is recovered and the right-handed (R) object then moves to the left because of the linearity of relations between the object velocity and the flow fields. One clearly sees that the drift motions of the chiral pair are identical under the parity-odd flow, leading to no chirality resolutions. On the contrary, if  operation is applied on an object of L in a parity-even flow ( Fig. 1(b)), the object handedness is converted into R and the drift direction is changed as well, while the flow itself remains invariant. This enables us to have the opposite motions of the chiral pair under the same ambient flow and thus demonstrates that the parity-even flow field is essential for chiral separation. According to Eq. (1), the translational velocity  v of an object includes two flow components U and E. It is obvious that U is parity-odd while E is parity-even as J remains invariant under  operation. Therefore, from the analysis above, we find that the drift velocity of a left-handed object is opposite to its chiral partner under a given rate-of-strain field: where ϕ′ denotes parity-inverted orientation of φ . It is to be noted that the angular motion remains invariant under  operation, and thus the probability distributions of orientation are symmetric, ϕ ϕ = − ′  v v ( ) ( ),(3)ϕ ϕ Φ =Φ ′( ) ( ) L R , at any time t (see the Supplemental Information for details). Accordingly, the orientation-averaged drift velocity of the left-handed object is given by ∫ ∫ ϕ ϕ ϕ ϕ ϕ ϕ = Φ = − ′ ′ Φ ′ = − Φ Φ    ˆˆv d v d v v ( ) ( ) ( ) ( ) ,(4)L L L R R R E E E E where the negative sign of the average drift velocity implies a possibility of chiral separation. In a similar way, the separation direction can be specified if a linear flow field has a mirror-reflection symmetry. Shear flow, γ = Û r yx ( ) with shear rate γ  , belongs to the case, if choosing the xy-plane as a mirror-symmetry plane. Regard a relation, ϕ ϕ ϕ ϕ = ″ − ″ ⊥ ⊥ ″ ′ˆˆv v v v ( ( ), ( )) ( ( ), ( )),(5)L L R R E E E E , , , , where the double prime indicates quantities transformed by mirror-reflection () operation. Here, α v E, and α ⊥ v E, are, respectively, the parallel and perpendicular components of a drift velocity α  v E to a chosen mirror-reflection plane. For a mirror-symmetric flow field (E″ = E), taking average over object orientations for ϕ ϕ Φ =Φ ′( ) ( ) L R , one has a relation, = − . Φ ⊥ Φ Φ ⊥ Φ ( ) ( ) v v v v , ,(6)L L R R E E E E , , , , Combining with Eq. (4), one finds that 〈 〉 α Φ v E, = 0, and the separation of objects with opposite chirality can occur along the direction perpendicular to the mirror-symmetry plane, as indeed in the case of the shear flow. We again note that Eqs (4) and (6)  v E to be well separated on macroscopic scales, at least, larger than experimental resolutions of microfluidic devices. Below we argue that this is achieved with quasi-two-dimensional flow fields. Eigenmode analysis. It is more intuitive to examine the equation for  v for the case of diagonalizable J (for a non-diagonalizable case the analysis can be performed in a similar way, leading to the qualitatively same conclusions; see the Supplemental Information for details). The equation for  v in Eq. (1) can be written in the diagonalizing basis of J, and one of its components along the direction of the q-th eigenvector of J reads as, λ ξ = + + λ α λ α λ α λ  r r v ,(7)q E, t q q q q where λ q is the q-th eigenvalue of J, and quantities with the subscript λ q symbolize the q-th component of transformed vectors = λ X X S ( ) q q with (SJS −1 ) q,q′ = λ q δ q,q′ and X = r, v E , ξ t . For notational simplicity, we shall drop the eigenvalue index q and chirality index α hereafter. Without loss of generality, we have set = U(0) 0. The formal solution of Eq. (7) is given by = + + Ξ . λ λ λ λ λ r t e r X t t ( ) (0) () () (8) t E, Here we define the displacements resulting from the drift motion, ∫ = ′ ′ λ λ λ − ′ X t dt e v t ( ) ( ) t t t E E ,0 ( ) , , and from the thermal noise ∫ ξ Ξ = ′ ′ λ λ λ − ′ t d t e t ( ) ( ) t t t 0 ( ) t . Taking average over the thermal noises and initial positions, we obtain the average displacement = λ λ r t X t ( ) ( ) (9) E, with the average of initial positions located at origin. Mean separation distance between particles having opposite chirality is given by  = − λ λ t r t r t ( ) ( ) ( ) R L , and separation precision can be quantified by  Σ t t ( )/ ( ) with Σ ≡ − λ λ t r t r t ( ) ( ) ( ) 2 2 2 . The chiral separation is achieved if the quantifiers satisfy following relations: σ Σ .   t t t ( )/ ( ) 1 and ( )/ 1 (10) 0   The first condition requires that the mean separation distance t ( )  should be much larger than the dispersion Σ (t) for efficient separation. The second condition states that the separation distance t ( )  ought to be much larger than the initial width of distribution σ 0 which roughly amounts to experimental resolution such as the distance between recovery outlets of devices for the separated particles. It is clear that an exact analysis of the separation conditions (10) requires to obtain the precise form of v E,λ (t) which is highly nonlinear and has no known solution to the best of our knowledge. Notwithstanding the difficulty, one can still put forward a reasonable analysis: Suppose that there exists a finite maximum drift velocity v m of v E,λ (t), viz. v E,λ (t) ≤ v m , which is reasonable because the drift velocity as given in Eq. (2) is determined by the product of the bounded quantities. It is also assumed that up to the leading order, the maximum drift velocity v m is proportional to the magnitude of flow field given as λ ∑ J i i 3 2 , and for a fixed |J|, only weakly depends on an individual λ i . This assumption may be supported by an observation that as increasing the flow gradient tensor J by a factor of a certain constant, the resulting drift velocity will be increased proportionally. Lacking in mathematical rigour, this ansatz is effective to extract an essential flow property to induce chiral separation, as evidenced later in our numerical simulations. Since ∫ ∫ ≤ dx f x dx g x ( ) ( ) for f(x) ≤ g(x), the maximum value of the mean separation is determined by r λ (t) in Eq. (8) with letting v E,λ (t) = v m as ≡ = 〈 〉 λ = λ d t D t r t ( ) max[ ( )] 2 [ ( )] (11) v t v ( ) m E, λ = − λ v e 2( / )( 1),(12) m t On the other hand, the positional dispersion can be written as σ σ σ λ Σ = + − = + − λ λ λ λ t t X t X t t e D e ( ) ( ) ( ) ( ) , ( ) ( /2 )( 1 ),(13)t t E E 2 2 , 2 , 2 2 0 2 2 2 where σ(t) determines the dispersion when v E,λ (t) is purely deterministic to annihilate fluctuations of X E,λ (t). Obviously, an inequality, Σ (t) ≥ σ(t), follows, which together with Eq. (10) then constitutes a necessary condition for chiral separation, (11) leads to σ Σ ≤ t t d t t ( )/ ( ) ( )/ ( )  and σ σ ≤ t d t ( )/ ()/ 0 0  . Equationσ σ   d t t dt ( )/ ( ) 1 and ( )/ 1,(14) 0 which is given in a greatly simplified form to allow an analytic approach and helps to extract essential flow factor inducing chiral separation. It should also be mentioned that in high Péclet number regime the drift velocity as a function of object orientation remains roughly constant, as suggested by Marcos et al. 11 . For the case, v m can be interpreted as the constant drift velocity and hence, conditions in Eq. (14) are equivalent to Eq. (10). In the following, we proceed our analysis of the separation conditions, (14) with d(t) and σ(t) determined by Eq. (12) and Eq. (13), respectively. Short time behavior. We examine the behavior of the separation precision for t ≪ t λ ≡ |λ| −1 , where t λ is the saturation time scale at which the separation precision with a non-zero λ approaches a constant value. In Fig. 2, we illustrate the characteristic behaviors of d/σ depending on λ. For λ = 0, d/σ grows unboundedly with time t, while for λ ≠ 0, it approaches a constant value after a time t λ . That is, in the case of λ ≠ 0, the feasibility of d/σ ≫ 1 depends on the system parameters such as initial dispersion and diffusion constant, even at large t. For t ≪ t λ , d is monotonically increasing function of time, and irrespective of the sign of the eigenvalue λ, d/σ is simplified as . From the separation conditions, Eq. (14), and the equation of the separation precision, Eq. (15), one can readily define the separation time scale t s at which both d/σ and d/σ 0 become of the order of unity (see Fig. 2 σ + σ d t t t t t t ( ) ( ) ,(15)) as ≡ σ t t t max( , ),(16) s D and therefore, an efficient and precise chiral resolution (i.e., d/σ ≫ 1 and d/σ 0 ≫ 1) occurs when . λ   t t t (17) s This constitutes the condition of macroscopic chiral separation for short times, i.e., t ≪ t λ . Long time behavior. In the long time regime of t ≫ t λ , if the eigenvalue is not zero (λ ≠ 0), d/σ is saturated to a certain value as σ λ λ                + > < . λ σ λ λ d t t t t t t ( 0) ( 0) (18) D D 2 When λ > 0, it is found from Eq. (18) that the condition of macroscopic separation (d/σ ≫ 1) is translated into . λ  t t (19) s This can also be intuitively understood from Fig. 2, i.e., only when t λ ≫ t s , the saturated value of d/σ can be much greater than one. Then, the condition of d/σ 0 ≫ 1 is always satisfied since σ − λ σ λ d t t e / ( / )( 1 ) t t 0 / . On the other hand, when λ < 0, the first condition of Eq. (14) leads to t λ ≫ t D , and the second condition does to t λ ≫ t σ . Remarkably, we find that in both of short-time and long-time regime, the macroscopic chiral separations are dictated by a single criterion, irrespectively of sign of λ: λ =             λ σ t t t t t or 1 min 1 , 1 ,(20) s s D which means that the eigenvalue of the velocity gradient tensor should be significantly smaller than the inverse of the separation time scale. We note that the eigenvalue index q has been omitted for notation simplicity. Therefore, Eq. (20) should be interpreted as conditions required to be satisfied by respective λ q in order to obtain the separation along the corresponding coordinate λ r q . In other words, the chiral separation may occur if at least one of the eigenvalues satisfies the condition. Dimensional analysis and estimates of parameters. Let us now express the separation condition, Eq. (20), in terms of physical parameters such as the linear size of the object  and magnitude of the flow velocity gradient V defined through J (for example, see Eqs (22) and (23) of the next section). One may define V also as ∑ = V J i j ij , 1 3 2 , where the proportionality constant is of the order of unity, and it is irrelevant to the present analysis. We introduce dimensionless parameters δ σ ε λ ≡ ≡ ≡   V c v V / , / , / , m 0 where δ is initial dispersion σ 0 in units of , and it seems reasonable to assume δ ≫ 1 in most cases of practical interest of small particles. ε is the amplitude of λ relative to the magnitude of the flow velocity gradient, which can characterize the dimensionality of the flow field; if ε = 0, the corresponding flow becomes a two-dimensional flow. c measures the chiral-dependent drift velocity relative to a variation of ambient flow velocity over . Since η D k T/ B with the solvent viscosity η, t σ = σ 0 /v m ~ δ/cV and η =  t D v k T c V / / D m B 2 2 2 3 . As a result, the separation condition of Eq. (20) can be written in dimensionless form as . Among the parameters, c affects both of t σ and t D , and its estimation is important for numerical evaluations of Eq. (21). The exact value of c, of course, relies on system details such as object shape and chirality, and it should be a tremendous task to obtain the general expression of c analytically. However, as we show in the Supplemental Information, a possible upper bound of c can be envisioned, which turns out to be of the order of 10 −2 . In addition, we numerically evaluate c for different objects and flow patterns considered in simulations, which is indeed found to be small as consistent with the proposed upper bound (see the next section of Simulation results). Combining these facts, we can take a conservative bound of |ε| for an efficient separation as  ε − ⪅ (10 ) 3 for various flow strengths and/or object sizes, even though a rather unrealistically narrow initial distribution (δ ~ 1, i.e., σ  0 ) is assumed. Considering the measurement and control accuracy of current microfluidic devices (to our knowledge, of the order of 0.1% at best), one might view this range of ε to be synonymous for a singular flow. It implies that the chiral separation is possible only by quasi-two-dimensional flows described by a velocity gradient tensor J with a vanishingly small eigenvalue. ε δ           c c Pe min ,(21) Before going further, we explicitly mention the validity range of our theory, especially, in terms of the relevant object size . Our theoretic formulation assumes the low Reynolds and high Péclet number conditions, which give the range of appropriate object size as η η ρ k T V V ( / ) (/ ) B 1/3 1 /2 . Note that ρ η =  v Re / where  v V and ρ is the fluid density. For a water at room temperature, µ µ . 0 01 m 1 m for V ~ 10 6 /s, and µ µ . 0 1 m 100 m for V ~ 10 2 /s. The present theory is based on the assumption of linear flow field which is hard to be realized in most cases over extended length and time scales. For example, the presence of hydrodynamic boundaries arising from confining walls of microfluidic devices leads to nonlinear flows, which might yield nontrivial effects on the separation. However, analysis on general nonlinear flow fields is beyond the scope of the present study, and the relevance of our linear flow analysis will be discussed later in more detail in section of Possible applications. Simulation results. In order to demonstrate the arguments, we perform Langevin dynamics simulations by integrating Eq. (1) with explicitly taking into account full hydrodynamic interactions among object elements at the level of the Rotne-Prager tensor 23 . As typical examples of enantiomers, helix and tetrahedral structures are concerned. We decompose the structures into arrays of N closely packed beads with radius a. The decomposition allows us to calculate their grand mobility tensors which in turn yield mobility tensors for a rigid body motion of Eq. (1), according to the conversion equations, Eq. (19) to (22) in ref. 23 (see the Supplemental Information for details). In simulations, we rescale all lengths by the bead radius a, giving linear size  of the helix and tetrahedral structure as 225a and 254a, respectively. Characteristic time scale is τ = 6πηNa 3 /k B T. The distribution of initial positions of objects is assumed to be a Gaussian centered at the origin with an initial width σ 0 . We consider two different types of flow fields, one with a diagonalizable velocity gradient tensor J A and the other with a non-diagonalizable tensor J B , parameterized by a dimensionless variable ε. First, flow-A is described by the velocity gradient tensor, ε ε ε ε =        −         V J ( ) 2 0 1 2 0 0 0 ,(22) A A which has distinct eigenvalues, λ 3 /V A = ε and λ ε ε 2 2 . If ε = 0, the eigenvalue λ 3 vanishes and the corresponding J A describes a two-dimensional flow. Flow-B is represented by = − ± + V / ( 8 9 )/2 A 1,ε ε ε ε =        − −        V J ( ) 1 0 0 0 0 0 2 ,(23)B B where a finite ε describes a deviation from the shear flow (ε = 0), and eigenvalues are degenerated, λ 3 /V B = 2ε and λ 1,2 /V B = − ε. Both flows are incompressible, i.e., J is traceless, and have a reflection symmetry about xy-plane. According to our symmetry argument, the chirality-dependent drift is expected to occur along the z-direction which is the eigenmode direction of λ 3 . Dimensionless flow velocities, τ πη = = ∼ V V Na V k T 6 / i i i B 3 for i = A, B, are set to 30 in order to realize high Pe; Pe ~ 3.2 × 10 4 for helix and Pe ~ 3.9 × 10 4 for tetrahedral. The magnitude of flow velocity gradient, V, can be defined in a basis independent way as = ∑ V EE i j ij ij 0 , . However, the current expression of V equals V 0 up to a prefactor of the order of one and does not lead to any qualitative difference in results. For a wide range of flow velocity V and object size , c is found to be small as − ⪅ c 10 3 for the helix and − ⪅ c 10 2 for the tetrahedral, in accord with the estimate on the possible maximum value of c (see the Supplemental Information). The previous criterion of Eq. (21) predicts that for separation to occur, the upper bound of |ε| is given as ε − ⪅ O(10 ) 3 even when a very narrow initial distribution of molecular size is assumed (δ ~ 1). We test this prediction through numerical simulations, as varying the value of ε of the flows considered above. Figure 3 explicitly shows the flow fields at different flow parameters, ε = 0 and ± 0.01, for the flow-B (a-c), and the time evolutions of corresponding probability distributions of R or L helices, along the z-direction, obtained from the simulations (d-f). Note that, in (b-c), the arrows indicating difference fields are magnified by 50 times for clear visibility. As consistent with Eq. (21), a small but finite value of ε leads to qualitatively different behaviors, despite the apparent similarity to the separable shear flow with ε = 0. For either small positive or negative ε, the probability distributions continue to substantially overlap even at t ≥ t λ , while for ε = 0, the two distributions are well discriminated and d increases with time, enabling complete separation. Now we propose to quantify the degree of the separation by the Jensen-Shannon divergence (JSD): ∫ ∑ || =     +     α α α p p dzp z p z p z p z JSD( ) 1 2 ( )ln 2 ( ) ( ) ( ) , L R L R where p α (z) represents the probability distribution to find an object with chirality α = L, R at a projected position z along a chosen axis of observation. Unlike d/σ, JSD is bounded as 0 ≤ JSD ≤ ln2, and its value depends on overlapping area between p L (z) and p R (z). If p L (z) = p R (z) (perfect overlap), JSD vanishes. If p L (z) has no overlapping region with p R (z) (complete separation), JSD reaches its maximum value ln 2. Any values of JSD less than ln 2 signal that finite overlap between p L (z) and p R (z) exists, and resulting separation is inaccurate. JSD therefore provides the well-defined scale of separation precision, taking into account shape details of p α (z). Fig. 4 is JSD as a function of ε for different flow patterns and chiral objects. It follows from Eqs (15) and (18) that when λ = 0, d/σ > 1 for t > t s , while for λ ≠ 0, it is saturated to a constant value after t λ . The occurrence of macroscopic separation can therefore be determined by measuring JSD at time t larger than t s (for λ = 0) and t λ (for λ ≠ 0) which are estimated as follows. For the high Pe regimes considered in this work, t s is given by t σ ~ δ/cV. In simulations, the upper bound of δ is (1)  , the lower bound of c is − , and the smallest value of |ε| other than zero is 0.01 in Fig. 4. Thus, JSD's are measured at  τ t (10 ) 3 for ε ≤ 0, τ (10 )  for 0 < ε ≤ 0.2, and  τ ( ) for ε > 0.2, which are long enough for JSD to reach its stationary value. For numerical evaluations of JSD, the probability distributions p(z) are discretized into histograms with the bin size of the order of the object size. The complete separation with the maximum JSD of ln 2 is achieved in all considered cases only when ε (and thus at least one of the eigenvalues) is vanishingly small. A finite value of JSD results for small , it is more pronounced that JSD yields very small values unless ε is vanishingly small. As shown in Fig. 4, the simulation results clearly demonstrate our claim that the complete separation indeed occurs by quasi-two-dimensional flows satisfying the condition, Eq. (21). Depicted in Possible applications. Finally, we discuss the relevance of our linear flow analysis in terms of possible practical applications. In most cases, it is challenging to realize linear flows persisting for extended length and time scales in microfluidic devices. A simple shear flow is a well-known exception. We exhibit here two other examples where the linearity of flows is easily assured and at the same time, the present analysis can be useful for practical purposes. In particular, we show that for chiral separations, it is enough to have a linear flow only in a localized region if a confining potential is applied together. Vortex flow. According to our theoretic formulations, the chiral separation occurs when both of the following conditions have to be satisfied: First, one of the eigenvalues of velocity gradient tensor should be much smaller than the inverse of the separation time scale (the eigenmode analysis). Secondly, there should exist a non-vanishing rate-of-strain field that induces a finite drift velocity (the parity-inversion argument). Now we present a salient example explicitly showing the indispensable role of the rate-of-strain field: consider a pure rotational flow with a perfect circular streamline, defined by the velocity gradient tensor, =        −        . V J 0 1 0 1 0 0 0 0 0( 24) This flow satisfies the first condition but obviously not the second condition because it has a vanishing rate-of-strain field, E = 0. Hence it cannot separate any kind of chiral pairs. On the contrary, a vortex flow deformed by a finite rate-of-strain field, e.g., with the following velocity gradient tensor, =        . − − .        V J 0 8 1 0 1 0 8 0 0 0 0 ( 25) fulfills the both conditions and could, therefore, lead to the separations, as indeed confirmed by the Langevin dynamics simulations (Fig. 5). The vortex flow can be an obvious solution for chiral separations, if not perfectly circular, and is another example of linear flows that persist for an extended period of time in microfluidic setups. As shown here, the vortex flows with circular streamlines have very distinct separation powers, depending on whether or not they have a finite rate-of-strain field. Our prediction on the separation power of vortex flows can be tested by a microfluidic four-roll mill device suggested by Lee et al. which can produce the entire spectrum of flow types, from purely rotational flow to purely extensional flow, by varying flow rate ratio 24 . Considering the dimensions of the device, the sub-micron helical objects ( µ  ⪅ m), for example, should be observed to be separated for elongated vortex flow but not for purely rotational flow. Also using various chiral objects of different shapes in this setup can be a feasible way to confirm our theory predicting that the flow properties rather than object shapes are essential to chiral separation. Confining potential. At low Reynolds numbers, the resistance formalism linearly relates the hydrodynamic net force and torque exerted on a rigid object to the flow parameters 21 . Due to the linearity of the relations, originating from the linearity of the Stokes equations, the external forces such as confining force can be separately added into the equations of motion (see the Supplemental Information). Consequently, even in the presence of external forces our analysis can be performed in a similar way and the separation criterion remains intact in general. In order to numerically verify this, we perform Langevin dynamics simulations with Flow-A and Flow-B previously defined, considering helical and tetrahedral objects under an external potential, φ(x, y) = k(x 2 + y 2 ). Here, we set ε = 0, and then Flow-A and B represent a two-dimensional extensional flow and a shear flow, respectively. The external potential plays a role as a two-dimensional confinement, constraining particle positions near to the origin in the xy plane. We note that for both flows, the separation occurs along the z-axis. The right panel of Fig. 6 shows the simulation results of temporal evolution of JSD in the presence of the potential while the left panel exhibits the results without the potential. As deduced, one can see that the separation behaviors remains the same qualitatively. This suggests that for practical applications of our analysis, a linear flow field does not necessarily have to persist over extended length and time scales. The two-dimensional confinement potential makes the separation (e.g., along the z-axis) take place only in a limited space of the xy-plane where the linearity of flows can be rather easily assumed for a time longer than the separation time scale. Discussion In practical applications of harnessing microfluidic devices for separations, one of the central questions will be to determine which flow has separation capability. Despite increasing attention to microfluidic chiral resolutions, most of the previous works are restricted to be considering an object of a specific shape in a given flow pattern. The complicated mathematical structure of the equations of motion present difficulties in understanding chiral separation phenomena even for a specific flow, and a comprehensive picture of common mechanisms and general conditions for chiral separations in terms of flow properties has been lacking. We have tackled this problem for an arbitrary linear flow using simple ideas, namely, considering symmetry properties and adopting the eigenmode analysis of flow fields. These enable us to draw an intuitive physical picture of the underlying mechanism of chiral separation dynamics. According to our results, the common features of separable flows are summarized as i) flows with a finite strain-rate tensor and ii) quasi-two-dimensional flows with small eigenvalues obeying Eq. (20). The typical examples satisfying both conditions are shear flow, vortex flow, and two-dimensional extensional flow, all of which are indeed demonstrated here to cause separations via the Langevin dynamics simulations. The present study thus provides a theoretical understanding of why two-dimensional flows such as shear and vortex flow are efficient to induce chiral separations. Our results provide simple criteria that would allow us to categorize and decide what kind of flows have a separation power or not. This is the prediction that cannot be easily made, without complicated numerical calculations or extensive simulations, from the theoretical studies known hitherto. It is also important to note that the separation criteria only concern the properties of flows, not of objects, so that they are applicable for objects of different shapes. In the presence of confining potential φ(x, y) = k(x 2 + y 2 ) with k = 8k B T/a 2 , JSD shows the very similar behaviors and the complete separation is also achieved for all cases. For helix and tetrahedral, the same objects as in Fig. 4 are considered, the number of ensembles is 10 5 , and the initial distributions are given as Gaussian centered at the origin with dispersion of 100a. Figure 1 . 1Schematic figure illustrating motions of a chiral object and its chiral partner under (a) parity-odd and (b) parity-even flows. The chiral objects are represented by the helices, the flow fields by blue arrows, and red arrows indicate the translational motions of objects induced by the flow. Scientific RepoRts | 6:35144 | DOI: 10.1038/srep35144 are obtained through the ensemble averages over time-dependent probability distributions of orientations. Separation criterion. The nonvanishing average drift velocity α Φ  v E is, though essential, only a necessary condition. For practical realizations of chiral separations, the position dependence of U should be analyzed. In other words, the chirality-independent drift by U r ( ) should not dominate the chirality-dependent drift α  v E by E. Furthermore, the positional distributions of particles should evolve by α σ ≡ σ 0 /v m is the time required for an object to travel a distance of initial positional dispersion σ 0 by the drift velocity v m , and ≡ t D v / D m 2 is the time scale at which the traveling distance by the drift motion is comparable to Figure 2 . 2Separation precision, d/σ, as a function of time t at three different eigenvalues: λ = 0 (solid line), small λ with t λ = |λ| −1 ≫ t s (dashed line), and large λ with t λ = |λ| −1 ≪ t s (dotted line). Here, t s indicates the separation time scale at which d/σ ≈ 1. For a vanishing eigenvalue λ (i.e., a singular velocity gradient tensor), d/σ monotonically increases with t, while for a non-zero eigenvalue, d/σ is saturated to a constant value after the saturation time scale, t λ . Figure 3 . 3Three-dimensional visualization of flow B with (a) ε = 0, and difference fields for (b) ε = 0.01 and (c) ε = − 0.01. Arrow length is proportional to the velocity magnitude, and the color varies from red to blue as U x changes from positive to negative. Arrows in (b,c) are enlarged by 50 times for visibility. (d-f) Probability distributions of R helix(solid lines) and L helix(dotted lines) are shown at different times for the corresponding flow fields. At each case, data are obtained from 10 6 ensembles of Langevin dynamics simulations with the initial distribution of width, σ 0 = 100a. Figure 4 . 4JSD as functions of flow parameter ε for (a) helix and (b) tetrahedral under flow A (solid lines) and B (dashed lines). Two different widths of initial distributions are considered, i.e., σ 0 = 10a (black lines) and 100a (red lines). The behaviors near ε = 0 are magnified in the insets. Shown in (c,d) are parts of the helix and tetrahedral considered in simulations. The helix consists of 25 turns with 6-beads per turn, the radius of 3a, and the helical pitch of 9a (N = 150). The tetrahedral structure has arm lengths of 200a, 104a, 48a, and 24a (N = 189). Scientific RepoRts | 6:35144 | DOI: 10.1038/srep35144positive ε from unrealistically narrow initial dispersions of particles (σ . Figure 5 . 5Two-dimensional flow streamlines for (a) an elongated vortex flow (Eq. (25)) and (c) a pure rotational flow (Eq. (24)) in the xy-plane. Probability distributions of right-handed (black) and left-handed (red) helices, along the z-direction (in units of a), at t = 500τ under (b) the elongated vortex flow and (d) the pure rotational flow. (e) JSDs as a function of time (in units of τ) for the respective flow patterns. The chiral separation is clearly achieved for the elongated vortex flow, but not for the pure rotational flow. Initial distributions are given as Gaussian centered at the origin with dispersion of 100a, the number of ensembles is 10 5 , and the same helical objects are considered as in Fig. 4. = ∼ V 30. Figure 6 . 6Temporal evolution of JSD for helix (black) and tetrahedral (red) under flow A (solid lines) and flow B (dashed lines). Here, we set ε = 0, σ 0 = 100a, = ∼ V 30, and time is in units of τ. (Left panel) In the absence of external potential, JSD reaches its maximum of ln 2, indicating a complete chiral separation, ) Scientific RepoRts | 6:35144 | DOI: 10.1038/srep35144 © The Author(s) 2016 AcknowledgementsAuthor ContributionsY.W.K. supervised the project, S.R. carried out the calculations under the guidance of J.Y. and Y.W.K. All authors discussed the results, wrote and reviewed the manuscript.Additional Information Chiral separations. A M Stalcup, Annu. Rev. Anal. Chem. 3341Stalcup, A. M. Chiral separations. Annu. Rev. Anal. Chem. 3, 341 (2010). A basis for sophisticated nonsense in pharmacokinetics and clinical pharmacology. E J Ariëns, Stereochemistry, Eur. J. Clin. Pharmacol. 26663Ariëns, E. J. Stereochemistry, A basis for sophisticated nonsense in pharmacokinetics and clinical pharmacology. Eur. J. Clin. Pharmacol. 26, 663 (1984). Separation of enantiomers: needs, challenges, perspectives. N M Maier, P Fronco, W Lindner, J. Chromatogr. A. 9063Maier, N. M., Fronco, P. & Lindner, W. Separation of enantiomers: needs, challenges, perspectives. J. Chromatogr. A. 906, 3 (2001). Mechanical selection of chiral crystals. P G De Gennes, Europhys. Lett. 46827De Gennes, P. G. Mechanical selection of chiral crystals. Europhys. Lett. 46, 827 (1999). 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Separation of chiral colloidal particles in a helical flow field. Soft Matter 9, 2525 (2013). Vortex flows impact chirality-specific lift forces. T M Hermans, K J M Bishop, P S Stewart, S H Davis, B A Grzybowski, Nat. Commun. 65640Hermans, T. M., Bishop, K. J. M., Stewart, P. S., Davis, S. H. & Grzybowski, B. A. Vortex flows impact chirality-specific lift forces. Nat. Commun. 6, 5640 (2015). Chiral separation in microflows. M Kostur, M Schindler, P Talkner, P Hänggi, Phys. Rev. Lett. 9614502Kostur, M., Schindler, M., Talkner, P. & Hänggi, P. Chiral separation in microflows. Phys. Rev. Lett. 96, 014502 (2006). Separation of chiral particles in micro-or nanofluidic channels. S Meinhardt, J Smiatek, R Eichhorn, F Schmid, Phys. Rev. Lett. 108214504Meinhardt, S., Smiatek, J., Eichhorn, R. & Schmid, F. Separation of chiral particles in micro-or nanofluidic channels. Phys. Rev. Lett. 108, 214504 (2012). Microfluidic sorting of stereoisomers. R Eichhorn, Phys. Rev. Lett. 10534502Eichhorn, R. Microfluidic sorting of stereoisomers. Phys. Rev. Lett. 105, 034502 (2010). Dynamical solutions for migration of chiral DNA-type objects in shear flows. P Chen, Q Y Zhang, Phys. Rev. E. 8456309Chen, P. & Zhang, Q. Y. Dynamical solutions for migration of chiral DNA-type objects in shear flows. Phys. Rev. E 84, 056309 (2011). Shear-induced chiral migration of particles with anisotropic rigidity. N Watari, R G Larson, Phys. Rev. Lett. 102246001Watari, N. & Larson, R. G. Shear-induced chiral migration of particles with anisotropic rigidity. Phys. Rev. Lett. 102, 246001 (2009). Transport of flexible chiral objects in a uniform shear flow. P Talkner, G L Ingold, P Hänggi, New. J. Phys. 1473006Talkner, P., Ingold, G. L. & Hänggi, P. Transport of flexible chiral objects in a uniform shear flow. New. J. Phys. 14, 073006 (2012). J Happel, H Brenner, Low Reynolds Number Hydrodynamics. HagueMartinus Nijhoff PublishersHappel, J. & Brenner, H. 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[ "Acoustic signal detection through the cross-correlation method in experiments with different signal to noise ratio and reverberation conditions", "Acoustic signal detection through the cross-correlation method in experiments with different signal to noise ratio and reverberation conditions" ]
[ "S Adrián-Martínez \nInstitut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain\n", "M Ardid *[email protected] \nInstitut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain\n", "M Bou-Cabo \nInstitut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain\n", "I Felis \nInstitut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain\n", "C Llorens \nInstitut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain\n", "J A Martínez-Mora \nInstitut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain\n", "M Saldaña \nInstitut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain\n" ]
[ "Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain", "Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain", "Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain", "Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain", "Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain", "Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain", "Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC)\nUniversitat Politècnica de València\nC/Paranimf 146730GandiaSpain" ]
[]
The study and application of signal detection techniques based on crosscorrelation method for acoustic transient signals in noisy and reverberant environments are presented. These techniques are shown to provide high signal to noise ratio, good signal discernment from very close echoes and accurate detection of signal arrival time.The proposed methodology has been tested on real data collected in environments and conditions where its benefits can be shown. This work focuses on the acoustic detection applied to tasks of positioning in underwater structures and calibration such those as ANTARES and KM3NeT deep-sea neutrino telescopes, as well as, in particle detection through acoustic events for the COUPP/PICO detectors. Moreover, a method for obtaining the real amplitude of the signal in time (voltage) by using cross correlation has been developed and tested and is described in this work.
10.1007/978-3-662-46338-3_7
[ "https://export.arxiv.org/pdf/1502.05038v1.pdf" ]
22,902,066
1502.05038
5a4556d7c67a6dadea10dd47d21c62e6be28d84d
Acoustic signal detection through the cross-correlation method in experiments with different signal to noise ratio and reverberation conditions S Adrián-Martínez Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC) Universitat Politècnica de València C/Paranimf 146730GandiaSpain M Ardid *[email protected] Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC) Universitat Politècnica de València C/Paranimf 146730GandiaSpain M Bou-Cabo Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC) Universitat Politècnica de València C/Paranimf 146730GandiaSpain I Felis Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC) Universitat Politècnica de València C/Paranimf 146730GandiaSpain C Llorens Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC) Universitat Politècnica de València C/Paranimf 146730GandiaSpain J A Martínez-Mora Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC) Universitat Politècnica de València C/Paranimf 146730GandiaSpain M Saldaña Institut d'Investigació per a la Gestió Integrada de Zones Costaneres (IGIC) Universitat Politècnica de València C/Paranimf 146730GandiaSpain Acoustic signal detection through the cross-correlation method in experiments with different signal to noise ratio and reverberation conditions Acoustic signal detectioncross-correlation methodprocessing techniquespositioningunderwater neutrino telescopesparticle detectors The study and application of signal detection techniques based on crosscorrelation method for acoustic transient signals in noisy and reverberant environments are presented. These techniques are shown to provide high signal to noise ratio, good signal discernment from very close echoes and accurate detection of signal arrival time.The proposed methodology has been tested on real data collected in environments and conditions where its benefits can be shown. This work focuses on the acoustic detection applied to tasks of positioning in underwater structures and calibration such those as ANTARES and KM3NeT deep-sea neutrino telescopes, as well as, in particle detection through acoustic events for the COUPP/PICO detectors. Moreover, a method for obtaining the real amplitude of the signal in time (voltage) by using cross correlation has been developed and tested and is described in this work. Introduction Acoustic signal detection has become an object of interest due to its utility and applicability in fields such as particle detection, underwater communication, medical issues, etc. The group of Acoustics Applied to Astroparticle Detection from the Universitat Politècnica de València collaborates with the particle detectors ANTARES [1], KM3NeT [2] and COUPP/PICO [3]. Acoustic technologies and processing analyses are developed and studied for positioning, calibration and particle detection tasks of the detectors. Acoustic emitters and receivers are used for the positioning systems of underwater neutrino telescopes ANTARES [4] and KM3NeT [5] in order to monitor the position of the optical detection modules of these telescopes. The position of optical sensors need to be monitored with 10 cm accuracy to be able to determine the trajectory of the muon produced after a neutrino interaction in the vicinity of the telescope from the Cherenkov light that it produces [6]. An important aspect of the acoustic positioning system is the time accuracy in the acoustic signal detection since the positions are evaluated from triangulation of the distances between emitters and receivers, which are determined from the travel time of the acoustic wave and the knowledge of the sound speed. The distances between emitters and receivers are of the order of 1 km. Therefore, the acoustic emitted signals suffer a considerable attenuation in the medium and arrive to the acoustic receivers with a low signal to noise ratio. The environmental noise may mask the signal making the detection and the accurate determination of its arrival time a difficult goal, especially for the larger future telescope KM3NeT with larger distances. On the other hand, an acoustic test bench has been developed for understanding the acoustic processes occurred inside of the vessels of the COUPP Bubble Chamber detector when a particle interacts in the medium transferring a small amount of energy, but very localized, to the superheated media [7]. This interaction produces a bubble through the nucleation process. Under these circumstances the distance from the bubble to the vessel walls are very short (cm order) and a reverberant field generated by multiple reflections in the walls takes place. With these conditions, the distinction of the direct signal from reflection is quite difficult to achieve, being also quite complex to determine the time and amplitude of the acoustic signal produced. The elaboration of protocols and post-processing techniques are necessary for the correct detection of the signals used in these tasks. Methods based on time and frequency analysis result insufficient in some cases. The first step consists of using the traditional technic of cross-correlation between the received signals and the emitted signals (expected) for localizing the source distance. In addition, the use of specific signals with wide band frequency or non-correlated such as sine sweep signals or Maximum Length Sequence (MLS) signal together with correlation methods increase the amplitude and the correlation peak narrows, this allows a better signal detection, improves the accuracy in the arrival time and the discernment of echoes. In this work the detection of acoustic signals with a unique receiver under a reverberant field or a high noise environment is shown. The correlation method has been studied and applied for this purpose. Moreover, a method for obtaining the real amplitude of the signal (voltage) by using cross-correlation technique has been developed. Its validation has been done by comparing the results with the ones obtained by analytic methods in time and frequency domain, achieving a high reliability for the accurate detection of acoustic signals and the analysis of them. The results obtained in these tests in different environments using different kind of signals are shown. In section 2 the cross-correlation technique is described, as well as the method proposed for signal detection. The application of the method under different situations: high reverberation, low signal-to-noise ratio (S/N) or very low S/N, is presented in section 3. Finally, the conclusions are summarized in section 4. 2 The cross-correlation method for signal detection Cross-correlation (or cross-covariance) consists on the displaced dot product between two signals. It is often used to quantify the degree of similarity or interdependence between two signals [8]. In our case, since all measurements were recorded using digital acquisition systems, all signals under study have been evaluated in discrete time, so that the correlation between two signals x and y with the same N samples length is expressed by the following expression:     N m n m y m x n y x Corr 1 ] [ · ] [ ] }[ , {(1) If we do y = x we obtain the autocorrelation of the signal x. Figure 1 shows the appearance of the signals used in these studies: tones, sweeps, and MLSs. On top, there are these ideal signals in the time domain, that is, the generated signals by the electric signal generator equipment. In the middle row, the spectrum of each signal can be seen, where the different bandwidths can be appreciated. At the bottom, the autocorrelations of each signal show that the higher bandwidth signals have a narrower correlation peak, so, in principle, they are easier to detect. To understand the importance and convenience of using these signals in each detector, the reader can look at articles [9,10]. It is worth to note that, in the cases shown, the correlation peak amplitude (Vmax,corr) is the same and equal to the number of samples of the signal in question (N). Therefore, it can be obtained the peak voltage of the signal (Vp) by the following expression: N V V corr p max, 2  (2) Furthermore, this ratio does not vary with the amplitude of the signal and is less susceptible to the presence of noise. However, the interest is the use of the method for the accurate detection of signals and the recorded signals will be influenced by reflections and noise that may vary the amplitude and profile of the direct signal detection. Figure 2 shows the case of a tone, a sweep and MLS received signals with a distance of 112.5 m between emission and reception (E-R). On the top, the receiving signals in time domain after applying a high order band pass filter are shown (the original recorded signal in time is so noisy that the receiving signal is completely masked). On the bottom, it can be seen the cross-correlation of each signal (without prefiltering) where direct signal reflections are easier and more effective to discern that working in the time or frequency domains, especially for high bandwidth signals (narrower autocorrelation peak). Nevertheless, using this it is only possible to locate the signal but cannot know a priori the peak amplitude of the signal. This is because trying to tackle the problem from both time and frequency domains is completely crucial windowing temporarily the direct signal avoiding reflections to obtain a reliable value of its amplitude, which is not always possible. Then, it would be important to obtain the corresponding relation between the maximum of the cross-correlation between received and emitted signal with the amplitude of the received signal avoiding reflections. This issue has been studied and has been found that if the amplitude of the signal sent (Vp,env), its number of samples (nenv) and the maximum correlation value (Vmax,corr) corresponding to the detection of this signal are known, then it is possible to obtain the peak-amplitude voltage of the received signal applying the following expression: env env p corr rec p n V V V 2 , max, , (3) In the following sections the results of applying this equation to the results of the correlations obtained and compared with values obtained applying time and frequency domain methods are presented. In addition, the improvements obtained by using this technique in terms of detection accuracy in different acoustical environments are also shown. Application The different conditions in which the measurements of acoustic detection were performed are: inside a small vessel, in a tank of acoustic test, in a pool, in the harbour of Gandia, and in ANTARES deep sea neutrino telescope. Although under different conditions of pressure, salinity and temperature, the acoustic propagation media in all the tests is water. Table 1 shows the relationship between the wavelength range associated with the studied signals (λ) and the geometrical dimensions of the places where acoustic processes occur (l). With this, it follows that conditions with higher ratio λ/l means working in a reverberant field, with a higher complexity, while configurations with a smaller λ/l ratio means that there is a less reverberant field, but usually a lower S/N ratio. As discussed below, both extreme situations make difficult the process of acoustic detection. Measure condition Characteristic distance l [m] λ / l The results obtained in these conditions, the acoustic systems used in transmission and reception, and the results in terms of improvement of signal detection and S/N using cross-correlation method are shown in the following sections. High reverberation conditions When emitter and receiver are close and the dimensions of the enclosure where the acoustic processes occur are comparatively small, both signal and reverberation are high. This is the case of the configurations shown in Figure 3 that corresponds to a part of the acoustic test bench for COUPP detector [11]. On the left, the two experimental setups are shown. The first one corresponds to acoustic propagation studies inside a vessel, and the second one was used to study the acoustic attenuation. On the right the transducers used are shown. The signal was emitted with the pre-amplified ITC 1042 transducer and received with the needle-like RESON TC 4038 transducer. -left shows that for the tones studied between 10 kHz and 100 kHz the accuracy of this method is quite good, with an error smaller than 10 %. Considering the characteristic dimensions of the problem and 1500 m/s as sound propagation speed, this uncertainly is of the same order of magnitude of the experimental uncertainly (1 mm). As expected, the maximum deviation corresponds to lower frequencies, and it seems there is some frequency dependent fluctuations. This can be another argument in favour of using broadband signals for cross-correlation techniques. The received amplitudes of the signals have been obtained using equation (3). The results are shown in Figure 5-right compared to the results obtained with standard techniques in time and frequency domains. It can be observed that the results are very similar. Low signal to noise ratio conditions The following configuration used is an intermediate step between high signal to noise ratio (section 3.1) and very low signal to noise (measurements in the ANTARES neutrino telescope, presented in the next section 3.3). This is the case of measurements taken on a pool as shown in figure 6 (left). In this experimental setup, the transmitter consists of an array of three transducers FFR SX83 (middle) and an electronic board to generate and amplify the different acoustic signals. This system can operate in three different modes: emitting with a single element, with the three elements connected in series and the three elements connected in parallel [10]. Our measures were made with the transducers connected in parallel so, in this embodiment, higher transmission power is obtained. The reception was performed using a FFR SX30 (right). Using tones between 10 kHz and 60 kHz in these conditions, we have calculated the emitter-receiver distances from flight times, as described above. The results are shown in figure 7 and compared with those obtained directly in time-domain method. In this case, we can see that the deviation of the measurements relative to a mean value is 5%, which corresponds to an uncertainty less than ± 20 cm. However, if we discard some out-layer measure (sine of 40 kHz) the deviation of the values is reduced to 2.3%, i.e., ± 9 cm. We think that a reason for the relatively large variation between different measurements at different frequencies might be the interference between the three emitters of the array, which depends on the frequency. Again here, the use of broadband signals with the cross-correlation method may help to mitigate this problem since it will average the response of the different frequencies. Emitter-receiver distances obtained by cross-correlation method using tones between 10 kHz to 60 kHz (considering 1500 m/s as the sound propagation speed). The plots of figure 8 show the results obtained by comparing the voltages (left) and the S/N ratios (right) both in cross-correlation method and time-domain method (in this case, since the signals can be windowed properly, avoiding the presence of reflections, values obtained in time and frequency domains are coincident). As before, using the Eq. 3 very similar results to the usual techniques are obtained. On the other hand, the S/N ratio increases considerably (at least 20 dB) for the set of signals used using correlation method. This improvement is crucial for a correct detection of the signals. Very low signal to noise ratio conditions The more complex environment in which this study has been performed is the acoustic measurements made in situ in deep-sea at the ANTARES site. In this case, the distance between emitter and receiver was about 180 m and the S/N ratio was quite low. Figure 9 shows on the left an artistic and schematic view of the telescope. The emitter was a FFR SX30 transducer, shown in the middle, with an electronic board designed specifically for this type of transducer to optimize and amplify the signal sent [10], and the receiving hydrophone was a HTI-08 transducer, shown on the right [12]. Signal to Noise ratio Time Correlation Fig. 9. View of the ANTARES neutrino telescope (left) and pictures of the emitter FFR SX30 (middle) and of the receiver HTI-08 (right) transducers. Since in this ANTARES test synchronization between transmitter and receiver was not available, it is not possible to calculate absolute flight times. However, the received amplitudes expression as well as the increase of the S/N ratio obtained by crosscorrelation method can be evaluated here, as shown in figure 10. In this case, sine signals of 20, 30 and 40 kHz, sweep signals between 20 to 48 kHz, and 28 to 44 kHz, and MLS signals were used. It can be concluded from these measurements that using the cross-correlation method is possible to obtain the signal amplitude accurately and obtain an increase of 15 to 20 dB in the S/N ratio, with a consequent improvement in the acoustic detection. Additionally, and with the aim of applying this technique for post-processing signals in the future KM3NeT neutrino telescope, simulations of propagation of signals measured in ANTARES over longer distances have been done. Figure 11 shows, the improvement in the S/N ratio as a function of the distance using the different signals and methods. Conclusions We have seen that, using different signal emission-acquisition systems, working on a wide range of distances and in very different environmental conditions, good acoustic detection through the technique of cross-correlation between the emitted and received signals can be obtained. This technique is more favourable for broadband signals (sweeps and MLS) because they have a narrower correlation peak and consequently they are easier to discern than others peaks. Furthermore, this technique is powerful in measurement conditions with a reduced S/N ratio, as the case in marine environments over long distances where the recorded signal is weak, or in environments with high background noise. In addition, we have obtained a relation between the peak value of the cross-correlation and the voltage value of the received signal, which synthesizes and optimizes the signal analysis. Fig. 1 . 1Signals used for acoustic studies: tone, sweep and MLS. Fig. 2 . 2Example of recorded signals at 112.5 m Emitter-Receiver distance in the harbour of Gandia. Fig. 3 . 3Experimental setups (left) and transducers used (right). Figure 4 4shows an example of a 30 kHz tone of 5 cycles of duration emitted and recorded under these conditions and their cross-correlation. It can be seen that the maximum of the correlation corresponds with the reception time of the received signal. Fig. 4 . 4Example of emitted signal, received signal, and cross-correlation. Figure 5 5Figure 5-left shows that for the tones studied between 10 kHz and 100 kHz the accuracy of this method is quite good, with an error smaller than 10 %. Considering the characteristic dimensions of the problem and 1500 m/s as sound propagation speed, this uncertainly is of the same order of magnitude of the experimental uncertainly (1 mm). As expected, the maximum deviation corresponds to lower frequencies, and it seems there is some frequency dependent fluctuations. This can be another argument in favour of using broadband signals for cross-correlation techniques. Fig. 5 . 5Left: distances obtained between emitter and receiver by cross-correlation with tones between 20 kHz to 100 kHz. Right: Received amplitudes through the cross-correlation method using Eq.3 and using time and frequency domain methods. Fig. 6 . 6Experimental setup (left), emitter (middle) and receiver (right) transducers. Fig. 7 . 7Fig. 7. Emitter-receiver distances obtained by cross-correlation method using tones between 10 kHz to 60 kHz (considering 1500 m/s as the sound propagation speed). Fig. 8 . 8Comparison of cross-correlation and time domain method to obtain the received voltage amplitude (left) and the S/N ratio (right). Fig. 10 . 10Received amplitude (left) and S/N ratio (right) both in cross-correlation and time domain method. Fig. 11 . 11S/N ratio obtained using time domain method (top) and cross-correlation method (bottom). Table 1. Characteristics of the acoustic conditions of the different measurements and tests.Vessel 0.02 2.2 Tank 0.05-1 0.22 Pool 4 0.022 Harbor 120 0.0005 Sea 200 0.0003 Sine 20k Sine 30k Sine 40k Sw.20-48kSw.28-44kMLS 10 Signal to Noise ratioSine 20kHz Sine 30kHz Sine 40kHz 0.05 0.1 0.15 0.2 0.25 Signals Amplitude [V pp ] Received voltage Time Correlation 20 30 40 50 60 Signals S/N [dB] Time Correlation Sine 20kHz Sine 30kHz Sine 40kHz 0.05 0.1 0.15 0.2 0.25 Signals Amplitude [V pp ] Received voltage Time Correlation Sine 20k Sine 30k Sine 40k Sw.20-48kSw.28-44k MLS 10 20 30 40 50 60 Signals S/N [dB] Signal to Noise ratio Time Correlation AcknowledgementsThis work has been supported by the Ministerio de Economía y Competitividad (Spanish Government), project ref. FPA2012-37528-C02-02, Multidark (CSD2009-00064). It has also being funded by Generalitat Valenciana, Prometeo/2009/26. Thanks to the ANTARES Collaboration for the help in the measurements made in the ANTARES deep-sea neutrino telescope. ANTARES: the first undersea neutrino telescope. M Ageron, ANTARES CollaborationNucl. Instr. And Meth. A. 656M. Ageron et al. (ANTARES Collaboration), ANTARES: the first undersea neutrino telescope, Nucl. Instr. And Meth. A, vol. 656 ( 2011) pp. 11-38. First dark matter search results from a 4-kg CF3I bubble chamber operated in a deep underground site. E Behnke, COUPP CollaborationPhys.Rev.D. 8652001E. Behnke et al. (COUPP Collaboration), First dark matter search results from a 4-kg CF3I bubble chamber operated in a deep underground site, Phys.Rev.D 86, 052001 (2012). Positioning system of the ANTARES neutrino telescope. M Ardid, Nucl. Instr. and Meth. A. 602M. Ardid, Positioning system of the ANTARES neutrino telescope, Nucl. Instr. and Meth. A, vol. 602 ( 2009) pp. 174-176. KM3NeT Acoustic position calibration of the KM3NeT neutrino telescope. G Larosa, M Ardid, Nucl. Instr. and Meth. A. 718G. Larosa and M. Ardid, KM3NeT Acoustic position calibration of the KM3NeT neutrino telescope, Nucl. Instr. and Meth. A, vol. 718 (2013) pp. 502-503. ANTARES: An Underwater Network of Sensors for Neutrino Astronomy and Deep-Sea Research. M Ardid, Ad Hoc & Sensor Wireless Networks. 8M. Ardid, ANTARES: An Underwater Network of Sensors for Neutrino Astronomy and Deep- Sea Research, Ad Hoc & Sensor Wireless Networks, vol. 8 (2009), pp. 21-34. Acoustic studies for alpha background rejection in dark matter bubble chamber detectors. M Bou-Cabo, M Ardid, I Felis, Proc. of the IV International Workshop in Low Radioactivity Techniques. AIP Conference Proceedings. of the IV International Workshop in Low Radioactivity Techniques. AIP Conference eedings1549M. Bou-Cabo, M. Ardid and I. Felis, Acoustic studies for alpha background rejection in dark matter bubble chamber detectors, Proc. of the IV International Workshop in Low Radioactivity Techniques. AIP Conference Proceedings, Vol. 1549, pp. 142-147 (2013). . J G G Proakis &amp; D, Manolakis, Digital Signal Processing. Prentice HallJ.G.Proakis & D.G.Manolakis, Digital Signal Processing, 3ed Prentice Hall (1996). Acoustic System development for the underwater neutrino telescope positioning KM3NeT. M Saldaña, Bienal de FísicaM.Saldaña. Acoustic System development for the underwater neutrino telescope positioning KM3NeT, Bienal de Física (2013). Acoustic Transmitters for Underwater Neutrino Telescopes. M Ardid, Sensors. 12M. Ardid et al., Acoustic Transmitters for Underwater Neutrino Telescopes, Sensors, vol. 12 (2012), pp. 4113-4132. Sistemas acústicos para la detección de Materia Oscura. I Felis, M Bou-Cabo, M Ardid, Bienal de Física. I.Felis, M.Bou-Cabo, M.Ardid, Sistemas acústicos para la detección de Materia Oscura, Bienal de Física (2013). Experimental Studies within ANTARES towards Acoustic Detection of Ultra High Energy Neutrinos in the Deep Sea. K Graf, FAU-PI1-DISS-08-001U. ErlangenPh.D. thesisK. Graf, Experimental Studies within ANTARES towards Acoustic Detection of Ultra High Energy Neutrinos in the Deep Sea, Ph.D. thesis, U. Erlangen (2008) FAU-PI1-DISS-08-001.
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[ "MORSE BOUNDARIES OF GRAPHS OF GROUPS WITH FINITE EDGE GROUPS", "MORSE BOUNDARIES OF GRAPHS OF GROUPS WITH FINITE EDGE GROUPS" ]
[ "Stefanie Zbinden " ]
[]
[]
In this paper we prove that the Morse boundary of a free product depends only on the Morse boundary of its factors. In fact, we also prove the analogous result for graphs of groups with finite edge groups and infinitely many ends. This is a generalization of the main result of[MS15]in the case of non-hyperbolic groups. arXiv:2110.14637v1 [math.GR] 27 Oct 2021Morse PropertiesIn this section unless stated otherwise, (X, e) is a pointed proper metric geodesic space and M a Morse gauge. For points p, q ∈ X we denote by [p, q] any geodesic from p to q. 2.1. Morse geodesics. Definition 2.1 (Morse gauge). A Morse gauge is a map M : R ≥1 × R ≥0 → R ≥0 . We denote the set of all Morse gauges by M. The set of all Morse gauges M has a partial order defined by M ≤ M if and onlyObserve that with this order, the maximum of a set of Morse gauges is the pointwise maximum.Remark 2.3. Observe that a geodesic segment γ is M -Morse, where M depends only on the length of γ. Namely, a (λ, ε)-quasi-geodesic with endpoints that are at distance at most |γ| has bounded length n, depending only on λ, ε and |γ|. Consequently, the quasi-geodesic has to stay in a C neighbourhood of γ where C depends only on λ, ε and |γ|. Definition 2.4 (δ M ). Let M be a Morse gauge. We define δ M = max{4M (1, 2M (5, 0)) + 2M (5, 0), 8M (3, 0)}.The value δ M is a measure of how close a point or geodesic is from some M -Morse geodesic γ, where all distances smaller than δ M are considered to be close.Morse geodesics have similar properties as geodesics in hyperbolic spaces. We cite some of these properties proven in [CS14] and[Cor17]. Additionally, we prove some more results. Lemma 2.5 (Adapted version of Lemma 2.5 of [CS14]). Let γ be an M -Morse quasi-geodesic in a geodesic metric space X.i) If ρ is a quasi-geodesic whose Hausdorff distance from γ is at most C, then ρ is M -Morse where M depends only on M and C. ii) If Y is a geodesic metric space and f : X → Y is a (λ, ε)-quasi-isometry, then f • γ is M -Morse where M depends only on λ, ε and M . iii) If γ is a (K, L)-quasi-geodesic, there exists C depending only on M, K, L such that for any two points x = γ(t) and y = γ(t ) on γ, the geodesic [x, y] has Hausdorff distance at most C from γ([t, t ]). Lemma 2.12. Let ξ : I → X be an M -Morse geodesic and J ⊂ I any interval. Then, the geodesic ξ| J : J → X is M S -Morse, where M S depends only on M . Proof. This follows directly from Lemma 2.1 of [Cor17]. Lemma 2.13. Let p, q ∈ X and let α, β : [0, ∞) → X be geodesics with bounded Hausdorff distance starting at p and q respectively. Let γ : [0, T ] → X be a geodesic from p to q. If two of the three geodesics α, β and γ are M -Morse, then the third one is M T -Morse, where M T depends only on M .In other words, Lemma 2.13 states that if we have a generalized triangle (α, β, γ) where two sides are M -Morse, then the third side is M -Morse.
10.1515/jgth-2021-0196
[ "https://arxiv.org/pdf/2110.14637v1.pdf" ]
240,070,992
2110.14637
4c76f41d55de56860dde1359af9e3e3eff8b1788
MORSE BOUNDARIES OF GRAPHS OF GROUPS WITH FINITE EDGE GROUPS Stefanie Zbinden MORSE BOUNDARIES OF GRAPHS OF GROUPS WITH FINITE EDGE GROUPS In this paper we prove that the Morse boundary of a free product depends only on the Morse boundary of its factors. In fact, we also prove the analogous result for graphs of groups with finite edge groups and infinitely many ends. This is a generalization of the main result of[MS15]in the case of non-hyperbolic groups. arXiv:2110.14637v1 [math.GR] 27 Oct 2021Morse PropertiesIn this section unless stated otherwise, (X, e) is a pointed proper metric geodesic space and M a Morse gauge. For points p, q ∈ X we denote by [p, q] any geodesic from p to q. 2.1. Morse geodesics. Definition 2.1 (Morse gauge). A Morse gauge is a map M : R ≥1 × R ≥0 → R ≥0 . We denote the set of all Morse gauges by M. The set of all Morse gauges M has a partial order defined by M ≤ M if and onlyObserve that with this order, the maximum of a set of Morse gauges is the pointwise maximum.Remark 2.3. Observe that a geodesic segment γ is M -Morse, where M depends only on the length of γ. Namely, a (λ, ε)-quasi-geodesic with endpoints that are at distance at most |γ| has bounded length n, depending only on λ, ε and |γ|. Consequently, the quasi-geodesic has to stay in a C neighbourhood of γ where C depends only on λ, ε and |γ|. Definition 2.4 (δ M ). Let M be a Morse gauge. We define δ M = max{4M (1, 2M (5, 0)) + 2M (5, 0), 8M (3, 0)}.The value δ M is a measure of how close a point or geodesic is from some M -Morse geodesic γ, where all distances smaller than δ M are considered to be close.Morse geodesics have similar properties as geodesics in hyperbolic spaces. We cite some of these properties proven in [CS14] and[Cor17]. Additionally, we prove some more results. Lemma 2.5 (Adapted version of Lemma 2.5 of [CS14]). Let γ be an M -Morse quasi-geodesic in a geodesic metric space X.i) If ρ is a quasi-geodesic whose Hausdorff distance from γ is at most C, then ρ is M -Morse where M depends only on M and C. ii) If Y is a geodesic metric space and f : X → Y is a (λ, ε)-quasi-isometry, then f • γ is M -Morse where M depends only on λ, ε and M . iii) If γ is a (K, L)-quasi-geodesic, there exists C depending only on M, K, L such that for any two points x = γ(t) and y = γ(t ) on γ, the geodesic [x, y] has Hausdorff distance at most C from γ([t, t ]). Lemma 2.12. Let ξ : I → X be an M -Morse geodesic and J ⊂ I any interval. Then, the geodesic ξ| J : J → X is M S -Morse, where M S depends only on M . Proof. This follows directly from Lemma 2.1 of [Cor17]. Lemma 2.13. Let p, q ∈ X and let α, β : [0, ∞) → X be geodesics with bounded Hausdorff distance starting at p and q respectively. Let γ : [0, T ] → X be a geodesic from p to q. If two of the three geodesics α, β and γ are M -Morse, then the third one is M T -Morse, where M T depends only on M .In other words, Lemma 2.13 states that if we have a generalized triangle (α, β, γ) where two sides are M -Morse, then the third side is M -Morse. Introduction The Morse boundary has been introduced in [Cor17] building on [CS14] and is a generalisation of the Gromov boundary for non-hyperbolic spaces. It captures hyperbolic properties of non-hyperbolic spaces and groups and, similar to the Gromov boundary, is a quasi-isometry invariant and visibility space. Morse boundaries inherit many properties of the Gromov boundary (see e.g. [CH17], [CCM19], [Mur19], [Liu21], [Zal18]) and [CCS20] gives a topological description of certain Morse boundaries. A summary of properties of the Morse boundary can be found in [Cor19]. For more information about hyperbolic spaces, we recommend [BH09] and for information about Cayley graphs and free products we recommend [SS12] although we will not follow the notation used there. Motivated by Dunwoody's accessibility [Dun85], that is, the fact that a finitely presented group G splits as a graph of groups with finite edge groups and one-ended vertex groups, we show that the Morse boundary of G only depends on the Morse boundaries of the vertex groups. Thus, this paper reduces the study of Morse boundaries (at least of finitely presented groups) to that of one-ended groups. However, our main result below also applies to non-finitely-presented groups, and when vertex groups are not one-ended: Theorem 1.1. For i = 1, 2, let G i be graphs of groups where all edge groups are finite and suppose that the groups π 1 G i have infinitely many ends. Denote by h(G i ) the set of homeomorphism types of Morse boundaries of the vertex groups in G i that are non-finite and not virtually cyclic. If h(G 1 ) = h(G 2 ), then the Morse boundaries ∂ * π 1 (G i ) are homeomorphic. In combination with [CCS20, Theorem 1.5], we note that Theorem 1.1 implies, for example, that all fundamental groups of 3-manifolds of the form M 1 #M 2 , where the M i are cusped hyperbolic 3-manifolds, have homeomorphic Morse boundaries. Our main result generalises the main result of [MS15], proving the same result for the Gromov boundary in the case of hyperbolic groups. Theorem 1.1 follows directly from Theorem 1.2 by applying Theorem 0.3 of [PW02]. Theorem 1.2. Let A 1 , A 2 and B 1 , B 2 be finitely generated groups that are infinite. Suppose ∂ * A 1 ∼ = ∂ * A 2 and ∂ * B 1 ∼ = ∂ * B 2 . Then, ∂ * (A 1 * B 1 ) ∼ = ∂ * (A 2 * B 2 ). Outline of proof: We follow the proof of [MS15] closely, adapting their ideas and definitions to nonhyperbolic groups. That is, we give a combinatorial description of the Morse boundary of a free product and then use the homeomorphism p : ∂ * A 1 → ∂ * A 2 to construct a bijectionp : A 1 → A 2 . In the construction of the bijection, we deviate from the proof of the hyperbolic case. In the hyperbolic case, the bijectionp has to be compatible with the homeomorphism p in the sense that any sequence (x i ) in A 1 that converges to some boundary element x in ∂A 1 satisfies that the sequence (p(x i )) converges to p(x). However, in our case, the bijectionp also has to satisfy that for every Morse gauge M , there exists a Morse gauge M such thatp(A 1 (M ) ) ⊂ A 2 (M ) . The main challenge in constructingp is that the sets A 1 (M ) do not necessarily have a total order. We managed to construct a bijection satisfying all conditions by picking a Morse geodesic line λ and then definingp(a) as one of the points far along p(a · λ). With the correct technical conditions of far away enough (which can be obtained by mapping the points back to A 1 ), the bijection satisfies all required conditions and induces a homeomorphism from ∂ * (A 1 * B 1 ) to ∂ * (A 2 * B 2 ) (see subsection 4.1 condition 7). Outline of paper: In Section 2 we recall definitions and properties around the Morse boundary and prove some technical lemmas about the Morse boundary. In Section 3 we give an alternate description of the Morse boundary of a free product. In Section 4 we show that the Morse boundary of a free product depends only on the Morse boundary of its factors. Finally in Section 5 we prove Theorem 1.1. The original version of Lemma 2.5 of [CS14] assumes that X is a CAT(0) space. However, as noted in [Cor17], the statement also holds if X is a geodesic metric space. Corollary 2.6 (Cor 2.5 and 2.6 of [Cor17] combined). Let X be a geodesic metric space and α : [0, ∞) → X be an N -Morse geodesic ray. i) Let β : [0, ∞) → X be a ray such that β(0) = α(0) and d(α(t), β(t)) < K for t ∈ [0, D] for some D ≥ 6K. Then d((α(t), β(t)) < δ N for all t ∈ [0, D − 2K]. ii) Let β : [0, ∞) be a ray such that d(α(t), β(t)) < K for all t ∈ [0, ∞). Then for all t ∈ [2K, ∞), (α(t), β(t)) < δ N + d(α(0), β(0)). In particular, if α(0) = β(0) then d(α(t), β(t)) < δ N for all t ∈ [0, ∞). Corollary 2.6 ii) shows that two geodesic rays that have the same direction are close from some point on. Part i) shows that two geodesics (where at least one of them is Morse) that are close at the beginning and not too far away at the end are close for a long time. The next Lemma proves a similar but slightly different statement. Lemma 2.7 (Lemma 2.7 of [Cor17]). Let X be a geodesic space and let α 1 , α 2 : [0, A] → X be N - Morse geodesics with α 1 (0) = α 2 (0). If d(α 1 (s), im(α 2 )) < K for some K > 0 and s ∈ [0, A], then d(α 1 (t), α 2 (t)) ≤ 8N (3, 0) < δ N for t < s − K − 4N (3, 0). Next, we want to prove that in a triangle, if two sides are M -Morse, then the third side is M -Morse, where M depends only on M . For triangles where all sides are finite, this is shown in [Cor17] and our proof will closely follow the proof from [Cor17]. As we have to concatenate quasi-geodesics and Morse geodesics, we first prove some technical lemmas. Lemma 2.8. Let p, q ∈ X and let γ : [0, T ] → X be a continuous (λ, ε)-quasi-geodesic. Let t, t ∈ [0, T ] such that γ(t) and γ(t ) are points on γ closest to p and q respectively. Finally, let α be a geodesic from p to γ(t) and let β be a geodesic from γ(t ) to q. The following hold: i) α · γ[t, T ] is a (3λ, ε) -quasi-geodesic. ii) If |t − t | ≥ 1 3λ (d(p, γ(t)) + d(q, γ(t ))), then α · γ[t, t ] · β is a (3λ, ε) -quasi-geodesic. Remark 2.9. The continuity assumption for γ is only needed for the existence of the points γ(t) and γ(t ). Proof. Assume α : [0, S] → X, the only thing left to prove for i) is that for s ∈ [0, S] and s ∈ [t, T ] we have 1 3λ (|S − s| + |s − t|) − ε ≤ d(α(s), γ(s )) ≤ 3λ(|S − s| + |s − t|) + ε.(1) Note that, due to the triangle inequality, the following inequalities hold: d(α(s), γ(s )) ≤ d(α(s), α(S)) + d(γ(t), γ(s )),(2)d(α(s), γ(s )) ≥ d(γ(s ), γ(t)) − d(α(s), α(S)).(3) As γ(t) is the closest point on γ to p we have d(α(s), γ(s )) ≥ d(α(s), α(S)). Inequality (2) combined with γ being a (λ, ε)-quasi-geodesic gives d(α(s), γ(s )) ≤ |S − s| + λ|s − t| + ε ≤ 3λ(|S − s| + |s − t|) + ε. Which proves the right part of (1). To prove the left part of (1) we make a case distinction. In the first case, assume that 2d(α(s), α(S)) ≥ d(γ(t), γ(s )). Then d(α(s), γ(s )) ≥ 1 3 d(α(s), α(S)) + 2 3 d(α(s), α(S)) ≥ 1 3 |S − s| + 1 3 d(γ(t), γ(s )) ≥ 1 3λ |S − s| + 1 3λ |s − t| − 1 3 ε ≥ 1 3λ (|S − s| + |s − t|) − ε. In the second case assume 2d(α(s), α(S)) ≤ d(γ(t), γ(s )). Then we use (3) to get d(α(s), γ(s )) ≥ 1 3 d(γ(s ), γ(t)) + 2 3 d(γ(s ), γ(t)) − d(α(s), α(S)) ≥ 1 3λ |s − t| − 1 3 ε + 1 3 d(α(s), α(S)) ≥ 1 3λ (|S − s| + |s − t|) − ε. This concludes the proof of i). Assume β : [0, S ] → X. To show ii) we can use i) and thus the only thing left to prove ii) is showing that for s ∈ [0, S] and s ∈ [0, S ] we have 1 3λ (|S − s| + |t − t| + s ) − ε ≤ d(α(s), β(s )) ≤ 3λ(|S − s| + |t − t| + s ) + ε. On one hand we have d(α(s), β(s )) ≤ d(α(s), α(S)) + d(γ(t), γ(t )) + d(α(0), α(s )) ≤ λ|t − t | + ε + |S − s| + s . On the other hand, we have d(α(s), β(s )) ≥ d(γ(t), γ(t )) − d(α(s), α(S)) − d(α(0), α(s )) ≥ 1 λ |t − t | − ε − (|S − s| + s ) ≥ 2 3λ |t − t | − ε ≥ 1 3λ (|S − s| + |t − t| + s ) − ε. Together, this concludes the proof. The following two lemmas state what happens to the Morseness of a geodesic when concatenating two Morse geodesics or conversely looking at a subsegment of a Morse geodesic. Proof. Let γ be a (λ, ε) -quasi-geodesic with endpoints on ξ. We have to show that γ stays in a bounded distance C from ξ, where C depends only on λ, ε, M and M . By Lemma 1.11 of [MS15] there exists a continuous (3λ, ε)-quasi-geodesic γ : [0, S] → X with the same endpoints as γ that has Hausdorff distance at most λ + ε from γ . It is enough to show that γ stays in a bounded distance C from ξ, where C depends only on λ, ε, M and M . Let γ(t) be a point on γ closest to ξ(T ) and let α be a geodesic from ξ(T ) to γ(t). By Lemma 2.8, the paths α · γ[t, S] and α · γ[0, t] are (3λ, ε)-quasi-geodesics. More importantly, α · γ[t, S] and α · γ[s, 0] have both endpoints on either ξ[0, T ] or ξ[T, ∞). The statement follows from the Morseness of ξ[0, T ] and ξ[T, ∞). Remark 2.11. The trick to tame γ by approximating it by a continuous quasi-geodesic γ can often be useful. In particular, when we want to show that a geodesic ξ is M -Morse, where M depends only on some set of parameters A, it is enough to show that any continuous (λ, ε)-quasi-geodesic with endpoints on ξ is contained in a C neighbourhood of ξ, where C depends only on λ, ε and A. Proof. Without loss of generality β is M -Morse. Let x be a point on α that has distance at most δ M from β. Such a point exists: Consider the path λ n = [q, α(t)] · α[t, s] · [α(s), β(n)], where α(t) and α(s) are the closest points on α to q and β(n) respectively. The length of [α(s), β(n)] is bounded by the Hausdorff distance K of α and β. The same holds for the length of [q, α(t)]. Therefore, if we choose n large enough, the conditions of Lemma 2.8 are satisfied and λ n is a (3, 0)-quasi-geodesic with endpoints on β. Thus d(α(s), β) ≤ M (3, 0) < δ M . In the following, assume n is chosen such that d(α(s), β(n)) < δ M . Consider Assume γ is M -Morse. Consider again the finite triangle ∆ 2 . By the same argument as above, we get that α[0, s] is M 3 -Morse, where M 3 depends only on M . In particular, M does not depend on the integer n we chose. By choosing n large enough we can get for arbitrarily large t that α[0, t] is M 3 -Morse. As any quasi-geodesic with endpoints on α has endpoints on α[0, t] for some large enough t, the geodesic α is M 3 -Morse itself. Setting M T = M 3 concludes the proof. 2.2. Morse boundary. As a set, the Morse boundary of X, denoted by ∂ * X, is the set of equivalence classes of Morse geodesic rays, where two Morse geodesic rays are equivalent if they have bounded Hausdorff distance. Elements z ∈ ∂ * X are called Morse directions. For a Morse geodesic ray ξ we denote its equivalence class by [ξ] ∈ ∂ * X. Definition 2.14 (Realization). Let z ∈ ∂ * X be a Morse direction. A geodesic ray γ : [0, ∞) → X with [γ] = z and γ(0) = e is called a realization of z. Similarly, for an element x ∈ X a geodesic segment γ starting at e and ending in x is called a realization of x. Notation. Let z ∈ ∂ M * X and let ξ be an M -Morse realization of z. We denote by G M e X the set of all M -Morse geodesic rays in X starting at e. For any positive n we define O M n (ξ) = {η ∈ G M e X | d(η(t), ξ(t)) < δ M for all t ∈ [0, n]}, U M n (ξ) = {[η] | η ∈ O M n (ξ)}, O M n (z) = {η ∈ G M e X | η ∈ O M n (ξ) for all M -Morse realizations ξ of z}, U M n (z) = {[η] | η ∈ O M n (z)}. It is shown in [Cor17] that {U M n (z)} n∈N is a fundamental system of neighbourhoods. It is clear that O M n (ξ) ⊂ O N n (ξ) if M ≤ N as then G M e X ⊂ G N e X. The same holds for U M n (ξ). However, we have to be a bit more careful with O M n (z) and U M n (z). This is because, if M ≤ N then G M e X ⊂ G N e X, but there could be more N -Morse realizations of z than there are M -Morse realizations and thus more conditions to satisfy. [Cor17] shows that for two M -Morse realizations ξ and ζ of some Morse direction z we have ξ ∈ O M n (ζ) for all n. Thus ξ ∈ O M n (z) which implies that z ∈ U M n (z). Remark 2.15. For n ≥ 12δ M We have O M n+4δ M (ξ) ⊂ O M n (z) ⊂ O M n (ξ). This holds because for two M - Morse realizations ξ, ζ of z we have d(ξ(t), ζ(t)) < δ M by Corollary 2.6 of [Cor17]. Thus if γ ∈ O M n+4δ M (ξ), Sometimes, we also want to make statements about finite geodesics that start at e and not only about geodesic rays. Therefore, we want to extend the notation to include finite geodesics. Let x ∈ X (M ) and let ζ be an M -Morse realization of x. For n ≤ d(e, x) we define the sets O M n (ζ), O M n (x), U M n (ζ) and U M n (x) as above. Furthermore, we sayG M e X is the set of all M -Morse geodesics in X (both rays and finite ones) starting at e. For an M -Morse realization ξ (either of a point x ∈ X or of an M -Morse direction x ∈ ∂ M * X) and some positive n (that, if ξ is a realization of a point x ∈ X, has to satisfy n ≤ d(e, x)) we defineÕ M n (ξ) = {η ∈G M e X | d(η(t), ξ(t)) < δ M for all t ∈ [0, n]}. The setsÕ M n (x),Ũ M n (ξ) andŨ M n (x) can be defined accordingly. Remark 2.17. The observation in Remark 2.15 also holds for the newly defined sets. The M -Morse boundary inherits properties of Gromov boundary of hyperbolic spaces due to the following homeomorphism proven in [CH17]. Theorem 2.18 (Thm 3.14 of [CH17]). If X is a proper geodesic metric space, then there is a homeomorphism between ∂X (M ) and ∂ M * X. Here, ∂X (M ) denotes the Gromov boundary of X (M ) . This is well defined as in Theorem A III) of [CH17] Cordes and Hume prove that X (M ) is hyperbolic. In [MS15] III, several properties of the Gromov boundary are shown. Important for us is that the Gromov boundary is compact and metrizable and thus, in particular, Hausdorff. With the homeomorphism above, we get that the M -Morse boundary of a proper geodesic metric space is compact and Hausdorff. Now we are ready to define a topology on the Morse boundary ∂ * X. Namely, consider ∂ * X as the direct limit of the M -Morse boundaries ∂ M * X. More precisely: ∂ * X = lim − − → M ∂ M * X, with inclusions i M,N : ∂ M * X → ∂ N * X, for all M ≤ N defined by the canonical inclusion ∂ M * X ⊂ ∂ N * X. Note that those inclusions are continuous, since, for any M -Morse direction z, we have U M n+4δ M (z) ⊂ U N n (z). Now ∂ M * X can be viewed both on its own with the topology induced by the fundamental system of neighbourhoods {U M n (z)} n∈N or as a subset of ∂ * X with the subspace topology. We show that those two topologies coincide. Lemma 2.19. The topology on ∂ M * X induced by the fundamental system of neighbourhoods {U M n (z)} n∈N coincides with the subspace topology of ∂ X * M ⊂ ∂ * X. Proof. To distinguish the two topologies on ∂ M * X we denote ∂ M * X with the topology induced by the neighbourhood basis by Y and ∂ M * X ⊂ ∂ * X with the subspace topology by Z. By definition of the direct limit topology, the inclusion i : Y → ∂ * X is continuous and thus if C ⊂ Z is closed, so is C ⊂ Y . Conversely, let C ⊂ Y be closed. Let N ≥ M be a Morse gauge. Then, as argued above, the inclusion i M,N : Y → ∂ N * X is continuous. As Y and ∂ N * X are compact and Hausdorff, i M,N (C) = C ⊂ ∂ N * X is closed. Now, let N be any Morse gauge and N = max{N, M }. Then C ∩ ∂ N * X = i −1 N,N (C) is closed. This concludes the proof, as it shows that C ⊂ ∂ * X is closed and thus also C ⊂ Z is closed. The Morse boundary is the direct limit of compact sets. However, a priori, as it is the direct limit of uncountably many compact sets it might not even be σ-compact. However, in [CD19] they characterized compact subsets of ∂ * X: Lemma 2.20 (Lemma 4.1 of [CD19]). If K ⊂ ∂ * X is non-empty and compact, then there exists M > 0 such that K ⊂ ∂ M * X. In Section 2.4, we will use this lemma to prove properties of homeomorphisms between Morse boundaries. Next, we prove two technical lemmas about the Morse boundary. The first one is about the Morseness of a realization of an M -Morse direction -a priori this might have nothing to do with the Morse gauge M . The second one is about properties of the neighbourhoods defined above. Lemma 2.21. Any realization ξ of an M -Morse direction z ∈ ∂ M * X is M C -Morse, where M C depends only on M . Proof. Let ζ be an M -Morse realization of z. Cor 2.6 of [Cor17] implies that d(ζ(t), ξ(t)) < δ M for all t ∈ [0, ∞). Lemma 2.5 (i) of [CS14] concludes the proof. Lemma 2.22. Let z ∈ ∂ M * X be an M -Morse direction. For any integer l ∈ N there exists a positive integer k such that for all y ∈Ũ M k (z) we haveŨ M k (y) ⊂Ũ M l (z). Remark 2.23. If we would allow a dependency of k on y, then the lemma is a direct consequence of thẽ U M l (z) being a fundamental system of neighbourhoods. The crucial part of this lemma is that there exists a uniform k, that is, there exists a choice of k that works for all y ∈Ũ M k (z). Proof. Set k = max{l + 2K, 6K}, where K = 2δ M . Let ξ be an M -Morse realization of z, let y ∈Ũ M k (z), and let η ∈Õ M k (z) be a realization of y. Furthermore, let λ ∈Õ M k (y). We have that d(ξ(t), η(t)) < δ M and d(η(t), λ(t)) < δ M for all t ∈ [0, k] and hence d(ξ(t), λ(t)) < 2δ M for all t ∈ [0, k]. Using Cor 2.5 of [Cor17] we get that d(ξ(t), λ(t)) < δ M for all t ∈ [0, l]. As this is true for any M -Morse realization ξ of z we have λ ∈Õ M l (z) and thus [λ] ∈Ũ M l (z) ⊂ U . 2.3. Morse geodesic lines. Let A be a finitely generated group with neutral element e. For any finite set of generators S ⊂ A, the Cayley-graph Γ(A, S) is a proper geodesic metric space. Furthermore, for two finite generating sets S 1 and S 2 , the two Cayley graphs Γ(A, S 1 ) and Γ(A, S 2 ) are quasi-isometric. In [Cor17] it is proven that the Morse boundary is a quasi-isometry invariant. Therefore, we can define the Morse boundary of A as the Morse boundary of Γ(A, S) for any finite generating set S and ∂ * A does not depend on the choice of S. Sometimes, by abuse of notation, we will implicitly identify A with its Cayley, assuming that we fixed a set of generators S. The base point of A (resp its Cayley graph) will be assumed to be its neutral element e. Group action on the Morse boundary. The left action of A by isometries on its Cayley graph induces a group action of A on the Morse boundary by a · [γ] = [a · γ]. Also observe that if γ is M -Morse, so is a · γ (but not necessarily [a · γ]). In the following, we assume |∂ * A| ≥ 2. Then, we fix a Morse geodesic line going through e, denoting it by λ(A) and denote by M 0 a Morse gauge such that λ(A) is M 0 -Morse. Remark 2.24. Note that by a geodesic line, we mean a bi-infinite geodesic. Such a geodesic line λ(A) exists: The Morse boundary is a visibility space (see [Cor17]) and thus having |∂ * A| ≥ 2 implies that there exists a Morse geodesic lineλ going through some point x ∈ A. Then x −1 ·λ is a Morse geodesic line going through the base point e. For any element x ∈ A we construct a Morse geodesic λ x corresponding to x as follows: Letλ denote the M 0 -Morse geodesic line x · λ(A) and let x 0 ∈ A be a point onλ closest to e. Define λ x to be any realization of [λ[0, ∞)] and call it the geodesic ray corresponding to x. The construction of λ x is depicted in Figure 1. Remark 2.25. In the following, we assume thatλ(0) = x 0 (if not reparametrizeλ) and that the integer t x such thatλ(t x ) = x is non-negative (if not replaceλ with its mirrored geodesic line, i.e. the geodesic lineλ that satisfiesλ (t) =λ(−t) for all t ∈ R ). Proof. The statements i) and ii) follow from Lemma 2.13, because ∆ = η,λ[0, ∞), λ x forms a generalized triangle. For any t ∈ [0, ∞) letλ(s t ) be the closest point onλ to λ x (t). For t large enough, this is a (3, 0)-quasigeodesic. Therefore, d(x, λ x ) < δ M λ . Thus, as η and λ x are geodesics, we have d(η(t), λ x (t)) < 3δ M λ for all t ∈ [0, d(e, x)]. Set C M (T ) = max{18δ M λ , T + 6δ M λ } and assume d(e, x) ≥ C M (T ). According to Corollary 2.5 of [Cor17], we have d(η(t), λ x (t)) < δ M for all t ∈ [0, T ]. This concludes the proof of iii). For any s ∈ [0, ∞) let λ x (t s ) be the point on λ x closest toλ(s). Define γ s = [λ(0), λ x (t 0 )] · λ x [t 0 , t s ] · [λ x (t s ),λ(s)]. By Lemma 2.8 the path γ t is a (3, 0)-quasi-geodesic for large enough s. Increasing s we have, by Lemma 2.1 of [Cor17], that d H ([λ(0), λ x (t 0 )]λ x [t 0 , ∞),λ[0, ∞)) ≤ 2M 0 (3, 0). Choose T 1 , T 2 large enough such that d(λ x (T 1 ),λ(T 2 )) ≤ 2M 0 (3, 0), then by Cor 2.6 of [Cor17] we have d(λ x (T 1 + s),λ(T 2 + s)) ≤ 2M 0 (3, 0) + δ M0 . So the Hausdorff distance between λ x [T 1 , ∞) andλ[T 2 , ∞) is bounded by some constant depending only on M 0 . Lemma 2.5 of [CS14] together with Lemma 2.12 conclude the proof of iv). In the proof we saw that C M (T ) depends only on M , M 0 and T but the lemma additionally allows a dependence on the space A itself. This is because below, we define λ x for groups A with |∂ * A| = 1 and show that Lemma 2.26 holds in that case as well. However, to do so we need the dependence on A. Let A be a group with |∂ * A| = 1, we say λ x = λ(A) for all x ∈ A, where λ(A) is a M 0 -Morse realization of the only Morse direction of A. Properties i) and ii) hold because (η, x · λ, λ x ) forms a generalized triangle. Property iv) holds trivially. Assume Property iii) does not hold, which implies that for some Morse gauge M , there exists an integer T and infinitely many x ∈ A that are less than T long δ M -close to λ x = λ(A). Let (x i ) i∈N be a sequence of those points. As they are all M -Morse, there exists a converging subsequence (x i k ) k∈N . As |∂ * A| = 1 they converge to λ(A), this is a contradiction to them not being T long close to λ(A). So, property iii) holds. 2.4. Homeomorphisms of Morse boundaries. Let A 1 and A 2 be finitely generated groups with basepoint e and p : ∂ * A 1 → ∂ * A 2 a homeomorphism. For any Morse geodesic ray ξ starting at the basepoint e we write p(ξ) for p([ξ]). Below, we discuss some properties of p. Lemma 2.28. There exists a map h : M → M such that p(∂ M * A 1 ) ⊂ ∂ h(M ) * A 2 and p −1 (∂ M * A 2 ) ⊂ ∂ h(M ) * A 1 . Proof. We show that there exists a map h 1 : M → M such that p(∂ M * A 1 ) ⊂ ∂ h1(M ) * A 2 for all Morse gauges M . As p is a homeomorphism, this implies the existence of a map h 2 : M → M such that p −1 (∂ M * A 2 ) ⊂ ∂ h2(M ) * A 1 for all Morse gauges M . We can then set h(M ) = max{h 1 (M ), h 2 (M )}. As ∂ M * A 1 is compact, so is p(∂ M * A 1 ). Lemma 2.20 then implies that p(∂ M * A 1 ) ⊂ ∂ N * A 2 for some Morse gauge N . Set h 1 (M ) = N . Conversely, let q : ∂ * A 1 → ∂ * A 2 be a map of which we want to prove that it is continuous. As ∂ * A 1 and ∂ * A 2 are constructed via direct limits, if there exists a map h : Proof. There exists an integer k such that Notation: For any set C ⊂ A * B we denote by Γ[C] the graph that contains all vertices of C and all edges that have at least one endpoint in C. M → M such that q| ∂ M * A1 : ∂ M * A 1 → ∂ h(M ) *p −1 (U h(M ) λ k (p(z))) ⊂ U h(h(M ) λ ) n (z).(4) A thorough discussion of amalgams and free products can be found in [SS12]. Important for us is that any element w ∈ A * B has a unique reduced representation w = a 1 b 2 · . . . · a k b k , where k ≥ 1, a i ∈ A and b i ∈ B for all 1 ≤ i ≤ k and furthermore a i = e for all i > 1, and similarly b i = e for all i < k. If b k = e we say the length of w is 2k − 1, otherwise, we say the length of w is 2k. Let w be an element of A * B. Assume that w = a 1 b 1 · . . . · a k b k has length l in its reduced representation. We denote by Γ w the graph Γ \ {w} and we analyze its connected components. Let C w ⊂ A * B be the set of all elements whose reduced representations have length ≥ l + 1 and start with a 1 b 1 · . . . · a k b k . Let C w = A * B \ (C w ∪ {w}). Observe that there exists no edge between C w and C w . Therefore, any geodesic between C w and C w goes through w. With the same argumentation, we get: Any path from e to w goes through a 1 , a 1 b 1 , . . . , a 1 b 1 · . . . · b k−1 and a 1 b 1 Lemma 3.1.· . . . · a k . Observe that for a = a ∈ A the sets C a and C a are disjoint. Therefore, we can partition A * B \ A into sets {C a } a∈A . This gives rise to a projectionπ : A * B → A defined byπ(a) = a for all a ∈ A and π(w) = u for the unique u ∈ A such that w ∈ C u . The projectionπ induces a projection π : Γ → Γ A defined by π(x) = x if x ∈ Γ A and π(x) =π(w) if x is connected to w ∈ A * B in Γ \ Γ A . This is well defined: if w and v are connected in Γ \ Γ A , thenπ(w) =π(v). Lemma 3.2. The projection π as defined above is continuous. Proof. We have to prove that for all x ∈ Γ and ε > 0 there exists δ > 0 such that if d(x, y) < δ, then d(π(x), π(y)) < ε. If x ∈ Γ A , choose δ = ε. This works as π(x) = x and any path from x to y goes through π(y). If x / ∈ Γ A choose δ = d(x, π(x)). Any y with d(y, x) < δ then satisfies that y and x are in the same connected component of Γ \ Γ A and thus π(x) = π(y). Lemma 3.3. Let γ be a continuous (λ, ε)-quasi-geodesic in Γ with endpoints in Γ A . Then π • γ : [a, b] → Γ A is a continuous (λ, ε)-quasi-geodesic with d H (γ, π • γ) ≤ λ 2 ε + ε. Proof. The map π • γ is continuous because π and γ are continuous. To show that π • γ is a (λ, ε)-quasigeodesic, we have to prove that 1 λ |t − s| − ε ≤ d(π • γ(t), π • γ(s)) ≤ λ|t − s| + ε,(5) for all t, s ∈ [a, b]. The right hand side follows from d(π • γ(t), π • γ(s)) ≤ d(γ(t), γ(s)), because γ is a (λ, ε)-quasi-geodesic. Note that (6) holds because if γ(t) and γ(s) are in the same connected component of Γ \ Γ A , then π • γ(t) = π • γ(s). If γ(t) and γ(s) are not in the same connected component of Γ \ Γ A , then any path (and thus any geodesic) from γ(t) to γ(s) goes through π • γ(t) and π • γ(s). To prove the left hand side of (5), we need the following observation: if π • γ(s) = a then there exists s 1 ≤ s ≤ s 2 such that γ(s 1 ) = x and γ(s 2 ) = x. This holds, as any path from γ(a) to γ(s) goes through x. Analogously any path from γ(s) to γ(b) goes through x. Let s, t ∈ [a, b] and s ≤ s ≤ t ≤ t such that γ(s ) = π(γ(s)) and γ(t ) = π(γ(t)). As γ is a (λ, ε)-quasi-geodesic, we have d(γ(s ), γ(t )) ≥ 1 λ |s − t | − ε. But d(γ(s ), γ(t )) = d(π • γ(t), π • γ(s)) and |s − t | ≥ |s − t|, which concludes the proof of the left hand side of (5). To show that d H (γ, π • γ) ≤ λ 2 ε + ε it is enough to show that d(γ(t), π • γ(t)) ≤ λ 2 ε + ε for all t ∈ [a, b]. Let t ∈ [a, b] and t 1 ≤ t ≤ t 2 such that γ(t 1 ) = π • γ(t) = γ(t 2 ). We have shown above that d(γ(t 1 ), γ(t)) ≤ λ|t 1 − t| + ε. We also have 0 = d(γ(t 1 ), γ(t 2 )) ≥ 1 λ |t 1 − t 2 | − ε. Therefore, |t 1 − t| ≤ |t 1 − t 2 | ≤ λε, which concludes the proof. Proof. Let γ ⊂ Γ be a continuous (λ, ε)-quasi-geodesic with endpoints on η. Lemma 3.3 shows that π • γ ⊂ Γ A is a (λ, ε)-quasi-geodesic with endpoints on η. Since η is M -Morse in Γ A , π • γ is in an M (λ, ε) neighbourhood of η. Thus γ is in an M (λ, ε) + λ 2 ε + ε neighbourhood of η. By Remark 2.11, this concludes the proof. 3.1. Construction of the combinatorial Morse boundary. Consider an M -Morse geodesic ray ξ in Γ starting at e. The geodesic ξ travels along an infinite sequence of edges (s i ) i∈N , each edge being from either S A or S B . One of the following happens: either (s i ) i∈N contains infinitely many edges from both S A and S B (we call this the infinite case) or the tail of (s i ) i∈N is contained completely in either S A or S B (we call this the finite case). Define c 1 = s 1 · . . . · s k1 such that k 1 is the maximal integer satisfying s i ∈ A for all 1 ≤ i ≤ k 1 . For i ≥ 2 iteratively define c i = s ki−1+1 · . . . · s ki where k i is the maximal integer such that s j ∈ G i%2 for all k i−1 < j ≤ k i , where G i%2 denotes A if i is odd and B otherwise. In the infinite case c 2i+1 ∈ A \ {e}, c 2i ∈ B \ {e} for all i ≥ 1 and ξ walks from c 1 to c 1 c 2 to c 1 c 2 c 3 and so on. In the finite case we have some i such that c i is defined as the product of infinitely many elements of G i%2 . Then, c i can not be viewed as an element of G i%2 but as a geodesic in Γ G i%2 starting at e and going along s ki−1+1 , s ki−1+2 , . . . and so on. This shows that in the infinite case, we can view ξ as an infinite sequence (c i ) i∈N and in the finite case we can view ξ as a sequence c 1 , . . . , c m and some geodesic ξ ∈ Γ G (m+1)%2 . It also motivates the following definition: Let M be a Morse gauge, define δ M T Γ =      (u i ) i∈N | u i ∈      A (M ) if i = 1, A (M ) \ {e} if i > 1 is odd, B (M ) \ {e} if i is even.      δ M f Γ =      (u 1 , . . . , u n ; z) | n ∈ N 0 , u i ∈      A (M ) if i = 1, A (M ) \ {e} if i > 1 is odd, B (M ) \ {e} if i is even. z ∈ ∂ M * G n%2      . Going back to our description from above we have in the infinite case (c i ) i∈N ∈ δ M S T Γ and in the finite case (c 1 , . . . , c i−1 ; [c i ]) ∈ δ M S f Γ. As a set, the combinatorial M -Morse boundary δ M * Γ is defined as δ M * Γ = δ M T Γ ∪ δ M f Γ. We will call elements of δ M * Γ combinatorial M -Morse geodesics and say that an element a ∈ δ M * Γ is: • Of infinite type if a ∈ δ M T Γ. • Of finite type if a ∈ δ M f Γ. Sometimes, we will talk about elements δ M * Γ without writing down its explicit representation. To nonetheless be able to talk about explicit parts of their representation, we introduce the following notation. Notation. Let a = (u 1 , . . . , u n ; z) be an element of δ M f Γ. We say the integer n is the length of a and we will denote it by l(a). For any integer 1 ≤ i ≤ l(a), u i (a) will denote u i and γ(a) will denote z. For an element b = (u i ) i∈N of δ M T Γ we will say that its length l(b) is infinite and for any positive integer i, u i (b) will denote u i . Furthermore, define v 0 (a) = e and v i (a) = u i v i−1 (a) for all 1 ≤ i ≤ l(a). Let M be a Morse gauge. We will define the topology on δ M * Γ by defining a fundamental system of neighbourhoods {V M k (a)} k∈N for all combinatorial M -Morse geodesics a ∈ δ M * Γ. Let k ∈ N be an integer and a ∈ δ M * Γ: • If a is of infinite type we will define Lemma 3.6. This is indeed a fundamental system of neighbourhoods. V M k (a) = {b ∈ δ M * Γ | l(b) ≥ k and u i (b) = u i (a) for all 1 ≤ i ≤ k}. • If a is of finite type, V M k (a) consists of all combinatorial M -Morse geodesics b ∈ δ M * Γ that satisfy: i) l(b) ≥ l(a), ii) u i (b) = u i (a), for all 1 ≤ i ≤ l(a), iii) u l(a)+1 ∈Ũ M k (γ(a)), if l(b) > l(a), iv) γ(b) ∈ U M k (γ(a)), if l(b) = l(a). Proof. Let a be a combinatorial M -Morse geodesic. We need to show the following three properties hold: (1) For all k ∈ N, we have a ∈ V M k (a). (2) For all i, j ∈ N there exists a positive integer k such that V M k (a) ⊂ V M i (a) ∩ V M j (a). (3) For all V M i (a) ∈ {V M n (a)} n∈N there exists V M j (a) ∈ {V M n (a)} n∈N such that for every b ∈ V M j (a) there exists k ∈ N such that V M k (b) ⊂ V M i (a) . Property 1 follows directly from the definition. Property 2 holds when setting k = max{i, j}. To prove Property 3, we will make a case distinction: Assume a is of infinite type. In this case, setting j = i + 1 and k = j works. Assume a is of finite type. Use Lemma 2.22 to get j ∈ N such that U M j (y) ⊂ U M i (γ(a)) for all y ∈ U M j (γ(a)). If b is of finite type, set k = j. If b is of infinite type, set k = l(a) + 1. Let M and N be two Morse gauges with M ≤ N . We have that both A (M ) ⊂ A (N ) and ∂ M * A ⊂ ∂ N * A and we have the same for B. Thus, the inclusion i M,N : δ M * Γ → δ N * Γ is well-defined. More importantly, for any combinatorial M -Morse geodesic a ∈ δ M * Γ, we have V M k (a) = V N k (a) ∩ δ M * Γ and hence the inclusion i M,N is not only well-defined but also continuous. Therefore, we can define the combinatorial Morse boundary as follows: Definition 3.7. The combinatorial Morse boundary of Γ is defined as the direct limit δ * Γ = lim − − → M δ M * Γ. Lemma 3.8. The topology on δ M * Γ induced by the fundamental system of neighbourhoods {V M k (a)} k∈N coincides with the subspace topology of δ M * Γ ⊂ δ * Γ. Proof. It is enough to show that any closed C ⊂ δ M * Γ satisfies that C ∩ δ N * Γ is closed for any Morse gauge N . In other words, we have to show that for any a ∈ δ N * Γ \ C there exists a neighbourhood V N k (a) such that V N k (a) ∩ C is empty. If there exists an integer k such that u k (a) / ∈ A (M ) ∪ B (M ) we are done, since in that case V N k (a) ∩ δ M * Γ is empty. If a ∈ δ M * Γ we are done, as V N k+l (a) ∩ δ M * Γ ⊂ V M k (a) for some large enough l and C is closed in δ M * Γ. Thus, the only case left to consider is when a is of finite type and a ∈ δ M * Γ. We may assume γ(a) ⊂ A. Define Y = {γ(b) | b ∈ C, v l(a) (b) = v l(a) (a)} ⊂ ∂ M * A. The set Y is closed in ∂ M * A because C is closed and thus Y ∩ ∂ N * A is closed in ∂ N * A. Therefore, there exists an integer k 1 such that U N k1 (γ(a)) ∩ Y is empty. If we manage to find an integer k 2 such that V N k (a) ∩ δ M f Γ ∩ C is empty we can set k = max(k 1 , k 2 ) and then V N k (a) ∩ C is empty. We show that such an integer k 2 exists by contradiction. Assume V N k (a) ∩ δ M f Γ ∩ C is nonempty for all positive integers k, then Z k = {u l(a)+1 (b) | b ∈ C ∩ V N k (a)} is non-empty. Consider the sequence (γ i ) {i∈N} such that γ i is an M -Morse realization of x for some x ∈ Z i . By Arzela-Ascoli, (γ i ) {i∈N} contains a subsequence (γ ij ) {j∈N} that converges uniformly on compact sets to some geodesic ray ξ. The proof of Lemma 2.10 of [Cor17] shows that ξ is M -Morse. Moreover, by the definition of (γ i ) {i∈N} , we have that ξ ∈ O N k (γ(a)) for all k and thus [ξ] = γ(a). This is a contradiction to a ∈ δ M * Γ, which concludes the proof. Below, we will show that the Morse boundary of Γ is homeomorphic to its combinatorial Morse boundary. This will allow us to, in the next sections, work with the combinatorial Morse boundary, which is somewhat tamer than the Morse boundary itself. 3.2. Homeomorphism. At the beginning of Section 3.1 we explained how, from a geodesic ξ in Γ starting at e we can get a combinatorial Morse geodesic. We will call the combinatorial geodesic constructed this way Ψ(ξ). Note: If ξ is M -Morse, then Ψ(ξ) is an M S -Morse combinatorial geodesic. On the other hand, let a ∈ δ * Γ be a combinatorial Morse geodesic. For all 1 ≤ i ≤ l(a), let p i be a geodesic from v i−1 (a) to v i (a). If a is of infinite type we define η a = p 1 p 2 . . . If a is of finite type we define η a = p 1 . . . p l(a) (v l(a) · ξ), where ξ is a realization of γ(a). In both cases we define Φ(a) = [η a ]. Proposition 3.9. The map Φ : δ * Γ → ∂ * Γ is well defined. Furthermore, if a ∈ δ M * Γ is an M -Morse combinatorial geodesic Φ(a) is M -Morse, where M depends only on M . Proof. Note that we first have to argue that η a is a geodesic. However, this follows from Lemma 3.1. Secondly, we show that when choosing appropriate geodesics p i and ξ, then η a is M -Morse where M depends only on M and lastly we show that [η a ] does not depend on the choice of representatives p i and ξ. For Let γ be a continuous (λ, ε)-quasi-geodesic with endpoints on η. Assume γ starts on p i for some i and ends on some p j for some j (the proof works analogously if one or more of them are on v l(a) (a) · ξ instead). Assume that i ≤ j, then by Lemma 3.1 γ goes through v i (a), . . . , v j−2 (a) and v j−1 (a). Thus we can split γ into multiple parts, each having endpoints on an M -Morse geodesic being a sub-geodesic of η a . Therefore, each part of γ is in an M (λ, ε) neighbourhood of η a . Using Remark 2.11, this shows that with this choice of p i and ξ, η a is an M -Morse geodesic, where M depends only on M . Now we have to show that [η a ] does not depend on the choice of p i and ξ. Or in other words, that if we choose differentp i andξ to construct a geodesicη a , thenη a and η a have bounded Hausdorff distance. As [ξ] = [ξ] it is clear that v l(a) · ξ and v l(a) ·ξ have bounded Hausdorff distance. Lemma 2.5 from [CS14] shows that p i andp i have Hausdorff distance at most C, where C depends only on M . Therefore, η a andη a have bounded Hausdorff distance, which concludes the proof. all 1 ≤ i ≤ l(a), choose p i such that p i is M -Morse in v i−1 (a) · Γ G i%2 ⊂ Γ. Such a choice is possible since u i (a) ∈ G i%2 (M ) . Furthermore, We can show that Φ is in fact a homeomorphism. Lemma 3.10. The map Φ is a bijection. Proof. Let [a] ∈ ∂ * Γ. We have Φ(Ψ(a)) = [a] and thus Φ is surjective. To prove injectivity, let a = a ∈ δ * Γ be combinatorial Morse geodesics, assume l(a) ≥ l(a ). We want to show that the to a and a constructed geodesics η a and η a do not have bounded Hausdorff distance. If there exists an integer i such that v i (a) = v i (a ) or if l(a) > l(a ), then for large enough t any geodesic between η a (t) and η a goes through v i . The distance d(v i (a), η a (t)) tends to infinity as t goes to infinity. Therefore, the Hausdorff distance between η a and η a is not First, we show that Φ M is continuous. To do so, let a ∈ δ M * Γ and let k be a positive integer. We claim that Φ M (V M k+c (a)) ⊂ U h(M ) k (Φ(a)), where c ≥ 4δ h(M ) + δ M + 4h(M )(3, 0) + 1. Let b ∈ V M k+c (a) . Assume a is of finite type and let D = d(e, v l(a) (a)). We have d(η b (k + c + D), η a (k + c + D)) < δ M , therefore, Lemma 2.7 of [Cor17] shows that for all t < k + D + 2δ h(M ) d(η b (t), η a (t)) < δ h(M ) . Therefore, η b ∈ O h(M ) k+4δ h(M ) (η a ) ⊂ O h(M ) k (Φ(a)) . Assume a is of infinite type. Note that D = d(e, v k+c (a)) ≥ k + c − 1, and that d(η a (D), η b (D)) = 0. Using Lemma 2.7 from [Cor17] again we get that η b ∈ O h(M ) k+4δ h(M ) (η a ) ⊂ O h(M ) k (Φ(a) ). This proves the claim and shows that Φ M is continuous. Secondly, we prove that Φ −1 M is continuous by showing that, for a ∈ ∂ M * Γ, Φ −1 M (U M l (a)) ⊂ V h(M ) k (Φ −1 (a)), where we will define l later. Assume Φ −1 (a) is of finite type and let b ∈ U M l (a). We use the following notation: • Φ −1 (a) =â and Φ −1 (b) =b • v l(â) (â) = w • D = d(b (t) ∈ Γ A . Then ζ b [t, ∞) ⊂ Γ[C ζ b (t) ] while ζ a ⊂ Γ[C ζ b (t) ]. Therefore, d(ζ b (t + δ M ), ζ a (t + δ M )) > δ M which is a contradiction to b ∈ U M l (a) if t < k + 4δ h(M ) . Hence ζ b [0, k + 4δ h(M ) ] ⊂ Γ A which concludes this case. At last, assume Φ −1 (a) is of infinite type and let b ∈ U M l (a). Set Φ −1 (a) =â and Φ −1 (b) =b. Furthermore, define • w = v k (â) • D = d(We have ξ a [D, ∞) ⊂ Γ[C w ], so ξ b ∩ Γ[C w ] = ∅, which impliesb ∈ V h(M ) k (â). Free product In this section, we prove Theorem 1.2. To do so, we use the following set up: Let p A : ∂ * A 1 → ∂ * A 2 and p B : ∂ * B 1 → ∂ * B 2 be homeomorphisms. We denote by p the map from ∂ * A 1 ∪ ∂ * B 1 to ∂ * A 2 ∪ ∂ * B 2 their union induces. We use p to construct bijectionsp A : A 1 → A 2 andp B : B 1 → B 2 that in the correct sense respect the continuity of p. This then induces a mapp from the combinatorial Morse boundary of A 1 * B 1 to the combinatorial Morse boundary of A 2 * B 2 , which we will prove to be a homeomorphism. We focus on the construction ofp A . The mapp B is then constructed analogously and thus all the results we prove aboutp A hold forp B as well. We will also drop the subscripts and talk only about mapŝ p and p instead ofp A and p A . Note that in the definition of p A , we use to x corresponding geodesics λ x , as defined in Section 2.3. In the case of |∂ * A 1 | ≥ 2, the geodesics λ x depend heavily on λ(A) and so will the map p A that we construct. However, p A has the desired properties no matter how we choose λ(A). 4.1. Bijections on the factors. If ∂ * A 1 = ∅ we letp be any bijection from A 1 to A 2 that satisfieŝ p(e) = e. So, for the rest of the section we assume that |∂ * A 1 | ≥ 1. In this case, we construct the bijectionp : A 1 → A 2 as follows: Setp(e) = e and enumerate the elements of A 1 and A 2 . Then, repeat the following two steps: • Step 1: Denote by x the lowest index element of A 1 which has not yet been matched. Assume λ x is M -Morse for some Morse gauge M and let η be an h(M )-Morse realization of p(λ x ). Let T 1 be as in Lemma 2.26 iv) and define M = h(h(M ) λ ). Set T = max{T 1 + 4δ M , 12δ M }. Let y = η(i) for some i ∈ N such that η(i) is still unmatched and satisfies p −1 (λ η(i) ) ∈ U M T ([λ x ]).(7) Match x to y, that is, setp(x) = y. We say that x is the initiator of the match and y is the target. • Step 2: Denote by y the lowest index element of A 2 which has not yet been matched. Repeat the process from Step 1, with the roles of A 1 and A 2 reversed, to construct a point x ∈ A 1 . Set p(x) = y. Note that in this case, y is called the initiator and x the target. Observe that Lemma 2.29 implies all but finitely many positive integers satisfy (7). Since also all but finitely many elements of A 2 are unmatched, an element η(i) with the desired properties indeed exists and hencep is well-defined. When we work with some x ∈ A 1 in the matching process, we take any Morse gauge M such that λ x is M -Morse and use it to construct the target y. We make no assumptions that the Morse gauge M is in any way "optimal". However, it is important for the well-definedness of later constructions, that if x is N -Morse, then y is N -Morse where N depends only on N, h and M 0 . This is why we posed the technical condition (7). It allows us to prove the following lemma: Proof. In this proof we use the notation from Step 1 of the construction ofp. Assume x ∈ A 1 (N ) is an initiator. Then, λ x is N λ -Morse and p(λ x ) is h(N λ )-Morse. Lemma 2.21 then implies that η is N = h(N λ ) C -Morse. Asp(x) lies on η it is N S -Morse. Note that N S depends only on N , M 0 and h. Therefore, part one of the lemma is true. It also shows that part two is true for any x ∈ A 1 that is a target. To prove the other direction, assume that x ∈ A 1 is an initiator and thatp(x) = η(i) is N -Morse. Observe that λ η(i) is N λ -Morse. Let ξ ∈ O M T ([λ x ]) be an M -Morse realization of p −1 (λ η(i) ). Note that [ξ] is h(N λ )-Morse and thus ξ is not only M -Morse but also N 1 -Morse, where N 1 = h(N λ ) C . Therefore, using Cor 2.5 of [Cor17], we get d(ξ(t), λ x (t)) < δ N1 for all t ∈ [0, T − 4δ M ]. Since T − 4δ M ≥ T 1 , the Hausdorff distance between λ x [0, T 1 ] and ξ[0, T 1 ] is bounded by δ N1 . Hence Lemma 2.5 i) of [CS14] implies that λ x [0, T ] is N 2 -Morse, where N 2 depends only on N 1 . Lemma 2.10 together with the fact that λ x [T 1 , ∞) is M 1 -Morse implies that λ x is N 3 -Morse, where N 3 depends only on M 1 and N 2 . By Lemma 2.26 ii), x is N 4 -Morse, where N 4 depends only on N 3 and M 0 . One can check step by step that N 4 only depends on N , M 0 and h, which is what we wanted to show. It also shows that part one is true for the case that x is a target. In the construction, we did not match an element x ∈ A 1 to another element y ∈ A 2 directly but first matched x to a geodesic, used p to get a geodesic in A 2 and then got a point y ∈ A 2 . Those intermediary geodesics are the connection between p andp and are important in proving thatp respects the continuity of p. This motivates the following definition: For any x ∈ A i for i ∈ {1, 2} which is an initiator define g(x), the geodesic matched to x, as [λ x ]. For any x ∈ A 1 which is a target, define g(x) as p −1 (g(p(x))). Similarly, for any y ∈ A 2 which is a target, define g(y) as p(g(p −1 (y))). Proof. It is enough to show the statement for x ∈ A 1 . If x is an initiator, then the statement holds due to Lemma 2.26 i). If x is a target, then Lemma 4.1 impliesp(x) is M 1 -Morse, where M 1 depends only on M , M 0 and h. Observe thatp(x) is a target, and thus g(p(x)) is M 2 -Morse, where M 2 depends only on M 1 and M 0 . As g(x) = p −1 (g(p(x))), it is h(M 2 )-Morse. The following notation will be useful: Let M be a Morse-gauge, we define M p ≥ M, M λ , h(M ) as a Morse gauge such that: • For all x ∈ A 1 that are M -Morsep(x), g(x) and λ x are M p -Morse. • For all y ∈ A 2 that are M -Morsep −1 (y), g(y) and λ y are M p -Morse. From now on, by abuse of notation, we denote by p the map A 1 ∪ ∂ * A 1 → A 2 ∪ ∂ * A 2 induced byp ∪ p. Our last goal of this subsection is to prove the following proposition: Note that M p 2 denotes (M p ) p . To do so, we need the following two technical lemmas. Lemma 2 . 10 . 210Let ξ : [0, ∞) → X be a geodesic ray such that ξ[0, T ] is M -Morse and ξ[T, ∞) is M -Morse for some Morse gauges M and M . Then, ξ is M -Morse, where M depends only on M and M . the triangle ∆ 1 = (β[0, n], [β(n), α(s)], [α(s), p]). By Remark 2.3, the geodesic [β(n), α(s)] is M 1 -Morse, where M 1 depends only on δ M and thus depends only on M . By Lemma 2.12 the geodesic β[0, n] is M S -Morse. As ∆ 1 is a finite triangle, [α(s), p] is M 2 -Morse, where M 2 depends only on M . Now assume instead that α is M -Morse. Consider the triangle ∆ 2 = ([α(s), p], γ, α[0, s]). As ∆ 2 is a finite triangle, and the Morseness of [α(s), p] and α[0, s] depends only on M , we have that γ is M T -Morse, where M T depends only on M . We denote by X (M ) ⊂ X the set of all points x ∈ X that have an M -Morse realization and we say that points x ∈ X (M ) are M -Morse. Note that Remark 2.3 implies that X = ∪ M ∈M X (M ) . Similarly, we define the set of all M -Morse directions ∂ M * X as the set of all Morse directions z ∈ ∂ * X that have an M -Morse realization. Intuitively speaking, the Morseness of λ(A) gives us a base Morseness M 0 and λ x captures the Morse properties of x up to the base Morseness. The lemma below states this more precisely: Lemma 2.26. Let x ∈ A and let η be an M -Morse realization of x. The following hold: i) The to x corresponding geodesic ray λ x is M λ -Morse, where M λ depends only on M and M 0 . Figure 1 . 1Construction of the to x corresponding geodesic ray λ x .ii) If λ x is N -Morse, then x ∈ X (N ) ,where N depends only on N and M 0 . iii) For any positive integer T , there exists a constant C M (T ), depending only on M , A, M 0 and T , such that if d(e, x) ≥ C M (T ), then d(λ x (t), η(t)) < δ M for all t ∈ [0, T ]. iv) There exists an integer T 1 such that λ x ([T 1 , ∞)) is M 1 -Morse, where M 1 depends only on M 0 . Remark 2.27. Lemma 2.26 implies that if d(e, x) ≥ C M (T ), then x is inŨ M λ T (λ x ) and λ x is in U M λ T (x). Together with Remark 2.15 we also have that if d(e, x) ≥ C M (T + 4δ M λ ), then x is inŨ M λ T ([λ x ]). A 2 2is continuous for all M , then so is q itself. The topology of ∂ M * A 1 and ∂ h(M ) * A 2 are given by a fundamental system of neighbourhoods, so it is enough to show that for all points x ∈ A 1 and positive integers k there exists an integer l such that q(U M l (x)) ⊂ U h(M ) k (q(x)). Lemma 2.29. Let z ∈ ∂ M * A 1 be an M -Morse direction and η be an h(M )-Morse realization of p(z). For any positive integer n, there exists an integer i 0 such that p −1 (λ η(i) ) ∈Ũ h(h(M ) λ ) n (z), for all i ≥ i 0 . Choose i 0 = 0C h(M ) (k + 4δ h(M ) λ ). For any i ≥ i 0 the geodesic η([0, i]) is an h(M )-Morse realization of η(i). Remark 2.27 shows that λ η(i) ∈ O h(M ) λ k (p(z)). The proof is concluded by (4). 3. The Morse boundary of the free product Let A and B be finitely generated groups. In this section, we give an alternative description of the Morse boundary of A * B, call it the combinatorial Morse boundary and show that it is homeomorphic to ∂ * (A * B). To do so, we fix Cayley graphs Γ A = Γ(A, S A ), Γ B = Γ(B, S B ) and Γ = Γ(A * B, S A ∪ S B ) for some finite generating sets S A ⊂ A and S B ⊂ B. With this notation, Γ A and Γ B can be isometrically embedded into Γ using the inclusions A → A * B and B → A * B. Lemma 3. 4 . 4Let η ⊂ Γ A be an M -Morse geodesic. Then, η considered as a geodesic in Γ is M -Morse, where M depends only on M . Remark 3 . 5 . 35Condition ii) is equivalent to the condition v l(a) (a) = v l(a) (b). if a is of finite type, choose ξ to be an M -Morse realization of γ(a) (again considered in Γ G i%2 ). Lemma 3.4 shows that the p i and ξ are M -Morse in Γ, where M depends only on M . We claim that η a is M -Morse, where M depends only on M . bounded and thus Φ(a) = Φ(a ). If l(a) = l(a ) and v i (a) = v i (a ) for all 1 ≤ i ≤ l(a), we have [γ(a)] = [γ(a )]. Thus, v l(a) · ξ and v l(a ) · ξ have unbounded Hausdorff distance and so do η a and η a . Proposition 3 . 11 . 311The map Φ is a homeomorphism. Proof. For a Morse gauge M , let h(M ) ≥ M, M S be a Morse gauge 1 respectively are continuous for all Morse gauges M . e, w) and l = k + D + 4δ h(M ) + δ M • ξ a and ξ b are M -Morse realizations of a and b respectively • w −1 · ξ a [D, ∞) = ζ a and w −1 · ξ b [D, ∞) = ζ b . Any realization of b goes through w, as ξ a [D, ∞) is a subset of Γ[C w ]. Therefore ζ a and ζ b are M -Morse geodesics starting at e. More importantly, ζ a ⊂ Γ A or ζ a ⊂ Γ B . We may assume that ζ a ⊂ Γ A and show in this case that ζ b [0, k + 4δ h(M ) ] ⊂ Γ A . Together with δ M ≤ δ h(M ) this then implies thatb ∈ V h(M ) k (â), which is what we needed to show. Assume ζ b [0, k + 4δ h(M ) ] ⊂ Γ A , let t ≥ 0 be the largest number such that ζ e, w) and l = D + δ M • ξ a and ξ b are M -Morse realizations of a and b respectively Figure 2 . 2Construction ofp(x). Lemma 4 . 1 . 41If x is N -Morse, thenp(x) is N -Morse, where N depends only on N , M 0 and h. Similarly, ifp(x) is N -Morse, then x is N -Morse, where N depends only on N, M 0 and h. Lemma 4 . 2 . 42Let x ∈ A i for i ∈ {1, 2} be M -Morse. Then, g(x) is M -Morse,where M depends only on M , M 0 and h. Proposition 4. 3 . 3For all positive integers l and M -Morse directions z ∈ ∂ M * A 1 there exists an integer k such that p(Ũ M k (z)) ⊂Ũ M p 2 l (p(z)). Lemma 4. 4 . 4Let k be a positive integer. For all x ∈ A 1 (M ) satisfying d(e, x) ≥ C M (k + 4δ Mp ) we have Proof. Assume x is an initiator. Then, g(x) = [λ x ]. Let η be an M -Morse realization of x. Remark 2.27 shows that η ∈Õ M λ k+4δ Mp (λ x ). This implies that η ∈Õ Mp k ([λ x ]) and that λ x ∈ O M λ k+4δ Mp (η). Consequently, λ x ∈ O Mp k (x). Assume x is a target. Then, there exists an M p -Morse realization ξ of g(x) that goes through x. Note that η = ξ[0, d(e, x)] is an M p -Morse realization of x satisfying η ∈Õ Mp d(e,x) (ξ) and ξ ∈ O Mp d(e,x) (η). Thus η ∈Õ Mp k (g(x)) and ξ ∈ O Mp k (x). Lemma 4 . 5 . 45Let l be a positive integer and let z ∈ ∂ M * A 1 be an M -Morse direction. There exists an integer k such that i) for all x ∈Ũ M k (z) ∩ A 1 we have g(x) ∈Ũ Mp l (z). ii) all x ∈ A 1 (M ) with d(e, x) ≥ k and g(x) ∈Ũ Acknowledgements: This paper was my Master Thesis. I want to thank my supervisors Alessandra Iozzi and Matthew Cordes for their helpful guidance throughout my work on the thesis. I also want to thank Alessandro Sisto for helping to find the topic of the thesis.Proof. Lemma 2.22 implies that there exists an integer k such that for all y ∈ŨSet k = C M (k + 2δ Mp ). Suppose x ∈Ũ M k (z) ∩ A 1 and let ξ be an M -Morse realization of z. We have ) and thus (8) also concludes the proof of ii).Proof of Proposition 4.3. Let z ∈ ∂ M * A 1 be an M -Morse direction and l a positive integer. First, use Lemma 4.5 ii) to get an integer k 1 such that all y ∈ A 1 (Mp) with d(e, y) ≥ k 1 and g(y) ∈Ũ. If x ∈ ∂ * A 1 , the statement follows directly. Therefore, we assume x ∈ A 1 . By choice of k 3 we have g(x) ∈ U Mp k2 (z) and by choice of k 2 we have p(g(x)) ∈ U M p 2 k1 (p(z)). Note that, by the choice of k, d(e, p(x)) ≥ k 1 and thus p(x) ∈Ũ M p 2 l (p(z)).Proposition 4.3 also holds if ∂ * A 1 = ∅, as any statement about elements of the empty set holds. Furthermore, if ∂ * A 1 = ∅, then ∂ M * A is finite for any Morse gauge M . Therefore, the existence of a Morse gauge M p such thatp(x) ∈ A 2 (Mp) for all x ∈ A 1 (M ) is clear. Those, apart fromp(e) = e, are the only properties aboutp that we will use later on. Thus, from now on, we do not have to make a case distinction whether ∂ * A 1 is empty or not.4.2.Proof of Theorem 1.2. We define the mapp : δ * Γ 1 → δ * Γ 2 viā p(u 1 , . . . , u n ; x) = (p(u 1 ), . . . ,p(u n ); p(x)) andp ((u i ) i∈N ) = (p(u i )) i∈N .The mapp is well defined as the image of any combinatorial M -Morse geodesic is a combinatorial M p -Morse geodesic (and asp(e) = e). The map is a bijection as p andp are a bijection. We are now ready to prove Theorem 1.2Proof of Theorem 1.2. It is enough to show that p is continuous since then, by symmetry,p −1 : δ * Γ 2 → δ * Γ 1 is continuous as well.To do so, we will show that for any Morse gauge M , x ∈ δ M * Γ 1 and l ∈ N there exists an integer k ∈ N such thatp(V M k (x)) ⊂ V M p 2 l (p(x)). Case 1: x is in δ M T Γ 1 : In this case, we can set k = l. Indeed: Let x ∈ V M l (x), then x is an M -Morse combinatorial geodesic with l(x ) ≥ l + 1 and u i (x ) = u i (x) for all 1 ≤ i ≤ l. Thusp(x ) is an M p -Morse combinatorial geodesic with l(p(x )) ≥ l + 1 and u i (p(x )) = u i (p(x)). Hencep(x ) ∈ V M p 2 l (p(x)). Case 2: x is in δ M f Γ 1 : We use Proposition 4.3 to get an integer k such that p(Ũ M k (γ(x))) ⊂ U M p 2 l (γ(p(x))). Let x ∈ V M k (x). Observe thatp(x ) is M p 2 Morse. Furthermore: i) l(x ) ≥ l(x) and thus l(p(x )) ≥ l(p(x)). ii) u i (x ) = u i (x) for all 1 ≤ i ≤ l(x) and thus u i (p(x )) = u i (p(x)) for all 1 ≤ i ≤ l(p(x)). if l(x ) = l(x), then γ(x ) ∈Ũ M k (γ(x)) and thus γ(p(x )) ∈Ũ M l (γ. if l(x ) = l(x), then γ(x ) ∈Ũ M k (γ(x)) and thus γ(p(x )) ∈Ũ M l (γ(p(x))). Therefore,p(x ) ∈ V M p 2 l. which concludes the proofTherefore,p(x ) ∈ V M p 2 l (p(x)), which concludes the proof. Papasoglu and K. Whyte stated below (see [PW02]). We will state a slight adaptation of the version used in. 15[MS15] and follows from the Theorem of P. Papasoglu and K. Whyte stated below (see [PW02]). We will state a slight adaptation of the version used in [MS15]. Let G, H be finitely generated groups with infinitely many ends and let G, H be their graph of groups decompositions with all edge groups finite. If G and H have the same set of quasi-isometry types of infinite vertex groups (without multiplicities) then G and H are quasiisometric. particular, if G 1. G m represent all quasi-isometry types of the infinite vertex groups of G which are not virtually cyclic. if m = 0, then G is quasi-isometric to the free group F 2Theorem 5.1 (Papasoglu-Whyte). Let G, H be finitely generated groups with infinitely many ends and let G, H be their graph of groups decompositions with all edge groups finite. If G and H have the same set of quasi-isometry types of infinite vertex groups (without multiplicities) then G and H are quasi- isometric. In particular, if G 1 , . . . , G m represent all quasi-isometry types of the infinite vertex groups of G which are not virtually cyclic, • if m = 0, then G is quasi-isometric to the free group F 2 ; • if m = 1, then G is quasi-isometric to G 1 * G 1. • if m = 1, then G is quasi-isometric to G 1 * G 1 ; • if m > 1, then G is quasi-isometric to G 1. * · · · * G M, • if m > 1, then G is quasi-isometric to G 1 * · · · * G m . If m = 0 for the group G, then m = 0 for the group H, as a group is quasi-isometric to a virtually cyclic group if and only if it is virtually cyclic itself. In the other two cases, G is quasi-isometric to G 1 * · · · * G m and H is quasi-isometric to H 1 * · · · * H m (here we set G 2 = G 1 if m = 1 and do the same for H). we get that ∂ * (G 1 * · · · * G m ) ∼ = ∂ * (H 1 * · · · * H m ). We can assume that the groups H 1 , . . . , H m are ordered in a way such that H i is quasi-isometric to G i . Therefore H i and G i have homeomorphic Morse boundaries. Using induction and Theorem 1.2 as base-case and step. The theorem follows from the Morse boundary being a quasi-isometry invariantProof of Theorem 1.1. The Morse boundary is a quasi-isometry invariant (see [Cor17]). If m = 0 for the group G, then m = 0 for the group H, as a group is quasi-isometric to a virtually cyclic group if and only if it is virtually cyclic itself. In the other two cases, G is quasi-isometric to G 1 * · · · * G m and H is quasi-isometric to H 1 * · · · * H m (here we set G 2 = G 1 if m = 1 and do the same for H). We can assume that the groups H 1 , . . . , H m are ordered in a way such that H i is quasi-isometric to G i . Therefore H i and G i have homeomorphic Morse boundaries. Using induction and Theorem 1.2 as base-case and step, we get that ∂ * (G 1 * · · · * G m ) ∼ = ∂ * (H 1 * · · · * H m ). The theorem follows from the Morse boundary being a quasi-isometry invariant. Metric Spaces of Non-Positive Curvature. Martin Bridson, Andr&apos;e Haefliger, Springer-Verlag319Berlin HeidelbergMartin Bridson and Andr'e Haefliger. Metric Spaces of Non-Positive Curvature, volume 319. Springer-Verlag Berlin Heidelberg, 01 2009. Quasi-mobius homeomorphisms of morse boundaries. Ruth Charney, Matthew Cordes, Devin Murray, Bulletin of the London Mathematical Society. 513Ruth Charney, Matthew Cordes, and Devin Murray. Quasi-mobius homeomorphisms of morse boundaries. Bul- letin of the London Mathematical Society, 51(3):501-515, Mar 2019. Complete topological descriptions of certain morse boundaries. Ruth Charney, Matthew Cordes, Alessandro Sisto, Ruth Charney, Matthew Cordes, and Alessandro Sisto. Complete topological descriptions of certain morse bound- aries, 2020. Boundary convex cocompactness and stability of subgroups of finitely generated groups. Matthew Cordes, Matthew Gentry Durham, International Mathematics Research Notices. 6Matthew Cordes and Matthew Gentry Durham. Boundary convex cocompactness and stability of subgroups of finitely generated groups. International Mathematics Research Notices, 2019(6):1699-1724, 2019. Stability and the morse boundary. Matthew Cordes, David Hume, Journal of the London Mathematical Society. 953Matthew Cordes and David Hume. Stability and the morse boundary. Journal of the London Mathematical Society, 95(3):963-988, 2017. Morse boundaries of proper geodesic metric spaces. Matthew Cordes, Groups Geom. Dyn. 114Matthew Cordes. Morse boundaries of proper geodesic metric spaces. Groups Geom. Dyn., 11(4):1281-1306, 2017. A survey on Morse boundaries and stability. Matthew Cordes, Beyond hyperbolicity. CambridgeCambridge Univ. Press454Matthew Cordes. A survey on Morse boundaries and stability. In Beyond hyperbolicity, volume 454 of London Math. Soc. Lecture Note Ser., pages 83-116. Cambridge Univ. Press, Cambridge, 2019. Contracting boundaries of cat(0) spaces. Ruth Charney, Harold Sultan, Journal of Topology. 81Ruth Charney and Harold Sultan. Contracting boundaries of cat(0) spaces. Journal of Topology, 8(1):93-117, 9 2014. M J Dunwoody, The accessibility of finitely presented groups. Inventiones mathematicae. 81M.J. Dunwoody. The accessibility of finitely presented groups. Inventiones mathematicae, 81:449-458, 1985. Dynamics on the Morse boundary. Qing Liu, Geom. Dedicata. 214Qing Liu. Dynamics on the Morse boundary. Geom. Dedicata, 214:1-20, 2021. Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries. Alexandre Martin, Jacek Swiatkowski, J. Group Theory. 182Alexandre Martin and Jacek Swiatkowski. Infinitely-ended hyperbolic groups with homeomorphic Gromov bound- aries. J. Group Theory, 18(2):273-289, 2015. Topology and dynamics of the contracting boundary of cocompact cat(0) spaces. Devin Murray, Pacific Journal of Mathematics. 2991Devin Murray. Topology and dynamics of the contracting boundary of cocompact cat(0) spaces. Pacific Journal of Mathematics, 299(1):89-116, Apr 2019. Quasi-isometries between groups with infinitely many ends. Panos Papasoglu, Kevin Whyte, Commentarii Mathematici Helvetici. 771Panos Papasoglu and Kevin Whyte. Quasi-isometries between groups with infinitely many ends. Commentarii Mathematici Helvetici, 77(1):133-144, 2002. . J Stilwell, J P Serre, Trees. Springer Monographs in Mathematics. SpringerJ. Stilwell and J.P. Serre. Trees. Springer Monographs in Mathematics. Springer Berlin Heidelberg, 2012. R Abdalrazzaq, Zalloum, A symbolic coding of the morse boundary. arXiv: Group Theory. Abdalrazzaq R. Zalloum. A symbolic coding of the morse boundary. arXiv: Group Theory, 2018.
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[ "Development of a low-background neutron detector array", "Development of a low-background neutron detector array" ]
[ "Y T Li \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "W P Lin \nInstitute of Nuclear Science and Technology\nKey Laboratory of Radiation Physics and Technology of the Ministry of Education\nSichuan University\n610064ChengduChina\n", "B Gao \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nJoint Department for Nuclear Physics\nInstitute of Modern Physics\nLanzhou University\nChinese Academy of Sciences\n730000LanzhouChina\n", "H Chen \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n", "H Huang \nJoint Department for Nuclear Physics\nInstitute of Modern Physics\nLanzhou University\nChinese Academy of Sciences\n730000LanzhouChina\n", "Y Huang \nInstitute of Nuclear Science and Technology\nKey Laboratory of Radiation Physics and Technology of the Ministry of Education\nSichuan University\n610064ChengduChina\n", "T Y Jiao \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "K A Li \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "X D Tang \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nJoint Department for Nuclear Physics\nInstitute of Modern Physics\nLanzhou University\nChinese Academy of Sciences\n730000LanzhouChina\n", "X Y Wang \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "X Fang \nSino-French Institute of Nuclear Engineering and Technology\nSun Yat-sen University\n519082ZhuhaiGuangdongPeople's Republic of China\n", "H X Huang \nChina Institute of Atomic Energy\n102413BeijingPeople's Republic of China\n", "J Ren \nChina Institute of Atomic Energy\n102413BeijingPeople's Republic of China\n", "L H Ru \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "X C Ruan \nChina Institute of Atomic Energy\n102413BeijingPeople's Republic of China\n", "N T Zhang \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "Z C Zhang \nInstitute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n" ]
[ "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Institute of Nuclear Science and Technology\nKey Laboratory of Radiation Physics and Technology of the Ministry of Education\nSichuan University\n610064ChengduChina", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Joint Department for Nuclear Physics\nInstitute of Modern Physics\nLanzhou University\nChinese Academy of Sciences\n730000LanzhouChina", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "Joint Department for Nuclear Physics\nInstitute of Modern Physics\nLanzhou University\nChinese Academy of Sciences\n730000LanzhouChina", "Institute of Nuclear Science and Technology\nKey Laboratory of Radiation Physics and Technology of the Ministry of Education\nSichuan University\n610064ChengduChina", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Joint Department for Nuclear Physics\nInstitute of Modern Physics\nLanzhou University\nChinese Academy of Sciences\n730000LanzhouChina", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Sino-French Institute of Nuclear Engineering and Technology\nSun Yat-sen University\n519082ZhuhaiGuangdongPeople's Republic of China", "China Institute of Atomic Energy\n102413BeijingPeople's Republic of China", "China Institute of Atomic Energy\n102413BeijingPeople's Republic of China", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "China Institute of Atomic Energy\n102413BeijingPeople's Republic of China", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina", "Institute of Modern Physics\nCAS Key Laboratory of High Precision Nuclear Spectroscopy\nChinese Academy of Sciences\n730000LanzhouChina", "School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina" ]
[]
A low-background neutron detector array was developed to measure the cross section of the 13 C(α,n) 16 O reaction, which is the neutron source for the s-process in AGB stars, in the Gamow window (Ec.m. = 190 ± 40 keV) at the China Jinping Underground Laboratory (CJPL). The detector array consists of 24 3 He proportional counters embedded in a polyethylene cube. Due to the deep underground location and a borated polyethylene shield around the detector array, a low background of 4.5(2)/hour was achieved. The 51 V(p, n) 51 Cr reaction was used to determine the neutron detection efficiency of the array for neutrons with energy En < 1 MeV. Geant4 simulations, which were shown to well reproduce experimental results, were used to extrapolate the detection efficiency to higher energies for neutrons emitted in the 13 C(α,n) 16 O reaction. The theoretical angular distributions of the 13 C(α,n) 16 O reaction were shown to be important in estimating the uncertainties of the detection efficiency.
10.1007/s41365-022-01030-0
[ "https://arxiv.org/pdf/2111.12552v3.pdf" ]
244,527,478
2111.12552
eab0e71dca0f9c7b89837e87641384f6ea28b3f5
Development of a low-background neutron detector array 17 Mar 2022 Y T Li Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina W P Lin Institute of Nuclear Science and Technology Key Laboratory of Radiation Physics and Technology of the Ministry of Education Sichuan University 610064ChengduChina B Gao Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina Joint Department for Nuclear Physics Institute of Modern Physics Lanzhou University Chinese Academy of Sciences 730000LanzhouChina H Chen Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina H Huang Joint Department for Nuclear Physics Institute of Modern Physics Lanzhou University Chinese Academy of Sciences 730000LanzhouChina Y Huang Institute of Nuclear Science and Technology Key Laboratory of Radiation Physics and Technology of the Ministry of Education Sichuan University 610064ChengduChina T Y Jiao Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina K A Li Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina X D Tang Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina Joint Department for Nuclear Physics Institute of Modern Physics Lanzhou University Chinese Academy of Sciences 730000LanzhouChina X Y Wang Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina X Fang Sino-French Institute of Nuclear Engineering and Technology Sun Yat-sen University 519082ZhuhaiGuangdongPeople's Republic of China H X Huang China Institute of Atomic Energy 102413BeijingPeople's Republic of China J Ren China Institute of Atomic Energy 102413BeijingPeople's Republic of China L H Ru Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina X C Ruan China Institute of Atomic Energy 102413BeijingPeople's Republic of China N T Zhang Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina Z C Zhang Institute of Modern Physics CAS Key Laboratory of High Precision Nuclear Spectroscopy Chinese Academy of Sciences 730000LanzhouChina School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina Development of a low-background neutron detector array 17 Mar 2022Underground laboratoryNeutron detectorLow background3 He Counter A low-background neutron detector array was developed to measure the cross section of the 13 C(α,n) 16 O reaction, which is the neutron source for the s-process in AGB stars, in the Gamow window (Ec.m. = 190 ± 40 keV) at the China Jinping Underground Laboratory (CJPL). The detector array consists of 24 3 He proportional counters embedded in a polyethylene cube. Due to the deep underground location and a borated polyethylene shield around the detector array, a low background of 4.5(2)/hour was achieved. The 51 V(p, n) 51 Cr reaction was used to determine the neutron detection efficiency of the array for neutrons with energy En < 1 MeV. Geant4 simulations, which were shown to well reproduce experimental results, were used to extrapolate the detection efficiency to higher energies for neutrons emitted in the 13 C(α,n) 16 O reaction. The theoretical angular distributions of the 13 C(α,n) 16 O reaction were shown to be important in estimating the uncertainties of the detection efficiency. I. INTRODUCTION The 13 C(α,n) 16 O reaction is the dominant neutron source for the slow neutron capture process (s-process), which synthesized roughly half of the elements heavier than iron in the Universe [1]. The main site of the s-process is the "He intershell" of asymptotic giant branch (AGB) stars where the temperature is 0.1 GK. This corresponds to a Gamow window of 190 ± 40 keV for the 13 C(α,n) 16 O reaction, which is far below the Coulomb barrier. Theoretical calculations predicted the cross section as low as ∼ 10 −14 barn at 190 keV [2,3]. Meanwhile, the 13 C(α,n) 16 O reaction is also the main neutron source for the intermediate neutron capture process (iprocess) with a typical temperature 0.2 GK [4,5]. A reliable direct measurement is still missing within the corresponding energy range from 0.2 to 0.54 MeV. To push direct measurements of the cross section towards the Gamow window, highintensity beams and low-background neutron detector array are indispensable. Present measurements performed at ground laboratories are limited by the cosmic-ray-induced background. With a typical background counting rate of a few hundred per hour (e.g. [6,7]), those measurements only reached a lower limit of E α ∼ 400 keV with a cross section of ∼ 10 −10 barn. The China JinPing underground Laboratory (CJPL) [8] with about 2400 m rock overburden (6700 m water equivalent) shields against most of the cosmic rays. The ultra-low background provides new opportunities for neutrino physics [9], search of dark matter [10,11] and direct cross section measurements of key reactions in nuclear astrophysics [12,13]. The Jinping Underground Nuclear Astrophysics experimental facility (JUNA) [12,13] has been initiated to measure important stellar nuclear reaction rates, taking advantage of the low-background environments in CJPL. The present JUNA project includes 12 C(α,γ) 16 O, 13 C(α,n) 16 O, 19 F(p,γ) 20 Ne and 25 Mg(p, γ) 26 Al reactions [12,13]. A high-intensity accelerator capable of delivering proton and α beams up to 10 mA in the energy range of 0.05 -0.4 MeV has been built for JUNA [12]. This made it possible to measure the 13 C(α,n) 16 O cross section at the region of the s-process Gamow window if combined with a high-efficiency and lowbackground neutron detection system. This paper describes the development and characterization of a low-background neutron detector array for the cross section measurement of the 13 C(α, n) 16 O reaction. II. DESIGN OF THE DETECTOR ARRAY The detector array consists of 24 3 He-filled proportional counters (manufactured by GE-Reuter Stokes) embedded in a 50 × 50 × 50 cm 3 polyethylene cube. Each proportional counter is filled with 3 He and argon gases, with pressures of 4 bar and 2.4 bar, respectively. The active length of each counter is 30 cm with a diameter of 2.54 cm. The sketch of the detector array is shown in Fig. 1. The 24 proportional counters are evenly distributed in two concentric rings with radius of R 1 = 8.5 cm and R 2 = 13 cm, respectively. The radius of the two rings were optimized using Geant4 simulations, following the same procedure as that in [14], to get the maximum detection efficiency for 2.0-MeV isotropic neutrons. A central bore hole with a diameter of 10 cm was created to accommodate the beam line and the target. Neutrons from the 13 C(α, n) 16 O reaction with energies of 2 -3 MeV are first moderated by the polyethylene cube and then captured by the proportional counters. To shield against environmental neutrons, the polyethylene cube is wrapped with a 5 cm-thick 7% borated polyethylene layer. The signal of each detector is processed by a chargesensitive pre-amplifier (model CAEN A1422). The output waveforms are digitized and recorded by a XIA Pixie-16 100 MHz card. Fig. 2 (a) shows a typical energy spectrum measured by the detector array using a 252 Cf source. The "Neutron & alpha" cut as shown in Fig. 2 (b) was used in the plot to select the neutron events (see the following sections for details). The peak at 0.76 MeV corresponds to full-energy events where the triton and proton produced from the 3 He + n → p + t reaction deposit all their kinetic energies in the sensitive region of the counter. If the proton or triton hits the wall of the detector before depositing all their kinetic energies in the sensitive region, the detector records only part of the reaction Q value. Those are wall-effect events which correspond to the flat region with lower energies starting from 0.18 MeV in the spectrum. The argon gas filled in the proportional counters has little effect on the neutron detection efficiency but can significantly decrease the track lengths of the produced proton and triton, and therefore reduce the wall effect. By defining a region of interest around the full energy peak between 0.64 -0.8 MeV (ROI) as indicated in Fig. 2 (a), we find 81.0±0.4% of the detected events are located in it. III. BACKGROUND MEASUREMENTS Stainless steel is chosen as the wall material of the proportional counters because it has much lower α emission rate comparing to aluminum [15,16]. The inset in Fig. 2 (a) shows an energy spectrum up to 7 MeV measured at the CJPL. The events located beyond the neutron full-energy peak are mainly due to α radioactivity from the wall material of the counters. The α spectrum has a relatively flat distribution below ∼ 3 MeV and shows a peak at 5 MeV, a typical energy of α decays from U/Th nuclides. Below 0.3 MeV energy region, lots of γ/β events exist, overlapping with the neutron events. Following the procedure described in Ref. [15], the β/γ events can be well separated from neutrons in the raise time versus energy two-dimensional plot as shown in Fig. 2 (b) taken at the CJPL. To characterize the background of the detector array, ground-and underground-measurements were performed at the China Institute of Atomic Energy and the A1 hall of the CJPL, respectively. The neutron events were selected by using the "neutron & alpha" cut from the raise time versus energy two-dimensional plot as shown in Fig. 2 (b). Such events also include contributions from α background. To disentangle the α backgrounds from the "neutron & alpha" events, a flat α energy spectrum is assumed [15,16] in the energy range 0.18 -2.0 MeV. The total background inside the "neutron & alpha" cut as shown in Fig. 2 (b) measured at CJPL for the whole array is 4.5(2)/hour, of which 1.94(5)/hour is from α background. The above α background is evaluated using the α counts in the energy range 1.0 -2.0 MeV which was assumed to have the same α emission rate as 0.18 -0.8 MeV. This assumption was verified using a 4 He proportional counter which will be discussed below. The total background is more than two orders of magnitude lower than the result of 1238(11)/hour obtained from ground-measurement. A 4 He counter, which has exactly the same parameters as the 3 He counters but filled with 4 He gas instead of 3 He, manufactured by the same supplier was used to investigate the α background in more detail. The 4 He counter is insensitive to neutrons but expected to have similar α background shape as the 3 He counters. The α background rate, with 28.9-hour measurement time in CJPL, was found to be 4.24 ± 0.48 and 4.22 ± 0.38 MeV −1 hour −1 , in 0.18 -0.8 and 1.0 -2.0 MeV energy regions, respectively. This justifies that the two energy regions have indeed the same α emission rates. However, we found an α emission rate of 2.63(30)/hour by using the same neutron cut as the 3 He counters in the raise time versus energy plot. This is about a factor of 30 higher than the 3 He counters. Note that the α emission rate of 1.94(5)/hour quoted above includes the background of the 24 3 He counters while the 2.63(30)/hour is only for one 4 He counter. This implies that the α background could differ significantly even for the same type of wall material. This has to be carefully investigated at the manufacturing stage in order to achieve low α background. To further reduce the α background, one can choose the events within the ROI only [see Fig. 2 (a)]. As a result, the α background is reduced by 74% at the cost of losing only about 19% of the neutron detection efficiency. This corresponds to a background of 2.5(1)/hour, of which 0.50(1)/hour is from α background. A better approach for α-background suppression is to use plastic scintillator as the moderator, where most of the α background can be eliminated via coincident measurements [17]. The α background is intrinsic properties of the proportional counters used in the array which does not change with environments. However, the neutron background may change with different environments and therefore should be characterized for each use. IV. EFFICIENCY CALIBRATION A. 51 V(p, n) 51 Cr Experiment and simulation Detection efficiency of the array was calibrated at the nuclear physics experiment (NPE) terminal of the 3-MV tandetron accelerator [18] at Sichuan university. The detection efficiency hereafter includes both full-energy and wall-effect events. Quasi Mono-energetic neutrons were produced using the 51 V(p, n) 51 Cr reaction (Q = -1534.8 keV) at incident energies between 1.7 and 2.6 MeV with a step of 0.15 MeV. The 51 V(p, n) 51 Cr reaction is widely used in the calibration of neutron detectors [14,16,[19][20][21][22] for its slow variation of neutron intensity and energy with angle, and the well-known target preparation and utilization. Fig. 3 shows the emitted neutron energy as a function of incident proton beam energy. When the incident energy is above 2.33 MeV, neutrons from transitions feeding the first excited state of 51 Cr (E x = 749 keV) are mixed with those feeding the ground state. However, as pointed out in Ref. [16], the contribution from transitions to the first excited state is negligible for incident energies up to 2.6 MeV [19], which was also confirmed in the present work (see discussions in the following sections). The proton beams were focused at the target with a diameter of less than 5 mm (full width at half maximum). The beam spots were monitored by a fluorescent target together with a beam position monitor located 1.5 m upstream the target for each beam energy. Vanadium targets with a thickness of 110 µg/cm 2 were used in the measurements. They were produced by evaporating natural vanadium on 1-mm thick tantalum disks with a diameter of 30 mm. The target thickness corresponds to an energy loss of 8∼11 keV for proton beams with energies of 1.7 -2.6 MeV. The beam intensity varied from 4 µA to 120 nA as the beam energy increased from 1.7 to 2.6 MeV, making the counting rate of the detector array stay around 10 4 /s with a dead time being less than 1% for all beam energies. A direct water cooling of the reaction target was used to reduce the sputtering and target loss. The possible contribution of beam induced background was investigated with a blank tantalum target at the beam energy of 2.0 MeV. The background contribution was found to be less than 1%. The total number of emitted neutrons was determined based on the activation method as described in Ref. [16]. The number of radioactive product 51 Cr equals to the number of emitted neutrons in the reaction. 51 Cr decays by electron capture with a half-life of T 1/2 = 27.7025(24) days and has a branching ratio of B = 9.91(1)% to decay to the first excited state of its daughter nuclei 51 V, which is followed by the emission of a 320-keV γ ray. The off-line measurement of this γ ray was carried out with a GEM series HPGe detector whose relative efficiency is 30%. The absolute efficiency of the HPGe detector was measured at a distance of 20 cm using 137 Cs [1.534(19)×10 5 Bq] and 152 Eu [5.72(6)×10 4 Bq] γ-ray sources. The distance of 20 cm is large enough to avoid pileups of cascading γ rays from the sources. A 51 Cr γ source was produced by the 51 V(p, n) 51 Cr reaction. Its activity was measured at the position of 20 cm and then used to determine the efficiency for the 320-keV γ line at the position of 10.2 cm (η 320 ), where all the irradiated targets were placed for off-line measurements. η 320 was determined to be 0.498(7)% at 10.2 cm. As beam current was stable within the irradiation time, the number of emitted neutrons was then determined by the off-line measurement of the 320 keV γ ray with the activation formula [16] N R = N γ B · η 320 · e λtw 1 − e −λtc · λ · t i 1 − e −λti ,(1) where N γ , t i , t c and t w are the number of detected 320-keV γ rays, the activation time, the counting time, and the waiting time elapsed between the end of irradiation and the start of the counting. λ = ln(2)/T 1/2 is the decay constant of 51 Cr. After dead time correction, the detection efficiency of the neutron detector is calculated as η n = N n N R ,(2) where N n is the detected number of neutrons by the detector array. Since the energy of the neutrons emitted in the 51 V(p, n) 51 Cr reaction is only up to ∼1 MeV, it is still far below that from 13 C(α, n) 16 O reaction. The Monte Carlo simulation code Geant4 [23,24] of version 10.4.6 was used to determine the neutron detection efficiency at neutron energies above 1 MeV. The detailed detector setup used the simulation is shown in Fig. 4. The beam pipe, target backing and water cooling loops were included in the simulation, which resembles the real physical setups. An overestimation of the neutron detection efficiency was obtained from the simulation compared to the experimental results. Similar overestimations have also been observed in several other setups [14,16]. This overestimation is interpreted as that some of the neutrons are absorbed by small contaminants in the moderating polyethylene. Instead of using a normalization factor, we added a small amount of boron into the moderating polyethylene in the simulation to take into account the neutron absorption effects. Fig. 5 (a) shows the reduced χ 2 , obtained by comparing the simulated and measured detection efficiencies, as a function of boron mass fraction. The minimum of χ 2 was found at the boron mass fraction of 0.054% by fitting the curve using a parabola. The simulated total, inner ring and outer ring detection efficiencies are shown in Fig. 5 (b). An estimation of the first excited state contributions was also carried out using the first excited state neutron component estimated from a statistic model calculation, calibrated using the experimental data [19], using TALYS [25]. The simulated total efficiencies with the first state neutron contributions are also shown as the blue short dashed curve in Fig. 5 (b). The maximum deviation between the simulation with and without the contribution of the first excited state is 0.9%, which is ineligible by comparing to the experimental uncertainty of 3.7%. The boron contaminant affects not only the total efficiency, but also the ratio R in/out of the detection efficiencies of the inner and the outer ring detectors. With the boron mass fraction obtained above, the measured ratio R in/out is also well reproduced by the two simulations (see Fig. 5 (c)). Except for the first point, the simulation results are 1%-3% lower than that of the experiment at E p < 2.4 MeV. Note that the isotropic angular distributions are used in the simulations. The discrepancy of R in/out between simulations and experiment is probably due to the non-isotropic distribution of the emitted neutrons. For E p = 2.45 and 2.6 MeV, the discrepancy between the simulated (with ground state neutrons) and measured R in/out ratios starts to increase to 1.6% and 5%, respectively. This is due to the opening of the decay channel to the first excited state in 51 Cr (see Fig. 3). Taking into ac- The reduced χ 2 , obtained by comparing the simulated and measured total detection efficiencies, as a function of boron mass fraction. The curve is a parabola fitting of the reduced χ 2 . (b) Measured and simulated total, inner ring and outer ring detection efficiencies as a function of proton beam energy in the 51 V(p, n) 51 Cr reaction. Blue short dashed curve represents the simulation with first excited state neutrons. The first excited state neutron component were estimated from a statistical model calculation, calibrated using the experimental data [19], using TALYS [25]. (c) Measured and simulated inner-to-outer-ring ratios R in/out at different proton beam energies. (d) Simulated total, inner ring and outer ring detection efficiencies for mono-energetic neutrons with an isotropic angular distribution. The neutron range indicated with 13 C(α, n) 16 O is for the α energy range Eα = 300 -800 keV. count the first excited state neutrons estimated from TALYS, the discrepancy is reduced to 0.6% and 2.3% for E p = 2.45 and 2.6 MeV, respectively, as shown in Fig. 5 (c). The contributions from the decay channel to the first excited state is negligibly small regarding the total detection efficiency though its effects on the R in/out is larger. The total, inner ring and outer ring detection efficiencies for monoenergetic neutrons up to 4.5 MeV with an isotropic angular distribution were also simulated and shown in Fig. 5 (d). The dependence of detection efficiency on the source position was measured by placing the detector array at different positions along the beam line (Z axis) using the 51 V(p, n) 51 Cr reaction at E p = 2 MeV. The position Z = 0 corresponds to the target being at the center of the array. Moving the detector array forward in the beam direction corresponds to positive Z values and backward corresponds to negative Z values. The relative efficiencies at different positions were normalized using integrated incident beam currents on the target. The results for the total, inner ring and outer ring detectors are shown in Fig. 6 together with the Geant4 simulations. In Fig. 6, the measured total efficiency at Z = 0 was normalized to the simulated value. Overall good agreements were found between the measurements and the simulations. Taking into account the average deviation between the experimental data and simulations between Fig. 5 (b) and Fig. 6, the difference between the Geant4 simulations and the experiment results is 2.8%, which reflects one of the systematic uncertainties of our simulation and is quoted as the uncertainty of the Geant4 simulations. B. Extrapolating the detection efficiency for the study of the 13 C(α, n) 16 O reaction at stellar energies It should be noted that the simulated efficiency in Fig. 5 cannot be directly applied to the 13 C(α, n) 16 O reaction in which the emitted neutrons are neither mono-energetic nor isotropic. The asymmetry in the efficiency curve shown in Fig. 6 indicates that the simulation predicts a slightly higher detection efficiency for neutrons emitted at backward angles than that at forward angles. Since the angular distribution of the 13 C(α, n) 16 O reaction is not measured at low energies close to the Gamow window, theoretically predicted angular distributions were used in our simulations. Legendre polyno- [26][27][28], the angular distribution effect on the efficiencies is corrected in our Geant4 simulations. Detection efficiencies were simulated using both isotropic and calculated angular distributions in the incident α energies between 0. 2 -3.15 MeV. In the simulation, the narrow resonances are ignored. Fig. 8 (a) shows simulated efficiencies using isotropic and calculated angular distributions. For α energies between 0.3 -0.8 MeV, the difference is 0.19% -2.28% which is relatively small. However, above 0.8 MeV, the difference becomes larger with a maximum of 4%. To evaluate the effect for the sharp resonance, we also simulated the detection efficiency at the energy where the a 1 coefficient reaches its maximum around E = 1.0534 MeV, as shown in Fig. 7 (a). The resulting efficiency is 11% less than the efficiency with an isotropic angular distribution. Therefore, the angular distribution effect needs to be evaluated carefully for the experiments aiming to a high precision. The effects of the angular distributions were further investigated by comparing the simulated and measured ratios of the inner ring and outer ring detection efficiencies R in/out . As shown in Fig. 8 (b), the R in/out is insensitive to the angular distribution and the simulations in both case agree reasonably well with the measured values from Ref. [30]. The difference between measured and simulated R in/out values is ∼ 4%. An alternative way of extrapolating the total detection efficiency was carried out as follows. The inner-ring detection efficiency is extrapolated to the neutron energy range relevant to the 13 C(α, n) 16 O reaction. Then the total detection efficiency is obtained using the measured R in/out values from Ref. [30]. The difference in the total detection efficiency between the two extrapolation methods is ∼ 2.5%, which is counted as another systematic uncertainty. Taking into account the uncertainties of the Geant4 simulations (2.8%), the extrapolation (2.5%), the neutron angular distributions (2.3%) and the detection efficiency of HPGe detector (1.5%), the overall uncertainty of the detection efficiency for the 13 C(α, n) 16 O reaction in the energy range E α = 300 -800 keV is determined to be 5%. Excluding the narrow resonances, the maximum uncertainty of neutron angular distributions is 4% if the energy range is extended up to 2.4 MeV. According to the ENDF angular distribution, the maximum uncertainty of neutron angular distributions at the energy beyond 2.4 MeV is 12%. Simulated ratios R in/out using both isotropic (red dashed line) and theoretically predicted (blue solid line) angular distributions. Solid triangles are the measured R in/out from Ref. [30]. V. SUMMARY A high-efficiency and low-background neutron detector array consisting of 24 3 He proportional counters embed-ded in a polyethylene moderator has been developed for the cross-section measurement of the 13 C(α, n) 16 O reaction at the China JinPing underground Laboratory. As a result of the deep underground location and a 5-cm thick borated polyethylene shield, a low background of 4.5(2)/hour was achieved, of which 1.94(5)/hour was from the internal α radioactivity. The 51 V(p, n) 51 Cr reaction was used to calibrate the neutron detection efficiency of the array for neutrons with energies E n < 1 MeV. For E n > 1 MeV, the Monte Carlo simulation code Geant4 was used to extrapolate the detection efficiency. Specifically, an energy dependent detection efficiency, which can be directly applied to the 13 C(α, n) 16 O reaction, was obtained from the simulation. The effects of the angular distribution of the 13 C(α,n) 16 O reaction on the detection efficiency, which were overlooked in previous works, were investigated in the present work and shown to be nonnegligible. Fig. 1 . 1(Color online) Schematic view of the detector array. In panel (a), the borated polyethylene shield (black brick), positions of the 3 He counters (red tubes) and the central bore hole are indicated. Traces of two neutron events are also shown as green curves. In panel (b), a three-dimensional view of the detector array is presented. Fig. 2 . 2(Color online) (a) Typical energy spectrum measured by the detector. A cut as shown in panel (b) was applied to select the neutrons (see text). The dashed lines indicate the regions of interest between 0.64 -0.8 MeV. The inset shows an energy spectrum in wider energy range without cuts. (b) Raise time versus energy twodimensional plot measured by the detector array. The β/γ-, neutron & alpha-and alpha-regions are indicated in the plot. Fig. 3 . 3The emitted neutron energies from the 51 V(p,n) 51 Cr reaction as a function of proton beam energies. The height of the bars represents the energy spread of the emitted neutrons. The two components of the neutron energies at Ep = 2.45 and 2.6 MeV correspond to transitions feeding the ground and the first-excited states of 51 Cr. Fig. 4 . 4(Color online) The detector setup used in the simulations for the 51 V(p, n) 51 Cr reaction. Fig. 5 . 5(Color online) (a) Fig. 6 . 6(Color online) Detection efficiencies of the total (solid squares), the inner ring (solid circles) and the outer ring (solid triangles) detectors as a function of the target position. The statistical errors are smaller than the symbol size. Solid, dashed and long dashed curves are those from the Geant4 simulations. mials up to the third order were used in calculating the angular distributions. Fig. 7 shows the Legendre polynomials coefficient a 1 used in the angular distributions of the 13 C(α, n) 16 O reaction as a function of beam energies in (a) and the representative angular distributions obtained at beam energies of 0.2, 0.7, 1.0534 and 1.2 MeV in (b). Adopting the angular distributions in Refs. Fig. 7 . 7(Color online) (a) Legendre polynomials coefficient a1 used in the angular distributions of the 13 C(α, n)16 O reaction as a function of beam energies. The red curve is from Paris/Hale ENDF-8[26] and the black dots are from Ref[27]. In order to match the excitation energy of17 O recommended by NNDC [29], the Legendre polynomials coefficients around 1.05 MeV and 1.33 MeV are shifted 2.4 keV and 1.7 keV, respectively, to higher energy. (b) Representative angular distributions obtained at beam energies of 0.2, 0.7, 1.0534 and 1.2 MeV. 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[]
[ "arXiv:physics/0412012v1 [physics.chem-ph] Extracting Work From A Single Heat Bath", "arXiv:physics/0412012v1 [physics.chem-ph] Extracting Work From A Single Heat Bath" ]
[ "Arbab A Khan \nDepartment of Electronics\nDepartment of Chemistry\nQuaid-i-Azam University\nIslamabadPakistan\n", "M S Ansari \nQuaid-i-Azam University\nIslamabadPakistan\n" ]
[ "Department of Electronics\nDepartment of Chemistry\nQuaid-i-Azam University\nIslamabadPakistan", "Quaid-i-Azam University\nIslamabadPakistan" ]
[]
We present here a machine that is capable of extracting work from a single heat bath. Although no significant temperature gradient is involved in the operation of the machine, yet the Carnot efficiency as high as one is achievable. Working of the machine is explained on the basis of a demon suggested by Maxwell. Utilizing the kinetic energy spectrum of the molecules in solution, the demon can send "hotter" molecules to a higher gravitational potential at the expense of their own energies.Difference in chemical potentials due to concentration gradients and use of semipermeable membranes ensure the continuing cyclic process. 0
null
[ "https://arxiv.org/pdf/physics/0412012v1.pdf" ]
93,147,464
physics/0412012
458382787d81570aaa6f33a10c18aff58f713ab0
arXiv:physics/0412012v1 [physics.chem-ph] Extracting Work From A Single Heat Bath 2 Dec 2004 Arbab A Khan Department of Electronics Department of Chemistry Quaid-i-Azam University IslamabadPakistan M S Ansari Quaid-i-Azam University IslamabadPakistan arXiv:physics/0412012v1 [physics.chem-ph] Extracting Work From A Single Heat Bath 2 Dec 2004 We present here a machine that is capable of extracting work from a single heat bath. Although no significant temperature gradient is involved in the operation of the machine, yet the Carnot efficiency as high as one is achievable. Working of the machine is explained on the basis of a demon suggested by Maxwell. Utilizing the kinetic energy spectrum of the molecules in solution, the demon can send "hotter" molecules to a higher gravitational potential at the expense of their own energies.Difference in chemical potentials due to concentration gradients and use of semipermeable membranes ensure the continuing cyclic process. 0 Perpetual Motion Machine (P MM) 1,2 , although hardly allowed by the laws of thermodynamics, has been a subject of great interest since long. P MM of first kind, in contrast to the law of conservation of energy, is supposed to create energy, thus violating the first law of thermodynamics. On the other hand P MM of second kind should be capable to convert total heat into useful work showing 100% efficiency of the heat engine, which is rather contrary to the second law of thermodynamics. Another manifestation that forbids P MM of second kind is that no work can be extracted from a single heat bath. This is because according to Carnot, greater efficiency of a heat engine is only possible if temperature of the energy source, T h is higher and the entropy sink is maintained at a lower temperature T c 2-4 whereby the Carnot efficiency, is given as η = 1 − T c /T h .(1) It is to note further that input of heat energy from a bath is essential for all heat engines but for a P MM of second kind it can take place without requiring any temperature difference and entropy sink. An additional requirement that prohibits the existence of P MM is pertaining to entropy, which for a spontaneous process must not decrease. Since P MM of second kind is capable of increasing order by separating high and low energy particles without any energy involvement, its existence is not allowed according to the entropy considerations as well. Maxwell's demon 5,6 , a hypothetical "creature", can sort hot and cold types of particles without any effort in measurement and consumption of energy, therefore it may also be considered equivalent to a P MM of second kind. Discovery of quantum non-demolition measurements and computing 7-12 have given an incentive and led to a refined review of thermodynamics demanding a more careful search for the existence of P MM. Recently in their excellent paper Scully et al. 13 working on a new kind of quantum heat engine using "phaseonium" 14 as fuel and utilizing lasing without inversion [15][16][17] , showed that 1 work can be extracted from a single heat bath but they had to consider vanishing quantum coherence to save laws of thermodynamics. In a recent communication Scully 18 used a Stern-Gerlach apparatus to sort hot and cold spin atoms thus acting as Maxwell's demon. To achieve a cyclic operation, the atomic center of mass was prepared in a well-defined quantum state, showing that the "cost" of preparing the center of mass wave packet was enough to preserve the second law of thermodynamics. Distribution of kinetic energy between the particles in a sample of gas or liquid at a given temperature is Gaussian 19,20 . In an isolated system, one possible way of sorting high and low energy particles can be moving up high-energy particles to a higher gravitational potential at the expense of their own heat energies. The above scheme may be employed for a cyclic process of mass-transfer to achieve useful work without involvement of an initial temperature gradient. At this point a system of two containers at different gravitational potentials may be considered as shown in Fig. 1. The container L at lower potential is filled with neat water, while the container H at higher potential is initially empty. The system is in thermal equilibrium at an absolute temperature T . The demon allows transfer of higher kinetic energy particles from L to H utilizing their own energies. Consequently, mass m of water may reach to H after gaining the gravitational potential energy mgh, where g is the (average) gravitational constant and h is the height to which water is raised from the surface of water in L. This will result in an overall decrease in temperature of the thermodynamically isolated system. This temperature decrease may be related with potential energy as ∆T = mgh/Mσ,(2) where M and σ are the mass and specific heat capacity of the whole system. The temperature of the system may again be raised back to T if water from H is allowed to drain into container L. If the setup is connected to a heat reservoir that can maintain temperature T , this demon will continue to pour more and more water into H by increasing the potential energy of water at the expense of heat taken from the bath. As a result, a state of dynamic equilibrium is reached. The osmotic pressure, Π may be related to the mole fraction of water, X A as Π = −(RT /V m )lnX A ,(3) where R is the gas constant, T is absolute temperature and V m is the molar volume of water. For dilute solutions, eq. (3) simplifies to well-known van't Hoff's expression that directly relates osmotic pressure with the solute concentration 21,?,? . It is evident that the column pressure, ρgh alone is responsible for the outwards flow of water through SP M. Hence in the event where the SP M is not immersed, water will come out of it under the influence of the column pressure. This is depicted in Fig. 2 (B), where shape of the SP M is slightly modified as per requirement of the experiment to be discussed later. We conclude that the setup shown in Fig. 3 behaves as a Maxwell's demon without itself consuming any energy or involving any change of state, but converts heat into potential energy. It is notable that the whole system is at a single temperature, T maintained by the heat bath of the surrounding. Further the system is non-temporal, i.e., all the parameters like concentration and pressure etc. are time independent. Figure Caption Fig. 1 . 1Maxwell's demon capable of separating higher energy water molecules to a container at higher potential. Kinetic energy spectrum of molecules in a solution requires the lighter solvent molecules to be more agile and hence an effective means of energy transfer. A semi-permeable membrane (SP M) allows only solvent molecules to pass through it but solute molecules, especially of larger size, are not allowed to cross. Statistically the solvent molecules may move across both sides of SP M. However, a net flux of solvent across the SP M may take place under the influence of gravitational and/or osmotic pressure. Consider the setup for osmosis as shown in Fig. 2. (A). A tube whose lower end is attached to an SP M contains sugar solution in it. If the SP M is immersed in neat water and the system is left alone to equilibrate, the level of the solution rises in the tube due to osmosis till a height h is finally attained at which the column level in the tube is maintained by a counterbalance of water-flow across the SP M. Under this condition the chemical potential of water on the two sides of the SP M becomes equal 21 . The column pressure, ρgh (ρ being the solution density in appropriate units) makes water to flow out of the tube through SP M while the osmotic pressure tends to maintain the level by an inwards flow. Fig. 2 . 2(A) Setup for osmosis showing a rise of the solution level due to inwards flow of water through the SPM. (B) Setup for the outwards flow of water under the influence of column pressure, ρgl.Now we join together the above two setups so that the sugar solution becomes enclosed in a tube whose both ends are attached to SP M as shown inFig. 3. Since the lower SP M remains immersed in water in L, osmosis continues and tends to maintain the column height h. The inward pressure at the lower SP M is still governed by eq. (3). On the other hand, the upper SP M has been kept hanging at an average height k above the water level in L.An outwards pressure on the upper SP M is an essential requirement for the outflow of water through it; keeping k < h ensures this working condition. A suitable arrangement for the escape of dissolved air or the air that enters into the tube through SP M may be made by providing an "air escape valve" above height h as shown inFig. 3. Water uptake through the lower SP M and release through the upper SP M makes reaching a dynamic equilibrium at which a steady flow of water from L to H is maintained. The neat water collected in H may be made to drain back into L thus completing the flow cycle and providing an opportunity to extract useful work from the system. We have successfully achieved the working of the setup shown inFig. 3at ambient temperature and also repeated the experiment. For this purpose, about 30 % by weight sugar solution was used and a natural SP M was employed. Fig. 3 . 3Setup showing a PMM of second kind, regarded as a Maxwell's demon, which can totally convert absorbed heat into useful work without any involvement of temperature difference.Functioning of the above P MM depends upon various factors related to its design and the working conditions. A steady flow rate, besides other factors, depends upon quality & size of the SP M, concentration of the sugar solution and temperature of the system. A number of factors tend to hinder the performance of the P MM. One important factor is the accumulation of water droplets at the outer surface of the upper SP M. This problem may be minimized by properly designing the shape of the SP M. Another factor is the concentration gradient developed within the solution due to induction of neat water. Diffusion of sugar 4 molecules and a slow rate of flow tend to nullify this gradient. The net effect of both the above factors is a reduction of the flow rate than expected. Fig. 1 1Maxwell's demon capable of separating higher energy water molecules to a container at higher potential. Fig. 2 ( 2A) Setup for osmosis showing a rise of the solution level due to inwards flow of water through the SPM. (B) Setup for the outwards flow of water under the influence of column pressure, ρgl. Fig. 3 3Setup showing a PMM of second kind, regarded as a Maxwell's demon, which can totally convert absorbed heat into useful work without any involvement of temperature difference. . S W Angrist, Scientific American. 2181114S.W. Angrist, Scientific American 218 (1) (1986) 114. W Kauzmann, Thermodynamics and Statistics. NYW.A. Benjamin, IncW. Kauzmann, Thermodynamics and Statistics (W.A. Benjamin, Inc., NY, 1967) pp. 123-8. . P W Atkins, Physical Chemistry. W.H. Freeman and Company4th ed.P.W. Atkins, Physical Chemistry 4th ed. (W.H. Freeman and Company, NY, 1990) pp. 96-8. K Annamalai, I Puri, Advanced Thermodynamic Engineering. Boca Raton, FlCRC PressK. Annamalai and I. Puri, Advanced Thermodynamic Engineering (CRC Press, Boca Raton, Fl, 2001). H C Baeyer, Warmth Disperses and Time Passes. New YorkRandom HouseH.C. von Baeyer, Warmth Disperses and Time Passes (Random House, New York, 1999). . M Brune, S Haroche, V Lefevre, J M Raimond, N Zagury, Phys. Rev. Lett. 65976M. Brune, S. Haroche, V. Lefevre, J.M. Raimond, N. Zagury, Phys. Rev. Lett. 65 (1990) 976. . M Brune, S Haroche, J M Raimond, L Davidovich, N Zagury, Phys. Rev. A. 455193M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, N. Zagury, Phys. Rev. A 45 (1992) 5193. . A Arbab, M S Khan, Zubairy, Phys. Lett. A. 254301Arbab A. Khan, M.S. Zubairy, Phys. Lett. A 254 (1999) 301. . N Imoto, H A Haus, Y Yamamoto, Phys. Rev. A. 322287N. Imoto, H.A. Haus, Y. Yamamoto, Phys. Rev. A 32 (1985) 2287. . J.-F Roch, K Vigneron, Ph, J Grelu, Ph, Poizat, Ph, Grangier, Phys. Rev. Lett. 78634J.-F. Roch, K. Vigneron, Ph. Grelu, J.-Ph. Poizat, Ph. Grangier, Phys. Rev. Lett. 78 (1997) 634. . A Arbab, M S Khan, Zubairy, Fortschr. Phys. 46417Arbab A. Khan, M.S. Zubairy, Fortschr. Phys. 46 (1998) 417. . M O Scully, M S Zubairy, G S Agarwal, H Walther, Science. 299862M.O. Scully, M.S. Zubairy, G.S. Agarwal, H. Walther, Science 299 (2003) 862. M O Scully, M S Zubairy, Quantum Optics. LondonCambridge PressM.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge Press. London, 1997). . O Kocharovskaya, Phys. Rep. 219175O. Kocharovskaya, Phys. Rep. 219 (1992) 175. . E Arimondo, Prog. Opt. 35257E. Arimondo, Prog. Opt. 35 (1996) 257. . S Harris, Phys. Today. 5036S. Harris, Phys. Today 50 (1995) 36. . M Scully, Phys. Rev. Lett. 87220601M. Scully, Phys. Rev. Lett. 87 (2001) 220601. F Reif, Statistical Physics. NYMcGraw-Hill, IncF. Reif, Statistical Physics (McGraw-Hill, Inc., NY, 1967) pp. 350-4. . U Lachish, J. Chem. Educ. 55369U. Lachish, J. Chem. Educ. 55 (1978) 369. D Tabor, Gases, liquids and solids and other states of matter. Cambridge University Press3rd ed.D. Tabor, Gases, liquids and solids and other states of matter, 3rd ed. (Cambridge University Press, 1993) pp. 276-80. . P W Atkins, Physical Chemistry. W.H. Freeman and Company4th ed.P.W. Atkins, Physical Chemistry 4th ed. (W.H. Freeman and Company, NY, 1990) pp. 171-2. J H Van&apos;t Hoff, Zeitsdhrift fur physikalische Chemie 1 (1887). 481J.H. van't Hoff, Zeitsdhrift fur physikalische Chemie 1 (1887) 481.
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[ "Accelerating Kaluza-Klein Universe in Modified theory of gravitation", "Accelerating Kaluza-Klein Universe in Modified theory of gravitation" ]
[ "S D Katore [email protected] \nDepartment of Mathematics\nSant Gadge Baba Amravati University Amravati-444602\nIndia\n", "S P Hatkar [email protected] \nDepartment of Mathematics, A. E. S. Arts, Commerce and Science College\nHingoli-431513India\n", "D P Tadas \nDepartment of Mathematics, Toshniwal Arts, Commerce and Science\nCollege Sengaon-431542India\n" ]
[ "Department of Mathematics\nSant Gadge Baba Amravati University Amravati-444602\nIndia", "Department of Mathematics, A. E. S. Arts, Commerce and Science College\nHingoli-431513India", "Department of Mathematics, Toshniwal Arts, Commerce and Science\nCollege Sengaon-431542India" ]
[]
The purpose of this paper is to study the Kaluza-Klein universe in the context of the f (R, T ) gravity theory using magnetized strange quark matter (MSQM). To obtain exact solutions of field equations, we assume two types of volumetric expansion: power law and exponential law volumetric expansions. The violation of energy conditions has been studied. The physical and geometrical properties of the examined model have also been investigated thoroughly.
10.1007/s10511-023-09773-3
[ "https://export.arxiv.org/pdf/2305.14925v1.pdf" ]
258,139,127
2305.14925
dac2c0b1a4811cf93d180aab1efe3ac778e4f012
Accelerating Kaluza-Klein Universe in Modified theory of gravitation S D Katore [email protected] Department of Mathematics Sant Gadge Baba Amravati University Amravati-444602 India S P Hatkar [email protected] Department of Mathematics, A. E. S. Arts, Commerce and Science College Hingoli-431513India D P Tadas Department of Mathematics, Toshniwal Arts, Commerce and Science College Sengaon-431542India Accelerating Kaluza-Klein Universe in Modified theory of gravitation Kaluza-Klein metricMagnetized Strange Quark MatterPower and Exponential lawf (RT ) gravity The purpose of this paper is to study the Kaluza-Klein universe in the context of the f (R, T ) gravity theory using magnetized strange quark matter (MSQM). To obtain exact solutions of field equations, we assume two types of volumetric expansion: power law and exponential law volumetric expansions. The violation of energy conditions has been studied. The physical and geometrical properties of the examined model have also been investigated thoroughly. Introduction: The study of cosmic accelerated expansion, which has been validated by various observations over the last two decades, is one of the most important cosmological egnima among cosmologists. The concept of accelerated expansion of the universe was first proposed by cosmological studies such as type Ia supernovae [1,2,3,4], and subsequently, measurements of the cosmic microwave background (CMB) from the wilkinson microwave anisotropy probe (WMAP) [5,6] and large-scale structure [7] have confirmed this idea. There are several modified theories proposed that can be found in the literature to explain the accelerating and expanding nature of the universe. Some of theme are f (R) theory [8,9,10], f (T ) theory [11,12,13], f (R, T ) theory [14,15], f (G) theory [16,17,18] and each theory has its own importance. Among the several modified theories of gravitation, the f (R, T ) theory of gravity proposed by Harko et al. [14] is an intriguing extension of general relativity (GR) that has received a lot of attention in recent years. The late time cosmic accelerated expansion of the universe can be explained by the f (R, T ) gravity theory. Houndjo et al. [15] used an auxiliary scalar field with two known forms of scale factor to reconstruct f (R, T ) = f (R) + f (T ) and obtained a transition from a matter-dominated phase to an accelerated phase. Sharif and Zubair [19] have studied the law of thermodynamics in f (R, T ) gravity theory. Using bulk viscous fluid, Chandel et al. [20] studied hypersurface homogeneous cosmological models with time-dependent cosmological terms. In the presence of the perfect fluid source, Reddy and Santhi [21] investigated LRS Bianchi-II space time. The LRS Bianchi-I bulk viscous cosmological models in f (R, T ) gravity have been investigated by Mahanta et al. [22]. Moreover, Harko and Lake [23] have investigated Kasner-type static, cylindrically symmetric interior string solutions using the f (R, L m ) theory of gravity. Pawar and Solanke [24] have explored prefect fluid LRS Bianchi-I cosmological models in f (R, T ) theory of gravity. In the context of f (R, T ) gravity theory, Singh and Singh [25] discussed the behaviour of a flat FRW cosmological model with a scalar field. Mishra et al. [26] have studied perfect fluid Bianchi-VI h space time in f (R, T ) theory of gravity. The Kaluza-Klein theory is a unification of Einstein's theory of gravitation and Maxwell's theory of electromagnetism by introducing compactified extra dimensions. It is regarded as a crucial forerunner to string theory and has received a lot of interest in recent years. Kaluza and Klein explained the role of electromagnetic field in the fremework of a five-dimensional space time. Recently, Mishra et al. [27] compared the Kaluza-Klein dark energy in the Lyra manifold with general relativity using magnetic field. Aktas [28] explored the behaviour of Kaluza-Klein massive and massless scalar field cosmological models with Λ in f (R, T ) gravity theory. Hatkar and Katore [29] have used Polytropic Equation of State in Lyra Geometry to examine the Kaluza-Klein space time. Strange quark matter (SQM) is a new kind of matter made up of many deconfined up (u), down (d), and strange (s) quarks [30,31] and its properties are studied in equilibrium with the weak interactions [32] using the MIT bag model [33]. Furthermore, in the context of the MIT bag model, the thermodynamical properties of SQM were examined in a strong magnetic field with quark confinement by density dependence quark masses, taking total baryon density, charge neutrality, and β-equilibrium at zero temperature into account [34,35,36]. Chakrabarty [37] used the conventional MIT bag model to investigate the effect of a strong magnetic field on the stability and properties of SQM. Singh and Beesham [38] have examined the LRS Bianchi-I cosmological model in f (R, T ) gravity using SQM. Also, Magnetism influences the anisotropies in CMB radiation, which also plays an important role in the formation of structures. Magnetic fields have been observed in the high redshift Lyman system, galaxies, clusters and stars. One of the most interesting areas of research is the relation between magnetic fields and SQM. The behaviour of magnetized strange quark matter (MSQM) for LRS Bianchi-I model has studied in f (R, T ) gravity by Sahoo et al. [39]. Aktas [40] have studied Bianchi I and V models with MSQM distributions in reconstructed f (R, T ) theory of gravity. Aktas and Aygun [41] have investigated FRW space time with MSQM solutions in f (R, T ) theory of gravity. Also, some authors like [42,43,44,45] have studied various aspects of MSQM cosmological models in f (R, T ) theory of gravitation. Recently, Khalafi and Malekolkalami [46] has studied the MSQM Bianchi-I space time for two different models of f (R, T ) thoery. In the present research work, we investigate MSQM distributions for the Kaluza-Klein metric in the context of f (R, T ) gravity theory, which is motivated by the dis-cussion above and a few MSQM studies in the literature. Section 2 is devoted to the metric and f (R, T ) gravity field equations for f (R, T ) = R + 2f 1 (T ). The solutions to the field equation for the power law and exponential volumetric expansion models are derived in sections 3 and 4, respectively. The energy condition and its interpretation of the investigated model are covered in Section 5. Finally, the last section is devoted to the conclusion. 2 Metric and f (R, T ) gravity field equations: In the Kaluza-Klein model, the additional dimension's contribution to the energy momentum tensor is often due to electromagnetic field stresses. We consider the fivedimensional Kaluza-Klein space time of the form as [47] ds 2 = −dt 2 + A 2 dx 2 + dy 2 + dz 2 + B 2 dψ 2 ,(1) where A and B are functions of t only. In the present study, we assume the matter contents described by energy momentum tensor for the MSQM is given in the following form as [39,41]: T ij = ρ + p + h 2 u i u j + h 2 2 − p g ij − h i h j (2) where ρ is energy density, P is pressure, h 2 is magnetic flux and u i = (0, 0, 0, 0, 1) is the velocity vector in comoving coordinate system satisfying the condition u i u i = −1. The magnetic flux h 2 = h i h i is chosen in the direction of x−axis satisfying h i u i = 0 [48, 49]. As the flux quantizes along x−axis, which gives the magnetic field in the yz−plane. The action for f (R, T ) gravity is expressed in the following form [14]: S = √ −g (f (R, T ) + L m ) d 4 x,(3) where f (R, T ) is an arbitrary function of the Ricci scalar R and the trace T of the stressenergy tensor of the matter T ij and L m denotes the matter Lagrangian density. The stress-energy tensor of the matter is defined as T ij = − 2 √ −g (δ √ −g L m ) δg ij (4) Assuming that the Lagrangian density L m of the matter depends only on the metric tensor components g ij and not on its derivatives, we obtain T ij = g ij L m − δ(L m ) δ g ij(5) The f (R, T ) gravity field equations are obtained by varying the action (3) with respect to the metric tensor components g ij , f R (R, T )R ij − 1 2 f (R, T )g ij + g ij ∇ i ∇ i − ∇ i ∇ j f R (R, T ) = 8πT ij −f T (R, T )T ij −f T (R, T )Θ ij (6) where ∇ i is the covariant derivative, Θ ij = −2T ij + g ij L m − 2g αβ ∂ 2 L m δg ij ∂g αβ , f R = ∂ f (R,T ) ∂R and f T = ∂ f (R,T ) ∂T . There are several theoretical models that can be used to represent various matter contributions to f (R, T ) gravity. However, Harko et al. [14] categorised these models into three distinct classes, which are as follows: f (R, T ) =                      R + 2f 1 (T ) f 1 (R) + f 2 (T ) f 1 (R) + f 2 (R)f 3 (T )(7) In order to analyse the exact solutions of the Kaluza-Klein universe, we consider the first model f (R, T ) = R + 2f 1 (T ), where f 1 (T ) is an arbitrary function of the trace of the stress-energy tensor of matter. We choose the arbitrary function f 1 (T ) of trace of the stress energy tensor of matter source given by f 1 (T ) = µT ⇒ f ′ 1 (T ) = µ,(8) where µ is a constant and dash ( ' ) denotes differentiation with respect to the argument. For this choice, the f (R, T ) gravity field equation (6) becomes R ij − 1 2 Rg ij = (8π + 2µ)T ij + (2µP + µT )T ij(9) Using co-moving coordinates, we get four independent field equations of f (R, T ) gravity for the given metric (1) are as follows: 2A 55 A + B 55 B + 2A 5 B 5 AB + A 5 2 A 2 = − (8π − 3µ) h 2 2 − (8π + 4µ) p + µρ(10)2A 55 A + B 55 B + 2A 5 B 5 AB + A 5 2 A 2 = (8π + 7µ) h 2 2 − (8π + 4µ) p + µρ (11) 3A 55 A + 3A 5 2 A 2 = (8π + 7µ) h 2 2 − (8π + 4µ) p + µρ(12)3A 5 B 5 AB + 3A 5 2 A 2 = (24π + 11µ) h 2 2 − 4µρ + (8π + 3µ) ρ(13) where the subscript '5' denotes differentiation with respect to t. The average scale factor a and volume V of universe are defined as V = a 4 = A 3 B(14) Equating equations (11) and (12), we get d dt A 5 A − B 5 B + A 5 A − B 5 B V 5 V = 0(15) on integrating above equation using the (14), the values of metric potentials A and B are as follows A = (c 2 V ) 1/4 exp c 1 4 dt V(16)B = c −3/4 2 V 1/4 exp −3c 1 4 dt V(17) Recently, Katore and Hatkar [47] and Sahoo et al. [50] have investigated some in-teresting results of Kaluza-Klein metric using MSQM distribution. Here, we follow Sahoo et al. [50] and Moraes [51]. Further, we have system of four independent equations (10) to (13) in five unknowns viz. A, B, h 2 , p and ρ. We need one more condition to get the exact solutions of field equations. We consider the two different volumetric expansions such as power law expansion and exponential law expansion. 3 Power law expansion model: Firstly, we consider the power law volumetric expansion as V = A 3 B = t 4m(18) where m is a positive constant. Using the equation (18), values of A and B are obtained as A = c 1/4 2 t m exp c 1 t 1−4m 4(1 − 4m)(19)B = c −3/4 2 t m exp −3c 1 t 1−4m 4(1 − 4m)(20) The metric potentials A and B both vanish at time t = 0, they start to increase with time and finally diverge to infinity as t → ∞. This is consistent with the big bang model. Now, the various cosmological parameters such as mean Hubble parameter H, Expansion scalar θ, Shear scalar σ obtained as follows: H = 1 4 3A 5 A + B 5 B = m t (21) θ = 3A 5 A + B 5 B = 4m t(22)σ 2 = 1 3 3A 5 A − B 5 B 2 = 3c 2 1 8t 8m(23) From the expression (21), (22) and (23), it is observed that the Hubble parameter H, exapansin scalar θ and shear scalar σ are decreasing function of time. They diverges to infinity as t → 0 and becomes zero at infinity. Moreover, as t → ∞, the ratio σ θ → 0. Hence, the investigated model approaches isotropy. Also, the deceleration parameter q is found to be q = d dt 1 H − 1 = −1 + 1 m(24) The sign of q indicates whether the universe is accelerating or decelerating. A positive sign of q implies a decelerating model, whereas a negative sign of q indicates a accelerating model. According to recent cosmic studies, the expansion of the universe is rapidly accelerating. From equation (24), it is clear that, the deceleration parameter q is negative for m > 1, thus the universe is accelerating. Now, the expression of pressure P , energy density ρ and magnetic flux h 2 for power law volumetric expansion model are obtained as follows: which is consistent with the observational value of EoS (ω) of DE obtained by [55,56]. P = − 3 2 c 1 2 (µ + 2π)t 2−8m + 3m(µ(4m − 3) + 8π(2m − 1)) 8 (8π 2 + 7πµ + 2µ 2 ) t 2 (25) ρ = 3 −c 2 1 πt 2−8m + 16(µ + π)m 2 − 4µm 8 (8π 2 + 7πµ + 2µ 2 ) t 2(26)h 2 = 0(27) Exponential volumetric expansion model: For exponential volumetric expansion law, we assume the volume factor as V = e 4nt(28) where n is constant. For this model, the values of A and B using the equation (28) are obtained as A = c 1/4 2 exp nt − c 1 e −4nt 16n(29)B = c −3/4 2 exp nt + 3c 1 e −4nt 16n(30) A and B are constant at t = 0, therefore the model has no singularity at t = 0. Also, as t → ∞, both A and B tends to infinity. Now, the various cosmological parameters such as mean Hubble parameter H, Expansion scalar θ and Shear scalar σ are found to be H = n(31)θ = 4n(32)σ 2 = 3 2 c 2 1 e 8nt(33) From the equations (31), (32) and (33), it is observed that the Hubble parameter (H), exapansin scalar (θ) are constant i.e. the rate of expansion of the universe is constant. Shear scalar (σ ) is decreasing function of time. Therefore, σ θ → 0 as t → ∞ i.e. the model approach to isotropy. Also, the deceleration parameter q is obtain as q = −1(34) The sign of deceleration parameter is negative, therefore the universe is accelerating. The quantities such as the pressure P , energy density ρ and magnetic flux h 2 for exponential volumetric expansion are obtained as P = − 3 2 c 2 1 (2π + µ)e −8nt + 12(4π + µ)n 2 8 (8π 2 + 7πµ + 2µ 2 )(35) ρ = e −8nt 48(π + µ)n 2 e 8nt − 3πc 2 1 8 (8π 2 + 7πµ + 2µ 2 ) (36) in figure 4 and 5. It is observe that, the energy density ρ is decreasing function time t. Initially energy density was constant near t = 0 and slightly increased to its maximum value and remain constant with increasing time. The pressure is negative which reveals that the matter behave like dark energy. h 2 = 0 (37) In this case, the figure 6 represents the behaviour of p/ρ with time t. We observed that, for different value of n, it begins to evolve in the phantom region and suddenly enters into the quintessence region with increasing time t. Recently, Scolnic et al. [57] combine the Planck 2015 CMB and SNe Ia measurements to calculate the best fit value for the EoS parameter ω = −1.026 ± 0.041. We observe that the value of p/ρ is in the range −1.162 ≤ ω ≤ −0.983, which is consistent with the observed vale of EoS of DE obtained by [58,57]. Energy Conditions: The energy conditions are essential for understanding the geometry of universe. The Raychaudhuri's equation is used to obtain the energy conditions, which are a set of linear pressure-density combinations that decribe the energy density can never be negative and gravity attracts always. These energy conditions are defined and stated as follows: • SEC ⇒ ρ + 3P ≥ 0, • NEC ⇒ ρ + P ≥ 0, • WEC ⇒ ρ ≥ 0, ρ + P ≥ 0 and • DEC ⇒ ρ > |P | ≥ 0. Therefore, for first power law volumetric expansion model, the expression for NEC, SEC and DEC are obtained by using the equations (25) and (26) as ρ + P = t −8m−2 6m(µ(12m − 1) + 8π)t 8m − 3c 2 1 (4π + µ)t 2 16 (8π 2 + 7πµ + 2µ 2 ) (38) ρ + 3P = − 3t −8m−2 c 2 1 (3µ + 8π)t 2 + 2m(8π(4m − 3) − µ(4m + 5))t 8m 16 (8π 2 + 7πµ + 2µ 2 ) (39) ρ − P = 3 c 2 1 µt 2−8m + 2m(µ(20m − 7) + 8π(4m − 1)) 16 (8π 2 + 7πµ + 2µ 2 ) t 2(40) Also, for Exponential volumetric expansion model, using equations (35) and (36) [50] has studied the energy condition and shows that only WEC and DEC satisfies for volumetric expansion model in f (R, T ) gravity. Also, Alvarenga et al. [52] shown that energy conditions are satisfied for suitable choice of inpute parameter in f (R, T ) gravity theory. Recently, Sahoo et al. [53] has observed that, NEC and DEC are satisfy but SEC is violated. It is important to note that, our invstigation are more relevant than [50,52] and resemble with the investigation of [53], because both models satisfies all energy condtions except SEC. 6 Conclusions: In this paper, we have studied Kaluza-Klein cosmological models with magnetized strange quark matter (MSQM) in f (R, T ) theory of gravity using power law and exponential volumetric expansions. • In power law expansion model, the shear and expansion scalars, as well as the Hubble parameter H, decrease over time. The rate of expansion of the universe is very high near t = 0. They diverge to infinity at t = 0 and vanish at infinite time. The energy density begins to decrease from a positive value, the pressure starts to increase from a negative value, and both approach zero at infinite time. It has been observed that the universe accelerates when m > 1. • In the exponential volumetric expansion model, it is observed that the pressure and the energy density of the universe gradually increase to their maximums and then remain constant with increasing time. For n > 0, the universe's expansion rate remains constant. Also, the negative value of the deceleration parameter q indicates that the universe is accelerating, which is consistent with recent findings. • It is important to note that, for both models, the negative pressure indicates that the matter behaves like dark energy, which causes the accelerated expansion of the universe. Moreover, the trajectory of p/ρ is lie in the range −1.38 ≤ ω ≤ −0.89 in the case first and −1.162 ≤ ω ≤ −0.983 in the second. The bounds of EoS parameter obtained by [55] is ω = −1.013 +0.068 −0.073 and [57] is ω = −1.026 ± 0.041. It seems that the obtained values of p/ρ in both cases are close to the observational bounds. Further, Aditya et al. [59] obtained the bounds of the EoS parameter in Lyra geometry as −1.3 ≤ ω ≤ −0.8. In Saez-Ballester theory, the bounds of the EoS parameter for the Kaluza-Klein model is −1.6 ≤ ω ≤ 0.2 [60]. When the values of the EoS parameter in scalar-tensor theories for the Kaluza-Klein model are compared to the range of p/ρ in the present model, the resultant value of p/ρ in f (R, T ) gravity is found to be more appropriate than that obtained for the EoS parameter in scalar-tensor theories [59,60]. Also, the magnetic flux vanishes for both models and It's worth noting that our findings are consistent with those of [40,41,46,54]. Moreover, both the power law and exponential law models satisfy the NEC and DEC, but the SEC is violated throughout the evolution. Figure 1 : 1Plot of pressure (P ) vs time t for c 1 = 1, µ = 0.1. Figure 2 : 2Plot of density (ρ) vs time t for c 1 = 1, µ = 0.1. The behaviour of pressure P and energy density ρ for the power law model is graphically depicted in figure 1 and 2 respectively. It is observe that the pressure is an increasing function of time t; it is very small near t = 0 and vanishes at infinite time t. It is worth noting that the pressure is negative for different values of parameter m, indicating that the SQM behaves like dark energy. Also, the energy density decreases as time t increases and it is infinte at t = 0 and vanishes as t → ∞. For small values of m, the density curve slowly approaches zero with increasing time. As parameter m becomes larger, the curve tends to a constant value. In the power law model, the graph of p/ρ shows the dynamical behaviour with increasing time t as depicted in figure 3. It is observed that, initially, it evolves in the phantom region p/ρ < −1, crosses the phantom divide line p/ρ = −1, suddenly enters into the quintessence region −1 < p/ρ < −1/3 and remains constants with increasing time. Suzuki et al. [55] combined all four probes, such as SNe Ia with BAO, CMB, and H 0 measurements, to determine the equation of state (EoS) parameter of dark energy (DE), which is ω = −1.013 +0.068 −0.073 for a flat wCDM model. In the present case, for m > 1, the value of p/ρ is in the range −1.38 ≤ ω ≤ −0.89, Figure 3 : 3The plot of ratio p/ρ vs time t for c 1 = 1, µ = 0.1. Figure 4 :Figure 5 : 45Plot of pressure (P ) vs time t for c 1 = 1, µ = 0.1.The graphical behaviour of pressure (P ) and energy density (ρ) vs time t is depicted Plot of density (ρ) vs time t for c 1 = 1, µ = 0.1. Figure 6 : 6The plot of ratio p/ρ vs time t for c 1 = 1, µ = 0.1. Figure 7 :Figure 8 : 78Plot of NEC, SEC and DEC vs time t for m = 2, c 1 = 1, µ = 0.1. Plot of NEC, SEC and DEC vs time t for n = 1, c 1 = 1, µ = 0.6.Figures 7 and 8 show the graphical representation of NEC, SEC and DEC vs time t for power law and exponential law, respectively. 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[]
[ "GNN-Geo: A Graph Neural Network-based Fine-grained IP geolocation Framework", "GNN-Geo: A Graph Neural Network-based Fine-grained IP geolocation Framework" ]
[ "Shichang Ding ", "Xiangyang Luo ", "Member, IEEEJinwei Wang ", "Fellow, IEEEXiaoming Fu " ]
[]
[ "IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING" ]
Rule-based fine-grained IP geolocation methods are hard to generalize in computer networks which do not follow hypothetical rules. Recently, deep learning methods, like multi-layer perceptron (MLP), are tried to increase generalization capabilities. However, MLP is not so suitable for graph-structured data like networks. MLP treats IP addresses as isolated instances and ignores the connection information, which limits geolocation accuracy. In this work, we research how to increase the generalization capability with an emerging graph deep learning method -Graph Neural Network (GNN). First, IP geolocation is re-formulated as an attributed graph node regression problem. Then, we propose a GNN-based IP geolocation framework named GNN-Geo. GNN-Geo consists of a preprocessor, an encoder, messaging passing (MP) layers and a decoder. The preprocessor and encoder transform measurement data into the initial node embeddings. MP layers refine the initial node embeddings by modeling the connection information. The decoder maps the refined embeddings to nodes' locations and relieves the convergence problem by considering prior knowledge. The experiments in 8 real-world IPv4/IPv6 networks in North America, Europe and Asia show the proposed GNN-Geo clearly outperforms the state-of-art rule-based and learning-based baselines. This work verifies the great potential of GNN for fine-grained IP geolocation.
10.1109/tnse.2023.3266752
[ "https://export.arxiv.org/pdf/2112.10767v7.pdf" ]
248,965,509
2112.10767
3ecc6aa74fb52941812936f70e775921c5359728
GNN-Geo: A Graph Neural Network-based Fine-grained IP geolocation Framework Shichang Ding Xiangyang Luo Member, IEEEJinwei Wang Fellow, IEEEXiaoming Fu GNN-Geo: A Graph Neural Network-based Fine-grained IP geolocation Framework IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING 110.1109/TNSE.2023.3266752Index Terms-Fine-grained IP geolocationGraph Neural NetworkComputer NetworkDeep Learning Rule-based fine-grained IP geolocation methods are hard to generalize in computer networks which do not follow hypothetical rules. Recently, deep learning methods, like multi-layer perceptron (MLP), are tried to increase generalization capabilities. However, MLP is not so suitable for graph-structured data like networks. MLP treats IP addresses as isolated instances and ignores the connection information, which limits geolocation accuracy. In this work, we research how to increase the generalization capability with an emerging graph deep learning method -Graph Neural Network (GNN). First, IP geolocation is re-formulated as an attributed graph node regression problem. Then, we propose a GNN-based IP geolocation framework named GNN-Geo. GNN-Geo consists of a preprocessor, an encoder, messaging passing (MP) layers and a decoder. The preprocessor and encoder transform measurement data into the initial node embeddings. MP layers refine the initial node embeddings by modeling the connection information. The decoder maps the refined embeddings to nodes' locations and relieves the convergence problem by considering prior knowledge. The experiments in 8 real-world IPv4/IPv6 networks in North America, Europe and Asia show the proposed GNN-Geo clearly outperforms the state-of-art rule-based and learning-based baselines. This work verifies the great potential of GNN for fine-grained IP geolocation. Measurement-based IP geolocation can geolocate a target IP after the probing host [2] obtains the measurement data (e.g., delay) to the target IP and landmarks [2,5]. Landmarks are IP addresses with known locations [2]. Generally, existing measurement-based fine-grained IP geolocation methods can be categorized into two kinds: rule-based and deep learning-based methods (referred as learning-based methods in this paper). For rule-based methods like SLG [5] and Corr-SLG [2], they assume that most hosts in one computer network are following some delay-distance rules. These hypothetical rules are proposed by human experts based on their observations. Then a target IP can be mapped to its geolocation based on the delay-distance rules. The performances of rule-based methods rely on the partition of hosts which follow the hypothetical rules. This limits the generalization capabilities of rule-based methods in networks where hosts don't follow their hypothetical rules. Recently, deep learning seems to be a hopeful solution to the generalization problem of IP geolocation methods. There are two possible advantages of learning-based IP geolocation. First, learning-based methods do not rely on hypothetical rules which are suitable for all computer networks. They can learn a "measurement data -locations" relationship suitable to a specific computer network based on this network's raw measurement data. Second, learning-based methods are capable of extracting nonlinear relationships, while non-linear rules are hard for human experts to observe. Encouraged by these possible advantages, researchers proposed several IP geolocation methods based on MLP (multi-layer perceptron), such as NN-Geo [8] and MLP-Geo [7]. Compared with NN-Geo, MLP-Geo is more accurate because it leverages not only delay but also routing information. However, it is hard for MLP to model the connection information of computer networks. MLP is not good at modeling non-Euclidean structured data. Nevertheless, computer networks are fundamentally represented as graphs, a typical non-Euclidean structured data [9,10]. MLP can only treat IP addresses as isolated arXiv:2112.10767v7 [cs.LG] 15 Apr 2023 instances, though hosts are non-independent and linked to each other through intermediate routers. This is why MLP-Geo can only leverage the router IDs (and the delay) between the probing host and the targets, ignoring all the other useful features, such as the links and delays between routers. Based on the experiments (shown in Table 1), MLP-Geo could be less accurate than rulebased methods in some actual networks. This indicates that IP geolocation needs a deep learning method more suitable for graph-structured data. Graph Neural Network (GNN) is an emerging deep learning method for graph-structured data presentation [11]. Recently, it has been successfully applied in different fields of graphstructured data, like recommendation system [12], social network [13], knowledge map [14], etc. However, little attention has been paid to introducing GNN into IP geolocation. As far as we know, this is the first work to explore the potential of GNN in finegrained IP geolocation. The main challenges of introducing GNN into IP geolocation tasks are at least three-folds. 1) How to formulate IP geolocation problem in the context of graph deep learning? 2) How to design a GNN-based Framework for the finegrained measurement-based IP geolocation task? 3) Which factors would clearly affect the performance of GNN in fine-grained IP geolocation performance? By tackling these challenges, the main contributions of our work can be summarized as follows. 1) Formulating IP geolocation as a graph learning task. We formulate the measurement-based fine-grained IP geolocation task as a semi-supervised attributed-graph node regression problem. First, IP addresses are mapped into graph nodes, and links are mapped into graph edges. Then, the features of IP addresses (e.g., delay) and links (e.g., the delay of link) are transformed into attributes of the graph nodes and edges. Finally, the IP geolocation problem is formulated as estimating the latitudes and longitudes of the target nodes based on the known locations of other nodes as well as the topology and attributes of the graph. This paves the road of introducing graph deep learning methods into IP geolocation. 2) GNN-Geo framework. We propose a GNN-based framework (GNN-Geo) to solve the node regression problem. It consists of four components: a preprocessor, an encoder, MP (message passing) layers, and a decoder. The preprocessor transforms the raw traceroute data of a computer network into an attributed-graph representation. The encoder generates the initial graph node/edge embeddings for MP layers. MP layers refine the node embeddings by propagating messages along the edges. Finally, the latitudes and longitudes of each node are decoded from the refined node embeddings. 3) Improving MLP-based decoder with rules. We highlight the limitation of the vanilla MLP-based decoder, which makes GNN model hard to converge. The convergence problem is relieved with an idea like "rule-based" methods: making GNN-Geo to find targets' fine-grained locations inside a possible coarse-grained area instead of the whole earth. Experiments show that this significantly reduces the error distances and training epochs of GNN-Geo. Take Hong Kong as an example, compared with the vanilla MLP-based decoder, the improved decoder reduces the error distance from more than 2,690 km to less than 8 km. 4) Large-scale experiment results outperform the best baselines. The experiments are carried out in 8 realworld IPv4/IPv6 networks including New York State, Hong Kong, Shanghai, Beijing, Tokyo, Tokyo (IPv6), Berlin (IPv6) and Munich (IPv6). The results show: (i) when the training dataset ratio is 70%, GNN-Geo outperforms the best baselines averagely by more than 21% w.r.t. both average error distance and median error distance; (ii) When training dataset ratio decrease from 70% to 10%, though its advantages are reduced on sparser dataset, GNN still outperform baselines in most cases. We also analyze how different factors (e.g., basic GNN models, GNN aggregation methods) would affect GNN-Geo's performance. These results validate the value of introducing GNN in fine-grained IP geolocation. The rest of the paper is organized as follows. Related works are introduced in Section 2. In Section 3, IP geolocation is formulated into a graph node regression problem. Then in Section 4, we discuss why GNN is worth to be tried in IP geolocation. In Section 5, we introduce the detail of proposed GNN-Geo. Section 6 shows and analyzes the experiment results in three real-world datasets. Section 7 discusses several possible future improvements for GNN-Geo. Finally, the paper is concluded in Section 8. RELATED WORK In this section, we introduce the works related to our study, mainly including fine-grained IP geolocation and Graph Neural Network. Fine-grained IP geolocation Besides measurement-based methods, there are two other kinds of fine-grained IP geolocation methods: database and data-miningbased method. IP geolocation databases can only provide the fine-grained locations for a small proportion of IP addresses, which could hardly satisfy people's needs [15]. Data-mining-based methods, like Checkin-Geo [6], can map IP addresses to locations based on a large amount of raw data which contains "IP-location" information such as users' logs. However, these kinds of raw data resources are only possessed by few internet giants like Microsoft Bing [4,15] and Tencent [6], which are not open to the public. For example, Checkin-Geo is a data-mining-based fine-grained method. First, it extracts users' names and their locations from the checkin [6] information collected on smartphone applications. Then it extracts users' names and users' IP addresses from the logins collected on PC applications. In the end, Checkin-Geo can map a user's PC IP address to his/her smartphone location. Both the PC and the smartphone applications are owned by a famous Chinese online social platform called "Tencent QQ". Recently, researchers from Microsoft build a large-scale IP-location groundtruth dataset, which is mined from the query logs of a commercial search engine. Based on the large-scale dataset, they can use machine learning methods to estimate the location of IP addresses from their reverse DNS hostnames [4]. We can see, these high quality large-scale ground truth data resources are hard to get for researchers outside Tencent or Microsoft. Besides this problem, these methods cannot work well in areas where those Internet giants are not serving, or the location-sharing service is forbidden. For example, Tencent QQ mainly serves in China and has been banned in India 1 . The limitations of database and data-mining-based methods require researchers to pay attention to measurement-based methods. In theory, measurement-based methods can geolocate any target IP if the delay or routing data between the target IP address and landmarks can be obtained. However, it is hard for measurementbased methods to get a fine-grained geolocation result. Because the relationship between measurement data and target IP's location is very complicated and different in various computer network environments. SLG is the first fine-grained measurement-based IP geolocation method [5]. It is also the first rule-based method as well as one of the most widely-used fine-grained IP geolocation baselines until now. It assumes that most hosts in a computer network are following a simple linear delay-distance rule: the shortest "relative delay" comes from the nearest landmark. Since the delay between a landmark and a target IP is hard to measure, "relative delay" is proposed by SLG as an approximation. Assume the delay from the probing host to a target IP is d pt , the delay from the probing host to a landmark is d pl , and the delay from the probing host to the closest common router [2,5] shared by the target IP and landmark is d pr , the "relative delay" between the target and the landmark is (d pt −d pr )+(d pl −d pr ). This rule originated from an observation of the relationship between relative delay and distance of 13 landmarks in New York City [5]. However, in further studies, researchers realized that this rule may be not valid for all intra-city networks [2]. Some researchers find that there is no clear relationship between relative delay and distance in cities like Zhengzhou (China) and Toronto (Canada) [2]. They explain that for the intra-city delay, the delay caused by geographical distance are not always the major constituent. The other factors like queuing delay and processing delay in routers could be more important. Thus the relationship between relative delay and distance is very complicated. In Corr-SLG, the relative-delay-distance correlation of all landmarks is calculated at first. Then all landmarks are divided into three groups. Corr-SLG leverages different delaydistance rules in each group. In the groups with a strong-positive relative-delay-distance correlation (close to 1), the relative delay increases as the distance increases. Thus the rule is similar to SLG, and Corr-SLG still map targets to the landmarks with the shortest relative delay. In the groups with a strong-negative correlation, they map targets to the landmarks with the largest relative delay (close to -1). However, Corr-SLG cannot decide how to handle the groups with weak correlation (close to 0). They simply map these targets to the average locations of landmarks. We can see the accuracy of the rule-based methods relies on the partition of hosts which follows their pre-assumed rules. Since the network environments may change in each city and at different times, the relationships between delay and distance may vary significantly. Thus, the performances of rule-based methods vary significantly. How to design a fine-grained method that can well generalize in different networks is an important problem. In recent years, deep learning has shown clear advantages in various networking problems [16][17][18][19][20][21]. People also begin to try deep learning methods in IP geolocation to improve the generalization capabilities. Both NN-Geo [8] and MLP-Geo [7] 1. www.xda-developers.com/india-bans-xiaomi-mi-browser-pro-qqinternational-app/ use MLP to estimate locations for target IP addresses. Compared with NN-Geo, MLP-Geo adds a new kind of useful information -the routers' IDs between probing hosts and the targets. Its performance is clearly better than NN-Geo since NN-Geo only utilizes the delay between probing hosts and the target IP. The aim of learning-based methods is not to find a rule which is suitable for all networks. Instead, they aim to present a method which can correctly find "rules" in different networks. Whether a method can find rules correctly mainly relies on its modeling ability towards the target networks. Actually, MLP is not so suitable for modeling the measurement data of computer networks. MLP is good at prepossessing Euclidean structure data. However, the topology of the internet is a non-Euclidean structure data [22]. MLP can only treat target IP addresses as isolated data instances while ignoring the connection information between targets. This would easily lead to suboptimal representations and performance impairment. We still need to improve the learning-based methods by utilizing deep learning methods more suitable to networks. Graph Neural Network (GNN) Graph Neural Network (GNN) is an emerging deep learning method, which is specially designed for graph-structured data [11,23,24]. GNN learns the representation of a graph node by recursively aggregating the information of its neighbor nodes. Then, the learned node representations can be used for node-level, edge-level or graph-level classification/regression tasks. There are several typical GNN architectures such as GCN (Convolutional Neural Network) [11], GAT (Graph Attention Network) [23], MPNN (Message Passing Neural Network) [24], etc. Most existing GNN-based methods can be seen as their variants or improved version. GCN is inspired by CNN's success in image processing area [25]. GCN uses multiple graph convolutional layers to aggregate information from neighbor nodes. GAT introduces attention mechanisms into information aggregation process. GCN usually aggregates the normalized sum of neighbors' information while GAT assigns attention weights to neighbors to get a weighted sum of neighbors' information. This can help GAT to differentiate the contributions of different neighbors and gain better performance. Both GCN and GAT mainly focus on aggregating node feature and ignore edge feature [26]. However, edge feature is crucial for IP geolocation tasks. For example, the delay or hop count between IP addresses have been verified as very important features in IP geolocation in previous methods [2,5]. Compared with GCN and GAT, MPNN is good at processing different kinds of edge features (including categorical and continuous). Thus in this paper, we use MPNN as the basic GNN model for the GNN-based IP geolocation framework. Due to its powerful spatial modelling ability, GNN has achieved impressive performances in various graph-type data areas, such as social network [27], knowledge map [28] and recommendation system [29]. Especially, there is also a kind of geolocation task in social network -user geolocation. User geolocation (UG) is to identify the geographic location of online social network (OSN) users, such as Twitter users [27]. Several works such as [27,[30][31][32] have shown GNN's advantages in user geolocation. For example, [27] presents Hierarchical Graph Neural Networks (HGNN) which leverages robust signals from geographically close crowds rather than individuals. Through exploiting both unlabeled nodes and isolated nodes, HGNN achieves better performance than traditional learning-based methods. Though computer network and OSN are both represented as graph-structured data, GNN-based UG methods cannot be simply transferred to IP geolocation. First, OSNs like Twitter usually contain many kinds of node features like user posts, location tags, pictures while node features in IP geolocation are quite limited, mainly relies on network measurement data (such as delay). Second, the links between OSN users usually do not contain crucial features like delay or hop count in IP geolocation. Thus, existing GNN-based UG methods usually ignore edge features and focus on exploiting rich node features. These GNN methods may lose important information when transferred to IP geolocation tasks. GNN-based IP geolocation methods should pay attention to both node and edge features. Besides, the accuracy of existing UG methods is mainly around city-level while existing fine-grained IP geolocation methods have already achieved street-level. The newly proposed GNN-based fine-grained IP geolocation methods should also achieve street-level. Recently, researchers also begin to try GNN in different problems of computer networks [22]. For example, network modeling is used to reconstruct the computer networks and predict the structure/topology of unseen computer networks. GNN has been introduced into network modelling in [33,34]. [35,36] discuss how to utilize GNN in network calculus analysis. GNN is also helpful in link delay prediction [37] and network traffic prediction [38][39][40]. GNN has been introduced into automatic detection for Botnets, which is important to prevent DDoS attacks [41]. However, most previous GNN-based works in computer networks are focusing graph-level or edge-level classification or regression tasks. Little attention has been paid to how to model the relationship between location signals from network measurement data for IP geolocation (node regression/classification) until now. IP GEOLOCATION FORMULATION To explore the potential of GNN in fine-grained IP geolocation, we need to formulate IP geolocation as a graph-based learning problem at first. Before introducing the formulation, we want to clarify several issues here. Since there have been many mature coarsegrained IP geolocation methods, in this paper, we assume the coarse-grained locations of targets have been obtained before finegrained geolocation. Thus, when mentioning a computer network in this work, it usually means a city-level network or at maximum state-level network. We also suggest researchers to select a most suitable coarse-grained method in each area. Since coarse-grained methods' performances may often change in different areas, and more accurate coarse-grained location results usually help a lot for following fine-grained IP geolocation activities. In this paper, the IP addresses IP i , i = 1, 2, ..., N IP and the links LK j , j = 1, 2, ..., N LK between IP addresses are used to represent the topology of a computer network. The link means a direct physical link between two IP addresses. The location of each IP address is represented as a pair of (latitude, longitude). Each IP address is associated with measured attributes, such as the delay from the probing host. Each link is also associated with measured attributes, such as the delay of the link. All IP addresses can be divided into four groups: the probing host IP address, the landmark IP addresses, the target IP addresses and the router IP addresses. The router IP addresses are the intermediate router IP addresses found by the probing host when tracerouting the landmarks and targets. As shown in Fig. 1, we map a computer network topology to an attributed undirected graph, G = (V, E, XV, XE). In the {loc vi , v i ∈ {V p , V l }}. The locations of targets are {loc vi , v i ∈ V t }. Herein, the measurement-based IP geolocation problem is formulated to learn a prediction function f (·), which can estimate the targets' location: [{loc vi , v i ∈ {V p , V l }}; G] f (·) → { loc vi , v i ∈ V t },(1) where the outputs are targets' estimated locations { loc vi , v i ∈ V t }, and the inputs are (i) the attributed undirected graph G = (V, E, XV, XE); (ii) the known locations of the probing host and landmarks {loc vi , v i ∈ {V p , V l }}. In this paper, we use the numerical values of (latitude, longitude) to represent the location of each node, thus this is an attributed graph node regression problem. If the location is categorical, like communities or streets, the IP geolocation problem can also be formulated into a node classification problem. Note that researchers usually do not have the location information of intermediate routers V r , thus this is also a typical semi-supervised problem. It is worth reporting here, it is also easy to estimate the location of intermediate routers by formulating IP geolocation into a graph node regression problem. Because the graph representations of router nodes can be naturally learned along with target nodes. THE POSSIBLE VALUE OF GNN FOR IP GEOLOCATION Generally speaking, the two most important phases in learningbased IP geolocation methods are: (i) target embedding and (ii) target location mapping. From previous methods like [2,5,7], we know raw network measurement data contains useful signals which can help in IP geolocation. The target embedding phase is responsible to extract all useful signals from raw network measurement data, and then compress them into vectorized representations of target IP addresses (i.e., targets' embeddings). Then the target location mapping phase is responsible to map targets' embeddings to targets' locations. The main problem of MLP-Geo is that its target embedding phase loses many useful signals from measurement data. MLP-Geo consists of three dense layers [7]. Here we assume its target embedding phase includes the first two layers. And its target location mapping phase is the last layer. Then the targets' embeddings E mlp 2 of MLP-Geo are generated as follows: E mlp 2 = f 2 (f 1 (Input)), (2) f i = σ(W i X i + b i ),(3) where Input is the input features of MLP-Geo, f i denotes the i-th-dense-layer of MLP-Geo, W i is the weight of the i-th-denselayer, and b i is the bias of the i-th-dense-layer. X i is the input of i-th-dense-layer and the output of the (i − 1)-th-dense-layer. Input is X 0 . In MLP-Geo, Input consists of the delay and router IDs from the probing host to the targets. From Formula 3, we can see the dense layer f i uses matrix multiplication to extract useful signals from the input data. In this way, it can only judge whether a router ID shows in the path between the probing host and a target. MLP-Geo cannot extract the sequence relationship of the routers between the probing host and targets. It also cannot describe the topology around an IP addresses (or the whole network). And it cannot leverage the delay between two directly-linked IP addresses, which have been proved useful by the rule-based methods like SLG. It is hard for MLP-Geo to extract these useful signals because it is not designed for non-Euclidean structure data like graph. The core concept of GNN (or more precisely, MPNN in this paper) is message passing [24]. As introduced in last section, all IP addresses are transformed into graph nodes. The target embedding phase of GNN is as follows: E gnn i = f i (. . . (f 2 (f 1 (E 0 )), (4) f i = {f message , f aggregate , f update },(5)M i = f message (E neighbor i−1 , E link ),(6)M S i = f aggregate (M i ),(7)E gnn i = f update (M S i , E gnn i−1 ),(8) where f i indicates the i-th-MP-layer, E gnn i is the node embeddings of the i-th-MP-layer, E 0 denotes the input (initial node embeddings). We can see both GNN and MLP learn new embeddings for IP addresses over multiple layers. GNN is a general framework. One MP layer f i consists of three functions: the message function f message , the aggregate function f aggregate and the update function f update . For a node, the message function f message passes its neighbor node's embedding (E neighbor i−1 ) along the edge/link (E link ) to it. A neighbor node means there is one edge between these two nodes. Then the aggregate function f aggregate collects the aggregation of all its neighbor nodes' messages (M S i ). The update function f update forms the node's new embedding E gnn i based on the aggregated messages M S i and the node's previous embeddings E gnn i−1 . From Formula 4-8, we can see the embedding of each graph node also aggregates with neighbor nodes' embeddings and its edge embeddings at the beginning of each MP layer. And in i-th-MP-layer, a node can get information from its i-hop neighbors. For example, in the 2nd-MP-layer, a node not only gets messages from its 1-hop neighbors but also gets information from 2-hop neighbors. Because the messages from its 2-hop neighbors are passed to its 1-hop neighbors in 1st-MP-layer. Through passing messages along the edges, after several MP layers, every node will gradually know the topology and node/edge attributes of the computer network. So compared with MLP, GNN can naturally describe the network topology in the targets' embeddings. And the other features like delays of links and IP addresses can also be utilized, because they can be easily combined into message function f message , or the embeddings of links E link . Thus, in some networks, if their measurement data (such as network topology, delays between routers and IP addresses) contain useful information for IP geolocation, and also is enough for GNN to learn (deep learning methods would overfit if dataset is too sparse), then GNN may show better performance in these networks. Next, to test how better GNN could be in real IP geolocation tasks, we design a GNN-based IP geolocation framework and carry out experiments to measure its performance in various real networks. OUR PROPOSED GNN-GEO FRAMEWORK We present the GNN-Geo framework in this section. Fig. 2 illustrates the framework, which consists of four components: a preprocessor, an encoder, MP layers, and a decoder. The preprocessor maps the raw measurement data of a computer network into its graph representation G = (V, E, XV, XE). The encoder generates the initial feature embeddings of G. The MP layers refine the initial feature embeddings with graph signals and output the refined embeddings for all nodes V . Finally, the decoder maps the refined node embeddings into the locations. We will describe each component one by one, followed by the optimization details. It is worth mentioning that by changing some details of each component, we can get different GNN-based IP geolocation methods from the GNN-Geo framework. The Preprocessor In this paper, the measurement data is mainly traceroute data from the probing host to landmarks and targets. The task of the preprocessor is transforming the raw traceroute data into the initial embeddings of G = (V, E, XV, XE) for the encoder. This mainly consists of two sub-tasks: (i) building the Graph G with the graph nodes V and the links E; (ii) extracting the node attributes XV and the link attributes XE. Building Graph for the Measured Computer Network Node. As mentioned in Section 3, we use the IP addresses and the links between IP addresses to represent the topology of a computer network. Thus all IP addresses in the raw traceroute data are transformed into the graph nodes. Each node v i is differentiated by a node ID. The ID is from 1 to N V . N V is the number of all IP addresses found in traceroute data. Edge. If the probing host finds a direct physical link between two IP addresses, then the link is transformed into a graph edge e j . During the traceroute, it is possible that the probing host cannot get the IP addresses and delays of some routers, which are called "anonymous routers". There are two kinds of methods to build an edge over an "anonymous router": (i) ignore the anonymous router and build an edge between the two routers before and after the anonymous router along the routing path; (ii) map the anonymous routers to an IP address found in other re-measured routing paths. Both solutions are tested in our early experiments. The difference in their final geolocation performances is not significant. This may be due to that GNN itself is a powerful feature extractor, which is not sensitive to such differences in the input graph. However, the first solution introduces more edges in G, which leads to higher computation complexity and greater memory needs in the optimization phase. Thus we leverage the second solution in this paper, which is introduced as follows. After tracerouting one IP address multiple times, we may get We assume that if two routing paths are the same as each other except for the anonymous routers, then the two paths are likely to be similar. Then we can map the anonymous routers in one path to the IP addresses in other similar paths. Our matching algorithm is shown in Algorithm 1. The raw path set refers to all the raw paths found by the traceroute. The completed path set refers to the paths of which the anonymous routers have been mapped to IP as much as possible. After using Algorithm 1, we can build edges based on the completed path set. We ignore any anonymous routers which are still not mapped to IP addresses. And then an edge is built between the two routers before and after the anonymous routers along the routing path. The number of all final edges is N E . It is worth mentioning here researchers can replace Algorithm 1 with other methods to check its influence on the final geolocation performance in future works. Algorithm 1: Map Anonymous Routers to IP addresses Input: Raw path set P r , Completed path set P c = ∅, Flag f lg = 1 Output: Completed path set P c 1 : for each P i r ∈ P r do 2 f lg = 1; 3 for each P j c ∈ P c do 4 if the hop count of P i r equals P j c then 5 if the hop of P i r equals P j c except anonymous routers then 6 Update anonymous routers in P j c with corresponding IP in P i r ; 7 Update P j c in P c with updated P j c ; Extracting Attributes Node Attributes. In this work, each graph node is associated with two kinds of node attributes: node delay and node IP address. The node delay is the direct delay from the probing host to the node. For each node, the delay is repeatedly measured by the probing host many times. The minimum one is selected as the node delay because it contains minimum congestion and is closer to the true propagation delay [2]. In the end, we combine the node IDs and node attributes as the initial feature of a node. v i denotes the initial features of node v i and V denotes the initial features of all nodes. Edge Attributes. Each edge can be associated with several kinds of features such as the edge delay, the head node IP address, and the tail node IP address. Here, the node nearer the probing host is called the head node of the edge, and the other one is the tail node. The edge delay is calculated by subtracting the node delay of the head node from the tail node. We keep edge delay even if it could be a negative value. Since we do not need to refine the edge representation in the MP layers, we do not need edge IDs to differentiate edges. The initial features of an edge only consist of its edge attributes. e j denotes the initial features of edge e j and E denotes the initial features of all edges. The Encoder The encoder aims to form initial low dimensional embeddings of graph nodes and edges for the MP layers. The embeddings of graph nodes and edges are generated from the initial node features V and the initial edge features E from the preprocessor. For each non-zero feature in V and E, we associate it with an embedding vector. We concatenate the embeddings of a node (or edge) into a vector to describe the node (or edge). To be specific, the low dimensional embeddings of a node v i and an edge e u are: h (0) vi = Q v T v i ,(9)h ej = Q e T e u ,(10) where Q v ∈ R N V ×G denotes the embedding matrix for all nodes V , Q e ∈ R N E ×K denotes the embedding matrix for all edges E. N V denotes the number of node features and G denotes the embedding size. N E denotes the number of edge features and K denotes the embedding size. Message Passing (MP) Layers The inputs of Message Passing (MP) layers are h V and h E from the encoder. MP layers will augment the initial graph node embeddings (h (0) V ) by explicitly modeling the edges between nodes and graph/edge attributes. Each MP layer consists of message, aggregate and update functions as follows. Message Function. For a node v i and one of its neighbour nodes v j , the message from v j to v i is defined as: m i←j = f message (h vj , e i←j ),(11) The inputs of f message are: (i) the embedding of the neighbour node h vj ; (ii) the embedding of the edge e i←j from v j to v i . In this paper, f message is implemented as: m i←j = f edge (h ei←j )h vj ,(12) where the edge network f edge is a two-layer MLP, which transforms the edge embedding h ei←j to a matrix W ei←j for a neighbor node's message h vj . The architecture of f edge is: W ei←j = W 2 edge (Relu(W 1 edge (e i←j ))),(13) where W ei←j is used as a weight to control the influence of the neighbor's message h vj to node i. W 1 edge and W 2 edge are the weight matrices of the two MLP layers. They first transform the initial edge embedding h ei←j (size K) to a temporary embedding (size 2K), then transform the temporary embedding into a onedimensional embedding. The one-dimensional weight embedding will be reshaped as a two-dimensional square weight matrix W ei←j (size G × G) to control the influence of neighbor node v j to node v i . The activation function of ReLU [42] is leveraged to catch the non-linear relationships. The edge embedding h ei←j is generated from the edge attributes like the edge delay and the IP addresses of two edge nodes. h vj contains the node ID and attributes (e.g., IP addresses, the delay to the probing host). It is reasonable to use the transformed edge embeddings as message weight. For example, if the edge delay is too large, it may reflect that v j is too far away, then its message is not so important to v i ; Or if the two IP addresses are very similar, it may reflect that the two nodes belong to one organization, then v i should pay more attention to v j . In this way, node v i will learn what is v j 's information (h vj ), and can decide how much information it want to receive from the relationship (f edge (h ei←j )). Aggregation and Update Function. U vi denotes the set of neighbor nodes of v i . The aggregation function f aggregate collects all the messages propagated from U vi to v i . The update function f update uses the collected messages A vi to update the embedding of v i . The aggregation and update function for v i are defined as: A vi = f aggregate v j ∈Uv i (m i←j ),(14)h (l) vi = f update (h (l−1) vi , A vi ),(15) where h vi is the initial node embedding from the encoder, as shown in Formula 9. The aggregation function f aggregate can be a simply symmetric function such as Mean [11], Max [43], or Sum [44]. Since no previous work has tested them in IP geolocation, we tried all three kinds of aggregators in experiments. The update function is implemented as follows: h (l) vi = ReLU(h (l−1) vi + A vi ),(16) where we update the node v i by summing up its previous embedding h (l−1) vi and its aggregated messages from neighbors A vi . Relu is used to do non-linear feature transformation since nodes and edges have rich semantic attributes in this paper. In this way, the new node embedding h (l) vi contains following information : (i) which nodes are its neighbors (topology nearby); (ii) its neighbors' attributes; (iii) its edges' attributes (e.g., IP address, delay); (iv) its own attributes (e.g., ID, IP address, delay). By stacking L MP layers, a node is capable of receiving the messages propagated from its L-hop neighbors. The final embeddings for all node h (L) V are then fed into the decoder for location estimation. The Decoder The decoder aims to estimate the locations from the nodes' refined embeddings h (L) V . We need to predict two numerical values (latitude and longitude) of all nodes. For problems in other research areas, a widely-used original decoder after MP layers consists of dense layers. For example, a typical two-dense-layers decoder is: loc = σ W 2 loc (W 1 loc h (L) V + b 1 loc ) + b 2 loc ,(17) where loc ∈ R N V ×2 denotes the estimated location matrix for all nodes V . The two dimensions ofl oc are the estimated numerical values of latitude and longitude for all nodes. W 1 loc and W 2 loc are the weights of the two dense layers. b 1 loc and b 2 loc are biases. σ is the non-linear activation function to endorse nonlinearity, such as Relu in Formula 16. Then we can train GNN-Geo by comparing the ground-truth locations loc train and the estimated locationŝ loc train . loc train (orl oc train ) means the real/estimated locations of the training dataset. However, this kind of decoder is not so suitable for IP geolocation: the output can be any data in R while the latitude and longitude range are (90S~90N, 180W~180E). Thus most output data is useless and significantly increases the difficulty of optimization. Actually, even finding a fine-grained location of targets among the whole earth is not necessary. As mentioned in Section 1, fine-grained IP geolocation is on the basis of coarsegrained geolocation. Thus, we usually already know a rough location of the target before estimating its fine-grained location. Inspired by this, we combine rules into the pure MLP-based decoder to improve the performance of GNN-based IP geolocation methods. First, we need to use two min-max scalers to transform the loc train (latitude & longitude) into loc t train . The scales of transformed latitude and longitude in loc t train are both [0, 1]. Then our decoder is implemented as follows: loc ts = Sigmoid(BatchNorm(W loc h (L) V + b loc )),(18) where loc ts is the estimated transformed location matrix for all nodes V . Sigmoid is an activation function of which output is [0, 1]. Sigmoid could become saturated if input values are too large, which makes learning even harder. BatchNorm refers to Batch Normalization [45]. Batch Normalization is applied here to ease the saturating problem of Sigmoid as well as preventing overfitting. Then we train GNN-Geo by comparing loc ts train and loc ts train . After training, loc can be scaled from loc ts by reversely using two min-max scalers. It is worth mentioning that we do not always need to scale the range of latitude and longitude of the training sets. The core idea is to find a rough area of targets, then re-scale the range of latitude and longitude of these areas to help the optimizer to converge. In this way, the decoder is actually based on a hypothetical rule that we could get a coarse-grained location of targets before finegrained geolocation. This hypothesis is common to previous finegrained methods. For example, SLG needs to estimate the coarsegrained location before estimating the fine-grained location of a target. And since SLG maps target IP addresses to landmarks, the largest estimated area is actually limited to the area that the known landmarks can cover. MLP-Geo also needs to cluster the landmarks and targets at first based on their measurement data, which is similar to coarse-grained IP geolocation. Corr-SLG also assumes that researchers already knew the city location of the targets and focus on finding fine-grained locations inside the city. This improved decoder can ease the burden of fine-tuning parameters by introducing some prior knowledge into GNN-based methods. Model Training To learn model parameters, we employ the Mean Square Error (MSE) loss between loc train andl oc train . In this work, we optimize the L 2 regularized MSE loss as follows: Loss = N (v)train i=1 (loc ts train − loc ts train ) 2 + λ Θ 2 ,(19) where N (v) train denotes all nodes in the training dataset. And Θ includes all the trainable model parameters of GNN-Geo. Deep learning methods usually suffer from overfitting. Besides Batch Normalization in the decoder, we also leverage L 2 regularization to prevent overfitting. Note that Batch Normalization is only used in training, and must be disabled during testing. λ controls the L 2 regularization strength. The Adam optimizer is employed to optimize the prediction model and update the model parameters by using the gradients of the loss function. PERFORMANCE ANALYSIS We evaluate experiments on 8 real-world datasets, aiming to answer the following research questions: • RQ1: how does GNN-Geo perform as compared with state-of-the-art rule-based and learning-based street-level IP geolocation methods? • RQ2: how does diffferent training-testing ratio affect GNN-Geo's performance? • RQ3: how can rules help to improve GNN-Geo's vanilla MLP-based decoder? • RQ4: how do different settings (e.g., basic GNN models, f aggregation ) affect the geolocating performance of GNN-Geo? Dataset Description To demonstrate the effectiveness and generalization capabilities of GNN-Geo, we plan to conduct experiments in various realworld computer networks. Due to privacy or intellectual property concerns, the datasets of most previous related works are not opened. So we need to collect our own datasets. Target Areas and Landmarks. To test the generalization capabilities of GNN-Geo in different network environments, we collect different landmark datasets from 8 real-networks: (i) the IPv4 networks of New York State, Hong Kong, Shanghai, Beijing, and Tokyo; (ii) the IPv6 networks of Tokyo, Berlin and Munich. In section 6, they are referred as NYS, HK, Shanghai, Beijing, Tokyo, Tokyo-IPv6, Berlin-IPv6 and Munich-IPv6 datasets, respectively. These networks belong to three continents (North America, Asia and Europe), including both developed countries and developing countries. The internet service providers (ISP) in these areas are also different. For example, the main ISPs are Spectrum and Verizon in NYS; Asia Netcom in HK; China Telecom and China Unicom in Shanghai and Beijing. The urban environment, and terrain of these areas are also different. For example, NYS consists of several cities, including New York City, Buffalo city, Rochester city, etc. HK mainly consists of Hong Kong Island, Kowloon Peninsula, New Territories, and some smaller islands. Shanghai consists of the main urban area on the Changjiang alluvial plain as well as rural areas on a large island called Chongming Island. All 8 networks cover densely populated urban areas and sparsely populated rural areas. The sizes of these areas are also different as shown in Table 1. We collect fine-grained IPv4 landmarks following the method in [5], which has been widely used by previous works. These landmarks are mainly famous organization website servers, whose IP addresses and organization locations are opened to the public. However, since more and more new website servers are deployed in the clouds, this collection method is not suitable for finding enough IPv6 fine-grained landmarks. Recently, Erik Rye and Robert Beverly [46] find many WiFi devices' IPv6 addresses and their geographical locations can be connected by their MAC address and WiFi BSSID (Basic Service Set Identifier). Based on this method, we collect IPv6 landmarks all over the world. In total, we find more than 350,000 IPv4/IPv6 landmarks. After verification and selection, there are 11,654 measurable fine-grained landmarks. Each landmark is an IP address with a pair of (latitude, longitude). Due to limited space, we only show the landmark distribution in NYS, HK, and Shanghai in Fig. 3. The covering area and density of landmarks also could affect the performance of IP geolocation methods [47]. The number of landmarks in each network are shown in Table 1. Engineering Applications of the Preprocessor and the Encoder We deploy one probing host in each target network. The probing host is IPv4 (or IPv6) if the target network is IPv4-based (IPv6based). Every probing host is responsible for getting the raw traceroute data in the area where it locates. The network graph of the area is then built based on the measurement data obtained by its probing host. For 10 days, one probing host probes all landmarks in its city every 10 minutes (1440 times in total). Each probing host is a server installed with network probing tools, of which function is similar to traceroute. The software can get the IP addresses of routers between the probing host and all landmarks. And it can also obtain the delays (accurate to milliseconds) between each router and the probing host. After the preprocessor, the number of nodes (IP addresses) The initial attribute feature for each node is a 15-dimensional vector of values. The first 5 dimensions are the raw numerical of node delay and IP address. The numerical node delay is then binned into 10 intervals to form the last 10 dimensions. Here we use K-means to cluster delay into 10 bins. 10 is empirically selected. IP addresses can also be transformed into categorical features. The initial attribute feature for each edge is an 11dimensional vector of values. The first dimension is the raw numerical data of the edge delay. The remaining 10 dimensions are the binned edge delay. Actually, it is easy to combine more kinds of features into the proposed GNN-Geo, such as the domain name of routers, the two IP addresses of an edge, the delay changing trends of a node or edge, etc. All these features can be transformed into node/edge attribute features by the preprocessor, and then naturally introduced into model training through the encoder. And efforts can also be put into more sophisticated pre-processing techniques. For example, we can observe the histogram distribution of delay, and manually decide the number of bins for each area instead of using empirical numbers. IP addresses can also be explicitly transformed into categorical features. However, these features or preprocessing techniques are not easy to be applied in previous baselines for comparison. In this work, we focus on testing the learning potential of GNN in IP geolocation. Thus we utilize a simple preprocessor in this work, leaving further investigation on the preprocessor to future research. Experiments Settings Evaluation Metrics. In each network, we use 4 kinds of training/validation/testing ratios to compare the performance of GNN-Geo with baselines: training-70%, training-50%, training-30%, and training-10%. For all ratios, the validation ratio is 20%. We mainly change the ratio of training dataset and testing dataset: for training-70%, 70% landmarks are randomly chosen for training, 20% for validation, and 10% for testing; for training-50%, 50% landmarks are randomly chosen for training, 20% for validation, and 30% for testing; for training-30%, 30% landmarks are randomly chosen for training, 20% for validation, and 50% for testing; for training-10%, only 10% landmarks are randomly chosen for training, 20% for validation, and 70% for testing. When answering RQ1 (Section 6.5 and 6.6), RQ3 (Section 6.8) and RQ4 (Section 6.9), we mainly use train-70% datasets. Training-50%, training-30%, and training-10% are mainly used in answering RQ2 (Section 6.7). We use the validation sets to do early stopping and hyperparameters fine-tuning. We compare performances on the test sets if not specially explained. For each training set, the latitude and longitude ranges of landmarks are extended by 0.1 decimal degree, before being scaled to the [0, 1]. For example, assume the latitude and longitude range of one training set is (30.61Ñ 31.73N, 121.14E~121.86E), then the range for scaling is (30.51N~31.83N, 121.04E~121.96E). 0.1 decimal degree is an empirical number, about 11.1 km for both longitude and latitude. We minimize the mean squared error loss following Formula 19. During the validation and test phase, we need to re-scale the output of the decoder into latitude and longitude values to calculate the geographical distance. The error distance of an IP address is the geographical distance between the ground-truth location and the estimated location. To evaluate the performance of IP geolocation, we adopt 2 evaluation metrics widely-used in previous works like [2,5]: average distance error, and median distance error. We also leverage Cumulative Distribution Function (CDF) to show the distribution of error distances. Parameter Settings. We implement GNN-Geo in Pytorch. Code and hashed datasets will be opened upon acceptance. The embedding sizes of the initial node ID feature in the encoder and the hidden node representation in the MP layers are searched between {32, 64, 128, 256}. We optimize the model with the Adam optimizer. We leverage the default initializer of Pytorch to initialize the model parameters and embeddings. For example, standard normal distribution N (0, 1) is used to initialize node ID embeddings. A grid search is applied to find the best hyper-parameters [48]. The learning rate is tuned amongst Baselines To evaluate the performance of GNN-Geo, we compare it with several state-of-the-art rule-based and learning-based street-level IP geolocation methods as follows. SLG [5]. SLG is a typical rule-based IP geolocation method and is also one of the most widely-used street-level IP geolocation baselines. Its core rule is to map a target IP to the landmark which has the smallest relative delay to the target IP. There is no need to set or tune hyper-parameters for SLG. Thus in each area, we combine the training set and validation set to geolocate the testing set. Corr-SLG [2]: Corr-SLG is a recent rule-based IP geolocation method. It divides landmarks into 1) Group A, strong-positive delay-distance-correlated landmarks; 2) Group B, strong-negative delay-distance-correlated landmarks; 3) Group C, weak delaydistance-correlated landmarks. The main hyper-parameters are C a and C b in Corr-SLG. They are used to divide all landmarks into Group A, Group B, and Group C based on delay-distance correlation. C a and C b are manually tuned among {0, 0.1, 0.2, . . . , 0.9} and {0.1, −0.2, . . . , −0.9, −1}, respectively. We split the landmarks into a training set, a validation set and a testing set (same to GNN-Geo). In each area, we use the validation set to tune C a and C b , and then report the results on the test set. The parameter searching methods are the same as [2]. MLP-Geo [7]. MLP-Geo is a learning-based IP geolocation method. Like GNN-Geo and Corr-SLG, we use validation sets to tune the hyperparameters of MLP-Geo in each area. The main hyper-parameters of MLP-Geo include the learning rate, the number of training epochs and the dimension size of the middle dense layer. A grid search is applied to find the best hyper-parameters. The learning rate is tuned amongst {0.1, 0.01, 0.001, 0.0001}. The dimension size is searched in {32, 64, 128, 256}. The number of training epochs is 20,000 to find the epoch corresponding to the best performance. The other settings are kept the same as suggested in [7]. For example, we set its beta to 30 when encoding the path. Geolocation Performance Comparison between GNN-Geo and Baselines The results of all methods on 8 datasets are shown in Table 1. The datasets are train-70% datasets. The numbers in Table 1 are averaged by 5 times of train-validation-testing (randomly changing seeds when splitting datasets). To achieve the best performance of all methods, parameter tuning of all methods is conducted as introduced in Section 6.4. From Table 1, we can see that the proposed framework GNN-Geo consistently outperforms all baselines w.r.t. average error distance, median error distance and max error distance on all three datasets. Please note that we leverage MSE Loss as an optimizer target function. MSE loss aims to minimize the whole loss on all data instances (not the median loss or the max loss). Thus we mainly discuss average error distance in this work. The best baseline (i.e., italic rows) in the following sections is the baseline with the smallest average error distance on the test sets if there is no special explanation. Accuracy. From Table 1, on training-70% datasets, GNN-Geo outperforms the best baselines by 28.76%, 16.61%, 26.14%, 21.33%, 23.16%, 5.18%, 25.46%, 46.73%, and 9.25% in NYS (Corr-SLG), HK (MLP-Geo), Shanghai (Corr-SLG), Beijing (SLG), Tokyo (Corr-SLG), Tokyo-IPv6 (MLP-Geo), Berlin-IPv6 (Corr-SLG), and Munich-IPv6 (MLP-Geo) w.r.t. average error distance, respectively. And GNN-Geo also outperforms the best baselines by 30.38%, 12.32%, 21.33%, 25.89%, 10.79%, 25.58%, 36.67%, and 5.44% in NYS (Corr-SLG), HK (Corr-SLG), Shanghai (SLG), Beijing (SLG), Tokyo (MLP-Geo), Tokyo-IPv6 (MLP-Geo), Berlin-IPv6 (Corr-SLG), and Munich-IPv6 (MLP-Geo) w.r.t. median error distance, respectively. The best baselines are different in different networks and are shown in brackets. So averagely, in these 8 IPv4/IPv6 networks, GNN-Geo outperforms the best baselines by 22.51% and 21.05% w.r.t. average error distance and median error distance, respectively. These results demonstrate that the geolocation accuracy of GNN-Geo is clearly higher than baselines, which validates the value of investigating GNN's potential in fine-grained IP geolocation. Generalization. It is interesting that the best baselines are all different in 8 networks. For average error distance, the best baseline is Corr-SLG for 4 times, MLP-Geo for 3 times, and SLG for 1 time; for median error distance, the best baseline is Corr-SLG for 3 times, MLP-Geo for 3 times, and SLG for 2 time. From the experiments, we can see the performances of baselines are not stable in different areas. Their performance varies significantly in different network environments. For SLG, it relies on the proportion of "strong-positive-delay-distance-correlated" landmarks. For Corr-SLG, the proportion of "strong-negative-delay-distancecorrelated" landmarks decides whether it can outperform SLG. Compared with previous baselines, GNN-Geo shows better generalization capabilities in different network environments. In NYS network, the max error distances are too large, overlapping the smaller cumulative probability with each other. Thus we magnify the picture (0 -10 km & 0.1 -0.6) in Fig. 4(a). From Fig. 4, we can see that: generally, the error distances of GNN-Geo are smaller than baselines in most phases of cumulative probability. This validates GNN-Geo's advantages in most cases. However, if we check the error distances less than 5 km, we can see SLG sometimes could be better than GNN-Geo in the very beginning. For example, in Shanghai and Beijing, SLG is clearly better than GNN-Geo in the smallest error distances (<2km). In fact, some smallest error distances of SLG are even 0 km. Why SLG could be better than GNN-Geo at the start and become worse in the latter part of cumulative probability? The rule of SLG is: "the shortest delay comes from the smallest distance". Thus SLG could get very accurate geolocation results for targets that follow this rule. Some error distances could be close to 0 if the landmarks are at the same location as the targets. This is why SLG performs well in the first part. However, the disadvantage of SLG is that its error distances could be very large for targets that follow the opposite rule -"the shortest delay comes from the longest distance". Besides, the error distances of SLG are also large for targets which have no clear linear delaydistance relationship at all. This is why the max error distances of SLG are very large in some datasets. If we observe the max error distances when probability reaches 100%, we can see SLG's max error distances are much larger than other baselines. For example, in Tokyo-IPv6, we could see when GNN-Geo's max error distance is just about SLG's top 85% max error distance. So the average error distances of SLG are not so good if there are many of these kinds of targets. The cumulative probability of SLG would quickly become worse in the latter part. GNN-Geo does not follow a specific linear delay-distance rule. Its "measurement data -location" mapping function is learned based on a supervised-learning method. The mapping function is optimized by minimizing the MSE loss of the whole training set. In other words, GNN-Geo considers the whole performance or average performance of all targets. Its mapping function tries to be suitable to as many targets as possible, instead of only focusing on a part of targets. In a dataset, if different targets follow different delay-distance or topology-distance rules, the mapping function of GNN-Geo may be less accurate for some targets than SLG. This is because the mapping function needs to get better performance for other targets which do not follow linear rules. Therefore GNN-Geo can outperform SLG in the latter part of cumulative probability. Actually, we can notice that MLP-Geo follows a similar trend with GNN-Geo. In Fig. 4(c), MLP-Geo is worse than SLG before 3 km and quickly becomes better afterward. This is because MLP-Geo is also a learning-based method and is optimized by MSE loss. Fig. 5 shows how different methods' performances are affected by changing training-validation-testing ratios in all datasets. Here, we select a typical method in rule-based learning (SLG) and learningbased learning methods (MLP-Geo) to compare with GNN-Geo. Parameters are re-tuned for learning-based methods when changing ratios. We notice that, though average error distances generally follow similar trends with median error distances, average error distances are easier to be affected by max error distance. Since the training ratio become smaller, some methods may have too large average/max error distances, which make other results in the same figures hard to distinguish. So, we mainly use median error distance to show the trends in Fig. 5. How training ratios affect GNN-Geo's Performance From Fig. 5, we can see that the performances of all methods generally become better when training ratio increases from 10% to 70%, validating that more landmarks in the same network usually lead to better geolocation accuracy for measurement-based methods. In 18 out of 24 cases (75%), GNN-Geo still clearly outperforms baselines with training ratio less than 70%. This shows that we still can use GNN-Geo even with relatively less landmarks. However, if the training ratio is too less (training-ratio is 10%), only in 50% cases, GNN-Geo is clearly better than other baselines. This shows that, compared with other methods, GNN's performances may be more easily affected with extremely smaller training datasets. This is because: (1) compared with SLG, as a learning-based method, GNN-Geo could be overfit on extremely sparse datasets; (2) compared with MLP-Geo, GNN-Geo has more parameters to train, so it is generally easier to overfit. So sometimes, if the training dataset is too small, and the signals (e.g., network topology) extracted by GNN-Geo are not so important to IP geolocation, then GNN-Geo's advantages could be decreased. Geolocation Performance Comparison over different Decoders As explained in section 5.4, GNN-based geolocation methods with the original decoders may be not suitable for IP geolocation. Here we compare the proposed decoder with several baseline decoders. All these decoders share the same preprocessor, encoder and MP layers. Pure MLP-based decoders do not limit the output data range. MLP-based decoders improved by rules scale the latitude and longitude range of the training set (+11 km) to [0,1]. For Vanilla MLP, the decoder consists of a dense layer with Relu and an output layer. For +BatchNorm, the decoder consists of a dense layer with Relu, a batch normalization layer and an output layer. For +Sigmoid, the decoder consists of a dense layer with Relu and an output layer with Sigmoid. For +BatchNorm+Sigmoid, the decoder consists of a dense layer with Relu, a batch normalization layer and an output layer with Sigmoid. +BatchNorm+Sigmoid is the proposed rule-based decoder used in GNN-Geo. The best geolocation performance w.r.t average error distance of all decoders on validation sets is shown in Table 2. Here we only show the results in first 3 networks (NYS, HK, Shanghai) with training-70% ratio as an example. For each dataset, the bold indicates the best average error distance for all decoders while the italic row indicates the best average error distance for all baseline decoders. "Epochs" indicate the training epoch number of the corresponding results. It shows how many epochs a decoder needs to converge. The training epochs are at most 10000 and early stopped if results are not improved in 1000 epochs. From Table 2, we can see the best average distance errors of two pure MLP-based decoders are more than 1,000 km even after 10,000 training epochs. This is because most of the output data belong to R and are far away from the real locations of the targets. It indicates that GNN is not a "silver bullet" to the IP geolocation problem. From Table 1, we can the average distances of GNN-Geo are between 8 km and 28 km, clearly outperforming all baselines. However, simply changing its decoder could lead to unacceptable performance. The original GNN cannot be directly used for IP geolocation. We still need to carefully design the structure of the whole GNN-Geo framework to gain better geolocation performance. The performances of decoders improved with rules are much better. We can find some trends: 1) +Sigmoid and +BatchNorm both can clearly increase the performance and significantly reduce the epoch numbers for convergence; 2) +Sigmoid is more accurate than +BatchNorm except Shanghai dataset; 3) +Sigmoid converges faster than +BatchNorm; 4) +BatchNorm+Sigmoid is better than only using +BatchNorm or +Sigmoid. We explain these trends as follows. Sigmoid limits the range of output into [0, 1], which eases the burden for the optimizer. All the output data must be in the area of the training datasets, which is a coarsegrained location estimation of the targets. However, only using Sigmoid could meet the saturating problem. Batch normalization can help to relieve this problem. Batch normalization itself can also facilitate neural network training [45,49]. Besides, it is noticed that rule-based Vanilla MLP is much better than the original Vanilla MLP, with its output still belonging to R. This is because our initializers sometimes would make the decoder's estimated data in the first several epochs close to 0. If this happens, its performance could be much better than the original MLP. These findings show that we can combine the advantages of rule-based and learning-based methods together to gain better performance. These findings and the advantages of SLG shown in Section 6.6 indicate that we should consider combining the advantages of rule-based and learning-based methods together to gain better performance. Further Investigation on GNN-Geo Properties In this section, we investigate the impact of different learning configurations on the performance of GNN-Geo. We start by exploring the influence of the depth of the MP layers. We then study how different message aggregation methods affect the performance. The effect of Regularization Coefficients is discussed later. Then, we analyze the influences of Graph Node Embedding Size. In Fig. 6 HK and Shanghai) as an example. However, we mainly analyze the effect of parameters on average error distance because the optimized target is MSE loss. GNN-Geo did not specifically seek a minimum median error distance or max error distance. We pick the best performance for each parameter on the testing sets. And no matter how parameters vary, the average error distances in Fig. 6 are consistently smaller than baselines in previous works for all datasets. It verifies the effectiveness of GNN-Geo, empirically showing that explicitly modeling the network connectivity can greatly facilitate the geolocation task. Finally, we also investigate how different GNN basic models would affect the performance of GNN-Geo. 6.9.1 Effect of the Depth of Message Passing Layers The depth of the MP layers is actually the number of MP layers. From Fig. 6(a), the best layer numbers are 3, 2, 2 in NYS, HK and Shanghai, reflectively. The worst layer numbers are 1, 5 and 1 in NYS, HK and Shanghai, respectively. Compared to the worst layer numbers, the improvements of the best layer numbers are 7.27%1 6.55%. Increasing the depth of GNN-Geo substantially enhances the geolocation performance. GNN-Geo-3 indicates GNN-Geo with three message propagation layers and similar notations for others. We can see that GNN-Geo-2, GNN-Geo-3 and GNN-Geo-4 achieve consistent improvement over GNN-Geo-1 across all datasets, which only consider the first-order neighbors only. When further stacking the propagation layer on the top of GNN-Geo-4, we find that GNN-Geo-5 leads to overfitting on the Hong Kong dataset. This might be caused by applying a too deep architecture that might introduce noises to the representation learning. The marginal improvements on the other two datasets verify that conducting 2-3 propagation layers are sufficient to model the network for IP geolocation. Effect of Aggregation Methods From Fig. 6(d), the best aggregation methods are Max, Mean, Mean in NYS, HK and Shanghai, respectively. The worst aggregation methods are Sum, Max, Max in NYS, HK and Shanghai, respectively. Compared to the worst aggregation methods, the improvements of the best aggregation methods are 9.58%, 1.92%, 18.95%, respectively. Neighborhood aggregation is a key operation, which distinguishes GNN from MLP as we discussed in section 4. Theoretically, all three methods have their limitation: 1) both Mean and Sum cannot treat each neighbor differently; 2) Max may only lose some neighbors' useful information. Previous work [44] has pointed out that Sum may be more powerful than Mean and Max for graphs without attributes. However, in three datasets of this paper, Mean is the best or the second-best, while Max is a little worse than Mean. Max could be the worst or the best in different datasets. The attributed graph is more complex because the attributes of edges and nodes are also influencing the performance besides the network structure. The best aggregation methods in attributed graphs for geolocation still need more depth analysis. 6.9.3 Effect of Graph Node Embedding Size From Fig. 6(g), all the best graph node embedding sizes are 128 in NYS, HK and Shanghai. All the worst graph node embedding sizes are 256 in NYS, HK and Shanghai. Compared to the worst graph node embedding sizes, the improvements of the best graph node embedding sizes are 5.71%, 5.68%, 14.49%, respectively. The influence patterns of Graph Node Embedding Size are stable for all three datasets. Increasing embedding size brings more representation power to the MP layers by introducing more complexity. This enables the model to learn more complicated patterns. So embedding size 128 performs better than 32 and 64 in all datasets. However, a too large embedding size like 256 could disturb the model with noises and result in overfitting, especially if there are not enough data instances for training. Effect of Regularization Coefficients From Fig. 6(j), the best Regularization Coefficients are 0.0005, 0.001, 0.001 in NYS, HK and Shanghai, respectively. The worst Regularization Coefficients are 0.01, 0.01, 0.0005 in NYS, HK and Shanghai, respectively. Compared to the worst Regularization Coefficients, the improvements of the best Regularization Coefficients are 31.90%, 1.47%, 6.36%. L 2 Regularization is the main method in GNN-Geo to overcome overfitting. However, too larger Regularization Coefficients could lead to underfitting. We find that the influence of Regularization Coefficients varies significantly in different datasets. The Regularization Coefficient is very important to NYS dataset since the improvement is larger than all the other parameters discussed in this section. However, the Regularization coefficient is the least influential parameter in the Hong Kong dataset, compared with the other parameters. Generally, the Regularization Coefficients 0.001 should be tried first because it is the best and second-best in all datasets. Effect of Regularization Coefficients From Fig. 6(j), the best Regularization Coefficients are 0.0005, 0.001, 0.001 in NYS, HK and Shanghai, respectively. The worst Regularization Coefficients are 0.01, 0.01, 0.0005 in NYS, HK and Shanghai, respectively. Compared to the worst Regularization Coefficients, the improvements of the best Regularization Coefficients are 31.90%, 1.47%, 6.36%. L 2 Regularization is the main method in GNN-Geo to overcome overfitting. However, too larger Regularization Coefficients could lead to underfitting. We find that the influence of Regularization Coefficients varies significantly in different datasets. The Regularization Coefficient is very important to NYS dataset since the improvement is larger than all the other parameters discussed in this section. However, the Regularization coefficient is the least influential parameter in the Hong Kong dataset, compared with the other parameters. Generally, the Regularization Coefficients 0.001 should be tried first because it is the best and second-best in all datasets. Effect of Different basic GNN models In section 5 and all previous experiments of Section 6, we select MPNN as the basic GNN model for message passing layers. However, besides MPNN, there are still other famous basic GNN models such as GCN [11] and GAT [23]. Can they be better than MPNN for IP geolocation tasks? To answer this question, GCN and GAT are also used for message passing layers, which are referred as GNN-Geo-GCN and GNN-Geo-GAT, respectively. We select training-70% and training-10% to test their performances. After parameters tuning, the average error distances and median error distances of GNN-Geo (GCN), GNN-Geo (GAT) and GNN-Geo (MPNN) are shown in Fig. 7. From Fig. 7, we can see for both average and median error distances, GNN-Geo (MPNN) outperforms the other two methods on training-70% and training-10% datasets. This validates the importance to explicitly model edge features for IP geolocation. GNN-Geo (GAT) is usually better than GNN-Geo (GCN), which may indicate that some neighbors should be more important than others, and attention mechanism should also be tried in improving GNN-Geo in the future. Besides, compared with train-10%, the advantages of MPNN are usually clearer in training-70% datasets. This may show that GNN-Geo (MPNN) is easier to overfit than other two methods on smaller training datasets. DISCUSSION In this paper, we mainly test whether GNN-Geo's accuracy outperforms previous fine-grained IP geolocation methods. From the experiments in Section 6.5, we can see that GNN-Geo is more accurate than previous methods due to its stronger modeling ability on non-linear relationships hidden in computer network measurement data. In this section, we will discuss what improvements GNN-Geo still needs to make it more practical for industrial-scale IP geolocation tasks. Increasing Accuracy with fewer Landmarks. GNN is a deep learning-based method. Like other deep learning-based methods, its performance is much easier to be affected than rule-based methods when the landmarks or training data instances are too few (as shown in Section 6.7). This is because deep learning methods are easier to overfit on sparse datasets. Thus, we need to consider how to improve GNN-Geo's performance in areas where the landmarks are hard to collect. Injecting explainability into GNN-Geo Compared with rulebased methods like SLG, deep learning-based methods often lack explainability for results. For example, for a target IP address in the Shanghai dataset, after geolocating it with SLG, it is easier to analyze why the error distance of this target is large. It is usually because its shortest delay does not come from its nearest landmark. However, it is hard to explain why GNN-Geo's error distance for this target is large. We need to discuss how to inject explainability into GNN-Geo's results for IP geolocation service users. CONCLUSION In this work, we first formulate the measurement-based fine-grained IP geolocation task into a semi-supervised attributedgraph node regression problem. Subsequently, we discuss why GNN is worth to be tested for IP geolocation. Then, we propose a GNN-based fine-grained IP geolocation framework (GNN-Geo) to solve the node regression problem. The framework consists of a preprocessor, an encoder, MP layers and a decoder. Moreover, we ease the convergence problem of GNN-based IP geolocation methods by scaling the range of output as well as combining the batch normalization and Sigmoid functions into the decoder. Finally, the experiments in 8 different real-world IPv4/IPv6 datasets verify the generalization capabilities of GNN-Geo for fine-grained IP geolocation. The main advantages of GNN-Geo are the strong modeling ability on the computer network measurement data and the powerful extraction ability of non-linear relationships. Compared with rule-based methods, its main disadvantage is explainability, which is also a common problem for deep learning-based methods. Compared with the previous deep learning-based methods, the stronger modeling ability of GNN-Geo also needs more computing resources for training. In future work, we plan to increase GNN-Geo's accuracy with fewer landmarks, inject explainability into GNN-Geo, lower the training cost of GNN-Geo and improve GNN-Geo with various mechanisms like attention. Shichang Ding received the Ph.D. degree from University of Göttingen, Germany, in 2020. He is currently a lecturer at State Key Laboratory of Mathematical Engineering and Advanced Computing. His research interests include cyberspace surveying and mapping, graph deep learning, and social computing. Xiangyang Luo received the Ph.D. degree from Information Engineering University, China, in 2010. He is currently a Professor and a Ph.D. Supervisor with State Key Laboratory of Mathematical Engineering and Advanced Computing. He has authored or coauthored more than 150 refereed international journal and conference papers. His research interests include multimedia security and cyberspace surveying and mapping. Jinwei Wang received the Ph.D. degree in information security from the Nanjing University of Science and Technology, China, in 2007. He is currently a Professor at the Nanjing University of Information Science and Technology. He has published over 50 papers. His research interests include multimedia copyright protection, multimedia forensics, multimedia encryption, and data authentication. Xiaoming Fu received the PhD degree in computer science from Tsinghua University, Beijing, China in 2000. He was then a research staff at Technical University of Berlin until joining the University of Göttingen, Germany in 2002, where he has been a professor in computer science and heading the computer networks group since 2007. He has spent research visits at Cambridge, Columbia, UCLA, Tsinghua University, Uppsala, and UPMC, and is an IEEE fellow and distinguished lecturer. His research interests include Internet-based systems, applications, and social networks. He is currently an editorial board member of IEEE Communications Magazine, IEEE Transactions on Network and Service Management, Elsevier Computer Networks, and Computer Communications, and has published over 150 peer-reviewed papers in renowned journals and international conference proceedings. Fig. 1 : 1Mapping a Computer Network into An Attributed Graph graph, a graph node v i ∈ V, i ∈ {0, 1, . . . , N V } represents an IP address IP i . Each link LK j is represented as a graph edge e j ∈ E, j ∈ {0, 1, . . . , N E }. Let xv i ∈ XV, i ∈ {0, 1, . . . , N V } and xe j ∈ XE, j ∈ {0, 1, . . . , N E } represent the attributes of the node v i and the edge e j , respectively. The latitude and longitude location of v i are loc vi . The two dimensions of loc vi are [−90, 90] and [−180, 180], respectively. Assume the nodes belong to the probing host, landmarks, targets and routers are V p , V l , V t and V r . Then the locations of the probing host and landmarks are Fig. 2 : 2Illustration of the GNN-Geo Framework several different routing paths. denotes the updated embedding of node v i after the l-th MP layer (l = 1, 2, 3, 4, . . . ). h (0) Fig. 3 : 3Landmark Distribution of Three Areas as an example {0.01, 0.001}, the coefficient of L 2 regularization is searched in {0.01, 0.001, 0.0005}. The depth of the MP layers L is searched in {1, 2, 3, 4, 5}. The number of hidden units of the edge network is searched in {4, 8, 16}. The dimension of initial edge embedding K is searched between {16, 8, 4}. The aggregation methods are searched among Mean, Sum and Max. Besides, the early stopping strategy is performed: training is stopped if the average distance error on the validation set does not increase for 1000 successive epochs. Usually, 4000 epochs are sufficient for GNN-Geo to converge. Fig. 4 :Fig. 5 : 45Error Distance CDF of 8 Networks (the maximum values of x-axis are GNN-Geo's max error distances) The Relationship between Error Distance and Training dataset/Full dataset 6.6 An Interesting Difference between Learning-based and Rule-based Methods Fig. 4 shows the cumulative probability of error distances of baselines and GNN-Geo in all 8 networks (training-70% datasets). Fig. 7 : 7Median Error Distances of Different GNN Basic Models in 8 Networks 5776) in Berlin-IPv6 and (380, 856) in Munich-IPv6. So, their edge/node ratio is 1.15, 1.20, 5.97, 5.66, 2.94, 2.16, 2.79 and 2.25, respectively.and the number of the links are (2872, 3303) in NYS, (3071, 3697) in HK, (2530, 15108) in Shanghai, (3310, 18742) in Beijing, (1498, 4408) in Tokyo, (2297, 4966) in Tokyo- IPv6, (2070, TABLE 1 : 1Performance (kilometers) Comparison of baselines and GNN-GeoAreas #Landmarks Size(km 2 ) Error Distance Rule-based Learning-based %Error Reduced SLG [5] Corr-SLG [2] MLP-Geo [7] GNN-Geo New York State 1705 141299 Average 38.176 38.303 43.274 27.198 28.76% Median 7.090 7.040 13.847 4.901 30.38% Hong Kong 2061 1108 Average 13.019 10.001 9.793 8.166 16.61% Median 11.933 7.523 9.140 6.596 12.32% Shanghai 1387 6340 Average 14.245 14.037 14.291 10.368 26.14% Median 11.056 11.281 11.659 8.698 21.33% Beijing 1835 16410 Average 12.394 13.855 12.992 9.523 23.16% Median 10.259 11.314 10.985 7.603 25.89% Tokyo 943 14034 Average 15.321 12.978 13.047 12.306 5.18% Median 13.206 10.217 9.701 8.654 10.79% Tokyo (IPv6) 1949 14034 Average 15.372 7.066 3.912 2.916 25.46% Median 10.813 5.702 1.806 1.344 25.58% Berlin (IPv6) 1503 883 Average 7.214 6.322 7.294 3.368 46.73% Median 5.141 5.100 6.300 3.230 36.67% Munich (IPv6) 271 310 Average 7.908 5.028 4.270 3.875 9.25% Median 7.890 4.312 3.416 3.230 5.44% 1 underline indicates the best metrics among baselines. TABLE 2 : 2Performance (kilometers) Comparison of Different Decoder Types Bold indicate the average error distance of GNN-Geo, underline rows indicate the validation average error distances among other decoder types. Note all these methods only differ in decoder types.2 "Epochs" indicates the training epoch number of the corresponding results.3 These results are the best average distance errors on validation sets.Decoder Type New York State Hong Kong Shanghai Average Epochs Average Epochs Average Epochs Pure MLP-based Decoders Vanilla MLP 3406.247 512 2690.566 10000 3266.079 10000 +BatchNorm 6010.929 26 2812.680 385 1087.136 20 MLP-based Decoders with Rules Vanilla MLP 69.663 6643 8.127 9828 12.504 3589 +BatchNorm 29.861 2073 7.735 4690 9.723 1808 +Sigmoid 26.445 1484 7.681 2265 9.843 1096 +BatchNorm+Sigmoid 25.165 3534 7.603 760 9.673 1439 1 , we show the effect of parameters on average error distance, median error distance and max error distance with training-70% datasets in the first 3 networks (NYS,Fig. 6: Parameter Study of GNN-Geo in New York State, Hong Kong, Shanghai(a) Average Error(km) (b) Median Error(km) (c) Max Error(km) (d) Average Error(km) (e) Median Error(km) (f) Max Error(km) (g) Average Error(km) (h) Median Error(km) (i) Max Error(km) (j) Average Error(km) (k) Median Error(km) (l) Max Error(km) ACKNOWLEDGMENTS Geocam: An ip-based geolocation service through fine-grained and stable webcam landmarks. 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[ "Unsupervised multi-object segmentation using attention and soft-argmax", "Unsupervised multi-object segmentation using attention and soft-argmax", "Unsupervised multi-object segmentation using attention and soft-argmax", "Unsupervised multi-object segmentation using attention and soft-argmax" ]
[ "Bruno Sauvalle [email protected] \nLa Fortelle Centre de Robotique Mines ParisTech PSL University\n\n", "Bruno Sauvalle [email protected] \nLa Fortelle Centre de Robotique Mines ParisTech PSL University\n\n" ]
[ "La Fortelle Centre de Robotique Mines ParisTech PSL University\n", "La Fortelle Centre de Robotique Mines ParisTech PSL University\n" ]
[]
We introduce a new architecture for unsupervised objectcentric representation learning and multi-object detection and segmentation, which uses a translation-equivariant attention mechanism to predict the coordinates of the objects present in the scene and to associate a feature vector to each object. A transformer encoder handles occlusions and redundant detections, and a convolutional autoencoder is in charge of background reconstruction. We show that this architecture significantly outperforms the state of the art on complex synthetic benchmarks.
10.1109/wacv56688.2023.00328
[ "https://export.arxiv.org/pdf/2205.13271v2.pdf" ]
249,097,913
2205.13271
2fd95baca57f0ed4323b56b8ade1fe3facec25a9
Unsupervised multi-object segmentation using attention and soft-argmax Bruno Sauvalle [email protected] La Fortelle Centre de Robotique Mines ParisTech PSL University Unsupervised multi-object segmentation using attention and soft-argmax We introduce a new architecture for unsupervised objectcentric representation learning and multi-object detection and segmentation, which uses a translation-equivariant attention mechanism to predict the coordinates of the objects present in the scene and to associate a feature vector to each object. A transformer encoder handles occlusions and redundant detections, and a convolutional autoencoder is in charge of background reconstruction. We show that this architecture significantly outperforms the state of the art on complex synthetic benchmarks. Introduction We consider in this paper the tasks of object-centric representation learning and unsupervised object detection and segmentation: Starting from a dataset of images showing various scenes cluttered with objects, our goal is to build a structured object-centric representation of these scenes, i.e. to map each object present in a scene to a vector representing this object and allowing to recover its appearance and segmentation mask. This task is very challenging because the objects appearing in the images may have different shapes, locations, colors or textures, can occlude each other, and we do not assume that the images share the same background. However the rewards of object-centric representations could be significant since they allow to perform complex reasoning on images or videos [11,42] and to learn better policies on downstream tasks involving object manipulation or localization [46,50]. The main issue with objectrepresentation learning today is however that existing models are able to process synthetic toy scenes with simple textures and backgrounds but fail to handle more complex or real-world scenes [29]. We propose to improve upon this situation by introducing a translation-equivariant and attention-based approach for unsupervised object detection, so that a translation of the input image leads to a similar translation of the coordinates of the detected objects, thanks to an attention map which is used not only to associate a feature vector to each object present in the scene, but also to predict the coordinates of these objects. The main contributions of this paper are the following: • We propose a theoretical justification for the use of attention maps and soft-argmax for object localization. • We introduce a new translation-equivariant and attention-based object detection and segmentation architecture which does not rely on any spatial prior. • We show that the proposed model substantially improves upon the state of the art on unsupervised object segmentation on complex synthetic benchmarks. The paper is organized as follows: In section 2, we provide some theoretical motivation for using attention maps and soft-argmax for object localization. In section 3, we review related work on unsupervised object instance segmentation. In section 4 we describe the proposed model. Experimental results are then provided in section 5. Motivation for using attention maps and soft-argmax for object localization It is widely recognized that the success of convolutional neural networks is associated with the fact that convolution layers are equivariant with respect to the action of the group of translations, which makes these layers efficient for detecting features which naturally have this property. It is also easy to show that linear convolution operators are the only linear operators which are equivariant with respect to the natural action of the translation group on feature maps. We introduce the following notations to describe the action of the translation group: We consider a grayscale image as a scalar-valued function ϕ(i, j) defined on Z 2 and an element of the group of translations as a vector (u, v) in Z 2 . The natural action T of the group of translations on an image can be described by the formula A model layer L is called equivariant with respect to translations if it satisfies L(T u,v ϕ) = T u,v (L(ϕ)).(2) Let's now consider a localization model M which takes as input an image ϕ(i, j) showing one object and produces as output the coordinates of the object present in this image. Such a model does not produce a feature map, so that the previous definition of translation equivariance can not be used for this model. We remark however that the group of translations acts naturally on Z 2 by the action T u,v (i, j) = i + u, j + v, and that the model M should have the equivariance property M (T u,v ϕ) = T u,v (M (ϕ)).(3) Indeed, if the complete image is translated by a vector (u, v), then the object present in this image is also translated, so that the associated coordinates have to be shifted according to the vector (u, v). It is not difficult to see that in the same way that convolutional operators are the only linear operators equivariant with respect to translations, it is also possible to fully describe which elementary operators follow this specific equivariance property. We first remark however that we have to restrict the space of possible input maps ϕ: if ϕ is a constant function, it does not change under the action of the translation group, so that the equivariance property 3 cannot be satisfied with such a function. We then suppose that ϕ satisfies p ϕ(p) = 1 and consider that the domain of the operator M is the corresponding affine space A. We also replace the linearity condition by an the following affinity condition: For all α i ∈ R, ϕ i ∈ A so that i α i = 1, we have M ( i α i ϕ i ) = i α i M (ϕ i ). We then have the following proposition: Proposition 2.1 An affine operator M which satisfies the equivariance property 3 has to be of the form M (ϕ) = C + p∈Z 2 ϕ(p)p(4) for some constant C in R 2 . Proof: We write the input map ϕ as a sum of spatially shifted version of the function δ ∈ A satisfying δ(p) = 1 for p = (0, 0) and δ(p) = 0 for p = (0, 0): ϕ(p) = q∈Z 2 ϕ(q)δ(p − q)(5) We then use the the affine property of M and equivariance property 3: M (ϕ) = M ( q ϕ(q)δ(p − q)) (6) = q ϕ(q)M (δ(p − q)) = q ϕ(q)(M (δ) + q) (7) = ( q ϕ(q))M (δ) + q ϕ(q)q (8) = M (δ) + q ϕ(q)q,(9) which proves the proposition since M (δ) is a constant. The proposition 2.1 can be interpreted as stating that in order to get an equivariant localization operator, the most straightforward method is to build a normalized attention map ϕ from the input image and compute the coordinates of the detected object using an attention mechanism with ϕ as attention map and pixel coordinates as target values. One remarks that it is precisely what the soft-argmax operator is doing: It takes an unnormalized scalar map φ as input, normalizes it using a softmax operator, and then perform localization using the same formula as in 2.1: soft-argmax(φ) = p∈Z 2 softmax(φ)(p)p = p∈Z 2 e φ(p) q∈Z 2 e φ(q) p(10) This operation is called soft-argmax because it allows to compute in a differentiable way an estimate of the coordinates of the maximum of the input map φ. Using softargmax then appears to be the most natural way to get an equivariant localization operator. Related work Unsupervised object detection and segmentation Unsupervised object detection and segmentation models are generally reconstruction models: They try to reconstruct the input image using a specific image rendering process which induces the required object-centric structure. In order to ensure that objects are properly detected, various objectness priors have been defined and implemented: • pixel similarity priors. Some models consider the task of object segmentation as a clustering problem, which can be addressed using deterministic [23,33] or probabilistic [15,20,44] methods: If the feature vectors associated to two different pixels of an image are very similar, then it is considered that these pixels should both belong to the same object or to the background. • independence priors. Some models assume that the images are sampled from a distribution which follows a probabilistic model featuring some independence priors between objects and the background, and use variational [19,16] or adversarial [7,3] methods to learn these distributions. • disentanglement of appearance and location. Foreground objects appearing in the scenes of a given dataset can have similar shapes and appearances but very different scales and locations. Object discovery is performed by disentangling the object appearance generation process, which is performed by a convolutional glimpse generator [1,30,10,40,26,25] or a learned dictionary [35,39], from the translation and scaling of the objects appearing in a scene, which is usually done by including a spatial transformer network [24] in the model. The model described in this paper belongs to this category and uses an convolutional glimpse generator. Object detection and segmentation without spatial prior State of the art supervised detection and segmentation models usually rely on predefined reference anchors or center points which are spatially organized according to a periodic grid structure. The use of periodic grids has also been proposed for unsupervised object detection [32,26,25,39]. Alternative detection methods relying on heatmaps produced by a U-net [37] or stacked U-nets [36] networks, which predict for each pixel the probability of presence of one object on this pixel have been implemented in the supervised setting [31,13]. For some specific applications such as human pose estimation or anatomical landmark localization [43], some supervised models predict one heatmap per object. The use of soft-argmax for converting heatmaps to object coordinates has been implemented in the supervised [41,34,6], semisupervised [22] and unsupervised settings [18,17] but has never been proposed for unsupervised object detection or segmentation. More recently, transformer-based [45] models using object [5,51,12] or mask [9,8] queries have been proposed which not not rely explicitly on a spatial grid. These models show that transformers are efficient in the supervised setting to avoid multiple detections of the same object. Description of proposed model Model architecture The overall architecture of the model is described in Fig 1. The proposed model is composed of a a foreground model and a background model. The background model is a deterministic convolutional autoencoder: We rely on the classical assumption [47] that background images lie on a low-dimensional manifold, and use the autoencoder to learn this manifold. The foreground model is also deterministic and associates to each object in the scene an appearance vector z what which is used to produce a glimpse of the object, which is then scaled and translated at the right position on the image using a spatial transformer network. The foreground encoding and reconstruction process can be described as follows: First, a high resolution feature map generator takes a color image of size h × w as input and produces a high resolution feature map Φ of dimension d Φ and several scalar attention logit maps A 1 , ..., A K . We will use in this paper the transformer-based Segformer model [49], which produces feature maps of size h * ×w * = h/4× w/4. The hyperparameter K is set to the maximum number of objects on a scene in the dataset. The scalar attention logit maps A 1 , ..., A K are transformed into a normalized attention maps A 1 , ..., A K using a softmax operator: A k (i, j) = e A k (i,j) i ,j e A k (i ,j )(11) We normalize the pixel indices (i, j) from the range [1, .., w * ] and [1, .., h * ] to the range [−1, 1] required by spatial transformer networks using the formulas x(i) = 2 i − 1 w * − 1 − 1 (12) y(j) = 2 j − 1 h * − 1 − 1,(13) and predict initial estimates x 0 k , y 0 k of the coordinates of the detected objects as the the center of mass of the attention maps A k : x 0 k = w * ,h * i=1,j=1 A k (i, j)x(i)(14)y 0 k = w * ,h * i=1,j=1 A k (i, j)y(j)(15) We also build K object query feature vectors φ 0 1 , ..., φ 0 K of dimension d Φ using the same attention maps A 1 , .., A K as weights and the feature map Φ as target values: φ 0 k = w * ,h * i=1,j=1 A k (i, j)Φ(i, j)(16) A transformer encoder then takes the K triplets (φ 0 k , x 0 k , y 0 k ) 1≤k≤K as inputs and produces a refined version (φ k , x k , y k ) 1≤k≤K taking into account possible detection redundancies and object occlusions. More precisely, we use Figure 1. Overview of proposed model. A High resolution feature map generator (Segformer model) is trained to produce a high resolution feature map Φ and K scalar attention maps (one per object query). These maps are used to predict the coordinates and scales of the detected objects and the associated feature vectors, which are refined by a transformer encoder and then used as inputs to a glimpse generator and a spatial transformer network to produce K object image layers and masks. A convolutional autoencoder is in charge of background reconstruction. a learned linear embedding to increase the dimension of the triplets (φ 0 k , x 0 k , y 0 k ) from d Φ + 2 to the input dimension d T of the transformer encoder, and a learned linear projection to reduce the dimension of the outputs of the transformer encoder from d T back to d Φ + 2. The transformer encoder does not take any positional encoding as input, considering that the transformation which has to be performed should not depend on the ordering of the detections. We force the final values of x k and y k to stay in the range [−1, 1] using clamping. Each transformed feature vector φ k is then split in three terms: φ k = (s k , α k , z what k ). • The first term s k is an inverse scaling factor. It is a scalar if objects in the dataset have widths and heights which are similar (isotropic scaling), or a pair of scalars s x k , x y k if this is not the case (anisotropic scaling). We force the values of s k to stay within a fixed range using a sigmoid function. The maximum value of this range ensures that a non-zero gradient will be available. The minimum value is set higher than 1 to make sure that the glimpse generator will not try to generate a full image layer. • The second term is a scalar which is assumed to predict the activation level α k of the object, which will be used to predict whether it is visible or not. We force this activation value to be positive using an exponential map. • The remaining coordinates form a vector z what k which codes for the appearance of the object. We then use a convolutional glimpse generator to build a color image o k of the associated object together with the associated scalar mask m k , using z what k as input. These images and masks are translated to the positions (x k , y k ) and scaled according to the inverse scaling factor s k using a spatial transformer network. We note L k and M k for k ∈ {1, .., K} the corresponding object image layers and masks, and L 0 the background image produced by the background model, so that we have a total of K + 1 image layers. We now have to decide for each pixel whether this pixel should show the background layer or one of the K object layers. In order to do this in a differentiable way, we multiply the predicted object masks M k with the associated object activation levels α k , and normalize the results to get one normalized weights distribution (w k ) 0≤k≤K per pixel: w k (i, j) = α k M k (i, j) k ∈0..K α k M k (i, j) ,(17) considering that the mask M 0 associated to the background is set to 1 everywhere and that it has a fixed learned activation factor α 0 . The final reconstructed imageX is then equal to the weighted sum of the various image layers using the weights w k :X (i, j) = K k=0 w k (i, j)L k (i, j)(18) During inference the segmentation map is built by assigning to each pixel the layer index k ∈ {0, .., K} for which w k (i, j) is the maximum. The background model is not needed to get the segmentation maps during inference. Model training 4.2.1 loss function In order to train the proposed model, we use a main reconstruction loss function and an auxiliary loss: reconstruction loss The local L 1 reconstruction error associated to the pixel (i, j) is l i,j = 3 c=1 |x c,i,j − x c,i,j |,(19) where x c,i,j andx c,i,j are the values of the color channel c at the position (i, j) in the input image and reconstructed image. The reconstruction loss is defined as the mean square of this reconstruction error. L rec = 1 hw w,h i=1,j=1 l 2 i,j(20) pixel entropy loss For a given pixel (i, j), we expect the distribution of the weights w 0 (i, j), .., w K (i, j) to be onehot, because we assume that the objects are opaque. We observe that a discrete distribution is one-hot if and only if it has a zero entropy, so that minimizing the entropy of this distribution would be a reasonable way to enforce a stick-breaking process. Considering however that the entropy function has a singular gradient near one-hot distributions, we use the square of the entropy function to build the loss function. We then define the pixel entropy loss as L pixel = 1 hw w,h i=1,j=1 ( K k=0 w k (i, j) log(w k (i, j) + )) 2 ,(21) where = 10 −20 is introduced to avoid any numerical issue with the logarithm function. This auxiliary loss is weighted using the weight λ pixel before being added to the reconstruction loss. During our experiments, we observed that the pixel entropy loss could prevent a successful initialization of the localization process during the beginning of the training. As a consequence, we smoothly activate this auxiliary loss during initialization using a quadratic warmup of the weight. The full loss function is then equal to L = L rec + min(1, step N pixel ) 2 λ pixel L pixel ,(22) where step is the current training iteration index and N pixel is a fixed hyperparameter. curriculum training The interaction between the background reconstruction model and the foreground model during training is a very challenging issue, because of the competition between them to reconstruct the image. We handle this problem as in [25] by implementing curriculum training. We will then evaluate two methods to train the proposed model: • baseline training (BT) : The background and foreground models are initialized randomly and trained simultaneously. • curriculum training (CT): The training of the model is split in three phases : 1. The background model is pretrained alone, using the methodology and robust loss function described in [38]. 2. The weights of the background model are then frozen and the foreground model is trained using the frozen background model. 3. The background and foreground models are then fine-tuned simultaneously. Experimental results Evaluation on public benchmarks We perform a quantitative evaluation of the proposed model on the following datasets: CLEVRTEX [29], CLEVR [27], ShapeStacks [21] and ObjectsRoom [28]. We implement on ShapeStacks, ObjectsRoom and CLEVR the same preprocessing as in [15]. We use the same hyperparameter values on these datasets, except for the hyperparameter K related to the number of object queries, which is set to the maximum number of objects in each dataset (i.e. 3 on ObjectsRoom, 6 on ShapeStacks and 10 on CLEVRTEX and CLEVR). We use isotropic scaling on CLEVR and ShapeStacks and anisotropic scaling on the other datasets. We use the versions B3 of the Segformer model, and rely on the Hugging Face implementation of this model, with pretrained weights on ImageNet-1k for the hierarchical transformer backbone, but random initialization for the MLP decoder which is used as a feature map generator. We use the standard Pytorch implementation of the transformer encoder. The architecture of the backgroud model autoencoder is the same as in [38]. The glimpse generator is a sequence of transpose convolution layers, group normalization [48] layers and CELU [2] non-linearities, and is described in the supplementary material. We use Adam as optimizer. The training process includes a quadratic warmup of the learning rate since the model contains a transformer encoder. We also decrease the learning rate by a factor of 10 when the number of training steps reaches 90% of the total number of training steps. The total number of training steps of the baseline training (BT) scenario is 125 000. In the curriculum training (CT) scenario, the number of training steps for background model pretraining (phase 1) is 500 000 on CLEVRTEX, ShapeStacks and ObjectsRoom, but 2500 on CLEVR, which shows a fixed background, as recommended in [38]. The number of training steps of phase 2 (training with frozen pretrained background model) is 30 000, and the number of training steps of the final fine-tuning phase (phase 3) is 95 000. Full implementation details and hyperparameter values are provided in the supplementary material, and the model code will be made available on the Github platform. In order to compare our results with published models, we compute the following evaluation metrics: mean intersection over union (mIoU) and adjusted rand index restricted to foreground objects (ARI-FG). We also provide the mean square error (MSE) between the reconstructed image and the input image, which provides an estimate of the accuracy of the learnt representation. We use the same definitions and methodology as [29] for these metrics. We provide the mean segmentation covering (defined in [16]) restricted to foreground objects (MSC-FG) on ObjectsRoom and ShapeStacks where mIoU baseline values are not available. We call AST-Seg (Attention and Soft-argmax with Transformer using Segformer) the proposed model, and AST-Seg-B3-BT, AST-Seg-B3-CT respectively the models using a Segformer B3 feature map generator trained under the baseline training or curriculum training scenarios. Table 1 and 2 provide the results obtained on these datasets with a comparison with published results. The proposed model trained under the baseline training scenario gets better average results than existing models on the CLEVR and CLEVRTEX dataset, but shows a very high variance. For example, on the CLEVR dataset, the model may fall during training in a bad minimum where the background model tries to predict the foreground objects. Using curriculum training allows to avoid this issue, get stable results on all datasets, and obtain a very significant mIoU improvement on the most complex datasets CLEVR and CLEVRTEX. Following the methodology proposed in [29], we also evaluated the generalization capability of a model trained on CLEVRTEX when applied to datasets containing out of distribution images showing unseen textures and shapes or camouflaged objects (OOD and CAMO datasets [29]). The results of this evaluation are provided in Table 3 and show that the proposed model generalizes well, although it is deterministic and does not use any specific regularization scheme. Some segmentation prediction samples are provided in Ablation study and additional experiments We provide in Table 4 results obtained using various ablations or modifications on the model architecture or loss function, which show that: • The model remains competitive if the transformer encoder is removed by setting (φ k , x k , y k ) 1≤k≤K = (φ 0 k , x 0 k , y 0 k ) 1≤k≤K . The results on the ShapeStacks and ObjectsRoom datasets are even improved with this simplified architecture, with a surprisingly strong improvement on the Shapestacks dataset, which shows the efficiency of the attention and soft-argmax mechanism. The transformer encoder is however necessary on the more complex CLEVR and CLEVRTEX datasets. • Training with a number of slots slightly higher than the maximum number of objects does not lead to significant changes in the results. A more substantial increase of the number of slots however leads to poor results on scenes with complex textures due to the increasing fragmentation the objects. This is very different from the situation observed on query-based supervised detection models like DETR, where the number of queries has to be very high compared to the number of objects. • It is possible to replace the Segformer high resolution feature map generator with any other generator. The proposed model was originally designed with a custom Unet feature map generator, which gets similar results as the Segformer model on CLEVR, ShapeStacks and ObjectsRoom, but underperforms on the more complex CLEVRTEX dataset. The architecture of this Unet is described in the supplementary material. • Using a pretrained backbone is necessary to get good performances with a Segformer feature map generator. • We tested an alternative training scenario where the background model remains frozen during the complete training of the foreground model (125 000 iterations). The main advantage of this scenario is that it is significantly faster and requires less memory, since the backgrounds of the training images can be pre-computed and memorized. The accuracy of the results is however lower than the curriculum training scenario proposed in this paper, except for the ObjectsRoom dataset. Table 2. Benchmark results on ObjectsRoom and ShapeStacks. Source: [15]. Model ObjectsRoom ShapeStacks ↑ARI-FG (%) ↑MSC-FG (%) ↑mIoU (%) ↓MSE ↑ARI-FG (%) ↑MSC-FG (%) ↑mIoU (%) ↓MSE MONet-g [4,15] 54 ± 0 33 ± 1 n/a n/a 70 ± 4 57 ± 12 n/a n/a Gen-v2 [15] 84 ± 1 58 ± 3 n/a n/a 81 ± 0 68 ± 1 n/a n/a SA [33] 79 ± 2 64 ± 13 n/a n/a 76 ± 1 70 ± 5 n/a n/a Computation time All experiments have been performed using a Nvidia RTX 3090 GPU and a AMD 7402 EPYC CPU. Some training durations are provided in Table 5. Conclusion We have described in this paper a new architecture for unsupervised object-centric representation learning and object detection and segmentation, which relies on attention and soft-argmax, and shown that this new architecture substantially improves upon the state of the art on existing benchmarks showing synthetic scenes with complex shapes and textures. We hope this work may help to extend the scope of structured object-centric representation learning from research to practical applications. Supplementary Material Hyperparameter values The hyperparameter values used for the proposed model are listed in Table 6. Pseudo-code for objects encoder and decoder The full encoding and rendering process is described in Algorithms 1 and 2. Additional implementation details The glimpse convolutional generator is described in Table 7. Synthetic datasets and preprocessing codes were downloaded from the following public repositories: • https://www.robots.ox.ac.uk/˜vgg/ data/clevrtex/ • https://ogroth.github.io/ shapestacks/ The Segformer pretrained weights were downloaded from the following link: https://huggingface.co/nvidia/mit-b3 The architecture of the U-net implemented for the ablation study is described in Table 8. It contains a sequence of downsample blocks which output feature maps of decreasing sizes, a center block which takes as input the feature map produced by the last downsample block, and upsample blocks, which take as input both the output of the previous upsample or center block and the feature map of the same size produced by corresponding downsample block. • A downsample block is composed of a convolutional layer with stride 2 and kernel size 4, with batch normalization and CELU, followed by a residual convolutional layer with stride 1 and kernel size 3 with batch normalization and CELU. • The center block is composed of a convolutional layer with stride 1 and kernel size 3 with batch normalization and CELU. • An upsample block is composed of a residual convolutional layer with stride 1 and kernel size 3 with batch normalization and CELU, followed by a transpose convolutional layer with stride 2 and kernel size 4, with batch normalization and CELU. Additional image samples Fig 2 . 2Other image samples are available in the supplementary material. The main limitation of the proposed model is the management of shadows, which may be considered by the model as separate objects or integrated to object segmentations. Figure 3 . 3Examples Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 . 45678Examples Examples Examples of segmentation predictions on ShapeStacks test dataset (using a model without transformer) Examples of segmentation predictions on CAMO test dataset using a model trained on CLEVRTEX only Examples of segmentation predictions on OOD test dataset using a model trained on CLEVRTEX only Table 1 . 1Benchmark results on CLEVR and CLEVRTEX. Results are shown (±σ) calculated over 3 runs. Source: [29] Model CLEVR CLEVRTEX ↑mIoU (%) ↑ARI-FG (%) ↓MSE ↑mIoU (%) ↑ARI-FG (%) ↓ MSE SPAIR [10] 65.95 ± 4.02 77.13 ± 1.92 55± 10 0.00 ± 0.00 0.00 ± 0.00 1101 ± 2 SPACE [32] 26.31 ± 12.93 22.75 ± 14.04 63± 3 9.14 ± 3.46 17.53 ± 4.13 298 ± 80 GNM [25] 59.92 ± 3.72 65.05 ± 4.19 43± 3 42.25 ± 0.18 53.37 ± 0.67 383 ± 2 MN [39] 56.81 ± 0.40 72.12 ± 0.64 75± 1 10.46 ± 0.10 38.31 ± 0.70 335 ± 1 DTI [35] 48.74 ± 2.17 89.54 ± 1.44 77± 12 33.79 ± 1.30 79.90 ± 1.37 438 ± 22 Gen-V2 [15] 9.48 ± 0.55 57.90 ± 20.38 158± 2 7.93 ± 1.53 31.19 ± 12.41 315 ± 106 eMORL [14] 50.19 ± 22.56 93.25 ± 3.24 33± 8 12.58 ± 2.39 45.00 ± 7.77 318 ± 43 MONet [4] 30.66 ± 14.87 54.47 ± 11.41 58± 12 19.78 ± 1.02 36.66 ± 0.87 146 ± 7 SA [33] 36.61 ± 24.83 95.89 ± 2.37 23± 3 22.58 ± 2.07 62.40 ± 2.23 254 ± 8 IODINE [19] 45.14 ± 17.85 93.81 ± 0.76 44± 9 29.17 ± 0.75 59.52 ± 2.20 340 ± 3 AST-Seg-B3-BT 71.92 ± 32.94 76.05 ± 36.13 51± 63 57.30 ± 15.72 71.79 ± 22.88 152 ± 39 AST-Seg-B3-CT 90.27 ± 0.20 98.26 ± 0.07 16± 1 79.58 ± 0.54 94.77 ± 0.51 139 ± 7 Table 3 . 3Benchmark generalization results on CAMO, and OOD for a model trained on CLEVRTEX.Results are shown (±σ) calculated Table 4 . 4Results of ablation study and additional experiments (results over 1 run, except for starred values, which are averages over 3 runs) Dataset CLEVRTEX CLEVR ShapeStacks ObjectsRoom mIoU ARI-FG mIoU ARI-FG mIoU ARI-FG mIoU ARI-FG full model AST-Seg-B3-CT (reference) 79.58* 94.77* 90.27* 98.26* 78.84* 79.34* 85.02* 87.23* model without transformer encoder 75.69 94.41 77.16 93.09 82.99* 82.29* 85.51* 88.49* K = 1 + maximum number of objects 79.11* 94.78* 91.03* 98.17* 78.87 80.05 82.90 86.45 K = 2 × maximum number of objects 62.10 89.96 90.56 98.29 54.88 65.16 66.78 78.58 using a Unet instead of Segformer feature generator 66.82 88.25 90.70 98.17 75.51 77.78 85.59 87.93 random initialization of Segformer backbone 61.74 80.22 88.94 97.77 62.73 68.40 77.71 79.23 training without pixel entropy loss 70.18 91.81 85.54 96.09 52.17 60.08 84.21 86.19 training using frozen pretrained background model 75.30 95.31 81.46 98.29 55.06 66.24 85.82 87.78 isotropic scaling 78.68 94.78 84.91 87.20 anisotropic scaling 87.21 98.53 45.47 36.43 Table 5. Training computation time with one Nvidia RTX 3090 GPU (curriculum training) Dataset image size background model pretraining (phase 1) full model training (phase 2 & 3) number of iterations training time number of iterations training time CLEVRTEX 128 × 128 500000 57 h 47 mn 125000 16 h 00 mn CLEVR 128 × 128 2500 20 mn 125000 12 h 03 mn ObjectsRoom 64 × 64 500000 14 h 57 mn 125000 6 h 31 mn ShapeStacks 64 × 64 500000 14 h 20 mn 125000 6 h 22 mn Table 6 . 6Hyperparameter values number of background model training iterations: -datasets with fixed backgrounds (CLEVR) 2500 -datasets with complex backgrounds (CLEVRTEX, ShapeStacks,ObjectsRoom) 500000 maximum value of inverse scaling factor s max 24 dimension of inputs and outputs of transformer encoder d T 256 number of heads of transformer encoder layer 8 dimension of feedforward transformer layer 512 number of layers of transformer encoder 6• https://github.com/deepmind/multi_ object_datasets • https://github.com/applied-ai-lab/ genesis.hyperparameter description Algorithm 1: EncodingInput: input image X Output: object latents {z what k , x k , y k , s k , α k } 1≥k≥K // feature and attention maps generation (Φ, A 1 , .., A K ) = Segformer(X)end // computation of positions and feature vectors before transformer refinement Attend, infer, repeat: Fast scene understanding with generative models. 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[ "https://github.com/deepmind/multi_", "https://github.com/applied-ai-lab/", "https://github.com/deepmind/multi-object-datasets/," ]
[ "Magnetoelectric coupling induced by interfacial orbital reconstruction", "Magnetoelectric coupling induced by interfacial orbital reconstruction" ]
[ "Bin Cui ", "Cheng Song [email protected] ", "Haijun Mao ", "Huaqiang Wu ", "Fan Li ", "Jingjing Peng ", "Guangyue Wang ", "Fei Zeng ", "Feng Pan ", "DrB C Cui ", "H J Song ", "F Mao ", "J J Li ", "G Y Peng ", "DrF Wang ", "Prof F Zeng ", "Pan ", "DrH Q Wu ", "\nSchool of Materials Science and Engineering\nInstitute of Microelectronics\nKey Laboratory of Advanced Materials (MOE)\nTsinghua University\n100084BeijingChina\n", "\nTsinghua University\n100084BeijingChina\n" ]
[ "School of Materials Science and Engineering\nInstitute of Microelectronics\nKey Laboratory of Advanced Materials (MOE)\nTsinghua University\n100084BeijingChina", "Tsinghua University\n100084BeijingChina" ]
[]
With the fast development of multifunctional devices, magnetoelectric (ME) couplinga bridge between magnetic and electric parameters plays a vital role in the integration and manipulation of microelectronics.[1][2][3][4][5][6][7]The artificial interface between ferromagnetic (FM) and ferroelectric (FE) oxides provides an ideal arena for electric-field control of magnetism with reduced energy consumption. Following the extensive studies, the proposed mechanisms responsible for the ME coupling could be summarized as follows: i) the electrostriction introduces strain variation in the FM layer, changing its lattice and concomitant magnetic properties;[8]ii) the delicate modulation of carrier density by polarization reversal in FE fieldeffect transistor motivates the FM/antiferromagnetic (AFM) phase transition;[9,10]iii) the use of multiferroic materials provides a route to the electrical control of spin arrangement by FM/AFM exchange coupling.[11][12][13]However the realization of electrostriction and exchange coupling needs high-quality thick FE layer (even FE substrates) and rare multiferroic
10.1002/adma201503115
[ "https://export.arxiv.org/pdf/1509.01798v1.pdf" ]
5,408,168
1509.01798
c820cc04fd024d5acbcb67619831c40f4081e6ba
Magnetoelectric coupling induced by interfacial orbital reconstruction Bin Cui Cheng Song [email protected] Haijun Mao Huaqiang Wu Fan Li Jingjing Peng Guangyue Wang Fei Zeng Feng Pan DrB C Cui H J Song F Mao J J Li G Y Peng DrF Wang Prof F Zeng Pan DrH Q Wu School of Materials Science and Engineering Institute of Microelectronics Key Laboratory of Advanced Materials (MOE) Tsinghua University 100084BeijingChina Tsinghua University 100084BeijingChina Magnetoelectric coupling induced by interfacial orbital reconstruction 1orbital reconstructionmagnetoelectric couplinginterface With the fast development of multifunctional devices, magnetoelectric (ME) couplinga bridge between magnetic and electric parameters plays a vital role in the integration and manipulation of microelectronics.[1][2][3][4][5][6][7]The artificial interface between ferromagnetic (FM) and ferroelectric (FE) oxides provides an ideal arena for electric-field control of magnetism with reduced energy consumption. Following the extensive studies, the proposed mechanisms responsible for the ME coupling could be summarized as follows: i) the electrostriction introduces strain variation in the FM layer, changing its lattice and concomitant magnetic properties;[8]ii) the delicate modulation of carrier density by polarization reversal in FE fieldeffect transistor motivates the FM/antiferromagnetic (AFM) phase transition;[9,10]iii) the use of multiferroic materials provides a route to the electrical control of spin arrangement by FM/AFM exchange coupling.[11][12][13]However the realization of electrostriction and exchange coupling needs high-quality thick FE layer (even FE substrates) and rare multiferroic 2 materials, respectively, while the carrier modulation only plays a significant role in the phase transition of materials near the FM/AFM critical points. Thus an elegant approach for the realization of ME coupling through a more generalized mechanism is intensely pursued. [14][15][16] The orbital occupancy determines the magnitude and anisotropy of the inter-atomic electron-transfer interaction and hence exerts a key influence on the electronic structure and magnetic ordering. [17] The interfacial covalent bond based on orbital reconstruction connects two distinct oxides at their interface, giving rise to novel electronic structure and performance. [18,19] In FE field-effect transistor, the interfacial orbital reconstruction could be effectively manipulated by the spatial movement of charged ion during polarization reversal, which is the essential behavior in all the FE materials without thickness limitation, thus providing a more generalized way for the ME coupling through orbital reconstruction. Here we realize the ME coupling by orbital reconstruction under electric field in model FE/FM heterostructures BaTiO 3 /La 2/3 Sr 1/3 MnO 3 (BTO/LSMO), which is expected to complete the mechanism accounting for ME coupling and to advance the development of orbital degree of freedom in the terrain of microelectronics. High-quality BTO (10 u.c.)/LSMO (t = 7, 10, 15, 25, and 50 u.c.) heterostructures were grown on SrTiO 3 (STO) substrates (Supporting Information Figure S1). The samples were then made into two types of devices (Supporting Information Figure S2 and Experimental section): a transistor device was used to carry out the transport measurements; while a film device was required for the synchrotron radiation measurements. A typical out-of-plane piezoresponse force microscopy (PFM) image of BTO (10 u.c.)/LSMO (15 u.c .) written by ±4 V is presented in Figure 1a, where the bright and dark regions stand for polarization upward and downward (P up and P down ), respectively. The in-plane compressive strain (Supporting Information Figure S3) guarantees the tetragonal BTO phase with good ferroelectricity. The as-grown BTO (outmost area of Figure 1a) exhibits a single domain with P up due to the polar discontinuity at the interface (Ref. 19). This is also concluded from the shift of local piezoelectric hysteresis and butterfly loops (Supporting Information Figure S4 and S5). The FE displacement of BTO measured in butterfly loops is around 15 pm (close to the theoretical value of 12 pm), [20] which would induce the orbital reconstruction of Mn in the heterostructure as described in Figure 1b. A highly overlapped out-of-plane orbital component (Figure 1c), strongly weakening the in-plane orbital occupancy. [21] The BTO in P up and P down states suppress and enhance the interfacial orbital hybridization, accompanied by the intensified and weakened Mn- By altering the orbital occupancy at the interface, FE polarization could modulate the relative weight of the interfacial channel with preferential Mn-2 2 y x d  occupancy. [22,23] The interfacial layer behaves a spin arrangement antiparallel to the bulk one, [22] and its modulation under electric field is bound to influence the magnetic and transport properties of LSMO. The temperature dependent resistance (R-T) curves of BTO/LSMO are displayed in Figure 1d. The thickness of interfacial layer is estimated to be around 3 u.c. for both BTO/LSMO and LSMO/STO interfaces according to the direct observations and transport measurements. [23,24] Thus the metallic-insulating phase separation in LSMO of 7 u.c. is remarkable, accompanied by distinctly semiconducting conductivity at low temperature. The resistance difference between P up and P down states is gradually eliminated by increased t and the ΔT C drops rapidly with ΔT C ∝ 1/t, reflecting the field effect is limited to the interface. It should be mentioned here that the existence of dead layer at the interface between LSMO and substrates shrinks the effective bulk thickness and enlarges the amplitude of resistance and T C variations, especially in the ultrathin films (7-15 u.c.). Nevertheless it keeps unchanged during the polarization reversal, thus it is not considered in our work. We now turn to directly explore the effect of the FE polarization on the interfacial magnetic properties by measuring the angular magnetoresistance (MR) of heterostructures at 10 K with magnetic field (H) rotating out-of-plane in Figure 3. The angle between magnetic field and normal direction of substrate is defined as θ, where θ = 0° (180°) and 90° (270°) correspond to the field perpendicular to and in the film plane, respectively (Supporting Information Figure S2). Remarkably, besides the typical anisotropic MR for FM manganites, a novel MR with maxima for the in-plane magnetic fields (referred to as pMR) is superimposed in the normalized angular MR [ Figure 3a (H = 5 T), 3b (H = 9 T) and 6 Supporting Information Figure S6]. The most eminent feature observed here is that the heterostructures with up-polarized BTO exhibit stronger pMR peaks [pMR = ( 0 90 R -R min )/ 0 0 R ] than those with the down-polarized ones: the pMR values of P up /P down are 0.070%/0.046% and 0.058%/0.034% for LSMO of 10 u.c. and 15 u.c. at 9 T, respectively. The pMR is highly related to the preferential x 2 -y 2 occupation at the interfacial layer under tensile strain, where the transport properties are dominated by the bands around the Fermi energy (3z 2 -r 2 orbital). The onsite energy on the 3z 2 -r 2 orbital increases with θ from 0° (outof-plane) to 90° (in-plane), which would narrow the band around the Г point and lead to a decreased velocity (increased resistance) for θ = 90° (pMR) under a magnetic field. [22] Then the variation of pMR peak with the switching of the FE polarization could be understood in terms of the orbital reconstruction: compared with the case of down-polarized state, the enhancement of Mn- The dependences of pMR on magnetic field and LSMO thickness are summarized in Figure 3c and 3d, respectively, where both the increased H and decreased t enlarge the pMR values. The height of the pMR peak rises with the enlarged magnetic field due to the band narrowing effect as shown in Figure 3c. The pMR value of P up state at 9 T rapidly reduces from 0.070% to 0.002% with the increase of LSMO thickness from 10 u.c. to 50 u.c. in Figure 3d, originating from the decreased relative weight of the interfacial layer. As a result, the difference between pMR values of P up and P down states is negligible in the thick film of 50 u.c., where the contribution of varied interfacial layer to the transport property is covered by the bulk signal. This electrical manipulation of transport properties is reversible (Supporting Information Figure S7). A XLD is, the larger relative occupancy of 2 2 y x d  ( 2 2 3 r z d  ) is. [25] The XLD results for LSMO thicker than 15 u.c. whose T C > 300 K are influenced by the FM signal and not shown here. [26] As LSMO grown on STO substrate is under tensile strain, the preferred orbital is (Figure 4d), respectively, for both scenarios of P up and P down . This can be explained by the fact that the in-plane preferential orbital occupancy introduced by STO substrate is reduced as the thickness of LSMO increases (Figure 4f). The dependence of valence and orbital occupancy on LSMO thickness in turn affirms that the modulation of electronic structure comes from the interface area. In previous studies, the role of 3d orbital in electric control of magnetism was investigated theoretically, [27][28][29][30] whereas our finding experimentally demonstrates the ME coupling induced by orbital reconstruction. The role of exchange coupling in ME coupling could be easily excluded here by the absence of multiferroic materials. And the role of strain in ME coupling could be ruled out by the following aspects: i) the ultrathin FE layer weakens the clamping effect of strain produced by polarization; ii) the in-plane lattices of (001) BTO are almost the same in P up and P down states. [31] However, it is generally accepted that the FE polarization would tune the carrier density of LSMO channel, which might also contribute to the changes in magnetic and electric performance. [3,9,10] Meanwhile, the entanglement between orbital reconstruction and carrier modulation makes it difficult to absolutely separate the role of orbital in this ME coupling. Figure 5a, the distance between Mn and Ti along the normal of substrate is enlarged without a connection between O. Thus the pMR peak of (110) heterostructure is higher than that of the (001) one due to the robust in-plane Mn orbital occupancy without the out-of-plane orbital hybridization, affirming the origin of pMR in turn. However, the FE poling of BTO (Supporting Information Figure S9) has a negligible influence on the pMR peak as shown in Figure 5b, because the orbital occupancy is stable in different polarization states. In addition, the small change of carrier density in La 2/3 Sr 1/3 MnO 3 (a robust and optimal FM phase) induced by polarization cannot obviously affect its electric properties without the assistance of orbital reconstruction (Supporting Information Figure S10). These observations in (110) heterostructure exclude the influence of carrier density variation on magnetotransport and reaffirm that the ME coupling observed in the (001) case is mainly caused by the orbital reconstruction rather than the carrier density change. Besides, we emphasize that the orbital reconstruction plays a dominant role in the ME coupling for the following reasons: i) the modulation of pMR signal is the direct reflection of orbital reconstruction under electric field; [22] ii) the delicate change of carrier density in the robust FM La 2/3 Sr 1/3 MnO 3 phase by polarization switching cannot obviously affect its magnetic and electric properties alone. [9,10,32,33] In this work, two additional elements might enhance the ME coupling effect: phase separation and oxygen vacancy migration. The metallic-insulating phase separation is widely accepted in ultrathin LSMO. As the LSMO were grown on STO with subtle tensile strain, the separated insulating phase should be x 2 -y 2 orbital ordering, which would increase the amount of controllable orbital and promote the manipulation effect additionally. Thus, it is reasonable to expect a larger ME coupling in Pr(Ca,Sr)MnO 3 , and (La,Ca)MnO 3 with a stronger phase separation tendency. [16,33] On the other hand, the inevitable oxygen vacancies with positive charges in oxides would migrate towards and away from the interface in the P up and P down states, respectively. [34,35] As the Ti-O-Mn bonding is based on the orbital hybridization among Ti, O, and Mn ions, [36] the P down state with less oxygen vacancies at the interface would stabilize the bonding and orbital reconstruction, while the P up state shows the opposite behavior. In summary, the interfacial orbital reconstruction of BTO/LSMO is driven by the shuttle displacement of Ti ion under FE polarization, displaying a direct and crucial role in manipulating the magnetism of heterostructures. With BTO polarized upward (downward), the orbital hybridization between Ti and Mn is suppressed (enhanced), followed by the interfacial preferential Mn- formation/distortion also provides a possible route to the reversible manipulation of novel magnetic state produced by the interfacial bonding. [21,23] Our findings not only present a broad opportunity to fill the missing member-orbital in the mechanism of magnetoelectric coupling, but also make the orbital degree of freedom straight forward to the application in microelectronic device. Experimental Section High-quality BTO (10 u.c.)/LSMO (t = 7, 10, 15, 25, and 50 u.c.) heterostructures were grown in layer-by-layer mode on TiO 2 -terminated STO substrates by pulsed layer deposition (PLD) from stoichiometric La 2/3 Sr 1/3 MnO 3 and BaTiO 3 targets. The LSMO was grown at 700 ℃ with an oxygen pressure of 200 mTorr while BTO was grown at 750 ℃ with an oxygen pressure of 4 mTorr. The growth was monitored in situ by RHEED (reflection highenergy electron diffraction) analysis, allowing the precise control of the thickness at the atomic level. Two types of devices were used in this work (Supporting Information Figure S2): a transistor device with a gate located in the vicinity of the channel was used to carry out the transport measurements in the physical property measurement system; while a film device of 2.5 mm × 5 mm was required for the XAS and XLD measurements in total electron yield mode at Beamline BL08U1A at Shanghai Synchrotron Radiation Facility. [33] A gate voltage of ±4 V was applied on the sample to switch the polarization and then removed before the measurements. The PFM measurements and ferroelectric switching were carried out by Cypher (Asylum Research). Supporting information Film growth Epitaxial heterostructures BaTiO 3 (BTO)/La 2/3 Sr 1/3 MnO 3 (LSMO) with different BTO and LSMO thicknesses were grown on (001) and (110) SrTiO 3 (STO) substrates. The growth dynamics was investigated by monitoring the intensity variations of the RHEED (reflection high-energy electron diffraction) patterns on (001) STO substrate. Figure S1 Device preparation There are two series of devices in this work. The transistor devices are used for the measurements of resistance versus temperature (R-T) curves and angular magnetoresistance (MR). The film was patterned into a transistor structure with a gate electrode located in the vicinity of channel by photo-lithography and wet etching as shown in Figure S2a. The effective size of the channel is 100 µm wide. A hard-baked photoresist covers the whole area except for the parts of electrodes and channel for electrical isolation between the gate and channel. Here the exposed channel is 70 µm wide, which is smaller than the whole one for preventing the leakage current at the side edge. A drop of silver paste is then used to connect the exposed channel and gate electrode. The rotation axis in the angular MR measurement is marked by the arrow in Figure S2a, while the magnetic field (H) is fixed. The film device with a large area of 2.5 mm × 5 mm is used for the measurements of x-ray absorption spectroscopy (XAS) and x-ray linear dichroism (XLD), where a metal-probe setup with a 50 µm tip was used to switch the polarization of the whole sample as shown in Figure S2b. [S1] A specific voltage is applied between gate electrode and heterostructure. Ferroelectricity of BaTiO 3 and its origin The x-ray diffraction (XRD) taken from BTO (200 u.c.)/LSMO (50 u.c.) on TiO 2terminated (001) STO substrate in Figure S3 Figure S4b, which exhibits a FE displacement of ~15 pm at ±4 V (close to the theoretical value of 12 pm [S3] ). Both the piezoresponse hysteresis and D-V butterfly loops shift toward the negative direction, reflecting a spontaneously upward polarization, due to the particular polar discontinuity in -(La,Sr)O-MnO 2 -BaO-TiO 2 -at the top interface of the heterostructures. [S4] These results also suggest that a gate voltage of ±4 V is large enough to switch the polarization. Figure S5b, reflecting the remnant polarization state of the film after +4 V poling, which is opposite to the case of as-grown one in Figure S5a. It is noteworthy that the phase contrast remains after repetitive measurements for many times. This result suggests that the approach we used to polarize the whole film is effective due to the high-quality of heterostructure under whole layer-by-layer growth mode, guaranteeing the ex situ XAS and XLD measurements. after polarization with +4 V. The angular magnetoresistance for BTO/LSMO The raw angular MR measurement results by rotating the magnetic field out-of-plane are shown in Figure S6. The normalized curves in Figure 3a and 3b are obtained by setting the difference between the maximum and minimum points in Figure S6 to be one. In contrast to the classic anisotropic MR peaks at θ = 0° and 180° for ferromagnetic manganites, the MR peaks due to the interfacial x 2y 2 preferential orbital occupancy called pMR (in-plane angular magentoresistance) are located at the angles of 90° and 270° (Ref. S5). Both the decrease of LSMO thickness and increase of measured field would enhance the pMR peak, as the former increases the interfacial proportion and the latter induces the narrowing of band structure. More importantly, the pMR peak could be manipulated by the polarization of BTO, that the BTO of P up state exhibits larger pMR values than P down due to the enhancement of interfacial x 2y 2 occupancy. Original XAS and XLD signals for all the samples The XAS and corresponding XLD are currently of great interest due to its promise of providing the ground-state electronic structure and orbital ordering with a high element resolution. The XAS spectra of Mn were normalized by dividing the spectra by a factor such that the L 3 pre-edge and L 2 post-edge have identical intensities for the two polarizations. After that, the pre-edge spectral region was set to zero and the peak at the L 3 edge was set to one. XLD, the difference between the two measurements (I ab -I c ), gives direct insight of the relative occupancies of 3z 2r 2 [P(3z 2r 2 )] and x 2y 2 [P(x 2y 2 )] orbital: the more negative (positive) A XLD is, the larger relative occupancy of r z d  ) is. [S6] The comparison of XLD in Figure S8 presents the information for the orbital occupancy of BTO/LSMO with different LSMO thicknesses under P up and P down . The A XLD is negative for the case of P up , suggesting that the preferential orbital occupancy of LSMO is x 2y 2 , while BTO of P down favors the 3z 2r 2 occupancy, regardless of the LSMO thickness. The variation in orbital occupancy is direct clue for the modulation of orbital hybridization under FE polarization. Note that the proportion of 3z 2r 2 orbital increases with the LSMO thickness as the tensile strain from substrate is gradually relaxed. Figure S8 Normalized XAS and XLD (the intensity is multiplied by 5 times) of BTO/LSMO heterostructures with different LSMO thicknesses and polarization states: 7 u.c. a) P up and b) P down , 10 u.c. c) P up and d) P down , and 15 u.c. e) P up and f) P down . The control experiment carried out on the (110) heterostructure The preparation of BTO/LSMO heterostructures on (110) STO in this control experiment is identical with the (001) BTO (10 u.c.)/LSMO (15 u.c.), as they were grown simultaneously in the same chamber. The (110) heterostructure exhibits obvious FE properties according to the out-of-plane PFM image in Figure S9a, although the contrast of this image is not as sharp as that of (001) one. However the sketch of (110) heterostructure in Figure S9b In contrast to the case of (001) sample, the resistance and Curie temperature (T C ) of (110) heterostructures are almost independent of the polarization state of BTO as shown in Figure S10a. A gate voltage of ±6 V was used to switch the polarization in (110) heterostructures. It is mainly because that the stoichiometric of LSMO is La 2/3 Sr 1/3 MnO 3 , at the center of FM phase region in the phase diagram, which should not be sensitive to the merely small changes of carrier density without variation of orbital occupancy under different polarization states. The (110) heterostructures exhibit stronger pMR peaks at 90° and 270° compared with the case of (001) in Figure S10b and S10c, as the interfacial in-plane x 2y 2 orbital occupancy is enhanced without out-of-plane orbital hybridization. Furthermore, the pMR peaks are stable as the polarization state of BTO switches from P up to P down . These results obtained from (110) heterostructure illustrate that the importance of orbital modulation in controlling the magnetism of metallic LSMO by FE polarization. Although the lattice distortions in (001) and (110) heterostructures might be somehow different, the magnetic and conductive properties of these two systems at P down states are quite close ( Figure S10a). The carrier density modulated by vertical upward/downward polarization (Figure S9a), if any, should be the same for the 25 two samples. Thus the control experiment provides additional evidence that orbital reconstruction accounts for the ME coupling. Figure S10 a) The temperature dependence of the resistance curves of (001) and (110) heterostructures with different polarization states. The angular MR for (110) BTO/LSMO under b) P up and c) P down states. Figure 1 a 1) Typical out-of-plane PFM images of BTO/LSMO heterostructure on TiO 2terminated (001) STO substrate. b) Sketch for the interfacial orbital reconstruction under different polarization states. The favored orbitals are dense while the disfavored ones are washy. c) Schematic for the orbital hybridization between Ti, O, and Mn. (d) Schematic of the LSMO channel composited by interfacial and bulk layer under P up and P down . d  orbital occupancy, respectively. The interfacial termination should not change the effect of polarization on interfacial orbital reconstruction as 4 both the (La,Sr)O and BaO could supply O ions for the Ti-O-Mn bonding. Here the LSMO layer could be treated as two parallel channels of bulk and interface, as shown in Figure 1d. Figure 2a . 2aThe Curie temperature (T C ) obtained from the metal-insulator transition temperature in Figure 2a and ΔT C = T C (P down ) -T C (P up ) are summarized as a function of LSMO thickness (t) in the left and right axis of Figure 2b, respectively. We firstly focus on the case of 7 u.c. LSMO. Compared with the case of P up , the resistance of LSMO with P down BTO decreases and the metallic conductive region is distinctly extended, followed by a T C enhancement of ~17 K. The improvement of electric and magnetic performance under P down state originates from the thinner interfacial channel with the depopulated Mn Figure 2 a 2) R-T curves for BTO (10 u.c.)/LSMO (t = 7 u.c., 10 u.c., 15 u.c., 25 u.c., and 50 u.c.) transistor devices. The solid curves are for P up and dot ones for P down . b) The T C of devices in P up and P down states (left axis) and their difference (ΔT C , right axis) as a function of LSMO thickness (t). orbital occupancy under up-polarized state favors the pMR. Figure 3 a 3) and b), Normalized angular MR for BTO (10 u.c.)/LSMO (t = 7 u.c., 10 u.c., 15 u.c., 25 u.c., and 50 u.c.) heterostructures at a magnetic field of 5 T and 9 T, respectively. The resistances are normalized with respect to (R -R min )/(R max -R min ). c) The pMR values as a function of magnetic field for LSMO of different thicknesses. d) The dependence of pMR values under P up and P down on LSMO thickness at 9 T.The electronic structures with different polarization states are then demonstrated by x-ray absorption spectroscopy (XAS) and x-ray linear dichroism (XLD) measurements at 300 K. Figure 4a 4ais the XAS L-edge of Ti. The energy differences between the two main peaks of the L 3 (ΔE L 3 ) and L 2 (ΔE L 2 ) are displayed as a function of the LSMO thickness in Figure 4b. All the ΔE L 3 and ΔE L 2 values of samples with different polarization states and LSMO thicknesses are smaller than that of the reference sample of STO substrate, implying the valence lower than Ti 4+ (Ref. 21). The extra electrons in Ti ions arise from Mn 3+ through the orbital hybridization. Remarkably, the Ti in the P down state exhibits a lower valence than that in P up one with the same LSMO thickness, indicating a robust orbital hybridization for the P down case due to larger orbital overlap. On the contrary, the L 3 peak of Mn (marked by the black triangle) shifts toward higher energies as the polarization state changes from up to down (Figure 4c and 4d). Such a shift suggests a higher Mn valence in LSMO, exactly in consistent with the lower Ti valence. The Mn-XLD curves in Figure 4e directly illustrate the variation of Mn orbital occupancy under different polarization states (more details in Supporting Information Figure S8). The area under XLD signal (I ab -I c ) around L 2 peak (647.5 eV-660.0 eV) (A XLD ) highlighted in Figure 4e is used to represent the relative orbital occupancy: the more negative (positive) the negative A XLD in the case of P up with the weak orbital hybridization.Conversely, the A XLD is positive is stabilized as the hybridization of out-of-plane orbital is enhanced with closer distance between Mn and Ti under P down state. The orbital occupancy altered by FE polarization are in line with the corresponding variation of occupancy, strongly supporting our understanding of ME coupling through the orbital reconstruction.With the increase of LSMO thickness, the ΔE L 3 and ΔE L 2 of Ti XAS reduces while MnL 3 peak position shifts to the higher energy direction, implying the gradual fall of Ti valence (Figure 4b) and the rise of Mn valence Figure 4 4Normalized a) Ti-XAS, c) Mn-XAS, and e) Mn-XLD for heterostructures with different LSMO thicknesses and polarization states (solid curves for P up and dot curves for P down ). The peak positions of Ti L 3 and L 2 edge are marked by solid and dashed straight lines and Mn L 3 peak is denoted by black triangle. b) The ΔE L 3 and L 2 in Ti-XAS, d) L 3 peak position in Mn-XAS and f) the A XLD of Mn-XLD as a function of LSMO thickness. Figure 5 a 5) Sketch for the atomic arrangement of (110) heterostructure. b) Normalized angular MR for (110) BTO/LSMO with different polarization states under magnetic fields ranging from 1 T to 9 T. We then design a control experiment using heterostructures on (110) STO substrate, which was simultaneously prepared with the (001) BTO (10 u.c.)/LSMO (15 u.c.), to exclude the interference from the modulation of carrier density under polarization reversal to some extent. On the basis of the atomic arrangement along [110] direction in rather than the changes in carrier density, in P up state increases the pMR values with suppressed magnetic and electric performances of FM metallic LSMO, occupancy of P down state does the opposite. This FE controlled bonding shows the typically clear RHEED oscillations recorded during the growth of BTO (10 u.c.)/LSMO (15 u.c.) and BTO (10 u.c.)/LSMO (50 u.c.), suggesting a continued layer-by-layer growth of heterostructures and -(La,Sr)O-MnO 2 -BaO-TiO 2 -top interface termination. The growth rates of LSMO and BTO are 0.77 nm/min and 0.60 nm/min, respectively. Figure S1 S1RHEED intensity oscillations of a) BTO (10 u.c.)/LSMO (15 u.c.) and b) BTO (10 u.c.)/LSMO (50 u.c.) grown on (001) TiO 2 -terminated STO substrate. Figure S2 a) The schematic of transistor device, where the source, drain, and gate electrodes are denoted as S, D, and G, respectively. The straight and bent arrows indicate the axis and direction of rotation, respectively. b) Sketch of the film device. reveals a c-axis orientation with no detectable impurity phase. The peak position of LSMO (002) at 47.92° demonstrates that a small inplane tensile strain of ~0.52% exists in the LSMO layer. Meanwhile the 2-theta angle for BTO (002) peak is around 44.17°, corresponding to c-axis lattice constant of 4.098 Å. This suggests that the BTO film undergoes in-plane compressive strain, producing a tetragonal ferroelectric (FE) phase. [S2] Figure S3 XRD data of the BTO (200 u.c.)/LSMO (50 u.c.) heterostructure on (001) STO substrate. The FE character of 10 u.c. BTO on 15 u.c. LSMO is confirmed by piezoresponse force microscopy (PFM) probe. The typical local piezoresponse hysteresis loops (Figure S4a) and well-shaped displacement-voltage (D-V) butterfly loops (Figure S4b) were measured in the as-grown area of out-of-plane PFM images in Figure 1a. The square hysteresis loops with sharply 180° FE switching in Figure S4a indicate that the heterostructure is of very high quality and possesses good FE properties. A positive voltage triggers the FE polarization towards the LSMO bottom layer, whereas a negative voltage polarizes the BTO upward. The FE displacement could be roughly estimated by D-V loops as shown Figure S4 a) Typical local piezoresponse hysteresis loops and b) displacement-voltage (D-V) butterfly loops of BTO (10 u.c.)/LSMO (15 u.c.) on (001) STO. The polarization states of BTO (10 u.c.)/LSMO (15 u.c.) before and after poling with +4 V using a film device are studied by the out-of-plane PFM shown in Figure S5a and S5b, respectively. The contrast of the image is correlated with the orientation of the FE polarization, with the dark (bright) region indicating downward (upward) FE polarization. Strikingly, the outmost area of image is polarized downward in Figure S5 S5Typical out-of-plane PFM image of BTO (10 u.c.)/LSMO (15 u.c.) a) before and b) Figure S6 S6The angular MR for BTO/LSMO with different LSMO thicknesses and BTO polarization states: 10 u.c. a) P up and b) P down , 15 u.c. c) P up and d) P down , 25 u.c. e) P up and f) P down , and 50 u.c. g) P up and h) P down . The measured magnetic field is marked at the top left corner.Typical results of reversible control (two P up /P down circles) of transport properties for BTO (10 u.c.)/LSMO (15 u.c.) are shown inFigure S7, where the angular MR measurements were carried out under a magnetic field of 5 T at 10 K. A series of gate voltage of -4 V → +4 V → -4 V → +4 V are applied on the sample to alter its polarization state as follows: P up -1 → P down -1 → P up -2 → P down -2. The two measurements of the same polarization state show good consistency, both in the temperature dependence of the resistance (Figure S7a) and angular MR curves (Figure S7band S7c), suggesting that the electrical manipulation of transport properties is reversible and reliable. Figure S7 S7Reversible control of a) temperature dependence of the resistance and angular MR for BTO (10 u.c.)/LSMO (15 u.c.) under b) P up and c) P down state. The angular MR measurements were carried out under a magnetic field of 5 T at 10 K. 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[ "Synthesis of built-in highly strained monolayer MoS2 using liquid precursor chemical vapor deposition", "Synthesis of built-in highly strained monolayer MoS2 using liquid precursor chemical vapor deposition" ]
[ "L Seravalli \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly\n", "F Esposito \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly\n\nDepartment of Mathematical, Physical and Computer Sciences\nUniversity of Parma\nParco Area delle Scienze 7/a43124ParmaItaly\n", "M Bosi \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly\n", "L Aversa \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nFBK Trento unit\nVia alla Cascata 56/C38123PovoTrentoItaly\n", "G Trevisi \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly\n", "R Verrucchi \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nFBK Trento unit\nVia alla Cascata 56/C38123PovoTrentoItaly\n", "L Lazzarini \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly\n", "F Rossi \nInstitute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly\n", "F Fabbri \nNEST\nIstituto Nanoscienze -CNR\nScuola Normale Superiore\nPiazza San Silvestro 1256127PisaItaly\n" ]
[ "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly", "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly", "Department of Mathematical, Physical and Computer Sciences\nUniversity of Parma\nParco Area delle Scienze 7/a43124ParmaItaly", "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly", "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nFBK Trento unit\nVia alla Cascata 56/C38123PovoTrentoItaly", "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly", "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nFBK Trento unit\nVia alla Cascata 56/C38123PovoTrentoItaly", "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly", "Institute of Materials for Electronics and Magnetism (IMEM-CNR)\nParco Area delle Scienze 37/a43124ParmaItaly", "NEST\nIstituto Nanoscienze -CNR\nScuola Normale Superiore\nPiazza San Silvestro 1256127PisaItaly" ]
[]
Strain engineering is an efficient tool to tune and tailor the electrical and optical properties of 2D materials. The built-in strain can be tuned during the synthesis process of a two dimensional semiconductor, as molybdenum disulfide, by employing different growth substrate with peculiar thermal properties. In this work we demonstrate that the built-in strain of MoS2 monolayers, grown on SiO2/Si substrate using liquid precursors chemical vapor deposition, is mainly dominated by the size of the monolayer. In fact, we identify a critical size equal to 20 m, from which the built-in strain increases drastically. The built-in strain is maximized for 60 m sized monolayer, leading to 1.2% tensile strain with a partial release of strain close to the monolayer triangular vertexes due to formation of nanocracks. These findings also imply that the standard method for evaluation of the number of layers based on the Raman modes separation becomes unreliable for monolayer with a lateral size above 20 m.
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[ "https://export.arxiv.org/pdf/2303.15881v1.pdf" ]
257,771,452
2303.15881
9e211ec0bb30e6092592f70b79a77ba263fbfd92
Synthesis of built-in highly strained monolayer MoS2 using liquid precursor chemical vapor deposition L Seravalli Institute of Materials for Electronics and Magnetism (IMEM-CNR) Parco Area delle Scienze 37/a43124ParmaItaly F Esposito Institute of Materials for Electronics and Magnetism (IMEM-CNR) Parco Area delle Scienze 37/a43124ParmaItaly Department of Mathematical, Physical and Computer Sciences University of Parma Parco Area delle Scienze 7/a43124ParmaItaly M Bosi Institute of Materials for Electronics and Magnetism (IMEM-CNR) Parco Area delle Scienze 37/a43124ParmaItaly L Aversa Institute of Materials for Electronics and Magnetism (IMEM-CNR) FBK Trento unit Via alla Cascata 56/C38123PovoTrentoItaly G Trevisi Institute of Materials for Electronics and Magnetism (IMEM-CNR) Parco Area delle Scienze 37/a43124ParmaItaly R Verrucchi Institute of Materials for Electronics and Magnetism (IMEM-CNR) FBK Trento unit Via alla Cascata 56/C38123PovoTrentoItaly L Lazzarini Institute of Materials for Electronics and Magnetism (IMEM-CNR) Parco Area delle Scienze 37/a43124ParmaItaly F Rossi Institute of Materials for Electronics and Magnetism (IMEM-CNR) Parco Area delle Scienze 37/a43124ParmaItaly F Fabbri NEST Istituto Nanoscienze -CNR Scuola Normale Superiore Piazza San Silvestro 1256127PisaItaly Synthesis of built-in highly strained monolayer MoS2 using liquid precursor chemical vapor deposition 1 Strain engineering is an efficient tool to tune and tailor the electrical and optical properties of 2D materials. The built-in strain can be tuned during the synthesis process of a two dimensional semiconductor, as molybdenum disulfide, by employing different growth substrate with peculiar thermal properties. In this work we demonstrate that the built-in strain of MoS2 monolayers, grown on SiO2/Si substrate using liquid precursors chemical vapor deposition, is mainly dominated by the size of the monolayer. In fact, we identify a critical size equal to 20 m, from which the built-in strain increases drastically. The built-in strain is maximized for 60 m sized monolayer, leading to 1.2% tensile strain with a partial release of strain close to the monolayer triangular vertexes due to formation of nanocracks. These findings also imply that the standard method for evaluation of the number of layers based on the Raman modes separation becomes unreliable for monolayer with a lateral size above 20 m. Introduction Strain engineering is an efficient tool to tune and tailor the electrical and optical properties of 2D materials since their electronic band structures are highly sensitive to mechanical deformation. [1][2][3][4][5][6][7][8] Among the different classes of 2D materials, semiconducting transition metal dichalcogenides (TMDs) have demonstrated the most interesting and surprising modifications of electronic properties induced by the application of strain. 9,10 For instance, strain can induce the indirect-to-direct bandgap transition in multilayer WSe2 flake 11 , while the opposite transition (the direct-to-indirect transition) can occur in monolayer TMDs 12,13 . With regard to the optical properties the local strain application can either induce exciton funneling 14,15 , efficient exciton to trion conversion 16 or the formation a new hybrid state of dark and localized excitons 17 . It is worth noting that TMD monolayers can withstand high tensile strains before breaking, as high as 10% for molybdenum based 18,19 and 19% for tungsten based 20 exfoliated monolayers. Different approaches have been employed for the application of external stress to 2D materials: the most employed one is based on the employment of bendable and stretchable polymeric substrates. [21][22][23][24][25][26][27] , where it is possible to apply mainly uniaxial strain to the 2D materials. This approach is maximized in the fabrication of the origami-like 28 and kirigami-like 29 MoS2 based devices, such as strain sensors or optoelectronic devices. A similar approach has been employed to fabricate human eye mimicking photodetectors. 30 Another widely used method is the transfer of the 2D material on a patterned nonplanar substrate for inducing localized strain, resulting into a local change of the band structure. The substrate can present insulating [31][32][33][34] or semiconducting 35,36 structures, namely stressors, that can apply a large local strain degree at their top. This large localized strain application modifies the band structure of 2D semiconducting materials, enabling efficient charge collection, desirable for bright single photon emission. 33,34 High degree of strain is also achieved by suspending 2D membranes on hole patterned substrates. 37 A local strain increase can be achieved on suspended membranes using different approaches: using an AFM tip 38 , applying an external gas pressure 39 or applying a gate voltage between a suspended monolayer membrane and an electrode below 17 . An additional novel approach to apply strain to TMDs heterostructures is the employment of polymeric artificial muscles, that taking advantage of low friction between different 2D materials can apply a tensile strain in van der Waals heterostructures. 40 The strain tuning during the chemical vapor deposition (CVD) process of 2D materials has been also demonstrated. 41,42 This method relies on the mismatch of thermal expansion coefficient (TEC) mismatch between the substrate and the 2D material. 4 For instance, monolayer WSe2 with built-in strains ranging from 1% tensile to compressive 0.2% was obtained using different growth substrates. The TEC approach has been also employed in order to synthetize strained WS2 on quartz with an oriented array of wrinkles. 43 In addition, strain induced buckling has been recently identified as a possible cause of threefold symmetric domain formation in hexagonal shaped WS2 monolayers. 44 In case of MoS2, the modification of strain and doping using different growth substrates has been demonstrated, where the maximum built-in tensile strain is found to be 0.4% with a SiO2 growth substrate. 45 In this work we demonstrate the dependence of the tensile strain as function of the lateral size of sharp-vertex shaped MoS2 monolayers grown on standard 300 nm thick SiO2/Si substrate using liquid precursor chemical vapor deposition. The built-in strain is demonstrated and evaluated by scanning Raman and photoluminescence spectroscopy. The monolayer MoS2, with lateral size below 20 m, presents a built-in compressive strain of 0.3%, while the flakes with the largest lateral size (60 m) are more strained, reaching an upper limit of 1.2%. It is worth noting that such flakes are affected by nanocracks close to the vertexes, revealing a partial release of the strain down to 0.7%. The builtstrain affects the optical properties; in particular, the highly strained flakes show a quenching and a red-shifting of the excitonic emissions of MoS2. Figure 1 illustrates the outcome of the CVD synthesis at increasing temperature using a solution containing molybdenum liquid precursor. In particular, the sketch (Fig 1a) shows the deposition of the precursor solution on the SiO2/Si growth substrate by spin coating, carried out for 30 s at 3000 rpm, of the precursor solution. This solution has three main components: a 3 mM solution of ammonium molybdate tetrahydrate (AMT), a 62.5 mM solution of sodium hydroxide (NaOH) and a 0.4 mM solution of iodixanol (Optiprep). The atomic force microscopy (AFM) is employed to study the number of layers composing the flakes and their lateral size. It is worth noting that the flakes obtained with the liquid precursor CVD have the sharp-vertex triangular shape. 46 The lateral size data are reported in Fig. 1b, that highlights the increase of the flake size with the growth temperature. The flakes, grown at 810°C, present an average lateral size of (15.1 ± 3.0) m, while the flakes synthetized at 820°C present an average lateral size of (24.1 ± 4.9) m. The maximum lateral size, (58.2 ± 5.7) m is obtained when the growth process is carried out at 830°C . Fig 1c, 1d and 1e present representative AFM topographical map of the flake obtained at increasing temperature. All the flakes, grown at the different temperatures, present fractal saw toothed edges, which are probably mediated by a diffusion-limited-aggregation (DLA) regime 47,48 In addition, the flake edges present nanoparticles decoration. Such effect is directly connected with the employment of the liquid precursor, since the nanoparticles are composed of precursor byproducts, as sodium oxide (Na2O). 49 Fig. S1, confirming the monolayer nature of MoS2 obtained at 830°C. It is worth noting that all the height profiles are acquired in area of the flake not affected by the nanoparticles and the flakes synthesized at 810°C and 820°C present a few nanometer thick step on the edge, similarly to WS2 flakes grown by CVD process using tungsten liquid precursor. 44 The nanoparticles decorating the flake edges present similar diameter, (26 ± 7) nm, while the height increases, increasing the growth temperature. In fact, the height of the nanoparticles varies from (8 ± 2) nm at 810°C to (18 ± 5) nm at 830°C, indicating that the temperature has an important role in the formation of precursor byproducts. The density of the nanoparticles increases in case of the growth carried out at 830°C with an average linear density of nanoparticles of 32 NPs/m, while the linear density decreases to 22 NPs/m for synthesis at lower temperatures. temperatures. Representative Raman spectrum (Fig. 3a), acquired at the center of the flake grown at 810°C (black line), presents the standard MoS2 Raman modes, E2g, at 383.6 cm -1 and A1g, at 403.5 cm -1 , with a Raman modes separation of 19.9 cm -1 . The A1g mode corresponds to the sulfur atoms oscillating in antiphase out-of-plane and the E2g mode is related to the sulfur and molybdenum atoms oscillating in antiphase parallel to the crystal plane. Results and discussion The separation of the Raman modes below 20 cm -1 is the standard benchmark of monolayer MoS2. 23,54,55 However, this analysis of the MoS2 monolayer grown at higher temperature demonstrates that both the MoS2 Raman modes are shifted. In fact, in case of the flake grown at 820°C (red line) the E2g and the A1g modes appear at 382.5 cm -1 and at 404 cm -1 , respectively. The separation of the Raman mode is 21.5 cm -1 , a value standardly reported for bilayer MoS2. 54,55 For the flakes grown at the 830°C (green line) the A1g mode is still set at 404 cm -1 , while the E2g presents an additional shift down to 380.2 cm -1 . The separation of the Raman modes is 23.8 cm -1 , the standard benchmark of few-layers MoS2. 54,55 The larger shift of the E2g mode is a symptom of the presence of strain because the A1g mode is less affected than the E2g mode, being the A1g mode related to the outof-plane vibration with respect to the E2g mode. In particular, the different response to strain of the two MoS2 Raman modes is reflected in the different Grüneisen parameter of each vibrational mode: γE2g= 0.68 and γA1g= 0.21. 26,39,56 Another clue of the presence of built-in strain applied to the MoS2 grown at high temperature is the broadening of the E2g mode, whose full width half maximum (FWHM) from 2.5 cm -1 (810°C grown monolayers) up to 10.4 cm -1 (830°C grown monolayer) 26 Raman suggests a different number of layers for the increasing growth temperature. This is clear in regards of the different shift of the two Raman modes when strain is applied to MoS2, because the E2g mode is more affected than the A1g mode, as previously explained. Therefore, we can assess that the presence of built-in strain in the MoS2 synthesis with liquid precursors, makes unreliable the Raman method for the evaluation of layers' number, using the separation of the vibrational modes. In order to quantify the built-in strain and the modification of doping, we employ the MoS2 correlation plot of the Raman shifts of the E2g and A1g modes, also known as the ε-n system. 57 This method allows to disentangle and to quantify the strain and doping variations and it is normally employed for studying the growth induced strain 58 , the effect of different growth substrates 45 or in MoS2 based van der Waals heterostructures 59,60 . The full lines represent the zero strain and zero doping lines, while the dashed lines correspond to iso-strain and the iso-doping lines, calculated following the insights of previous works. 39,61 In addition, it is worth mentioning the importance of the origin of the ε-n system: we set the zero strain and charge neutrality phonon frequencies 385 cm -1 for the E2g mode and 405 cm -1 for the A1g mode, evaluated in case of CVD grown MoS2 suspended monolayer membrane. 39 The data regarding the MoS2 monolayers grown at 810°C present a round distribution revealing an average value of (0.29 ± 0.06)% of tensile strain and an average positive charge concentration of (0.23 ± 0.22) x 10 13 cm -2 , while in case of the monolayer synthetized at 820°C, the data distribution is less dispersed and the average tensile strain increases up to (0.44 ± 0.12)% and the charge concentration decreases close to neutrality, (0.06 ± 0.10) x 10 13 cm -2 . The average strain reaches its maximum in case of the MoS2 monolayer obtained at 830°C, (1.15 ± 0.5)%, albeit the data present a linear distribution along the iso-doping line relative to an electron charge concentration -0.45 x10 13 cm -2 . In order to clarify the data spreading in the A1g versus E2g correlation plot, in case of the monolayers obtained at 830°C, the spatial distribution of strain is shown in Fig. 4. The methodology for the development of the strain maps is reported in Fig. S5 with the strain maps of the flakes grown at 810°C and 820°C. The Raman spectra in the amorphous carbon range are reported in Fig. S3 showing the absence of any amorphous carbon related peaks, that can arise due to organic compounds (i.e. iodixanol) in the precursor solution. In addition, the data regarding the MoS2 MLs transferred on a clean SiO2/Si substrate are reported in Fig. S4. The strain map of the 830°C grown monolayer (Fig. 4a) reveals that the strain is partially released in the triangular vertexes of the flake. This finding supports the spreading of the data reported in Fig. 3c. In fact, while in the central part of the flake is above 1%, the strain evaluation drops to 0.7% -0.8%, close to the vertexes of the triangular monolayer. The strain release mechanism is based on the formation of nanocracks in such regions, demonstrated by AFM phase analysis (Fig 4b). The AFM phase analysis is employed to maximize the contrast of the nanocracks, the topographical map is reported in Fig S6 of the Supporting Information. A similar strain release mechanism has been previously reported in case of MoS2 monolayer exposed to hydrogen or oxygen plasma. 62 Another method for evaluating the strain in atomically thin semiconducting TMDs is the analysis of the photoluminescence (PL) emission. Several previous works have demonstrated the strain dependence of the PL emission energy and intensity. 21,22,27,39,42,64,65 In case of MoS2 monolayer, the standard PL spectrum presents two main peaks attributed to the A and B excitons. The emission energy of the A exciton is reported to vary between 1.82 eV and 1.89 eV 21,27,65,66 , while the B exciton energy varies between 1.97 eV and 2.05 eV 21,27,42,65,66 . In our particular case, Figure 5 presents the representative PL spectra of the MoS2 monolayer grown at increasing temperatures, acquired at the center of the flake. The PL spectrum of 810°C grown ML shows one sharp peak at 1.86 eV and a faint shoulder in the high energy side, namely at 2.01 eV. Increasing the growth temperature at 820°C, the A exciton peaks suffers of 58% quenching of the integrated PL intensity and a slight red-shift of the emission energy (0.01 eV), being peaked at 1.85 eV. The red shift is even higher and the quenching more serious, in case of the 830°C grown monolayers, in fact the PL emission energy shifts down to 1.78 eV and quenching increases to 62%. It is worth noting that the quenching effect is limited by the broadening of the A excitonic PL peak. Fig. 5b resumes the PL A exciton emission energy and the PL integrated intensity as function of the growth temperature of the MoS2 MLs. We consider a PL energy gauge factor of -99 ± 6 meV/%, measured in case of biaxial strain of CVD grown MoS2 monolayer 39 , that is good agreement with theoretical prediction of 105 meV/% 67 . The biaxial strain, obtained by the PL shift, is 0.4% in case of the ML grown at 810°C, while it increases up to 0.5% for the 820°C synthetized MLs. The biaxial strain is maximized in case of the ML grown at 830°C with a value equal to 1.2%. It is worth noting that these values are calculated considering the peak energy of the A exciton for unstrained MoS2 ML at 1.9 eV. 39 (Fig S8-S11), related to the use of liquid precursors. This demonstrates that the TEC mismatch has a limited effect on the built-in strain in MoS2 flakes obtained with liquid precursors, while the main effect of the temperature is to increase the size of MoS2 MLs. Therefore, the main cause of large built-in strain in MoS2 ML grown at 830°C is related to the large lateral size of the sharp-vertex flakes. In fact, tensile strain-size dependence has been previously reported in case of MoS2 ML grown at a temperature above 800°C, where the critical size limit for the appearance of strain is 17 m. 46 In addition, the increase in size beyond the critical limit leads to spatial inhomogeneity of the built-in strain, where the edges 71,72 and the vertexes 46,73 are more affected. The possible mechanism is, therefore, resumed in Fig. 6b, where the MLs, obtained at 810°C presents a limited amount of homogeneous strain due to the size close to the critical limit of 17 m. The MoS2 MLs, presenting a size slightly greater than 17 m, show an increasing strain in the body of the flake with an enhancement on the flake vertexes (See Fig. S5 for the strain map of 820°C grown flakes). The built-in strain is maximized with large inhomogeneity in the vertexes when the lateral size of the flake is much larger than 17 m. The built-in strain value is more than two times the expected values for theoretical MoS2/SiO2 TEC mismatch induced strain, demonstrating that the size dependent strain is highly enhanced in case of large-area MoS2 ML. It is worth noting that, in our particular case, the expected built-in strain in the triangular vertexes should be even higher despite the formation of nanocracks partially releases the accumulated strain. The dependence of the built strain as function of the ML size is reported in Fig. 6c. The polynomial fitting reveals that the semi-empirical law of the average built-in strain as function of the average size has parabolic behavior with the following parameters: Built-in= A•d 2 + B•d [1] Where Built-in is the built-in tensile strain, d is the lateral size of the ML, and A and B are the parameters obtained from the fitting procedures, imposing that the intercept is equal to zero and they are: A=7.3 x 10 -5 %/m 2 , and B= 0.017 %/m. Following this semi-empirical prediction, a MoS2 ML with a lateral size of 100 m should be affected by a 2.4% of strain in the central body of the flake. The built-in strain dependence on the ML size and its inhomogeneity can be attributed to the particular DLA growth regime, that gives rise to the fractal saw toothed edges. This dependence and inhomogeneity can be also enhanced by the particular growth temperature range (above 800°C) and the use of liquid precursors. Conclusion In conclusion, we demonstrate that the built-in strain of MoS2 monolayers, grown on SiO2/Si substrate using liquid precursors chemical vapor deposition, is mainly dominated by the size of the Figure S12 shows a schematic illustration of the CVD reactor, i.e. an open tube with a diameter of about 1 inch heated in a two zones furnace. The sulfur boat is positioned in a low temperature zone (T = 180° C), while the growth substrate, after the spinning process of the Mo precursor solution, is placed in the high temperature zone (810°C < T < 830°C). 74,75 AFM topography and phase maps were collected using a Bruker AFM operated in the Scan assist mode. Scanning Raman and photoluminescence spectroscopies were carried out with a Renishaw InVia system, equipped with confocal microscope, a 532 nm excitation laser and a 2400 line/mm grating (spectral resolution <1 cm -1 ). All the analyses were performed with an 100X objective (NA=0.85), excitation laser power 500 W, acquisition time 4s for each spectrum and pixel size of 1m × 1m. Cross-sectional TEM analysis of a MoS2 flake grown at 830 °C was performed on a TEM-lamella prepared by a Zeiss Auriga Compact Focused Ion Beam (FIB) system. In order to protect the atomically thin layer of MoS2 from ion beam damaging 76 , the specimen was coated with amorphous Carbon in a Balzers CED-010 setup prior to apply the standard procedure of FIB-lamella preparation. The lamella was then observed in a JEOL JEM 2200-FS microscope, operated at 200 kV. We performed Xray Photoelectron Spectroscopy (XPS) in an ultra-high-vacuum (UHV) chamber using a VSW HA100 hemispherical electron energy analyzer with a PSP power supply and control. 77 We used a non-monochromatized Mgk X-Ray source (photon at 1253.6 eV), with a final energy resolution at 0.86 eV. The Au4f 7/2 peak at 84.0 eV has been used as calibration for the binding energy (BE) scale. Core level lineshape analysis has been performed using Voigt functions with a Gaussian to Lorentian ratio of 30%, after the subtraction of a Shirley background. The typical precision for each component's energy position was ±0.05 eV. while for the area evaluation it was approximately ±2%. In order to clarify the effect of the substrate on the built-in strain, we transfer the MoS2 MLs, grown at 820°C. It is worth mentioning that in the transfer of the 830°C synthetized MLs is hindered due to the presence of the nanocracks at the edge of the flakes. The transfer process is a modified dry viscoelastic stamping method [2][3][4][5] , where the standard polydimethylsiloxane (PDMS) stamp is replaces with a polyethylene terephthalate (PET) tape with an acrylic adhesive. A similar approach has been widely employed in the transfer of graphene 6 and III-nitrides [7][8][9] . Supporting Information The transfer is based on different steps (Fig S4a). First, the PET tape is gently attached on the SiO2/Si surface where the MoS2 MLs are grown. Then the tape is peeled off very slowly and the MoS2 flakes remain attached to the tape. Then the tape is then attached on the acceptor surface, that in our case is a clean SiO2/Si substrate. The specimen with the tape on tops is then annealed at 125°C in order to melt the acrylic adhesive and the tape is then gently peeled off, releasing the MoS2 MLs on top of the new substrate. Then the specimen is rinsed in warm acetone and IPA for the removal of the acrylic adhesive residuals. This particular novel transfer method is developed because the standard transfer approach, employing the poly(methyl methacrylate) (PMMA) spin coating on the specimen surface, is hindered by presence of the byproduct nanoparticles on the edge of the flakes. The AFM topological map, Fig S4b, reports the presence of some areas of the flake affected by thick dhesive residuals, nevertheless a large part of the nanoparticles decorating the flakes edge are absent. The comparison of the Raman spectra before and after the transfer reveals that the appearance of different Raman modes in the 1000-1600 cm -1 range, related to the acrylic adhesive residuals (Fig. S4c). 10,11 The analysis of the MoS2 Raman modes (Fig.S4d), depicted in the A1g versus E2g Raman shift correlation plot, reveals that after the transfer process the strain is partially released in the MoS2 ML, The strain map (Fig S5a) of the MoS2 ML grown at 810°C shows a tensile strain varying between 0.2% and 0.4%. In case of the MoS2 ML grown at 820°C (Fig S5b), the strain map reveals that an increase of the tensile strain in the vertexes of the triangular monolayer, that reaches the maximum close to 0.5%, while in the body of the ML the tensile strain varies between 0.3% and 0.4%. Figure S6: Topographical map of the MoS2 ML affected by nanocracks. We performed X-Ray Photoelectron Spectroscopy (XPS) on the Mo precursor solution spunned on SiO2/Si substrate after a high temperature treatment at 830°C to identify the surface chemical state of the compounds formed during such treatment. As a reference, we analyzed Na2SiO3 pure powder, purchased from Sigma-Aldrich, Merck Life Science S.r.l.. Despite the formation of Na2MoO4 is cited as possible reaction product between MoO3 and NaOH at high temperature 16 , we do not detect any evidence of sodium molybdate from XPS. Na2MoO4 pure powder was analyzed for reference in the same UHV apparatus and compared to results from the treated surface ("Mo precursor solution" sample), but lineshape analysis is not consistent with this compound. Long range spectrum of the treated substrate (Fig. S7, black curve) shows the presence of sodium (Na1s), oxygen (O1s), carbon (C1s), molybdenum (Mo3d), and silicon (Si2p). Mo3d core levels revealed the presence of two doublets (Mo3d 5/2, 3/2 due to spin orbit coupling) located at 232.2-235.3eV and 230.4-233.5eV, related to MoO3 and MoO2 molybdenum oxides (Fig. S8a) 17,18 . Si2p shows a main feature at 103.5 eV, due to SiO2 and a minor component at 102.5 eV, that is consistent with the presence of Na2SiO3 ( Figure S8b) and S8c), bottom curves). The Na/Si ratio is 2.1±0.05 and Na1s-Si2p energy difference is 969.4 eV (Na1s is located at 1072.1 eV), in good agreement with the results of sodium silicate analyzed as reference. The schematic illustration of the CVD reactor is shown in Fig. S11. The low temperature zone is heated up using a heating belt to reach the desired temperature. The sulfur powder is placed on an alumina boat while the growth substrate, where the Mo precursor solution is spun, is positioned on a graphite susceptor for a uniform heating. Figure 1 . 1a) Illustrative sketch of the substrate preparation prior to the growth process. b) Resuming graph of the MoS2 monolayer lateral size versus the growth temperature of the CVD process. c), d) and e) Representative AFM topographic maps of the MoS2 flake grown at different temperatures. The lateral size of the flake (white arrow) is 15.3 m, 19.2 m and 63.8 m in case of the 810°C, 820°C and 830 °C growth temperatures, respectively. Figure 2 . 2AFM topographic maps of the edges of the flakes grown at increasing temperatures: a) 810°C, b) 820°C and c) 830°C. The height profiles are obtained where the yellow line is reported on each map. The AFM topographic maps, acquired on the edge of the flakes, are reported in Fig. 2. The main purpose of such analysis is the evaluation of the number of layers of the MoS2 flake and the morphological analysis of the nanoparticles (NPs) decorating the flake edges. The topographical analysis reveals the presence of nanoparticles of smaller size inside the MoS2 flake and on the SiO2 substrate. All the MoS2 flakes are monolayer: the height profiles reported on the different panels, reveal a flake thickness of 0.7 -0.8 nm, the standard thickness of MoS2 monolayer. 50-53 Crosssectional TEM analysis is reported in Figure 3 . 3a) Representative Raman spectra of the MoS2 monolayers obtained at increasing temperature: 810°C (black line), 820°C (red line), 830°C (green line). The spectra are vertically shifted for sake of clarity. The Raman mode separation is indicated for each spectrum. b) Histogram of the Raman mode separation for the different growth temperature. c) MoS2 doping/strain correlation plot. Figure3 presents the Raman characterization of the monolayer MoS2, synthetized at increasing , in fact the application of strain induces the degeneracy breaking of the E Raman mode. The breaking of the degeneracy causes the splitting of the E2g mode in case of the presence of uniaxial strain larger than 0.8%. 21,27 Figure 3b reports the statistical analysis of the Raman modes separation, obtained by the Raman maps of a single flake shown in Supplementary Figure S2. The Gaussian distribution of the Raman modes separation is obtained by fitting the histogram data. In the case of the flakes synthetized at 810°C, the distribution is centered at 20.4 cm -1 with a FWHM of 0.7 cm -1 and it becomes even sharper in case of the synthesis at 820°C with a FWHM of 0.5 cm -1 , being centered at 21.3 cm -1 . The largest distribution of the Raman mode separation is a FWHM of 1.4 cm -1 in case of the 830°C grown monolayers, where the Gaussian fitting reveals that the center is at 24.8 cm -1 . The statistical analysis of the Raman modes separation is necessary to clarify the discrepancy with the AFM analysis reported in Fig. 2, in fact all the flake analyzed are monolayer in nature whereas the Figure 4 4Representative strain map of the MoS2 monolayers obtained at 830°C (panel a) and the AFM phase map of the edge of the flakes (panel b). mainly on the formation of six-fold hexagonal patterned net of nanocracks. The angle distribution of the lines relative to the edge of the MoS2 crystal has preferential orientations of 0°, 60°, and 120° as shown inFig. 4b, which indicates that the line patterns are associated with the crystallographic orientation of the MoS2. This effect is important because it demonstrates that MoS2 monolayer can sustain a limited amount of built-in tensile strain during the growth process, while the strain applied by external procedures can reach the 10%.18 Figure 5 . 5a) Representative PL spectra of the MoS2 monolayers synthetized at 810°C (black line), 820°C (red line) and 830°C (green line), acquired at the center of the flake. b) Resuming behaviors of the A exciton emission energy (black dots) and PL intensity (red triangles) as function of the growth temperature. The dashed lines are guides for the reader eyes. , 66 Figure 6 666. a) Temperature dependent built-in strain values with a direct comparison with theoretical TCE mismatch strain of MoS2/SiO2 interface b) Sketch depicting the mechanism of the strain generation in the MoS2 ML with dependence on the flake size. c) Dependence of the strain on the lateral size of the flake, reporting the polynomial fit. Figure 6 6proposes a mechanism for the large variation of the MoS2 built-in strain with the reduced tuning of the growth temperature. This model is based on three main aspects: 1) the thermal expansion coefficient (TEC) mismatch between the two dimensional material and the substrate during the growth process,4 2) the employment of the liquid precursors in the CVD process,49,58 3) the strain dependence on the size of the MoS2 ML synthetized at temperature beyond 800°C. 46,68 The direct comparison of the built-in strain values as function of the temperature with the theoretical TEC induced mismatch for the synthesis of MoS2 on thermal silicon dioxide is shown in Fig. 6a. Thermal silicon dioxide presents a poor linear TEC equal to a = 0.24 x 10 6 K -1 69 while MoS2 TEC is equal to a = 7.6 x 10 6 K -1 . 70 Based on this TEC mismatch, a degree of built-in tensile strain in the MoS2 ranging from 0.577% at 810°C to 0.593% at 830°C, can be expected. Albeit, the experimental values show that the applied built-in strain are lower than the theoretical values of the TEC mismatch induced built-in strain in case of the synthesis at 810°C and 820°C. The partial release of strain can be due the presence of an interfacial layer of Na2SiO3, as demonstrated by the XPS analysis reported in the supporting information monolayer. Using correlative analyses of AFM, Raman and PL, we are able to highlight an inconsistency in the number of layers in MoS2 flakes obtained by liquid precursors CVD. This leads to the identification of built-in strain dependent on the lateral size of MoS2 monolayer. The built-in strain values are close to the prediction of thermal expansion coefficient mismatch for monolayer with a lateral size lower than 20 m. The built-in strain is drastically increases for 60 m sized monolayer, leading to 1.2% tensile strain with a partial release of strain close to the monolayer triangular vertexes due to formation of nanocracks. A semi-empirical model is defined for the possible prediction of the expected built-strain for a certain lateral size of MoS2 monolayer without considering the possible release of strain due to the material cracking. The built-in strain dependence on the ML size and its inhomogeneity are possible attributed to the diffusion limited aggregation regime of the growth process, that gives rise to the fractal saw toothed edges of the MoS2 monolayers. Experimental Section The chemicals composing the Mo liquid precursors are AMT (Sigma Aldrich purity 99.98%), NaOH (Carlo Erba) and Optiprep, a iodixanol based component provided by Serumwerk Bernburg AG normally employed in cell cultures. MoS2 flakes were synthesized at atmospheric pressure in an open tube using S powder with nitrogen as carrier gas. The employed substrates are commercial 300 nm thick SiO2 coated highly conductive silicon wafer (Siltronic A.G.). 3 Institute of Materials for Electronics and Magnetism (IMEM-CNR), FBK Trento unit, Via alla Cascata 56/C, 38123 Povo (Trento), Italy 4 NEST, Istituto Nanoscienze -CNR, Scuola Normale Superiore, Piazza San Silvestro 12, 56127 Pisa, Italy List of the Figures: S1 Cross-sectional TEM analysis of 830°C grown MoS2 ML S2 Raman mapping of the MoS2 modes separation and of the intensity of the E2g mode. S3 Raman spectra in the range of the amorphous carbon. S4 Transfer process and characterization of the MoS2 MLs after transfer. S5 Strain maps of the MoS2 MLs grown at 810°C and 820°C. S6 Topographical map of the MoS2 ML affected by nanocracks. S7-S8 XPS characterization of Mo precursor solution spunned on SiO2/Si substrate undergone a high temperature treatment at 830°C. S9 Schematic illustration of the CVD reactor. Figure S1 : S1Cross-sectional TEM analysis of 830°C grown MoS2 ML.Fig. S1. A representative TEM image acquired at the interface, where MoS2 is embedded between the amorphous SiO2 and the protective Carbon coating. As shown from the intensity profile in the inset, a layer thickness of 0.65 nm is measured, that corresponds to the characteristic monolayer thickness of MoS2. Figure S2 : S2Raman mapping of the MoS2 mode separation and of the intensity of the E2g mode. Fig. S2 a), b) and c) Raman modes separation maps of the MoS2 ML grown at 810°C, 820°C and 830°C, respectively. d), e) and f) E2g intensity maps of the MoS2 ML grown at 810°C, 820°C and 830°C, respectively. The Raman mode separation maps reveal an enhanced mode separation, with increasing the growth temperature, as reported in Fig (S2a, S2b and S2c). The data reported here are employed for the statistical study shown in Fig. 3b. The intensity of the E2g mode are shown in Fig S2d, S2e and S2ffor increasing growth temperature. The main effect is reported inFig. S2fwhere the intensity of the E2g mode appears fainter in the vertexes of the triangular structure, an effect that can be related to the presence of defects, as the nanocracks reported inFig. 4b. Figure S3 : S3Raman spectra in the range of the amorphous carbon. Fig. S3 1 Figure S4 : S31S4Raman spectra of the MoS2 MLs grown at increasing temperature in the range of amorphous carbon. The Raman spectra in the range of amorphous carbon reveal no peaks that can be assigned to the presence of amorphous carbon in all the specimens in analysis. This result was previously reported for similar CVD process. Transfer process and characterization of the MoS2 MLs after transfer.Figure S4. a) Graphical representation depicting the transfer process. b) AFM topological map of the MoS2 ML, grown at 820°C, after the transfer. c) Representative Raman spectra before (full line) and after (dashed line) the transfer. d) MoS2 doping/strain correlation plot, where the full dots represent the data before the transfer while the open dots are acquired after the transfer. 13 Figure S5 : 13S5decreasing down to 0.35% and with a concurrent neutralization of the free carrier concentration. This second effect can be related to the presence of the adhesive residuals on the MoS2 surface, as previously demonstrated in case of graphene. 12,Strain maps of the MoS2 MLs grown at 810°C and 820°C. Fig. S5 a) and b) Strain map of the MoS2 ML synthetized at 810°C and 820°C with the same color code reported in Fig. 4 of the manuscript. The white arrows indicate the increase of strain at the vertexes of the vertex sharp triangular flake.The strain maps are obtained using the standard method proposed by A. Michail et al. Fig. S6 S6Topological map of the MOS2 ML area affected by nanocracks, indicated by yellow arrows.The detection of the nanocracks in AFM topographic mode is hindered by the presence of the byproduct nanoparticles inside the MoS2 monolayer. The yellow arrows highlight the nanocracks reported inFig. 4b. Figure S7 - S7S8: XPS characterization of Mo precursor solution spunned on a SiO2/Si substrate undergone a high temperature treatment at 830°C Fig. S7 : S7Long range XPS spectra of the Mo precursor solution on SiO2 surface after a treatment at 830°C (black curve), and pure Na2SiO3 for reference (green curve). Fig. S8 S8XPS core level spectra of Mo3d a), Na1s b), and Si2p c) of the Mo precursor solution on SiO2 surface after a treatment at 830°C. 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[ "Angle-dependent Magnetoresistance of an Ordered Bose Glass of Vortices in YBa 2 Cu 3 O 7−δ Thin Films with a Periodic Pinning Lattice *", "Angle-dependent Magnetoresistance of an Ordered Bose Glass of Vortices in YBa 2 Cu 3 O 7−δ Thin Films with a Periodic Pinning Lattice *" ]
[ "Bernd Aichner \nFaculty of Physics\nUniversity of Vienna\nA-1090WienAustria\n", "Lucas Backmeister \nFaculty of Physics\nUniversity of Vienna\nA-1090WienAustria\n", "Max Karrer \nPhysikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany\n", "Katja Wurster \nPhysikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany\n", "Reinhold Kleiner \nPhysikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany\n", "Edward Goldobin \nPhysikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany\n", "Dieter Koelle \nPhysikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany\n", "Wolfgang Lang \nFaculty of Physics\nUniversity of Vienna\nA-1090WienAustria\n" ]
[ "Faculty of Physics\nUniversity of Vienna\nA-1090WienAustria", "Faculty of Physics\nUniversity of Vienna\nA-1090WienAustria", "Physikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany", "Physikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany", "Physikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany", "Physikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany", "Physikalisches Institut\nCenter for Quantum Science (CQ)\nLISA +\nUniversität Tübingen\nD-72076TübingenGermany", "Faculty of Physics\nUniversity of Vienna\nA-1090WienAustria" ]
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The competition between intrinsic disorder in superconducting YBa2Cu3O 7−δ (YBCO) thin films and an ultradense triangular lattice of cylindrical pinning centers spaced at 30 nm intervals results in an ordered Bose glass phase of vortices. The samples were created by scanning the focused beam of a helium-ion microscope over the surface of the YBCO thin film to form columns of point defects where superconductivity was locally suppressed. The voltage-current isotherms reveal critical behavior and scale in the vicinity of the second-order glass transition. The latter exhibits a distinct peak in melting temperature (Tg) vs. applied magnetic field (Ba) at the magnetic commensurability field, along with a sharp rise in the lifetimes of glassy fluctuations. Angle-dependent magnetoresistance measurements in constant-Lorentz-force geometry unveil a strong increase in anisotropy compared to a pristine reference film where the density of vortices matches that of the columnar defects. The pinning is therefore, dominated by the magnetic-field component parallel to the columnar defects, exposing its one-dimensional character. These results support the idea of an ordered Bose glass phase.
10.3390/condmat8020032
[ "https://export.arxiv.org/pdf/2303.15912v1.pdf" ]
257,771,527
2303.15912
0b95ae33f5d2f0916be5da89311c87bb0801d04e
Angle-dependent Magnetoresistance of an Ordered Bose Glass of Vortices in YBa 2 Cu 3 O 7−δ Thin Films with a Periodic Pinning Lattice * Bernd Aichner Faculty of Physics University of Vienna A-1090WienAustria Lucas Backmeister Faculty of Physics University of Vienna A-1090WienAustria Max Karrer Physikalisches Institut Center for Quantum Science (CQ) LISA + Universität Tübingen D-72076TübingenGermany Katja Wurster Physikalisches Institut Center for Quantum Science (CQ) LISA + Universität Tübingen D-72076TübingenGermany Reinhold Kleiner Physikalisches Institut Center for Quantum Science (CQ) LISA + Universität Tübingen D-72076TübingenGermany Edward Goldobin Physikalisches Institut Center for Quantum Science (CQ) LISA + Universität Tübingen D-72076TübingenGermany Dieter Koelle Physikalisches Institut Center for Quantum Science (CQ) LISA + Universität Tübingen D-72076TübingenGermany Wolfgang Lang Faculty of Physics University of Vienna A-1090WienAustria Angle-dependent Magnetoresistance of an Ordered Bose Glass of Vortices in YBa 2 Cu 3 O 7−δ Thin Films with a Periodic Pinning Lattice * 10.3390/condmat8020032 The competition between intrinsic disorder in superconducting YBa2Cu3O 7−δ (YBCO) thin films and an ultradense triangular lattice of cylindrical pinning centers spaced at 30 nm intervals results in an ordered Bose glass phase of vortices. The samples were created by scanning the focused beam of a helium-ion microscope over the surface of the YBCO thin film to form columns of point defects where superconductivity was locally suppressed. The voltage-current isotherms reveal critical behavior and scale in the vicinity of the second-order glass transition. The latter exhibits a distinct peak in melting temperature (Tg) vs. applied magnetic field (Ba) at the magnetic commensurability field, along with a sharp rise in the lifetimes of glassy fluctuations. Angle-dependent magnetoresistance measurements in constant-Lorentz-force geometry unveil a strong increase in anisotropy compared to a pristine reference film where the density of vortices matches that of the columnar defects. The pinning is therefore, dominated by the magnetic-field component parallel to the columnar defects, exposing its one-dimensional character. These results support the idea of an ordered Bose glass phase. I. INTRODUCTION Magnetic flux penetrates a type-II superconductor in flux quanta of Φ 0 = h/(2e), where h is Planck's constant and e is the elementary charge. In the ideal situation of a defect-free material, these flux quanta, also known as fluxons or Abrikosov vortices, form the densest packing possible, a hexagonal lattice of flux cylinders. However, in real-world materials, disorder is omnipresent and promotes the nucleation of defects. As a result, they play an essential role in anchoring vortices, and so in preventing their unwanted motion. This motion causes dissipation and spoils the functionality of most superconducting systems. With the advent of copper-oxide high-temperature superconductors (HTS), there has been renewed interest in the interaction between defect topography and the collective arrangement of vortices. One obvious cause is the strong anisotropy of HTS and their extremely short Ginzburg-Landau coherence lengths. Defects must have dimensions comparable to or greater than the coherence length-smaller ones have a minimal impact. As a result, a short coherence length makes the material more sensitive to microscopic imperfections, such as point defects on the scale of a unit cell or less. These novel properties of the HTS give rise to a menagerie of miscellaneous defects with various dimensionalities, hierarchies, correlations, and characteristic length scales. Many different thermodynamic vortex phases can develop when a magnetic field B a is applied to a superconductor, depending not only on the specific defect topography, but also on the magnitude of the magnetic field, its orientation with respect to the main crystallographic axes, and the temperature. The very rich phase diagrams of HTS that resulted have stimulated various analyses of this so-called "vortex matter" [1]. The fluxons in a clean superconductor are structured regularly with long-range order and three-dimensional (3D) correlation, forming a hexagonal Abrikosov vortex lattice that can be termed a vortex crystal. It melts through a first-order transition. Indeed, signatures of a first-order transition have been reported in pure untwinned YBa 2 Cu 3 O 7−δ (YBCO) single crystals [2]. The inclusion of a sufficiently large density of uncorrelated zero-dimensional (0D) point defects results in a vortex glass (VG) [3]. It is akin to a frozen liquid, as the name implies. In such a VG, the vortex cores form bent and even entangled threads hitting as many point defects as possible to minimize their energy. The VG melts via a second-order transition which can be characterized by critical exponents [4]. A key characteristic of a VG is its zero resistance state below the vortex glass melting temperature T g in the limit of vanishing applied current density j → 0. Randomly arranged one-dimensional (1D) pinning centers, such as columnar defects (CDs) formed by swift heavy-ion irradiation [5], lead to the Bose glass (BG), in which vortex threads are straight and pinned by the columnar defects [6,7]. However, intrinsic twin boundaries and screw dislocations in YBCO that are oriented parallel to the crystallographic c axis can also cause BG behavior. This has led to inconsistent observations of VG [8] or BG [9] behavior in YBCO films, depending on the details of the material's morphology. As the VG and the BG have similar scaling laws for their voltage-current (V -I) characteristics near the glass temperature, unam-biguous discrimination from these measurements alone is impossible. To resolve this issue, transport measurements at different angles of the magnetic field were performed [10], since they indicate the dimensionality of the pinning centers. Frustrating disorder in a BG leads to the vortex Mott insulator (MI) [7]. Although the MI also melts through a first-order transition, it differs significantly from the vortex crystal: The latter assembles at any magnetic field in the Shubnikov state between the lower and upper critical magnetic field, whereas the MI is attached to commensurability with a periodic pinning lattice and so emerges only at discrete magnetic fields. Differential resistance measurements in metallic superconductors with periodic pinning potential landscapes have revealed the presence of a MI-to-vortex-liquid transition [11,12]. Finally, in a more realistic experimental setting, the coexistence of an intentionally generated periodic CD lattice and intrinsic defects of various hierarchies might result in the ordered Bose glass (OBG) [13], which is the topic of this research. We will briefly review some of the experimental hallmarks of an OBG before presenting measurements of angular magnetoresistance in nanopatterned YBCO films, which show the predominant pinning along the periodic CDs and support the OBG picture. II. RESULTS AND DISCUSSION The competition between the above-mentioned theoretically proposed vortex phases in real materials controlled by temperature, magnetic field, and disorder has long been an issue [14][15][16], and alternative models of vortex melting have been proposed [17]. Previous attempts to analyze this problem were hampered by the fact that both columnar and point defects were randomly distributed in the material. Only recently has it become feasible to create regular CD patterns with spacings smaller than the London penetration depth using masked [18][19][20][21] ion irradiation. The discoveries of artificial vortex ice [22] and anomalous metallic states [23,24] in YBCO are prominent examples. Even narrower patterns have been reported using focused light-ion irradiation [13,25,26]. When the applied magnetic field matches the density of CDs in such densely patterned samples, pronounced vortex commensurability effects can be observed down to low temperatures at so-called matching fields, given by B m = m 2Φ 0 √ 3a 2 ,(1) where m is a rational number and a denotes the lattice constant of a triangular CD lattice. The collective vortex properties are significantly altered at the matching fields B m [27]. The most notable commensurability effects are at m = 1. Such a vortex-matching effect is illustrated in Figure 1a in a YBCO bridge subjected to focused 30 keV He + irradiation in a helium ion microscope. The fabrication process is described in detail elsewhere [13,25]. The irradiation produces a triangular lattice of CDs with distances a = (30 ± 0.6) nm. Superconductivity is completely suppressed inside these CDs, which span the entire thickness of the YBCO film. In perfect agreement with Equation (1), the critical current density j c (B) as a function of the magnetic field applied orthogonally to the sample surface displays a peak at the first matching field B 1 = 2.65 T and a hump at B 2 = 5.31 T. In the inset of Figure 1a, the temperature dependence of the resistance in the zero magnetic field of the patterned YBCO film is compared to an unirradiated reference bridge prepared on the same substrate. While the pristine YBCO bridge has a T c = 88.7 K (defined as the inflection point), irradiation reduces it to T c ∼ 77 K and increases the normal-state resistance. Some ions scatter away from their direct paths within the planned CDs, resulting in a small number of point defects between the CDs. Such small crystal lattice distortions are known to reduce T c [28][29][30]. The rising offset of a linear extrapolation of the normal state resistivity, which is substantially larger in the irradiated sample, further supports this. Contrarily, the irradiation has only a marginal effect on the slope, indicating that the charge carrier density in the inter-CD areas is only minimally modified. The VG and BG theories both anticipate a significant qualitative change in the nonlinear V -I isotherms at the magnetic-field-dependent glass-melting temperature T g (B) < T c , as shown in Figure 1b for the case of B a = B 1 . The green line denotes the power-law behavior of the V -I curves near T g and marks the bifurcation between two sets of the characteristics. On the left of the green line at T > T g (B 1 ), an ohmic behavior in low currents is connected with the thermally assisted flux motion (TAFF) [31] commonly observed in cuprate superconductors. This suggests that even with vanishing current density j → 0, the material has finite resistance. In sharp contrast, the isotherms to the right of the green line at T < T g (B 1 ) have negative curvature and point to a zero-resistance state at T < T g (B 1 ) at j < j c . Notably, the glass theories anticipate critical scaling of various physical parameters at the continuous secondorder phase transition. As one prominent example, this may be tested by re-plotting the V -I isotherms according to (2) for 3D vortex correlations. In this case, ν and z are critical parameters that describe the divergence of the glass correlation length and the lifetimes of glassy fluctuations at T g , respectively, [3]. In a Bose glass, the situation is analogous, and the respective critical parameters can be easily converted [8] to be compatible with Equation (2). (V /I)|1 − T /T g | ν(1−z) = F ± [(I/T )|1 − T /T g | −2ν ], The scaling collapse of the V -I isotherms onto two material-dependent functions F + above and F − below T g has been proven in a number of HTS with varying defect landscapes. The marginal dependence of the critical parameters ν and z on the applied field B a , however, was a common observation [32]. The situation is very different in YBCO films patterned with a dense triangular CD lattice, as demonstrated in Figure 2a, where scaling collapses at the matching field B 1 = 2.65 T and at an off-matching field B = 4 T are compared. It is instructive to compare the slopes of the bifurcations, represented by dotted lines. The bifurcation line at T g not only divides the branches F ± but also obeys a power law (V /I) | T =Tg ∝ I (z−1)/2 ,(3) yielding a direct estimate of the dynamic scaling parameter z. It is obvious that z and the qualitative behavior of F ± are sensitive to the magnetic field in our nanopatterned samples. While the critical parameter ν = 1.3 ± 0.2 is similar in many HTS [32] and is equally insensitive to B a here, T g and z show a strong magnetic-field dependence, as determined by an extensive set of scaling data at various magnetic fields [13]. Figure 2b illustrates the decline of the glass temperature T g with the applied magnetic field, interrupted by maxima at B 1 and B 2 . In contrast, pristine YBCO exhibits a nearly linear reduction of T g with the magnetic field [33]. As long as the applied magnetic field is not commensurable, a similar trend, represented by the dotted line, can be observed in the nanopatterned sample. As a linear decrease in T g is theoretically predicted for a BG with random defects [34], the observed peaks are a distinguishing feature of the OBG. The sharp peak of z at the matching field B 1 seen in Figure 2b is an even more exciting feature. Remarkably, both VG and BG theories predict, and experiments confirm, z to be roughly in the range z ∼ 4 . . . 6. The observed value z = 9 definitely suggests a more ordered arrangement of vortex matter, the ordered Bose glass, which is addressed in greater detail elsewhere [13]. The rather weak rise in z at B 2 could be explained by the additional vortices being located at interstitial places, where pinning is dominated by magnetic caging from trapped flux in the CDs and randomly arranged intrinsic pinning sites between the CDs. An analogous weakening of a BG has been theoretically discussed [15]. The previously discussed results leave one important question unanswered. As the collapse of the V -I curves and the resulting parameters cannot reveal the dimensionality of the pinning centers, angular-dependent measurements of resistivity in a magnetic field can provide further information. When the magnetic field is tilted by an angle α off the crystallographic c axis, 0D pinning sites should cause only moderate angle dependence, and 1D columnar defects would effectively pin solely the magnetic field component parallel to them. According to the scaling approach for anisotropic uniaxial superconductors [35], resistivity curves in magnetic fields applied at an angle α to the c axis collapse into one curve when plotted as a function of a reduced magnetic field: B * = B a γ −2 sin 2 α + cos 2 α),(4) where the anisotropy factor γ = m c /m ab , and m ab and m c are the effective masses along the ab and c directions, respectively. Figure 3a shows relevant data on a reference bridge patterned on the same substrate. The temperature T = 84.5 K corresponds to a reduced temperature t = T /T c = 0.96. It was chosen to cover a significant portion of the resistive transition in the applied magnetic field. The magnetic field's tilt plane was oriented perpendicular to the current direction to achieve a constant Lorentz force condition, as portrayed in the inset of Figure 3b. As is commonly observed [36], the resistivity at all magnetic fields decreases systematically as α increases. Figure 3b demonstrates the excellent scaling of the data using γ = 4.5 as the only adjustable parameter. In this case, γ is slightly lower than estimated from other methods, which could be attributed to the fact that YBCO is a layered superconductor and therefore not perfectly described by an anisotropic model. V / I | 1 -T / T g | ν( 1 -z ) ( Ω) ( I / T ) | 1 -T / T g | -2 ν ( A / K ) Despite numerous studies on the angular dependence of transport properties in superconductors with random 0D and 1D defects, these are rare for periodic CDs in metallic [37,38] and copper-oxide superconductors [16,26]. Figure 4a presents the angular dependence of our nanopatterned sample at T = 74 K, which is the same reduced temperature t = 0.96 as used for the pristine reference sample. In a magnetic field applied parallel to the CDs and the c axis (α = 0 • ), a distinct minimum of the resistivity at B 1 , indicates the commensurability of a plastic flow of vortices [39] with the CD lattice. A second weaker minimum can be seen at B 2 . Both attributes correspond to the maxima of j c shown in Figure 1a. When a magnetic field B a = B 1 is applied that is tilted away from the c axis, commensurable locking of fluxons into the CDs is no longer possible, and the paths of magnetic flux lines are distorted. We are not aware of any theoretical work for the specific experimental situation discussed here, but we can draw on findings for randomly arranged CDs [14,15]. Several competing effects must be considered in an inclined magnetic field [40]. (i) The flux lines can be oriented straight along the applied magnetic field; (ii) they can transit through the film along the shortest path (i.e., the c direction), which coincides with being fully locked into the CDs; or (iii) they can maintain their average directions along B a and proliferate through the material in vortex segments trapped in the CDs, which are connected by Josephson strings along the CuO 2 planes [14], as sketched in the inset of Figure 4a. The movement of these Josephson strings is assisted by inherent point defects between the CDs. The Lorentz forces on fluxon segments and Josephson strings point in different directions, causing kinks in the vortex lines to wander, eventually leading to fluxon hopping to neighboring CDs. This process is partially similar to vortex propagation by flux cutting, which has been observed in vicinal YBCO films [41]. As a result, as the number of kinks increases at larger α, the resistivity rapidly rises, as shown by the open circles in the inset of Figure 4c. For α > 60 • , a counteracting reduction in resistivity is presumably caused by intrinsic pinning of Josephson strings between the CuO 2 planes [42]. An attempt to scale the data to B * using Equation (4) fails partially, as illustrated in Figure 4b. While the highfield data B a > B 2 appear to converge to a universal line (highlighted in yellow), the minima at B 1 appear at different values of B * depending on the tilt angle. Furthermore, γ = 2.1 is much lower than in the pristine YBCO film. The latter is not surprising, and it is well-known that nanocomposites in YBCO films significantly reduce anisotropy [43], even down to γ = 1.4 for a YBCO film with 13% BaZrO 3 (BZO) content [44]. At high B a > B 2 , the scaling of the ρ(B * ) curves appears to indicate that the OBG behavior breaks down and intrinsic point defects dominate the resistivity. In this instance, the anisotropic scaling law is recovered. The angular-dependent resistivity in samples with a periodic columnar pinning landscape should differ from that in an anisotropic superconductor, as previously found in wider square arrays of CDs [16,26]. It can be modeled by formally increasing γ to infinity in Equation (4), resulting in B || = B a cos α.(5) In this case, only B || , the component of B a parallel to the CD's axes, determines the matching condition of Equation (1). Indeed, when the data are scaled to B || , the matching minima in the resistivity are perfectly aligned according to the adapted Equation (1); i.e., B || = B 1 , as shown in Figure 4c. This finding proves that magnetic flux is trapped predominantly within the CDs even when an oblique magnetic field is applied. As a result of minimizing flux-line kinks, dissipation caused by wandering Josephson strings and fluxon hopping is avoided. However, this idealized picture must be supplemented by intrinsic defects between the CDs, which, in conjunction with thermal fluctuations, promote vortex line bending. Furthermore, deviations from a path aligned with the applied magnetic field increase the fluxon's energy. At larger angles, these processes become more effective, resulting in enhanced dissipation. The resistivity minimum at B || = B 1 remains visible, but its base value rises. The filled triangles in the inset of Figure 4c show that in oblique magnetic fields, resistivity at the matching minima rises only moderately for α ≤ 40 • but increases progressively at larger angles. Remarkably, the scaled positions of the matching minima remain in place even at large angles α ≤ 70 • . It has previously been established that 2D-glass theories cannot account for the V -I data [13] and that the OBG relies on 3D vortex correlations, which appear to be stable up to high tilt angles. Indeed, in 2D vortex systems, the vortex lattice has been found to be fragile in low applied magnetic field [45], and melting was found to be better described by the vortex molasses scenario [17]. Note that close to α = 90 • , additional influences, such as the lock-in of vortices between the CuO 2 planes [42,46] and possible smectic vortex glass behavior [47], come into play. Our results cannot distinguish whether there are two flux quanta inside each CD at B 2 or if one is trapped at an interstitial position. However, the the scaling of the second minimum suggests that vortices are aligned in parallel to the CDs, even in the latter case. The specific scaling of the angle-dependent resistivity in the mixed state of the nanopatterned film affirms the picture of an ordered Bose glass [13]. The OBG is a vortex arrangement with order between the vortex Mott insulator and the Bose glass. It is stimulated by a defect landscape composed of both periodically arranged 1D columnar defects, capable of trapping one or more vortices along the entire thickness of the sample, and disordered defects of diverse dimensionality between the CDs. The latter comprise point defects already present in the pristine material, such as vacancies and intermixing of atom species, and Frenkel defects created by a few ion trajectories scattered off the incident beam direction. They are commonly responsible for VG behavior. On the other hand, 1D screw dislocations, grain boundaries, and mosaics of 2D twins are examples of intrinsic random defects oriented along the c axis in YBCO that cause behavior. III. MATERIALS AND METHODS The experiments were carried out with very thin YBCO films, epitaxially grown on (LaAlO 3 ) 0.3 (Sr 2 AlTaO 6 ) 0.7 (LSAT) substrates by pulsed laser deposition (PLD). Laue oscillations at the YBCO (001) Bragg peak indicated a YBCO film thickness of t = (26.0 ± 2.4) nm. The full width at half maximum (FWHM) of 0.08 • of the YBCO (005) peak's rocking curve confirmed the excellent c-axis orientation of the films. Following PLD, a 20 nm thick Au film was evaporated in situ with an electron beam. Both the Au and the YBCO films were partially removed using Ar ion milling to form bridge structures 8 µm in width and 40 µm in length with voltage probes separated by 20 µm. Then, a a window in the Au layer was removed with Lugol's iodine to allow direct access to the YBCO layer for irradiation while protecting the sample's contact areas. The prepatterned YBCO microbridges were introduced into the Zeiss Orion NanoFab He-ion microscope (HIM) and aligned under low ion fluence. The HIM focused 30 keV He + ion beam was set to a spot control value that resulted in an estimated 9 nm FWHM average diameter for He + ion trajectories within the film. An area of 36 × 16 µm 2 was irradiated with a triangular spot lattice with distances a = (30 ± 0.6) nm, covering the entire width of the bridge and extending beyond the voltage probes. A total of 10 4 ions per spot were required to completely suppress superconductivity in nanopillars that crossed the entire thickness of the YBCO film. A too-narrow ion beam can cause unwanted amorphization at the YBCO film's surface, as discussed elsewhere [25]. In fact, irradiation of a 26.5 nm thin La 1.84 Sr 0.16 CuO 4 (LSCO) film with a nominally 0.5 nm wide He-FIB beam revealed orders-of-magnitude-larger damaged areas [48]. Contrarily, a comparable experiment in thin-film YBCO bridges resulted in operational Josephson junctions employing an ion fluence, at which no amorphization was detected [49]. The relative weakness of the copper-oxygen bonds in YBCO as compared to LSCO could explain these differences. Moderate ion doses can suppress the T c of YBCO while preserving the crystallographic framework. However, in LSCO, a higher fluence is required to convert the material into an insulator, which causes significantly more damage. For electronic transport measurements, the contact pads were connected by 50 µm thick Au wire and Ag paste to the sample holder of a Physical Properties Measurement System (PPMS) equipped with a 9 T superconducting solenoid and a variable-temperature insert (Quantum Design). At fixed temperatures and in stable magnetic fields parallel to the crystallographic c axis, a large number of V -I curves, limited to 100 µV to avoid heating effects, were collected. A voltage criterion of 200 nV was used to define the critical current. A horizontal rotator mounted in the PPMS was utilized to measure angle-dependent resistance. The α = 90 • setting of the dial was calibrated by minimizing the resistance due to the intrinsic vortex lock-in transition when the magnetic field was oriented precisely parallel to the ab planes. All measurements were performed in both current polarities to eliminate spurious thermoelectric signals. IV. CONCLUSIONS We studied the interaction of vortices in a landscape of a triangular pinning array of 1D CDs and intrinsic defects in thin YBCO films. Measurements of V -I isotherms and resistivity at various temperatures and in oblique magnetic fields revealed a second-order glass transition that we call ordered Bose glass. Its characteristics are magnetic-field commensurability effects which are represented by peaks in the glass-melting temperature and the lifetimes of glassy fluctuations. The the latter exceed theoretical predictions and previous experiments on disordered Bose glasses. The frustrated disorder was revealed further when the angular magnetoresistivity was compared to that of a pristine reference sample. Magnetoresistivity scales well in plain YBCO films using the scaling approach for uniaxial anisotropic superconductors. Contrarily, the matching signatures in the nanopatterned sample are determined solely by the magnetic-field component parallel to the 1D pinning channels, indicating that the magnetic flux is trapped within these defect nanopillars. These findings identify the ordered Bose glass as a topological phase intermediate between the vortex Mott insulator and the Bose glass. It differs from the Mott insulator by second-order melting of vortex matter and from the Bose glass by apparent commensurability effects. The designed periodic pinning landscapes are an excellent test-bed for studying vortex matter in copper-oxide superconductors with their ubiquitous intrinsic defects. They could be useful in experimentally exploring and scrutinizing theoretical predictions of the complex behavior of vortex matter. Author contributions W.L., D.K., E.G., and R.K. conceived and supervised the experiments; K.W. grew the film; M.K. patterned the film and performed the focused ion beam irradiation; B.A., L.B., and W.L. performed the transport measure-ments; B.A., L.B., and W.L. analyzed the data; and all authors discussed the results and contributed to writing the paper. All authors have read and agreed to the published version of the manuscript. Funding This research was funded by a joint project of the Austrian Science Fund (FWF), grant I4865-N; and the German Research Foundation (DFG), grant KO 1303/16-1. It is based upon work from COST Actions CA21144 (SuperQuMap), CA19108 (Hi-SCALE), and CA19140 (FIT4NANO), supported by COST (European Cooperation in Science and Technology). Data availability The data presented in this study are available on reasonable request from the corresponding author. Conflicts of interest The authors declare no conflict of interest. current density at 5 K of the irradiated YBCO bridge. The broken lines represent the number of vortices per unit cell of the triangular CD lattice, which correspond to the matching fields Bm = m×2.65 T, as calculated from Equation(1)with the nominal geometry of the irradiation pattern. Inset: Resistivities of a YBCO thin film with a triangular array of 30 nm spaced columnar defects and an unirradiated reference bridge fabricated on the same substrate. (b) V -I isotherms at 66 K (red), 65.5 to 52 K in 0.25 K steps, and 50 to 30 K (blue) in 2 K steps of the nanopatterned sample at the matching field B1 = 2.65 T. The green line accentuates the isotherm at 59.5 K, which is closest to the glass temperature Tg = 59.4 K. Figures adapted from[13]. FIG. 2 . 2(a) Comparison of the collapsed V /I vs. I curves using Equation (2) at the first matching field B1 = 2.65 T and an off-matching field B = 4 T. The dotted lines indicate the bifurcation and have different exponents. The colors are identical to those in Figure 1b. (b) Scaling parameters of the nanopatterned YBCO film. The representative error bars illustrate the uncertainty caused by interdependence of the fit parameters Tg and z. Solid lines are guides to the eye. Figures adapted from[13]. FIG. 3 . 3(a) The angular dependence of the unirradiated reference sample's resistivity as a function of the applied magnetic field, Ba. (b) Data scaling to the reduced magnetic field (B * ) using Equation(4). The experimental configuration is sketched in the inset. FIG. 4 . 4Angular dependence of the resistivity ρ of the nanopatterned YBCO thin film at 74 K in various magnetic fields, tilted by an angle α. The color code is displayed in the center panel. 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[ "Published for SISSA by Springer Discrete R-symmetries and anomaly universality in heterotic orbifolds", "Published for SISSA by Springer Discrete R-symmetries and anomaly universality in heterotic orbifolds" ]
[ "Nana G Cabo Bizet \nCentro de Aplicaciones Tecnológicas y Desarrollo Nuclear\nCalle 30, esq.a 5ta Ave6122Miramar, La HabanaCuba\n", "Tatsuo Kobayashi [email protected] \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n", "Damián K Mayorga Peña \nBethe Center for Theoretical Physics and Physikalisches Institut\nUniversität Bonn\nNussallee 1253115BonnGermany\n", "Susha L Parameswaran [email protected] \nDepartment of Mathematics and Physics\nLeibniz Universität Hannover\nWelfengarten 130167HannoverGermany\n", "Matthias Schmitz [email protected] \nBethe Center for Theoretical Physics and Physikalisches Institut\nUniversität Bonn\nNussallee 1253115BonnGermany\n", "Ivonne Zavala [email protected] \nCentre for Theoretical Physics\nUniversity of Groningen\nNijenborgh 49747 AGGroningenThe Netherlands\n" ]
[ "Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear\nCalle 30, esq.a 5ta Ave6122Miramar, La HabanaCuba", "Department of Physics\nKyoto University\n606-8502KyotoJapan", "Bethe Center for Theoretical Physics and Physikalisches Institut\nUniversität Bonn\nNussallee 1253115BonnGermany", "Department of Mathematics and Physics\nLeibniz Universität Hannover\nWelfengarten 130167HannoverGermany", "Bethe Center for Theoretical Physics and Physikalisches Institut\nUniversität Bonn\nNussallee 1253115BonnGermany", "Centre for Theoretical Physics\nUniversity of Groningen\nNijenborgh 49747 AGGroningenThe Netherlands" ]
[ "JHEP02" ]
We study discrete R-symmetries, which appear in the 4D low energy effective field theory derived from heterotic orbifold models. We derive the R-symmetries directly from the geometrical symmetries of the orbifolds. In particular, we obtain the corresponding R-charges by requiring that the couplings be invariant under these symmetries. This allows for a more general treatment than the explicit computations of correlation functions made previously by the authors, including models with discrete Wilson lines, and orbifold symmetries beyond plane-by-plane rotational invariance. The R-charges obtained in this manner differ from those derived in earlier explicit computations. We study the anomalies associated with these R-symmetries, and comment on the results.
10.1007/jhep02(2014)098
[ "https://web.archive.org/web/20170922153449/http:/www.openaccessrepository.it/record/6228/files/main.pdf?version=1" ]
53,997,902
1308.5669
fdb82c99075ad20373266f5d68f6fe32d9fc784b
Published for SISSA by Springer Discrete R-symmetries and anomaly universality in heterotic orbifolds 2014 Nana G Cabo Bizet Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear Calle 30, esq.a 5ta Ave6122Miramar, La HabanaCuba Tatsuo Kobayashi [email protected] Department of Physics Kyoto University 606-8502KyotoJapan Damián K Mayorga Peña Bethe Center for Theoretical Physics and Physikalisches Institut Universität Bonn Nussallee 1253115BonnGermany Susha L Parameswaran [email protected] Department of Mathematics and Physics Leibniz Universität Hannover Welfengarten 130167HannoverGermany Matthias Schmitz [email protected] Bethe Center for Theoretical Physics and Physikalisches Institut Universität Bonn Nussallee 1253115BonnGermany Ivonne Zavala [email protected] Centre for Theoretical Physics University of Groningen Nijenborgh 49747 AGGroningenThe Netherlands Published for SISSA by Springer Discrete R-symmetries and anomaly universality in heterotic orbifolds JHEP02 98201410.1007/JHEP02(2014)098Received: October 15, 2013 Revised: January 23, 2014 Accepted: January 29, 2014Contents 1 Introduction 1Superstrings and Heterotic StringsDiscrete and Finite SymmetriesConfor- mal Field Models in String TheoryAnomalies in Field and String Theories ArXiv ePrint: 13085669 We study discrete R-symmetries, which appear in the 4D low energy effective field theory derived from heterotic orbifold models. We derive the R-symmetries directly from the geometrical symmetries of the orbifolds. In particular, we obtain the corresponding R-charges by requiring that the couplings be invariant under these symmetries. This allows for a more general treatment than the explicit computations of correlation functions made previously by the authors, including models with discrete Wilson lines, and orbifold symmetries beyond plane-by-plane rotational invariance. The R-charges obtained in this manner differ from those derived in earlier explicit computations. We study the anomalies associated with these R-symmetries, and comment on the results. Introduction Discrete symmetries are often imposed in the context of particle physics model building beyond the Standard Model in order to forbid unwanted terms in the Lagrangian. For example, they have been invoked in order to guarantee the absence of certain operators leading to exceedingly fast proton decay. They have also been very useful for flavor physics to generate textures of quark and lepton masses and mixings. From a stringy perspective, discrete symmetries are expected to appear, either as discrete remnants of a broken gauge symmetry [1][2][3] or to be an inherent property of the compactification from ten to four dimensions. In this respect, heterotic orbifold compactifications [4,5] provide a phenomenologically promising UV-complete framework [6][7][8][9][10][11][12][13][14][15][16][17][18] that is rich in discrete symmetries, which moreover have intuitive geometric interpretations. One of the features which makes them appealing for phenomenology is the presence of R-symmetries. These R-symmetries can be understood as elements of the Lorentz group in the compact space since they treat 4D bosons and fermions in a different manner. Therefore they are expected not to commute with the generator of the 4D N = 1 SUSY algebra. For the specific case of orbifold compactifications, one expects those rotations in SO (6) which are symmetries of the orbifold to manifest as R-symmetries of the low energy effective field theory (LEEFT). The identification of R-symmetries in heterotic models appeared first in [19] for the Z 3 orbifold, where they were associated to orbifold isometries which in this case were the twists acting on a single plane. Later on, more general expressions were found in [6]. However, it was recently pointed out by the authors [20], that in general, the R-charges JHEP02(2014)098 defined in [6], receive a non-trivial contribution from the so-called gamma-phases. In the same way as gauge invariance and other selection rules have been derived [4,5,21,22], the analysis in [20] was made via the explicit calculation of vanishing correlation functions, in the absence of discrete Wilson lines. A classification of the symmetries observed in orbifold geometries was presented in the appendix of [20]. There it was observed that such orbifold symmetries are exhibited by the classical instanton solutions. Thus one expects them to induce R-symmetries in the low energy. In the present note we determine the form of the expected R-charges of the physical states, by assuming that the orbifold isometries are also manifest symmetries of the LEEFT. This allows us to include discrete Wilson lines in the analysis, as well as to consider more general cases. The R-charges obtained in this way turn out to differ in the sign of the gamma-phase contribution, with respect to the former derivation. We provide some possible interpretations of these two results. Anomalies of discrete symmetries have important implications in 4D field theory [23][24][25], in particular string-derived LEEFTs. Such anomalies are expected to cancel via the Green Schwarz mechanism [26]. In heterotic orbifold models, there is only one axion available and hence, all anomalies must be universal, i.e. to cancel up to a common axion shift. 1 In [28,29], anomalies were studied for the discrete R-charges corresponding to [6]. Here, we study explicitly the anomaly conditions for the two types of R-charges: the one derived in this paper and the one obtained in [20]. We determine their universality for several orbifolds, factorizable and non-factorizable. While anomaly universality could be expected for the R-symmetries, there exists no proof that this should be indeed the case. We discuss this issue and its implications. The paper is organized as follows: in section 2 we review briefly the orbifold CFT; in section 3 we present a derivation of R-symmetries from orbifold isometries, which can be applied to the more general cases in which discrete Wilson lines are present. In section 4 we compute the anomalies for the R-symmetries in several explicit factorizable and nonfactorizable examples. In section 5 we study further possible R-symmetries which appear in the low energy effective field theory due to orbifold symmetries that do not leave the fixed points invariant. We discuss these symmetries, as well as the more familiar R-symmetries of section 3, by using an explicit model in 6. In section 7 we conclude with the discussion of our results. Note added: while this paper was in preparation, [30] appeared on the arXiv. There the R-symmetries of the Z 6II orbifold are derived and the contribution of the discrete Wilson lines is considered for the first time. The R-charges derived there also differ in the sign of the gamma phase contribution compared to our previous result [20]. The results presented here were obtained independently and hence confirm (and extend) those in [30]. String orbifold CFT We begin by briefly describing the relevant elements of the string orbifold conformal field theory. Our focus will be on symmetric, six-dimensional orbifolds constructed by modding JHEP02(2014)098 out a non-freely acting, Abelian isometry of the torus O 6 = T 6 P = C 3 /Λ P = C 3 S , where P is the point group and S = P Λ is the space group. In order to retain N = 1 supersymmetry in four dimensions, the point group must be Z N or Z N × Z M . Here we discuss orbifolds in the first class, but our results can easily be generalized to models of the second type. For Z N orbifolds, the generator of the point group θ can be brought to the diagonal form θ = diag(e 2πiv 1 , e 2πiv 2 , e 2πiv 3 ) , (2.1) where the coordinates of the internal space have been taken in a complexified basis X i ,X i (i = 1, 2, 3). The twist vector v = (v 1 , v 2 , v 3 ) is constrained by N = 1 supersymmetry to satisfy v 1 + v 2 + v 3 = 0 mod 1. In the orbifold background, strings can close up to twist and lattice identifications. The closed string boundary conditions involving only lattice identifications give rise to the untwisted sector, where the relevant conformal primary fields are the identity operator; the left-moving oscillator fields ∂X i , ∂X i (plus their complex conjugates); the sixteen extra left-movers contributing the Cartans ∂X I (I = 1, . . . , 16) and the roots e ip·X of the E 8 ×E 8 gauge symmetry; and the exponential e iq (a) ·H , with H a five-dimensional vector of free fields corresponding to the bosonized right-moving fermions and q (a) is either a bosonic (a = 1) of a fermionic (a = 1/2) weight of SO(10), known as H-momentum. The boundary conditions of the internal space bosonic coordinates for twisted strings are of the form X i (e 2πi z, e −2πiz ) = (gX) i (z,z) = (θ k X) i (z,z) + λ i ,(2.2) for any constructing element g = (θ k , λ) ∈ S, where z,z are the complexified worldsheet coordinates and λ i ∈ Λ. These boundary conditions correspond to strings in the k-th twisted sector. Notice that strings closed by g and hgh −1 |h ∈ S are physically equivalent, that is, physical twisted states are associated with conjugacy classes and not space group elements. The standard way to deal with the branch singularities in (2.2) is to introduce twist fields σ(z,z) [21,22] which serve to implement the local monodromy conditions, ∂X i (z,z)σ(w,w) = (z − w) −k i τ + . . . , ∂X i (z,z)σ(w,w) = (z − w) −k iτ + . . . , (2.3) where k i = kv i mod 1 andk i = (1 − k i ) mod 1, such that 0 ≤ k i ,k i < 1, and τ,τ are excited twist fields. The conformal dimensions of σ are given by ∆ σ =∆ σ = 1 2 3 i=1 k i (1 − k i ) . (2.4) The twist fields for worldsheet fermions can be written in terms of the bosonized fermions as e ikv·H . This leads to the definition of a shifted H-momentum: q (a) sh = q (a) + k · (0, 0, v 1 , v 2 , v 3 ) ,(2. JHEP02(2014)098 so that the primaries in the vertex operators take the familiar form e iq (a) sh ·H . Furthermore, modular invariance requires the twist to be embedded in the gauge degrees of freedom. In fact, not only the twist but the full space group can be embedded as a shift: g = (θ k , n α e α ) → V g = kV + n α W α ,(2.6) where {e α }, α = 1, . . . , 6 spans Λ, V is the embedding of θ and the discrete Wilson lines W α are related to the lattice shifts [31][32][33]. Note that both the group laws of S as well as modular invariance impose several non-trivial constraints on the choice for the embedding vectors, which are summarized for example in [34]. The relevant primary for the twisted vertices is e 2πip sh ·X , with p sh = p + V g . To summarize, the vertex operators describing the emission of twisted states are given by: V −a = e −aφ 3 i=1 (∂X i ) N i L (∂X i )N i L e iq (a) sh ·H e ip sh ·X σ , (2.7) where φ is the superconformal ghost and the integers N i L andN i L count, respectively, the number of left-moving holomorphic and anti-holomorphic oscillators present in the state. Untwisted vertex operators have the same form as those presented before but with all momenta unshifted and with the twist field replaced by the identity operator. Note that in writing (2.7), we have taken the four-dimensional momentum to zero, and neglected cocycle [35] and normalization factors, which are unimportant for our purposes. Furthermore, invariance of the vertex operators under the full space group needs to be satisfied. Considering a certain space group element h, the bosonic fields transform according to ∂X i h − → e 2πiv i h ∂X i , X I h − → X I + V I h , H i → H i − v i h . (2.8) In order to see how h acts on the twist fields it is convenient to decompose them into a sum of auxiliary twists σ g , one for each element in the conjugacy class [g] σ ∼ g ∈[g] e 2πiγ(g ) σ g , (2.9) where the phasesγ(g ) will be determined presently. For the auxiliary twists one has the following transformation behavior [34]. This leads to σ g h − → e 2πiΦ(g,h) σ hgh −1 , (2.10) with the vacuum phase Φ(g, h) = − 1 2 (V g · V h − v g · v h )σ h − → e 2πi[γ h +Φ(g,h)] σ ,(2.11) where we have defined γ h = γ(g ) −γ(hg h −1 ) mod 1, for some g ∈ [g] ,(2. JHEP02(2014)098 together with the condition −γ(hg h −1 ) +γ(g ) = −γ(hgh −1 ) +γ(g) mod 1 , (2.13) for any pair of elements g , g ∈ [g]. Now we can finally see how the vertex (2.7) transforms under h, V −a h − → exp{2πi[p sh · V h − v i h (q (a) i sh − N i L +N i L ) + γ h + Φ(g, h)]} V −a . (2.14) From (2.12) one can see that if there exists an element g ∈ [g] which commutes with a certain h, then γ h = 0 mod 1, such that eq. (2.14) becomes a projection condition p sh · V h − v i h (q (a) i sh − N i L +N i L ) + Φ(g, h) = 0 mod 1 , (2.15) which are the so-called orbifold GSO projectors responsible for an N = 1 supersymmetric spectrum. In all other cases, the gamma-phases can be found by demanding the transformation of V −a to be trivial [33,36,37], i.e. γ h = −p sh · V h + v i h (q (a) i sh − N i L +N i L ) − Φ(g, h) =γ(g) −γ(hgh −1 ) mod 1 . (2.16) In this way, space group invariance fixes allγ(g ) except for one, which can be reabsorbed as an overall phase in σ. Notice that there is generically more than one physical twist field σ for each conjugacy class, given by the different linear combinations of auxiliary twist fields, in which the different gamma-phase coefficients are determined in terms of the other quantum numbers of the physical state. Discrete R-symmetries from orbifold isometries In this section we identify discrete R-symmetries which are expected to appear in the low energy effective field theory of a given orbifold compactification, due to symmetries in the orbifold geometry. Examples of such discrete R-symmetries were explicitly verified in [20], by computing correlation functions. Our present approach will instead be to assume that symmetries in the orbifold geometry give rise to R-symmetries in the effective field theory and -given this assumption -infer the corresponding charge conservation laws. This allows us to be more general, including symmetries beyond the plane-by-plane independent twist symmetries, as well as models with Wilson lines. The absence of a coupling between L chiral superfields Φ α (α = 1, . . . , L) in the superpotential of the LEEFT can be deduced from the vanishing of the tree-level L-point correlator ψψφ L−3 . It is easy to see that the vertex operators V −1 and V −1/2 corresponding to the bosonic and fermionic fields in a left chiral supermultiplet are related by a shift in their fermionic weights q (1) sh = q (1/2) sh + ± 1 2 , ± 1 2 , − 1 2 , − 1 2 , − 1 2 . (3.1) JHEP02(2014)098 The computation of the tree-level amplitude requires the emission vertices to cancel the background ghost-charge of two on the sphere. Thus it is necessary to shift the ghost picture of some of the vertex operators according to V 0 = e φ T F V −1 , (3.2) where T F is the worldsheet supersymmetry current [38,39] T F =∂X iψi +∂X i ψ i ,(3.3) with ψ j = exp{i q j · H} and q i j = δ i j . This picture-changing operation allows one to write bosonic vertices with zero ghost-charge, at the price of introducing the right-moving oscil-lators∂X i and∂X i and additional H-momentum. The correlator can then be written as F = V −1/2 (z 1 ,z 1 )V −1/2 (z 2 ,z 2 )V −1 (z 3 ,z 3 )V 0 (z 4 ,z 4 ) . . . V 0 (z L ,z L ) , (3.4) where each V α = V (z α ,z α ) represents a certain physical state from the massless spectrum. It is possible to infer several selection rules from the explicit form of the correlator. 2 In the following we will make use of the space group selection rule 1 ⊂ L α=1 [g α ] , (3.5) gauge invariance L α=1 p I sh α = 0 ,(3.6) and H-momentum conservation L α=1 q (1) i sh α = −1 − N i R ,(3.7) where N i R counts the number of holomorphic right-moving oscillators in the correlator. Using the above rules the correlator (3.4) can be rewritten in the form F = L α=1 3 i=1 (∂X i ) N i L α (∂X i )N i L α (∂X i ) N i R σ α . (3.8) Let us now deduce the coupling selection rules arising from symmetries of the orbifold geometries. A classification of these symmetries was drawn in the appendix of [20]. In particular, we are interested in rotations of the torus lattice, which leave the fixed-point structure of the orbifold invariant. This subgroup of automorphisms was called D in [20]. Denoting these automorphisms by , they satisfy By definition, given a g ∈ S, (g) is conjugate to g, and hence there exists a space group element h g such that θ = θ −1 and (g) ∈ [g] ∀g ∈ S ,(3.(g) = h g gh −1 g . (3.11) Writing g = (θ k , λ) and (g) = (θ k , λ), h g = (θ l , µ) can be determined by finding a solution to the equation µ = (1 − θ k ) −1 ( − θ l )λ . (3.12) In analogy to (2.10), the most general transformation behavior for the auxiliary twist fields under ρ is given by σ g − → e 2πiΦ (g) σ (g) ,(3.13) which, for the physical twist fields described in eq. (2.9), implies σ − → g ∈[g] e 2πi[−γ( (g ))+γ(g )+Φ (g )] e 2πiγ( (g )) σ (g ) . (3.14) Since preserves conjugacy classes, the vertex operators have to be invariant up to phases. This means that we have to require the structure of σ to be preserved, which is guaranteed if the following condition is satisfied γ(g) −γ( (g)) + Φ (g) =γ(hgh −1 ) −γ( (hgh −1 )) + Φ (hgh −1 ) mod 1 . (3.15) Using (hgh −1 ) = (h) (g) (h) −1 and the definition (2.12) we find Φ (hgh −1 ) = Φ (g) + γ h − γ (h) mod 1. (3.16) Note that this equation implies that once we know the phase Φ (g), the phases for all other elements of the conjugacy class are automatically fixed. Note moreover that the phase, Φ (g), acquired by the auxiliary twist field σ g must depend only on the space-group element g, whereas the gamma-phases associated with the physical twist field σ depend, via (2.16), on the quantum numbers of the corresponding state. Therefore, if (3.16) is to be fulfilled for all physical states, the vacuum-phases and gamma-phases must independently fulfill Φ (hgh −1 ) − Φ (g) = 0 mod 1 , γ h − γ (h) = 0 mod 1 (3.17) for all space group elements g, h. Now, plugging (3.11) into (3.15) and using (3.17) we find γ hg = γ h g mod 1 ,(3.ξ i L α=1 N i L α −N i L α + N i R α + L α=1 γ hg α + Φ (g α ) = 0 mod 1 . (3.20) The phases Φ can be removed from the previous equation, since the space group selection rule together with the OPEs for the twist fields imply 3 L α=1 Φ (g α ) = 0 mod 1 . The invariance condition for the correlator can now be written in terms of well known quantities when combined with H-momentum conservation (3.7) and reads L α=1 3 i=1 ξ i q (1) i sh α − N i L α +N i L α − γ hg α = − 3 i=1 ξ i mod 1 . (3.22) In the case i ξ i = 0 mod 1, this condition looks precisely like the coupling selection rule originating from an R-symmetry. In this case, take M to be the smallest integer such that R ≡ −M i ξ i (3.23) is an integer. Then eq. (3.22) takes the more familiar form L α=1 r α = R mod M , with r α = 3 i=1 M ξ i q (1) i sh α − N i L α +N i L α − M γ hg α . (3.24) Thus, by imposing the symmetry of the orbifold generated by ∈ D on the correlation function, we have derived a quantity that can be readily interpreted as a Z R M discrete symmetry of the low energy effective field theory in which R denotes the charge of the superpotential and r α are the charges of the fields. 4 Surprisingly, the discrete symmetry defined in (3.24) does not coincide with the explicit R-symmetry result derived in [20], due to the sign in the last term, which gives the contribution of the gamma-phase to the R-charges. In the following section we discuss this discrepancy in terms of the anomalies for both results. Universal R-symmetry anomalies In this section we compute the anomalies for the Z R M -symmetry derived in the previous section and compare them with the result for the Z R M -symmetry derived in [20]. In heterotic 3 Using the space group selection rule, the leading term in the OPE of all auxiliary twist fields involved in the coupling is proportional to the identity, which transforms trivially under . 4 For a comprehensive summary of R-charge conventions we refer to [41]. JHEP02(2014)098 orbifold compactifications, there is only one axion available to cancel would-be anomalies via the Green-Schwarz mechanism, so that one typically expects anomalies to be universal. Exceptions to this are the anomalies of discrete target-space modular symmetries, which in many cases can be made universal only after including contributions from one-loop threshold corrections [42,43]. For anomalies involving U(1) factors the universality holds up to Kač-Moody levels, so we focus only on gravitational and non-Abelian anomalies for which the levels are all equal to 1. Under a Z R M transformation the path integral measure transforms as DψDψ → DψDψ exp − 2πi 1 M a A G 2 a −Z R M · 1 16π 2 tr{F a ∧ F a } +A grav. 2 −Z R M · 1 284π 2 tr{R ∧ R} , (4.1) as can be seen from applying Fujikawa's method [44,45], where the Pontryagin indices T (N a ) 16π 2 tr{F a ∧ F a } and 1 2 1 284π 2 tr{R ∧ R} (4.2) are integer valued [46,47], and here T (N a ) denotes the Dynkin index of the fundamental representation. The corresponding anomaly coefficients are given by [23-25, 28, 29] 5 A G 2 a −Z R M = C 2 (G a ) R 2 + α r α − R 2 T (R α a ) , (4.3) A grav. 2 −Z R M = − 21 − 1 − N T − N U + a dim{adj(G a )} R 2 + α r α − R 2 · dim{R α } ,(4.4) with C 2 (G a ) being the quadratic Casimir of G a , α running over left chiral matter representations and T (R α a ) its corresponding Dynkin index. In eq. (4.4), the contributions of −21 and −1 correspond to the gravitino and dilatino respectively, N T and N U are the number of T -and U -modulini and a runs over all gauge factors (including U(1)'s). If anomalies are cancelled by the same axion shift, given two gauge factors G a,b the so-called universality conditions must hold A G 2 a −Z R M mod M T (N a ) = A G 2 b −Z R M mod M T (N b ) , (4.5) A G 2 a −Z R M mod M T (N a ) = 1 24 A grav. 2 −Z R M mod M 2 . (4.6) Let us focus on the orbifolds presented in table 1, with the isometries discussed in [20]. As examples, we give the space group elements h g needed to calculate the gamma-phases JHEP02(2014)098 Table 1. Summary of point groups studied with their corresponding lattices and corresponding orbifold isometries. The charge of the superpotential R and the order of the symmetry M are also given. Orbifold Lattice Twist R M ξ 1 ξ 2 ξ 3 Z 4 SO(4) 2 × SU(2) 2 1 4 , 1 4 , − 2 4 1/4 1/4 0 −1 2 1/2 0 0 −1 2 0 0 −1/2 +1 2 Z 4 SU(4) 2 1 4 , 1 4 , − 2 4 1/2 0 0 −1 2 0 1/2 0 −1 2 Z 6I G 2 × G 2 × SU(3) 1 6 , 1 6 , − 2 6 1/6 1/6 0 −1 3 0 0 −1/3 +1 3 Z 6II G 2 × SU(3) × SU(2) 2 1 6 , 2 6 , − 3 6 1/6 0 0 −1 6 0 1/3 0 −1 3 0 0 −1/2 +1 2 Z 8I SO(9) × SO(5) 1 8 , − 3 8 , 2 8 1/4 −3/4 0 +1 2 0 0 1/2 −1 2 Z 8II SO(8) × SO(4) 1 8 , 3 8 , − 4 8 1/8 3/8 0 −1 2 0 0 −1/2 +1 2 Z 12I SU(3) × F 4 4 12 , 1 12 , − 5 12 1/3 0 0 −1 3 0 1/12 −5/12 +1 3 Z 12II for the R-symmetries identified for the Z 4 and Z 6II orbifold in the appendix. We used the C++ orbifolder [48] to compute the spectrum and the corresponding anomalies for all of the embeddings classified in [49,51] without Wilson lines, with the R-charge assignment given in eq. (3.24) for the factorizable Z 4 , Z 6I and Z 6II , as well as the non-factorizable Z 8I orbifold. In all models the R-anomalies satisfy universality conditions. When taking the R-charges without the gamma contribution [6], universality is particular to very few models. The same is observed when using the opposite sign for the γ phases, as derived in [20]. Furthermore we considered models with Wilson lines. For each of the allowed shift embeddings we randomly generated 10 000 Wilson line configurations and found that in these cases the Rcharges computed from eq. (3.24) show universality relations for all orbifolds studied. This is an overwhelming result and a strong hint that the R-charges derived here are correct. However, the reason for the opposite sign in front of the gamma-phase contribution derived in [20] remains to be understood. Here, we discuss a possible way out. We have assumed the auxiliary twist fields σ g to transform according to (2.10). Suppose instead that the JHEP02(2014)098 auxiliary twist fields σ g have the (albeit counter-intuitive) transformation law σ g h − → e 2πiΦ(g,h) σ h −1 gh , (4.7) where the role of h and h −1 is interchanged compared to (2.10). The resulting R-charge assignment (3.22), (3.24) is independent of this change. Indeed, if one goes through the derivations in section 3 using this transformation behavior one arrives at σ − → e 2πi[γ h −1 g +Φ (g)] σ instead of (3.19), while (2.16) gets modified to γ h = p sh · V h − v i h (q (a) i sh − N i L +N i L ) + Φ(g, h) , so that the resulting R-charge conservation law remains precisely (3.24). Meanwhile, the same R-charge conservation law (3.24) would be derived by the explicit computations in [20]. Further R-symmetry candidates Having observed universality for the R-symmetries derived above, let us now elaborate on an additional set of symmetries which was already introduced in [20]. There we observed that some lattice automorphisms exchange certain fixed points of the same twisted sector. We denoted the subgroup of lattice automorphisms satisfying this property by F . At first it seems that this kind of isometry has nothing to do with the string orbifold compactification. In some cases, however, one observes that the fixed points which get mapped to each other under a certain ζ ∈ F allocate identical matter representations, and hence it gives rise to a symmetry in the low energy effective field theory. 6 Now we are concerned with the computation of the charges of the fields under this new type of symmetry. We will use the fact that, for all cases considered, the elements in F can be written in a block diagonal form as ζ = diag(e 2πiη 1 , e 2πiη 2 , e 2πiη 3 ). (5.1) As expected, for those vertex operators which are eigenstates of ζ, the charges are similar to (3.24): r α = 3 i=1 M η i q (a) i sh α − N i L α +N i L α − M γ hg α . (5.2) Let us therefore consider vertices of states located at non-invariant fixed points. For simplicity, let us assume that ζ only interchanges certain conjugacy classes, i.e. [g] ζ ←→ [g ] , g g . JHEP02(2014)098 This implies that a vertex V from [g] gets mapped to its counterpart V , where V and V share the same quantum numbers. 7 Writing σ ∼ g∈[g] e 2πiγ(g) σ g , σ ∼ g ∈[g ] e 2πiγ (g ) σ g , the twist fields involved in these vertices will then transform according to 8 σ ζ − → exp{2πi[γ(g) −γ (ζ(g)) + Φ ζ (g)]}σ , σ ζ − → exp{2πi[γ (g ) −γ(ζ(g )) + Φ ζ (g )]}σ ,(5.4) and therefore V ζ − → exp{2πi[−η i (q (a) i sh − N i L +N i L ) +γ(g) −γ (ζ(g)) + Φ ζ (g)]}V , V ζ − → exp{2πi[−η i (q (a) i sh − N i L +N i L ) +γ (g ) −γ(ζ(g )) + Φ ζ (g )]}V . (5.5) Recall that q (a) i sh , N i L andN i L are the same for both V and V . Note that, a priori, the transformation phases in (5.4), (5.5) cannot be related to physical gamma-phases since g and ζ(g) belong to different conjugacy classes. Since V and V differ only in their conjugacy classes and carry identical quantum numbers, one expects couplings involving either V and V to differ only by constant phases. One can write the vertices V and V in a basis of eigenstates of ζ V (s) = V + e 2πi(δ+s) V , s = 0 , 1 2 , (5.6) in which δ is a phase fixed so that the operators V (s) transform indeed only up to a phase under ζ. Using equations (5.5) and (5.6), we can fix δ and write the transformation behavior of V (s) under ζ as V (s) ζ − → exp 2πi −η i (q (a) i sh − N i L +N i L ) + 1 2 γ hg + γ h g + s V (s) . (5.7) Here the space group elements h g and h g are defined such that ζ(g ) = h g gh −1 g , ζ(g) = h g g h −1 g ,(5.8) for any combination of representatives g and g in the same way as described in section 3, and recall that γ hg =γ(g) −γ(h g gh −1 g ) and γ h g =γ (g ) −γ (h g g h −1 g ). From the transformation property of the V (s) we can now read off their corresponding R-charges r (s) = M 3 i=1 η i (q (a) i sh − N i L +N i L ) − 1 2 M γ hg + γ h g − M s ,(5.9) 7 Note that although V and V are associated with different conjugacy classes, the conjugacy class is not a good quantum number to distinguish the states. Moreover, note that if the coupling V1 . . . VL−1V is allowed by all selection rules, then so is V1 . . . VL−1V . 8 In general one can allow for vacuum phases for the twist fields under this transformation. However they turn out to be irrelevant for our discussion in the same way as already observed in section 3. JHEP02(2014)098 where M is the smallest integer such that R ≡ −M 3 i=1 η i ∈ Z . (5.10) When writing the low energy effective field theory in terms of the ζ-eigenstates, the corresponding R-charge conservation law implies that any coupling must vanish unless L α=1 r α = R mod M . (5.11) The result we just obtained for the elements in F , at least for the case of factorizable orbifolds, has remarkable implications. R-symmetries of the LEEFT are not only due to those remnants of the Lorentz group which leave the fixed points invariant. Even those automorphisms mapping different fixed point conjugacy classes to each other can source Rsymmetries in the field theory. In contrast to those emerging from symmetries in D, these novel R-symmetries can be broken by Wilson line configurations that spoil the degeneracy of the states located at the non-invariant fixed points. Note that in our derivation we assumed that ζ at most interchanges pairs of conjugacy classes, but in principle more intricate transformation patterns can emerge, particularly in the case of non-factorizable orbifolds. We expect that in those cases, the charges can be computed in a similar fashion. 6 An explicit example: Z 4 on SO(4) 2 × SU(2) 2 Here we illustrate our results by discussing in detail the Z 4 orbifold on the lattice of SO(4) 2 ×SU(2) 2 with the twist as given in table 1. One easily sees that a basis of generators for the group D is given by 1 4 , e 2πi 1 4 , 1) , 2 = (θ 1 ) 2 = (e 2πi 1 2 , 1, 1) , 1 = θ 1 θ 2 = (e 2πi3 = θ 3 = (1, 1, e −2πi 1 4 ) . (6.1) As stressed before, each of these symmetries leads to universal anomalies in all of the models studied. As an example let us discuss the following shift embedding and Wilson line configuration V = −1, − 3 4 , 0, 0, 0, 0, 0, 1 4 , 0, 0, 0, 0, 0, 0, 0, 1 2 , Recall the identifications for the Wilson lines: figure 1 where the Wilson line configuration for this model is shown. This embedding leaves the following gauge symmetry unbroken W 1 = W 2 = 7 4 , 1 4 , − 3 4 , − 1 4 , 1 4 , 1 4 , 5 4 , 1 4 , − 7 4 , − 1 4 , − 1 4 , − 1 4 , − 1 4 , − 1 4 , 1 4 , 7 4 , W 3 = W 4 = − 1 2 , − 3 2 , − 3 2 , 1, − 3 2 , 3 2 , 3 2 , 1, − 1 4 , − 7 4 , − 5 4 , − 1 4 , 1 4 , 1 4 , 3 4 , − 7 4 , W 5 = 0, − 1 2 , 3 2 , − 3 2 , 3 2 , 3 2 , 1, 3 2 , 1, −2, 0, 1, 1 2 , 2, 1, − 3 2 , W 6 = 0 .W 1 ∼ W 2 , W 3 ∼ W 4 , seeSU(4) 1 × SU(2) 1 × SU(2) 2 × SU(4) 2 × SU(2) 3 × U(1) 7 ⊂ E 8 × E 8 . The anomaly coefficients obtained for this specific orbifold model are A grav. 2 − 1 = −76 , A grav. 2 − 2 = 94 , A grav. 2 − 3 = 84 , A SU(4) 2 1 − 1 = −3 , A SU(4) 2 1 − 2 = 3 , A SU(4) 2 1 − 3 = −1 , A SU(2) 2 1 − 1 = −5 , A SU(2) 2 1 − 2 = 1 , A SU(2) 2 1 − 3 = 5 , A SU(2) 2 2 − 1 = −11 , A SU(2) 2 2 − 2 = 6 , A SU(2) 2 2 − 3 = 5 , A SU(4) 2 2 − 1 = −3 , A SU(4) 2 2 − 2 = −1 , A SU(4) 2 2 − 3 = −1 , A SU(2) 2 3 − 1 = −11 , A SU(2) 2 3 − 2 = 3 , A SU(2) 2 3 − 3 = 5 . (6.3) One can straightforwardly check that all of these values satisfy the universality conditions (4.5) and (4.6). This model also serves to discuss the effects of the new R-symmetries emerging from F . Note that ζ = θ 1 = (e 2πi 1 4 , 1, 1) ∈ F , (6.4) interchanges the fixed points z g = e 2 + e 3 2 ζ ←→ z g = e 2 + e 4 2 ,(6.5) of the second twisted sector T 2 , which are generated by space group elements g and g from different conjugacy classes. This is illustrated in figure 2. Note that as we have W 1 = W 2 , W 3 = W 4 , the transformation ζ respects the Wilson line structure. Hence the spectrum contains identical states V and V sitting at each of the relevant fixed points. As an example consider the states specified by the following quantum numbers p sh = − 3 4 , 1 4 , − 1 4 , − 1 4 , − 1 4 , − 1 4 , − 1 4 , − 1 4 , 0, 0, − 1 2 , 1 2 , 0, 0, 0, 0 , (6.6) with no left-moving oscillators. The p sh presented is the highest weight of the representation (1, 2, 2, 1, 2) with all U(1) charges equal to zero. Two identical copies of this state live at the fixed points under consideration. The elements h g and h g needed to compute the R-charges are given by h g = (θ, e 3 ) and h g = (θ, 0) , (6.8) and the corresponding gamma-phases are γ(g) = γ(g ) = 3/4. With this information we can compute the R-charges for the eigenstates of ζ to be q sh = 0, − 1 2 , − 1 2 , 0 ,(6.r (s) = −7/2 − 4s . (6.9) We also computed the anomaly coefficients for the R-symmetry ζ, with a scan of over 100.000 randomly generated models. In all cases the anomalies turned out to be universal. Similar results are to be expected for all orbifolds for which the group F is non-trivial. Note that, in our example, ζ 2 = 2 and one can show that the R-charges under 2 are twice those under ζ up to multiples of 2. This implies that one can safely take ζ and 1 as a basis for all R-symmetries in the factorizable Z 4 orbifold. Discussion In this work we have derived R-symmetries expected in the low energy effective field theory of heterotic orbifold compactifications, directly from the symmetries observed in the orbifold geometry. In particular, by imposing that the string correlation functions are invariant under such symmetries, we were able to infer the R-charges that are conserved in the low energy theory. This approach allowed us to be more general than the explicit computations of vanishing correlation functions pursued in [20]. For example, we were presently able to treat orbifold models with discrete Wilson lines. Moreover, we identified new R-symmetries, which arise from rotations which interchange inequivalent fixed points supporting the same physical states. The conserved R-charges associated with rotations preserving the fixed point structure of the orbifold were derived in explicit models for all Z N orbifold models, with and without discrete Wilson lines. The corresponding anomalies were then computed, and a scan of thousands of randomly generated models showed that the anomalies were universal. This is also the case for the R-symmetries conjectured for non-factorizable orbifolds, even though JHEP02(2014)098 there are some non-trivial steps still missing for the full understanding of their CFT. Further, we identified an additional source for R-symmetries, namely those isometries under which certain fixed points (that support the same twisted matter) get exchanged. An example was given for a Z 4 orbifold. It is remarkable that the corresponding R-symmetry anomalies were also found to satisfy universality relations. The universality of the R-symmetry anomalies is certainly a beautiful and compelling result. On the other hand, the R-charges that were obtained from the explicit computation of vanishing string correlation functions [20], have the opposite sign in the gamma-phase contribution, and do not always lead to the universal anomalies. It remains an essential open question to understand the reason behind this mismatch, although we have pointed out a possible origin for the discrepancy. Moreover, it seems important to bear in mind the following observations. Anomaly universality does not necessarily guarantee that a symmetry is an exact symmetry. Examples in which anomalies are universal, but the symmetry is explicitly broken by non-perturbative effects, are some continuous targetspace modular symmetries [42,53]. Meanwhile, anomalies which are non-universal might be partially cancelled by one-loop threshold effects, as sometimes observed for discrete target-space modular invariance [42,43]. Finally, as discrete symmetries are by definition global there is no inconsistency if they happen to be anomalous. Simply, this would imply that they are not symmetries in the full quantum theory. Despite the fact that the lattices studied here are the simplest possibilities, we expect similar results for the more general orbifold models discussed in [54][55][56]. It remains to be discussed how these redefined R-charges affect the phenomenology of MSSM like models found all over the orbifold landscape. In those models where the top Yukawa coupling is purely untwisted, one can guarantee its survival. However, in order to address issues such as Yukawa textures, decoupling of the exotics and proton decay it is necessary to look at explicit models. An important question concerns the effects of the new R-charge redefinitions in the particular context of Z 2 × Z 2 [17,18], especially in those models where the famous Z R 4 symmetry of [57,58] could be realized. Another interesting issue has to do with the fact that the R-charges now receive contributions from the gauge part of the theory, so it is worth studying how the gauge bundle information enters the R-symmetries that one also expects to see in the orbifold phase of gauged linear sigma models, as well as in partial blow-ups. JHEP02(2014)098 A Space group elements h g for the Z 4 and Z 6II orbifolds In this appendix we present the values of h g for Z 4 and Z 6II used in the main text, in tables 2 and 3. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. F 4 × 4SO Figure 1 . 1Wilson line configuration for the Z 4 orbifold studied in the text. Figure 2 . 2Representation of the θ 1 action on the T 2 sector fixed points of the Z 4 orbifold studied. 12) and in general, take a block diagonal form, i.e.9) JHEP02(2014)098 = diag(e 2πiξ 1 , e 2πiξ 2 , e 2πiξ 3 ) . (3.10) See[27] and references therein for such universality conditions for U(1) gauge anomalies. For a review on these selection rules we refer to[40]. Recall that gauginos and matter fermions both contribute to the anomaly. The charge of the fermions can be inferred from the piece θψ ⊂ Φ: if the charge of the multiplet Φ is denoted by r, then the charge of the fermion is r − R/2. Analogously, the gauginos appear in the vector multiplet in the form θθθλ, so that their charge is R/2. Analogous symmetries were considered in[52], but as these were permutation symmetries rather than rotational symmetries, they did not correspond to R-symmetries. AcknowledgmentsWe would like to thank M. Blaszczyk, S. Förste, P. Oehlmann and F. Rühle for useful discussions. N. G. C. 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[ "RELATION BETWEEN HÉNON MAPS WITH BIHOLOMORPHIC ESCAPING SETS RATNA PAL", "RELATION BETWEEN HÉNON MAPS WITH BIHOLOMORPHIC ESCAPING SETS RATNA PAL" ]
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Let H and F be two Hénon maps with biholomorphically equivalent escaping sets, then there exist affine automorphisms A1 and A2 in C 2 such that
10.1007/s00208-023-02630-w
[ "https://export.arxiv.org/pdf/2106.11661v2.pdf" ]
235,592,885
2106.11661
121d25a934107a44d8381585cdbf522c60f94d35
RELATION BETWEEN HÉNON MAPS WITH BIHOLOMORPHIC ESCAPING SETS RATNA PAL 24 Apr 2023 RELATION BETWEEN HÉNON MAPS WITH BIHOLOMORPHIC ESCAPING SETS RATNA PAL 24 Apr 2023arXiv:2106.11661v2 [math.CV]2020 Mathematics Subject Classification Primary: 32H02 ; Secondary 32H50 Key words and phrases Hénon mapsescaping setsBöttcher function Let H and F be two Hénon maps with biholomorphically equivalent escaping sets, then there exist affine automorphisms A1 and A2 in C 2 such that Introduction In the complex plane the simplest examples of holomorphic dynamical systems with nontrivial dynamical behaviour are the polynomial maps of degree greater than or equal to 2. The linear polynomials, in other words, the automorphisms of the complex plane, generate trivial dynamics. In contrast, the class of polynomial automorphisms of C 2 is large and possesses rich dynamical features. A dynamical classification of these maps was given by Friedland-Milnor [7]. They showed that any polynomial automorphism of C 2 is conjugate to one of the following maps: • an affine map; • an elementary map, i.e., the maps of the form (x, y) → (ax + b, sy + p(x)) with as = 0, where p is a polynomial in single variable of degree strictly greater than one; • a finite composition of Hénon maps, where Hénon maps are the maps of the form (1.1) (x, y) → (y, p(y) − δx) with δ = 0 and p a polynomial in single variable of degree d ≥ 2. The degree of a single Hénon map H of the form (1.1) is defined to be the degree of the polynomial p. The degree of composition of Hénon maps H n • · · · • H 1 is defined to be d n · · · d 1 where d i = deg(H i ), for 1 ≤ i ≤ n. Hénon maps are generalization of classical real quadratic Hénon maps introduced by astronomer Michel Hénon. The relevance of these maps in complex dynamics became apparent in the above-mentioned classification theorem of Friedland-Milnor. Moreover it turned out that the composition of Hénon maps are the only dynamically non-trivial polynomial automorphisms in C 2 , which naturally drew attention of many foremost researchers towards these maps. The pioneering work on Hénon maps was done by Bedford-Smillie ( [1], [2], [3]), Fornaess-Sibony ( [8]) and Hubbard-Oberste-Vorth ( [9], [10]). As in the case of polynomials in the complex plane, the orbit of any point in C 2 under the iterations of a Hénon map H (or more generally, finite composition of Hénon maps) either diverges to infinity or always remains bounded. The collection of points I + H ⊆ C 2 which escape to infinity are called the escaping set of H and the collection of points K + H ⊆ C 2 whose orbits remain bounded are called the non-escaping set of H. The union of escaping set I + H and the interior of non-escaping set K + H is the largest set of normality of the sequence of maps {H n } n≥1 , that is, I + H ∪intK + H is the Fatou set of H. The escaping sets and the interiors of non-escaping sets of Hénon maps can be thought of as analogues of the unbounded components and the bounded components of Fatou sets of polynomials in the complex plane. However, the non-escaping sets of Hénon maps are not bounded in C 2 . The common boundary set J + H of the escaping set I + H and the non-escaping set K + H is the Julia set of H. The present article addresses a rigidity property of Hénon maps of the form (1.1). To what extent do the escaping sets of Hénon maps determine the Hénon maps? In other words, if the escaping sets of a pair of Hénon maps H and F of degree d are biholomorphically equivalent, then are these two Hénon maps closely related? This question is first studied in a recent work of Bonnot-Radu-Tanase ( [4]), where they prove that H and F coincide, for d = 2. Further, they produce examples to show that for d ≥ 3, H and F might not be even conjugate to each other. In this article we establish a precise relation between H and F of any degree d ≥ 2 with biholomorphic escaping sets (Theorem 1.2). The rigidity question raised here is conceived based on an explicit description of analytic structure of the escaping sets given by Hubbard-Oberste-Vorth in [9]. A convenient description of the escaping set I + H of a given Hénon map H is given in terms of logarithmic rate of escape function, the so-called Green's function G + H of the Hénon map H. One can show that I + H is precisely where the Green's function is strictly positive. Further, G + H : I + H → R + is a pluri-harmonic submersion and the level sets of G + H are three dimensional manifolds, which are naturally foliated by copies of C. The Green's function G + H is inextricably related to the Böttcher function φ + H of H, which is one of the key ingredients in describing the analytic structure of I + H . The Böttcher functions of Hénon maps can be considered as analogues of Böttcher functions of polynomials in C near infinity and they are defined in appropriate neighbourhoods of a point at infinity [0 : 1 : 0] in P 2 . To be a bit more precise, for R > 0 sufficiently large, φ + H is defined on the open set V + R = (x, y) ∈ C 2 : |y| > max {|x|, R} ⊆ I + H by means of approaching the point [0 : 1 : 0] by the n-fold iteration H n , then returning back by appropriate d n -th root of the mapping y → y d n and finally taking the limit as n → ∞. Consequently, the range of φ + H lies inside C \D, where D is the unit disc in C and G + H simply straightens out. The Hénon map H lifts as a mapH : C × (C \D) → C × (C \D) and one can write downH explicitly (see (2.6)). H ≡ log φ + H in V + R . The following result relies on two main ingredients, also used by Bonnot-Radu-Tanase in [4]: The above-mentioned explicit description of the coveringĨ + H of I + H and a method given by Bousch in [5]. Theorem 1.1. Let H(x, y) = (y, p H (y) − δ H x) and F (x, y) = (y, p F (y) − δ F x) be a pair of Hénon maps, where p H (y) = y d + d−2 i=0 a H i y i and p F (y) = y d + d−2 i=0 a F i y i . Let I + H and I + F be escaping sets of H and F , respectively and let I + H and I + F be biholomorphically equivalent. Then βp H (y) = αp F (αy), for some α, β ∈ C with α d+1 = β and β d−1 = 1. Further, we have δ H = γδ F , with γ d−1 = 1. Therefore, (1.2) F ≡ L • B • H • B, where B(x, y) = (γαβ −1 x, α −1 y) and L(x, y) = (γ −1 βx, βy), for all (x, y) ∈ C 2 . Now note that p H and p F in Theorem 1.1 are monic and centered (next to highest coefficients vanish), whereas for an arbitrary Hénon map the associated polynomial in one variable is not necessarily monic and centered. However, it is not hard to see that up to conjugation any arbitrary Hénon map is of the form as in Theorem 1.1. Here goes a brief justification. Up to conjugation by an affine automorphism of C, any polynomial in one variable is a monic and centered polynomial of the same degree. In particular, there exists an affine automorphism σ H of C such that σ −1 H •p H •σ H =p H in C, wherep H is monic and centered. Thus if we consider the affine automorphism A H (x, y) = (σ H (x), σ H (y)), for (x, y) ∈ C 2 , then A −1 H • H • A H =Ĥ, wherê H(x, y) = (y,p H (y) − δ H x), for all (x, y) ∈ C 2 with deg(p H ) = deg(p H ). Clearly, A H (I + H ) = I + H . Similarly, there exist an affine map A F and a Hénon mapF such that A −1 F • F • A F =F and A F (I + F ) = I + F , whereF (x, y) = (y,p F (y) − δ F x) , for all (x, y) ∈ C 2 withp F monic and centered. Therefore, once we prove Theorem 1.1, the following result is obtained immediately. Theorem 1.2. Let H and F be two Hénon maps with biholomorphically equivalent escaping sets, then there exist affine automorphisms A 1 and A 2 in C 2 such that F = A 1 • H • A 2 in C 2 . Acknowledgement: The author would like to thank the referees for making helpful comments. Preliminaries Let (2.1) H(x, y) = (y, p(y) − δx) be a Hénon map, where p is a monic and centered polynomial in single variable of degree d ≥ 2 and δ = 0. In this section, we see a few fundamental definitions and a couple of known results on Hénon maps pertaining to the theme of the present article. Filtration: For R > 0, let V + R = {(x, y) ∈ C 2 : |x| < |y|, |y| > R}, V − R = {(x, y) ∈ C 2 : |y| < |x|, |x| > R}, V R = {(x, y) ∈ C 2 : |x|, |y| ≤ R}. This is called a filtration. For a given Hénon map H, there exists R > 0 sufficiently large such that H(V + R ) ⊂ V + R , H(V + R ∪ V R ) ⊂ V + R ∪ V R and H −1 (V − R ) ⊂ V − R , H −1 (V − R ∪ V R ) ⊂ V − R ∪ V R . Escaping and non-escaping sets: The set I + H = (x, y) ∈ C 2 : H n (x, y) → ∞ as n → ∞ isI + H = C 2 \ K + H = ∞ n=0 H −n (V + R ). Any Hénon map H can be extended meromorphically to P 2 with an isolated indeterminacy point Green's function: The Green's function of H is defined to be G + H (x, y) := lim n→∞ 1 d n log + H n (x, y) , for all (x, y) ∈ C 2 , where log + (t) = max{log t, 0}. It turns out that G + H is non-negative everywhere in C 2 , plurisubharmonic in C 2 , pluriharmonic on C 2 \ K + H and vanishes precisely on K + H . Further, G + H • H = dG + H in C 2 . The Green's function G + H has logarithmic growth near infinity, i.e., there exist R > 0 and L > 0 such that log + |y| − L ≤ G + H (x, y) ≤ log + |y| + L, for (x, y) ∈ V + R ∪ V R . Böttcher function: Let H n (x, y) = ((H n ) 1 (x, y), (H n ) 2 (x, y)), for (x, y) ∈ C 2 . Note that y n = (H n ) 2 (x, y) is a polynomial in x and y of degree d n . The function φ + H (x, y) := lim n→∞ y 1 d n n = y. y 1 1 d y . · · · . y 1 d n+1 n+1 y 1 d n n · · · , for (x, y) ∈ V + R , defines an analytic function from V + R to C\D and is called the Böttcher function of the Hénon map H. Further, φ + H • H(x, y) = φ + H (x, y) d , for all (x, y) ∈ V + R and φ + H (x, y) ∼ y as (x, y) → ∞ in V + R . Comparing the definitions of Green's function and Böttcher functions it follows instantly that G + H = log|φ + H | in V + R . For a detailed treatment of the above discussion, the inquisitive readers can see [1], [2], [3], [9] and [11]. Now we present one of the technical ingredients required for the proof of Theorem 1.1. A series of change of coordinates we see here is a part of the standard theory of Hénon maps. However, most of the results presented here are paraphrasing of a couple of lemmas appearing in the beginning of the appendix of [4]. However the genesis of these results goes back to [6] and [9]. For M > 0 sufficiently large, let U + R = (x, y) ∈ V + R : |φ + H (x, y)| > M max{R, |x|} . One can easily check that U + R ⊆ V + R and H(U + R ) ⊆ U + R .on U + R such that • ψ H • H(x, y) = (δ/d)ψ H (x, y) + Q φ + H (x, y) , for all (x, y) ∈ U + R , where Q is a monic polynomial of degree d + 1; • the map Φ H = (ψ H , φ + H ) : U + R → C 2 is an injective holomorphic map. The map (x, y) → (x, ζ) = (x, φ + H (x, y)) maps U + R biholomorphically to (x, y) ∈ C 2 : |y| > M max{R, |x|} . Let (x, ζ) → (x, y(x, ζ)) be the inverse map. It turns out that ψ H (x, y) = ζ x 0 ∂y ∂ζ (u, ζ)du, for (x, y) ∈ U + R and Q(ζ) is the polynomial part of the power series expansion of (2.3) ζ d y(0,ζ) 0 ∂y ∂ζ (u, ζ d )du (see proof of [11,Lemma 7.3.7] for details). Next we consider the following change of coordinate near p = [0 : 1 : 0]: T : (x, y(x, ζ)) = (x, y) → x y , 1 y = (t, w). Note that T 2 = Id and H takes the following form in (t, w)-coordinate: (t, w) → (x, y) → (y, p(y) − δx) → y p(y) − δx , 1 p(y) − δx = w d−1 w d p(1/w) − δtw d−1 , w d w d p(1/w) − δtw d−1 .(2.4) 1/ζ = w(1 + wα(t, w)), where α(t, w) is a power series in t, w. Proof. Note that 1 ζ = 1 φ + H (x, y) = lim n→∞ 1 (H n (x, y)) 2 1 d n = lim n→∞ (T H n T −1 (t, w)) 2 1 d n . Equivalently, 1 ζ = w. w d w d p( 1 w ) − δtw d−1 /w d 1 d . w d 1 w d 1 p( 1 w 1 ) − δt 1 w d−1 1 /w d 1 1 d 2 · · · = w. 1 w d p( 1 w ) − δtw d−1 1 d . 1 w d 1 p( 1 w 1 ) − δt 1 w d−1 1 1 d 2 · · · = wX(t, w) (here the d n -th roots, for n ≥ 1, are taken to be the principal branches of roots). One can check that the above series converges. Further, since φ + H (x, y)/y is bounded in V + R , it follows that X(t, w) is also bounded in its domain of definition, which is a subset of {|t| < 1} × {0 < |w| < 1/R}. Thus X(t, w) has a power series expansion. Also note that X(t, 0) = 1, for all t. So X(t, w) = 1 + wα(t, w), where α(t, w) is a power series in t and w. Therefore, finally we obtain (2.4). Next we consider the following change of variables: (t, w) → (r, s) = (x/ζ, 1/ζ), which is a biholomorphism and is tangent to the identity (see [4,Lemma 4.4]). Therefore, t = r + T 2 (r, s) and w = s + S 2 (r, s), where T 2 and S 2 both are power series with monomials of degree at least 2. Lemma 2.3. With the above notations, we have y = ζ (1 + C/ζ + U (x/ζ, 1/ζ)) , where C ∈ C and U is a power series in two variables with monomials of degree at least 2. Proof. By Lemma 2.2, we have 1/ζ = w(1 + wα(t, w)). Thus y/ζ = (1 + wα(t, w)). Therefore, by replacing t = r + T 2 (r, s) and w = s + S 2 (r, s), we get y = ζ (1 + (s + S 2 (r, s))(α(r + T 2 (r, s), s + S 2 (r, s)))) = ζ (1 + sβ(r, s) + S 2 (r, s)β(r, s)) , where β(r, s) = α(r + T 2 (r, s), s + S 2 (r, s)). Now since (r, s) = (x/ζ, 1/ζ), we have y = ζ (1 + C/ζ + U (x/ζ, 1/ζ)) , where C ∈ C is the constant term of the power series expansion of β and U is a power series in two variables without constant term. Therefore, y(x, ζ) = ζ + C + A 0 −1 ζ + · · · +x A 1 −1 ζ m 1 + · · · +x 2 A 2 −1 ζ m 2 + · · · +· · ·+x n A n −1 ζ mn + · · · +· · · , where m 1 = 1, m n = n − 1, for n ≥ 2 and A 0 −1 , A 1 −1 , A 2 −1 and so on are in C. The following lemma is one of the main takeaways from this section. Lemma 2.4. With the above notations, we have y(0, ζ) = ζ + D 1 ζ + D 2 ζ 2 · · · , and ζ(0, y) = y + L 1 y + L 2 y 2 · · · , where D i 's and L i 's are constants. Further, Q(ζ) is given by the polynomial part of ζ d y(0, ζ) and therefore, Q is centered and monic. Proof. Since p is monic and centered in (2.1), it follows from [9, Prop. 5.2] that ζ(0, y) = y + L 1 y + L 2 y 2 · · · . Now since y = y(0, ζ(0, y)), y = y(0, ζ(0, y)) = ζ(0, y) + D 0 + D 1 ζ(0, y) + · · · = y + L 1 y + L 2 y 2 + · · · + D 0 + D 1 y + L 1 y + L 2 y 2 + · · · −1 + · · · . Thus D 0 = 0 and consequently, we have y(0, ζ) = ζ + D 1 ζ + D 2 ζ 2 + · · · . The last assertion follows immediately from (2.3). As we shall see in Section 3 and in Section 4, the particular forms of the power series representations of y(0, ζ) and ζ(0, y) as obtained in Lemma 2.4 and the knowledge of analytic structure of the escaping sets of Hénon maps are the key players in unravelling the relation between any pair of Hénon maps with biholomorphic escaping sets. Here we give a brief description of analytic structure of Hénon maps. For a detailed account of analytic structure of escaping sets of Hénon maps the inquisitive readers can look at [6], [9] and [11]. For any Hénon map H, it turns out that the Riemann surface of the Böttcher function φ + H is a covering space of the escaping set I + H and it is isomorphic to C × (C \D). For a Hénon map H of degree d, the fundamental group of I + H is Z [1/d] = {k/d n : k, n ∈ Z} and the covering C × (C \D) of I + H arises corresponding to the subgroup Z ⊆ Z [1/d]. Therefore, I + H is a quotient of C × (C \D) by some discrete subgroup of the automorphisms of C × (C \ D) isomorphic to Z [1/d]/Z. For each element [k/d n ] ∈ Z[1/d]/Z, there exists a unique deck transformation γ k/d n from C × (C \D) to C × (C \D) such that (2.5) γ k/d n z ζ =   z + d δ n−1 l=0 d δ l Q(ζ d l ) − Q e 2kπi d n ζ) d l e 2kπi d n ζ   , for n ≥ 0 and k ≥ 1. Further, H lifts as a holomorphic map (2.6)H(z, ζ) = (δ/d)z + Q(ζ), ζ d from C × (C \D) to C × (C \D). A brief idea of the proof of the main theorem In this section we sketch the main idea of the proof of Theorem 1.1. Let a be a biholomorphism between I + H and I + F . Any biholomorphism between I + H and I + F , which induces identity as an isomorphism between the fundamental groups of I + H and I + F , can be lifted as a biholomorphism from C × (C \D) to C × (C \D). Now since any group isomorphism of Z[1/d] is of the form x → ±d s x, for some s ∈ Z, the map a up to pre-composition with some n-fold iterates of F (or F −1 ), i.e., the map F ±n • a, for some n ∈ N, induces identity map between the fundamental groups of the escaping sets I + H and I + F . Thus, it is harmless to assume that a lifts as a biholomorphism A from C × (C \D) to C × (C \D). Therefore, π • A = a • π and consequently, the fiber of any point p ∈ I + H of the natural projection map π H : C × (C \D) → I + H maps into the fiber of the point a(p) ∈ I + F of the projection map π F : C × (C \D) → I + F by the biholomorphism A. Fiber of any point of the projection maps π H and π F are captured by the group of deck transformation of C × (C \D), which is isomorphic to Z[1/d]/Z. Thus, it turns out that if (z, ζ) ∈ C × (C \D) is in the fiber of any point p ∈ I + H (or I + F ), then any other point in the same fiber will be of the form γ k/d n (z, ζ), for k ≥ 1 and n ≥ 0, where γ k/d n is the deck transformation corresponding to [k/d n ] ∈ Z[1/d]/Z. An explicit description of this fibers can be written down (see (2.5)). Now adapting an idea of Bousch ([5]), one can narrow down the possible forms of A by comparing the fibers of p and a(p). In fact, in our case, A has a very simple form (u, v) → (u + γ, αv), with |α| = 1 and γ ∈ C. Thus we obtain an explicit expression for the image of the foliatioñ φ + H = ζ 0 under the map A. These foliations come down to the corresponding escaping sets and induce rigidity on escaping sets. As a consequence, one expects a close relation between H and F , which is validated in our main theorem. Let H(x, y) = (y, p H (y)−δ H x) and F (x, y) = (y, p F (y)−δ F x) be The above-mentioned idea was employed by Bonnot-Radu-Tanase in [4] in establishing the relation between H and F with biholomorphic escaping sets when deg(H) = deg(F ) = 2. They attempts to extract the relation between the coefficients of p H and p F by directly comparing the fibers of p and a(p). Although their approach works for lower degrees (for d = 2, 3), since precise computations can be carried out in these cases but as degree increases, to extract the relation between the coefficients of p H and p F seems very difficult by performing such direct calculations. We take a different approach towards this problem. For two Hénon maps H and F with biholomorphic escaping sets, we first establish the relation between coefficients of Q H and Q F (see Section 3). The explicit relation between Q H and Q F is used to obtain a neat relation between the polynomials p H and p F , namely, βp H (y) = αp F (αy), for all y ∈ C with α d+1 = β and β d−1 = 1 (see Section 4). To establish the relation between H and F , we are yet to investigate the relation between the Jacobians δ H and δ F . It turns out that δ H = γδ F , with γ d−1 = 1, thanks to [4]. It is noteworthy that all our calculations in Section 3 and in Section 4 are performed under the assumption that δ H = δ F . In Section 5, we outline how to handle the case when δ H and δ F are different. Relation between Q H and Q F Let H and F be two Hénon maps as in Theorem 1.1 with bihholomorphic escaping sets I + H and I + F , respectively. As indicated in the Introduction, for now we assume δ H = δ F = δ. Later (in Section 6), we handle the case when δ H = δ F . π 1 (a) : Z [1/d] → Z [1/d] is ±Id (identity maps). It is easy to see that any group isomorphism of Z[1/d] is of the form x → ±d s x, for some s ∈ Z. Thus, there exists n ∈ N such that F ±n • a induces identity on Z[1/d] and thus lifts as an automorphism of C × C \D . Therefore, without loss of generality, we assume that a lifts as an automorphism A of C × C \D . C × C \D C × C \D I + H I + F A π H π F aγ k/d n from C × (C \D) to C × (C \D) of the form (4.1) γ k/d n z ζ =   z + d δ n−1 l=0 d δ l Q H (ζ d l ) − Q H e 2kπi d n ζ) d l e 2kπi d n ζ   . The same holds for the Hénon map F . Thus ifp = (z, ζ) ∈ C × C \D lies in the π −1 H (p) for some p ∈ I + H , then π −1 H (p) = {γ k/d n (z, ζ) : n ≥ 0, k ≥ 1}. Form of lifts: First note that any automorphism A of C × (C \D) is of the form: (4.2) A(z, ζ) = (A 1 (z, ζ), A 2 (z, ζ)) = (β(ζ)z + γ(ζ), αζ), where |α| = 1 and β, γ are holomorphic maps from C \D to C * and C, respectively (see [5,Section 3]). Note that since a•π H = π F •A, if (z, ζ), (z ′ , ζ ′ ) ∈ π −1 H (p), for some p ∈ I + H , then A(z, ζ), A(z ′ , ζ ′ ) ∈ π −1 F (a(p)). Now let (z, ζ) and (z ′ , ζ ′ ) be in the same fiber of π H , then using (4.1), we have (ζ ′ /ζ) d n = 1 and z ′ = z + d δ n−1 l=0 d δ l Q H (ζ d l ) − Q H e 2kπi d n ζ) d l , for some n ∈ N. Therefore the difference between the first coordinates of A, i.e., A 1 (z ′ , ζ ′ ) − A 1 (z, ζ) = β(ζ ′ ) − β(ζ) u + β(ζ ′ ) d δ n−1 l=0 d δ l Q H (ζ d l ) − Q H e 2kπi d n ζ) d l + γ(ζ ′ ) − γ(ζ) . (4.3) Now since A(z, ζ) and A(z ′ , ζ ′ ) are in the same fiber of π F , the difference A 1 (z, ζ) − A 1 (z ′ , ζ ′ ) is a function of αζ and αζ ′ . Thus it follows from (4.3) that β(ζ) = β(ζ ′ ), i.e., β(ζ) = β ζ.e 2πik d n , for all k ≥ 1 and for all n ≥ 0. Therefore, β(ζ) ≡ β in C. Thus it follows form (4.3) that A 1 (z ′ , ζ ′ ) − A 1 (z, ζ) = ∆ H (ζ, ζ ′ ) + γ(ζ ′ ) − γ(ζ), (4.4) where ∆ H (ζ, ζ ′ ) = β d δ n−1 l=0 d δ l Q H (ζ d l ) − Q H e 2kπi/d n ζ) d l . On the other hand since A(z, ζ) and A(z ′ , ζ ′ ) are in the same fiber of π F , (4.5) A 1 (z ′ , ζ ′ ) − A 1 (z, ζ) = ∆ F (αζ, αζ ′ ) = d δ n−1 l=0 d δ l Q F (α d l ζ d l ) − Q F e 2kπi d n αζ) d l . Since γ is a holomorphic function on C \D, comparing (4.4) and (4.5), it follows that for a fixed ζ ∈ C \D, the modulus of the difference between ∆ H (ζ, ζ ′ ) and ∆ F (αζ, αζ ′ ) is uniformly bounded for all v ′ = e 2kπi/d n ζ with n ≥ 0 and k ≥ 1. Note that for ζ ′ = e 2kπi/d n ζ, the difference ∆ H (ζ, ζ ′ ) − ∆ F (αζ, αζ ′ ) is a polynomial of degree (d + 1)d n−1 and it can be written as β n−1 l=0 d δ l Q H ζ d l − Q H e 2kπi d n ζ d l − n−1 l=0 d δ l Q F α d l ζ d l − Q F e 2kπi d n αζ d l = n−1 l=0 d δ l βQ H ζ d l − Q F α d l ζ d l − n−1 l=0 d δ l βQ H e 2kπi d n ζ d l − Q F e 2kπi d n αζ d l = d n−1 δ n−1 βζ d n−1 (d+1) − (αζ) d n−1 (d+1) − β e 2πi d n ζ d n−1 (d+1) + αζe 2πi d n d n−1 (d+1) + R(ζ) = d δ n−1 β − α d n−1 (d+1) 1 − e 2πi d ζ d n−1 (d+1) 1 +R(ζ) , where R(ζ) = O ζ d n and R(ζ) = R(ζ)(d/δ) −n+1 β − α d n−1 (d+1) −1 1 − e 2πi d −1 ζ −d n−1 (d+1) . We claim that α d n−1 (d+1) → β, as n → ∞. If not, then there exists a subsequence {n l } l≥1 such that (4.6) α d n l −1 (d+1) − β > c > 0, for all l ≥ 1. Thus, if |d/δ| ≥ 1, then (note that R is a polynomial of degree at most d n ) |R(ζ)| ≤ 4nK 1 |ζ| d n−1 , for some K 1 > 1. On the other hand, if |d/δ| < 1, then |R(ζ)| ≤ 4nK 2 |ζ| d n−1 δ d n−1 , for some K 2 > 1. Therefore, since the modulus of ∆ H ζ, e 2πi d n ζ − ∆ F αζ, αe 2πi d n ζ is uniformly bounded, for all n ≥ 1, we get a contradiction if (4.6) holds. Thus (4.7) α d n (d+1) → β, as n → ∞. Also, (4.8) α d n+1 (d+1) → β as n → ∞ and thus dividing (4.8) by (4.7) and taking the limit, we get (4.9) α (d+1)(d−1)d n → 1, as n → ∞. Therefore, we get β d−1 = 1. On the other hand, it follows from (4.7) that α d+1 d n → β and also note that β is a repelling fixed point for the map z → z d . Therefore, (4.7) holds if and only if α d+1 = β. Thus α d n (d+1) − β = 0, for all n ≥ 1, which in turn gives γ(ζ) = γ e 2kπi d n ζ , for all k ≥ 1 and n ≥ 0. Thus γ ≡ γ 0 , for some γ 0 ∈ C. Relation between Q H and Q F : Note that it follows from Lemma 2.4 that next to the highest degree coefficients of Q H and Q F vanish. Let (4.10) Q H (ζ) = ζ d+1 + A H d−1 ζ d−1 + A H d−2 ζ d−2 + · · · + A H 1 ζ + A H 0 and (4.11) Q F (ζ) = ζ d+1 + A F d−1 ζ d−1 + A F d−2 ζ d−2 + · · · + A F 1 ζ + A F 0 . Since γ is identically constant in the complex plane, taking k, n = 1, it follows from (4.4) and (4.5) that (4.12) β d δ Q H (ζ) − Q H e 2πi d ζ = ∆ H (ζ, ζ ′ ) = ∆ F (αζ, αζ ′ ) = d δ Q F (αζ) − Q F e 2πi d αζ . Comparing coefficients of both sides of (4.12), we get 1) and since β = α d+1 , equivalently we get 1). Note that the relation between the constant terms of Q H and Q F , i.e., the relation between A H 0 and A F 0 cannot be extracted from (4.12). However, as we are going to see in the next section that the explicit relation between A H d−i and A F d−i , for 1 ≤ i ≤ (d − 1), obtained in (4.13) is sufficient to track down the relation between p H and p F . βA H d−k = α d−k A F d−k , for 1 ≤ k ≤ (d −(4.13) A H d−k = α −(k+1) A F d−k , for 1 ≤ k ≤ (d − Relation between p H and p F It follows from Lemma 2.4 that there exist (5.1) y H (ζ) ≡ y H (0, ζ) = ζ + D H 1 ζ + D H 2 ζ 2 + · · · + D H d−1 ζ d−1 + O 1 ζ d and (5.2) y F (ζ) ≡ y F (0, ζ) = ζ + D F 1 ζ + D F 2 ζ 2 + · · · + D F d−1 ζ d−1 + O 1 ζ d , with D H i , D F i ∈ C for i ≥ 1, such that Q H (ζ) and Q F (ζ) are polynomial parts of ζ d y H (0, ζ) and ζ d y F (0, ζ), respectively. Therefore, the precise relation obtained in (4.13) determines a similar relation between the first few corresponding coefficients of the power series expansion of y H (ζ) and y F (ζ), namely (5.3) D H k = α −(k+1) D F k , for 1 ≤ k ≤ (d − 1). Since we assume that the Jacobian determinants of H and F are the same, that is, δ H = δ F = δ, understanding the relation between the polynomials p H and p F is sufficient to capture the relation between H and F . We show in this section that the relation between the coefficients of p H and p F can be extracted via the Böttcher coordinates of H and F , namely via the functions ζ H and ζ F , respectively. Let (5.4) ζ H (y) ≡ ζ H (0, y) = y + L H 1 y + L H 2 y 2 + · · · + L H d−1 y d−1 + O 1 y d and (5.5) ζ F (y) ≡ ζ F (0, y) = y + L F 1 y + L F 2 y 2 + · · · + L F d−1 y d−1 + O 1 y d . Recall from Section 2 that ζ H • y H (0, ζ) = ζ and ζ F • y F (0, ζ) = ζ, for all ζ ∈ C with |ζ| > R, where R > 0 is sufficiently large. Thus implementing (5.1), (5.2), (5.4) and (5.5) together, we get D H 1 ζ + D H 2 ζ 2 + · · · + D H d−1 ζ d−1 + O 1 ζ d + L H 1 ζ 1 + D H 1 ζ 2 + D H 2 ζ 3 + · · · + D H d−1 ζ d−1 + O 1 ζ d −1 + L H 2 ζ 2 1 + D H 1 ζ 2 + D H 2 ζ 3 + · · · + D H d−1 ζ d−1 + O 1 ζ d −2 + · · · = 0 (5.6) and D F 1 ζ + D F 2 ζ 2 + · · · + D F d ζ d + O 1 ζ d + L F 1 ζ 1 + D F 1 ζ 2 + D F 2 ζ 3 + · · · + D F d−1 ζ d−1 + O 1 ζ d −1 + L F 2 ζ 2 1 + D F 1 ζ 2 + D F 2 ζ 3 + · · · + D F d−1 ζ d−1 + O 1 ζ d −2 + · · · = 0. (5.7) Note that expanding (5.6) and (5.7), one can express L H k and L F k , for any k ≥ 1, in terms of D H 1 , D H 2 , . . . and D F 1 , D F 2 , . . ., respectively. In fact, using the relation between D H k and D F k , for 1 ≤ k ≤ (d − 1), obtained in (5.3), one can establish an explicit relation between L H k and L F k for 1 ≤ k ≤ (d − 1). Claim L : For 1 ≤ k ≤ d − 1, L H k = α −(k+1) L F k . Note that implementing claim L to (5.4) and (5.5), we obtain (5.8) αζ H (y) − ζ F (αy) = O 1/y d . As we shall see shortly that the above relation between ζ H and ζ F plays the key role in establishing the relation between p H and p F . For now assuming claim L is true and thus assuming (5.8) holds, let us first determine the relation between p H and p F . We see a proof of the claim L in the end of the present section. Relation between p H and p F : Recall from Section 2 that for |y| > R, with R large enough, one can write (5.9) ζ H (y) ≡ ζ H (0, y) = y. p H (y) y d 1 d p H (y 1,H ) y d 1,H 1 d 2 · · · and (5.10) ζ F (y) ≡ ζ F (0, y) = y. p F (y) y d 1 d p F (y 1,F ) y d 1,F 1 d 2 · · · , where y 1,H = (H(x, y)) 2 = p H (y)−δx and y 1, F = (F (x, y)) 2 = p F (y)−δx. Thus ζ H (y)−O(1/y d ) and ζ F (y) − O(1/y d ) are determined by y. p H (y) y d 1 d and y. p F (y) y d 1 d , respectively. The power series expansion of the holomorphic function (1 + z) 1/d (principal branch) is 1 + (1/d)z + ((1/d − 1)/2d) z 2 + · · · , for z ∈ D = {z ∈ C : |z| < 1}. Now note that y. p H (y) y d 1 d = y. y d + a H d−2 y d−2 + · · · + a H 1 y + a H 0 y d 1 d = y   1 + 1 d a H d−2 y 2 + · · · + a H 0 y d + 1 2d 1 d − 1 a H d−2 y 2 + · · · + a H 0 y d 2 + · · ·   (5.11) and αy. p F (αy) (αy) d 1 d = αy. (αy) d + a F d−2 (αy) d−2 + · · · + a F 1 (αy) + a F 0 (αy) d 1 d = αy   1 + 1 d a F d−2 α 2 y 2 + · · · + a F 0 α d y d + 1 2d 1 d − 1 a F d−2 α 2 y 2 + · · · + a F 0 α d y d 2 + · · ·   , (5.12) for all y ∈ C with |y| large enough. Since (5.8) holds, comparing (5.11) and (5.12), we get a F d−2 = α 2 a H d−2 . Let us assume that a F d−k = α k a H d−k , up to some k, where 2 ≤ k ≤ (d − 1). Now it follows from (5.11) that the coefficient of 1/y k in the expansion of αζ H (y) is (5.13) c H k+1 = α a H d−(k+1) /d + G a H d−2 , · · · , a H d−k , where G is defined as follows. The function G is the polynomial in (k − 1) complex variables determined by the coefficient of 1/y k+1 of the power series expansion of (5.14) p H (y) y d 1 d − 1 d a H d−2 y 2 + · · · + a H 0 y d − 1 = 1 2d 1 d − 1 a H d−2 y 2 + · · · + a H 0 y d 2 + · · · (see 5.11). In other words, if the coefficient of 1/y k+1 in (5.14) is n i=1 g i Π k j=2 (a H d−j ) i(j) with g i ∈ C, then G(x 1 , x 2 , . . . , x k−1 ) = n i=1 g i Π k j=2 x j i(j) . Similarly, it follows from (5.12) that the coefficient of 1/y k in the expansion of ζ F (αy) is (5.15) c F k+1 = a F d−(k+1) α k d + αG a F d−2 α 2 , · · · , a F d−k α k = a F d−(k+1) α k d + αG(a H d−2 , · · · , a H d−k ). Since c H k+1 = c F k+1 , comparing (5.13) and (5.15), we have a F d−(k+1) = α k+1 a H d−(k+1) . Therefore, the following relation between the corresponding coefficients of p H and p F holds: (5.16) a F d−k = α k a H d−k , for 2 ≤ k ≤ d. Now since β = α d+1 and (5.16) holds, we have p F (y) = y d + a F d−2 y d−2 + · · · + a F 1 y + a F 0 = y d + α 2 a H d−2 y d−2 + α 3 a H d−3 y d−3 + · · · + α d−1 a H 1 y + α d a H 0 = y d + βα −(d−1) a H d−2 y d−2 + βα −(d−2) a H d−3 y d−3 + · · · + βα −2 a H 1 y + βα −1 a H 0 = βα −1 (α −1 y) d + a H d−2 (α −1 y) d−2 + a H d−3 (α −1 y) d−3 + · · · + a H 1 (α −1 y) + a H 0 = (βα −1 )p H (α −1 y). In other words, we obtain (5.17) αp F (αy) = βp H (y), for all y ∈ C, with α d+1 = β and β d−1 = 1. Before giving a formal proof of the claim L, let us establish the relation between L H k and L F k with bare hands when the common degree d of the Hénon maps H and F is small. To start with, first note that the coefficients of 1/ζ k vanish in both (5.6) and (5.7), for all k ≥ 1. • Let d = 2. Thus by (5.3), we have D H 1 = α −2 D F 1 . The coefficients of 1/ζ in (5.6) and (5.7) are D H 1 + L H 1 and D F 1 + L F 1 , respectively and both vanish. Thus L H 1 = −D H 1 = −α −2 D F 1 = α −2 L F 1 . • Let d = 3. By (5.3), we have D H i = α −(i+1) D F i , for i = 1, 2. Since the coefficients of 1/ζ vanish in (5.6) and (5.7), as before we can show that L H 1 = α −2 L F 1 . Using the fact that coefficients of 1/ζ 2 , which are D H 2 + L H 1 .0 + L H 2 and D F 2 + L F 1 .0 + L F 2 , vanish, we have L H 2 = −D H 2 = −α −3 D F 2 = α −3 L F 2 . • Let d = 4. By (5.3), we have D H i = α −(i+1) D F i , for 1 ≤ i ≤ 3 . Now since the coefficients 1/ζ i vanish in (5.6) and (5.7), as before we can show that L H i = α −(i+1) L F i , for 1 ≤ i ≤ 2. Now the coefficients of 1/ζ 3 in (5.6) and (5.7) are D H 3 + L H 1 . C 1 D H 1 + L H 2 .0 + L H 3 and D F 3 + L F 1 . C 1 D F 1 + L F 2 .0 + L F 3 , respectively for some C 1 ∈ C and they vanish. Therefore, L H 3 = −D H 3 − L H 1 . C 1 D H 1 , which implies L H 3 = −α −4 D F 3 − α −4 L F 1 . C 1 D F 1 = α −4 L F 3 . • Let d = 5. Thus we have D H i = α −(i+1) D F i , for 1 ≤ i ≤ 4. Using the same arguments as before we can show L H i = α −(i+1) L F i , for 1 ≤ i ≤ 3. Now the coefficients of 1/ζ 4 in (5.6) and (5.7) are D H 4 + L H 1 . C 2 D H 2 + L H 2 . C 3 D H 1 + L H 3 .0 + L H 4 and D F 4 + L F 1 . C 2 D F 2 + L F 2 . C 3 D F 1 + L F 3 .0 + L F 4 , for some C 2 , C 3 ∈ C and since they vanish, a simple calculation as above gives that L H 4 = α −5 L F 4 . • Let d = 6. Thus we have D H i = α −(i+1) D F i , for 1 ≤ i ≤ 5. As before we can show L H i = α −(i+1) L F i , for 1 ≤ i ≤ 4. The coefficients of 1/ζ 5 in (5.6) and (5.7) are M X 5 . Here for X = H, F , M X 5 = D X 5 + L X 1 . C 4 D X 3 + C 5 (D X 1 ) 2 + L X 2 . C 6 D X 2 + L X 3 .(C 7 D X 1 ) + L X 4 .0 + L X 5 , where C 4 , C 5 and so on are from C. Since M H 5 and M F 5 vanish, calculations as above give us L H 5 = α −6 L F 5 . • Let d = 7. Thus D H i = α −(i+1) D F i , for 1 ≤ i ≤ 6. Hence we can show L H i = α −(i+1) L F i , for 1 ≤ i ≤ 5 as before. The coefficients of 1/ζ 6 in (5.6) and (5.7) are M X 6 . Here for X = H, F , M X 6 = D X 6 + L X 1 . C 8 D X 4 + C 9 D X 1 D X 2 + L X 2 . C 10 D X 3 + C 11 (D X 1 ) 2 + L X 3 .(C 12 D X 2 ) +L X 4 .(C 13 D X 1 ) + L X 5 .0 + L X 6 , where C 8 , C 9 and so on are from C. Since M H 6 and M F 6 vanish, we get L H 6 = α −7 L F 6 . • Let d = 8. So D H i = α −(i+1) D F i , for 1 ≤ i ≤ 7. Hence we can show L H i = α −(i+1) L F i , for 1 ≤ i ≤ 6 just as before. Now the the coefficients of 1/ζ 7 in (5.6) and (5.7) are M X 7 . Here for X = H, F , M X 7 = D X 7 + L X 1 . C 14 D X 5 + C 15 D X 1 D X 3 + C 16 (D X 2 ) 2 + C 17 (D X 1 ) 3 + L X 2 . C 18 D X 4 + C 19 D X 1 D X 2 + L X 3 . C 20 D X 3 + C 21 (D X 1 ) 2 + L X 4 .(C 22 D X 2 ) + L X 5 .(C 23 D X 1 ) + L X 6 .0 + L X 7 , where C 14 , C 15 and so on are from C. A simple calculation as above gives us L H 7 = α −8 L F 7 . As promised earlier, now we see a proof of the claim L. Proof of Claim L. First we make a few observations, which follow immediately by chasing the expression of the coefficients of 1/ζ s in (5.6) and (5.7), for s ≥ 1 and then using the fact that all of them vanish. (O1) For any s ≥ 1, the coefficients of 1/ζ s in (5.6) and (5.7) R H s = s−1 i=1 L H i R H i,s and R F s = s−1 i=1 L F i R F i,s . Let 2 ≤ i ≤ s − 1. It follows from (5.6) that R H i,ζ d−1 + O 1 ζ d −(i−1) . Therefore, for 2 ≤ i ≤ s − 1, if (5.18) R H i,s = C(i, s, 1)D H (i, s, 1) + C(i, s, 2)D H (i, s, 2) + · · · + C(i, s, i s )D H (i, s, i s ) and (5.19) R F i,s = C(i, s, 1)D F (i, s, 1) + C(i, s, 2)D F (i, s, 2) + · · · + C(i, s, i s )D F (i, s, i s ), where for 1 ≤ j ≤ i s , D H (i,1 + D H 1 ζ 2 + D H 2 ζ 3 + · · · + D H d−1 ζ d−1 + O 1 ζ d −1 . Similarly, R H 1,s+1 is the coefficient of 1/ζ s of the power series expansion of 1 + D H 1 ζ 2 + D H 2 ζ 3 + · · · + D H d−1 ζ d−1 + O 1 ζ d −1 . Therefore, if (5.23) R H 1,s = 1s j=1 C(1, s, j)D H s j 1 D H s j 2 · · · D H s j m(s j ) , then (5.24) R H 1,s+1 = 1s j=1 C (1) (1, s, j)D H s j 1 +1 D s j 2 · · · D H s j m(s j ) + C (2) (1, s, j)D H s j 1 D H s j 2 +1 · · · D H s j m(s j ) + · · · + C (m(s j )) (1, s, j)D H s j 1 .D s j 2 · · · D H s j m(s j ) +1 + C s (D H 1 ) s 2 , with C s , C (k) (1, s, j) ∈ C, for 1 ≤ j ≤ 1 s and 1 ≤ k ≤ m(s j ). Further, C s = 0, if s is not divisible by 2 and C s = 0, if s is divisible by 2. The same relation holds between R F 1,s and R F 1,s+1 . Let d ≥ 4. Note that we have already established claim L for d = 2, 3 just before starting the present proof. Let 1 ≤ k ≤ d − 2. Then we prove that if for 1 ≤ s ≤ k, (•) L H s = α −(s+1) L F s , and (••) for each fixed pair (i, s) satisfying 2 ≤ i ≤ s − 1, D H (i, s, j) = α −(s−i) D F (i, s, j), for 1 ≤ j ≤ i s and R H 1,s = α −(s−1) R F 1,s (which in turn gives R H i,s = α −(s−i) R F i,s , for 1 ≤ i ≤ s − 1), then L H k+1 = α −(k+2) L F k+1 . Now observe that performing the same calculations, which we did immediately before starting this proof, (•) and (••) hold for a first few k's with 1 ≤ k ≤ d − 2. The coefficients of 1/ζ k+1 in (5.6) and (5.7) are (5.25) D H k+1 + L H 1 R H 1,k+1 + L H 2 R H 2,k+1 + · · · + L H k R H k,k+1 + L H k+1 , and (5.26) D F k+1 + L F 1 R F 1,k+1 + L F 2 R F 2,k+1 + · · · + L F k R F k,k+1 + L F k+1 , respectively (note that both vanish). Let 2 ≤ i ≤ k. Then by (5.18), it follows that (5.27) R H i,k+1 = C(i, k + 1, 1)D H (i, k + 1, 1) + · · · + C(i, k + 1, i k+1 )D H (i, k + 1, i k+1 ), and thus by (5.20) (5.28) R H i−1,k =C(i, k + 1, 1)D H (i, k + 1, 1) + · · · +C(i, k + 1, i k+1 )D H (i, k + 1, i k+1 ). Therefore, clearly for 1 ≤ j ≤ i k+1 , D H (i, k + 1, j) = D H (i − 1, k, r j ), for some 1 ≤ r j ≤ (i − 1) k . Similar conclusion holds for D F (i, k+1, j), i.e., for 1 ≤ j ≤ i k+1 , D F (i, k+1, j) = D F (i−1, k, r j ). Therefore, using (••) for s = k, we have D H (i, k + 1, j) = D H (i − 1, k, r j ) = α −(k−i+1) D F (i − 1, k, r j ) = α −(k−i+1) D F (i, k + 1, j), for 2 ≤ i ≤ k and for 1 ≤ j ≤ i k+1 . Therefore, (5.29) R H i,k+1 = α −(k+1−i) R F i,k+1 , for 2 ≤ i ≤ k. By (••), we have R H 1,k = α −(k−1) R F 1,k . Thus comparing (5.23) and (5.24) along with using the fact that D H k = α −(k+1) D F k , for 1 ≤ k ≤ d − 1 (see (5.3)), we obtain R H 1,k+1 = α −k R F 1,k+1 . (5.30) Now as mentioned before (5.25) and (5.26) both are equal to zero. Therefore using (5.29), (5.30) and (5.3), we get L H k+1 = − D H k+1 + L H 1 R H 1,k+1 + L H 2 R H 2,k+1 + · · · + L H k R H k,k+1 = −α −(k+2) D F k+1 + L F 1 R F 1,k+1 + L F 2 R F 2,k+1 + · · · + L F k R F k,k+1 = α −(k+2) L F k+1 . Thus the claim follows. Hénon maps with different Jacobians Let H(x, y) = (y, p H (y) − δ H x) and F (x, y) = (y, p F (y) − δ F x) be such that δ H = δ F .H(z, ζ) = (δ H /d)z + Q H (ζ), ζ d . A simple calculation givesH d−1 (z, ζ) = (δ H /d) d−1 z +Q H (ζ), ζ d d−1 , whereQ H (ζ) = (δ H /d) d−2 Q H (ζ) + (δ H /d) d−3 Q H (ζ d ) + · · · + Q H ζ d d−2 . Note that degH = degF =d = for each n ≥ 0, k ≥ 1 and for all (z, ζ) ∈ C × (C \D). Thus, ifp = (z, ζ) ∈ C × C \D is a point in the fiber of p ∈ I + H = I + H , then the other points in the fiber are precisely γ k/d n (z, ζ) with n ≥ 0 and k ≥ 1. Note that the main hindrance to run a similar set of calculations as in Section 3 for any arbitrary pair of Hénon maps H and F of the same degree is that the Jacobians of H and F are possibly different. But if we work with H d−1 and F d−1 , there is no such issues. Since (6.2) holds, a moment's thought assures that exactly similar calculations as in Section 3 run smoothly for the maps H d−1 and F d−1 . Therefore, if a : I + H → I + H is a biholomorphism and if A : C × (C \D) → C × (C \D) is a lift, i.e, A is of the form (z, ζ) → (β(z)ζ + γ(ζ), αζ), then as in Section 3, we can show • β(ζ) ≡ β in C; • α d d−2 (d+1)d n−1 → β as n → ∞ and consequently, α d d−2 (d+1) = β with β d−1 = 1. Thus as in (4.12) we obtain (6.3) βd δ Q H (ζ) −Q H e 2πĩ d ζ =d δ Q F (αζ) −Q F e 2πĩ d αζ , for all ζ ∈ C \D, with α d d−2 (d+1) = β and β d−1 = 1. Expanding (6.3), we get β δ d−2 H d d−2 Q H (ζ) − Q H e 2πĩ d ζ + β δ d−3 H d d−3 Q H (ζ d ) − Q H e 2πĩ d ζ d + · · · + β Q H ζ d d−2 − Q H e 2πĩ d ζ d d−2 = δ d−2 F d d−2 Q F (αζ) − Q F e 2πĩ d αζ + δ d−3 F d d−3 Q F (αζ) d − Q F e 2πĩ d αζ d + · · · + Q F (αζ) d d−2 − Q F e 2πĩ d αζ d d−2 . (6.4) While comparing the coefficients of the polynomials appearing in the right hand side and in the left hand side of (6.4), note that if ζ d r l = ζ d s m for 0 ≤ r, s ≤ d − 2 and 1 ≤ l, m ≤ d + 1 with r = s (without loss of generality r > s, say), then m = d r−s l. Since d ≥ 2 and 0 ≤ r, s ≤ d − 2 and 1 ≤ l, m ≤ d + 1, clearly r = s + 1 and l = 1. Thus since next to the highest coefficients of both Q H and Q F vanish (Q H and Q F are of the form (4.10) and (4.11), respectively), we obtain (6.5) β Q H ζ d d−2 − Q H e 2πĩ d ζ d d−2 = Q F (αζ) d d−2 − Q F e 2πĩ d αζ d d−2 . Comparing both sides of (6.5), we get βA H d−k =α d−k A F d−k , for 1 ≤ k ≤ (d − 1), whereα = α d d−2 . Equivalently, we get (6.6) A H d−k =α −(k+1) A F d−k for 1 ≤ k ≤ (d − 1), withα d+1 = 1. Note that (6.6) is an analogue of (4.13) and therefore, using the same set of arguments as in Section 4, we get (6.7) βp H (y) =αp F (αy), for all y ∈ C, withα d+1 = β and β d−1 = 1. Thus, (7.2) A 1 (x, y) = µ 1 B 1 (x, y) and A 2 (x, y) = µ 2 B 2 (x, y), for some µ 1 , µ 2 ∈ C, with µ d−1 1 = µ d−1 2 = 1. Now comparing (7.1) and (7.2), we obtain µ 1 = γ −1 β and µ 2 = β. Therefore, A(x, y) = L µ • B(x, y), for all (x, y) ∈ C 2 , where L µ (x, y) = (µ 1 x, µ 2 y). Thus, F ≡ L µ • B • H • B. Proof of Theorem 1.2: Implementing the idea discussed just after stating the Theorem 1.1 in the Introduction, proof of Theorem 1.2 follows immediately once we prove Theorem 1.1. It would be interesting to investigate the converse of Theorem 1.2. Question: If H and F are two Hénon maps such that F = A 1 • H • A 2 in C 2 , where A 1 and A 2 are affine automorphisms in C 2 , then how are the escaping sets of H and F related? [0 : 1 : 0]. In fact, one can prove that the points in I + H under iteration of H converges to the point [0 : 1 : 0] uniformly (on compacts). Lemma 2 . 2 . 22With the above notations, near p = [0 : 1 : 0] we have Lifting biholomorphisms between escaping sets: Recall from Section 2 that the fundamental group of escaping set of any Hénon map of degree d is Z[1/d] and C × (C \D) is the covering of the escaping set corresponding to the subgroup Z ⊆ Z[1/d]. Let π H : C × (C \D) → I + H and π F : C × (C \D) → I + F be the covering maps. Now note that a biholomorphism a from I + H to I + F can be lifted as an automorphism A of C × (C \D) if and only if the induced group isomorphism ∈ s, j) and D F (i, s, j) are products of powers of D H l 's and D F l 's (1 ≤ l < s), respectively, with C(i, s, j) ∈ C, then(5.20) R H i−1,s−1 =C(i, s, 1)D H (i, s, 1) +C(i, s, 2)D H (i, s, 2) + · · · +C(i, s, i s )D H (i, s, i s ) and (5.21) R F i−1,s−1 =C(i, s, 1)D F (i, s, 1) +C(i, s, 2)D F (i, s, 2) + · · · +C(i, s, i s )D F (i, s, i s ), withC(i, s, j) ∈ C, for 1 ≤ j ≤ i s .(O3) Let s ≥ 1 and i = 1. Then (5.22) R H 1,s = C(1, s, 1)D H (i, s, 1) + C(1, s, 2)D H (i, s, 2) + · · · + C(1, s, 1 s )D H (1, s, 1 s ), where for 1 ≤ j ≤ 1 s , D H (1, s, j) j 1 , s j 2 , . . . , s j m(s j ) ∈ {1, 2, . . . , s − 1} (s j 1 , s j 2 , . . . , s j m(s j ) are not possibly all distinct), i.D H 1 , . . . , D H s−1 . Now R H 1,s is the coefficient of 1/ζ s−1 of the power series expansion of F (αy) = βp H (y), with α d+1 = 1 and δ F = γδ H , with α d 2 −1 = 1 and β d−1 = γ d−1 = 1. Thus if (7.1) A(x, y) = (αx, βα −1 y) and B(x, y) = (γαβ −1 x, α −1 y), then a simple calculation gives F ≡ A • H • B in C 2 . Now note that if we write A = (A 1 , A 2 ) and B = (B 1 , B 2 ), then (A 1 (x, y)) d−1 = (B 1 (x, y)) d−1 and (A 2 (x, y)) d−1 = (B 2 (x, y)) d−1 . Although the Böttcher function φ + H does not extend analytically to I + H , it extends along curves in I + H starting in V + R and defines a multi-valued analytic map in I + H . LetĨ + H be the covering of I + H obtained as the Riemann domain of φ + H and let φ + H lifts as a single-valued holomorphic functionφ + H :Ĩ + H → C \D. Further,Ĩ + H is biholomorphically equivalent to the domain {(z, ζ) : z ∈ C, |ζ| > 1}. By construction ζ =φ + H and thus in this new coordinate the level sets ofφ + called the escaping set of H and the setK + H = (x, y) ∈ C 2 : the sequence {H n (x, y)} n≥1 is bounded is called the non-escaping set of H. One can prove that K + H ⊂ V R ∪ V − R and (2.2) There exists a holomorphic function ψ HLemma 2.1. ([11, Lemma 7.3.7], [6, Prop. 2.2]) a pair of Hénon maps of degree d. Thus the fundamental groups of both I + H and I + F are Z[1/d]. It follows from (2.5) that for H there exists a polynomial Q H of degree d + 1 such that for each element [k/d n ] ∈ Z[1/d]/Z, with n ≥ 0 and k ≥ 1, there exists a unique deck transformation For s ≥ 1, one can writevanish. Further, they are of the form D H s + R H s + L H s and D F s + R F s + L F s , respectively, where R H s and R F s are linear combinations of products of powers of D H i 's, L H j 's and D F i 's, L F j 's, respectively, with 1 ≤ i, j < s. (O2) As before, we assume that deg H = deg F = d ≥ 2. It follows from [4, Thm. 2.7] that if I + H and I + Now instead of working with H and F , we work withH = H d−1 andF = F d−1 . Note that since (6.1) holds the Jacobians ofH andF are the same. Further, I + H d−1= I + F d−1 . IfH : C × (C \D) → C × (C \D)is the lift of H : I + H → I + H , then clearlyH d−1 : C × (C \D) → C × (C \D) is the lift of H d−1 : I + H → I + H . It follows from (2.6) thatF are biholomorphic, then (6.1) δ d−1 H = δ d−1 F = δ. d d−1 and det J H = det J H = δ, where J H and J F are the Jacobian matrices of H and F , respectively. Also it is easy to see that Z[1/d n ] = Z[1/d] for all n ≥ 1, and in particular, Z[1/d] = Z[1/d d−1 ] = Z[1/d]. 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Systems 9 (1989), no. 1, 67-99. Sibony: Complex Hénon mappings in C 2 and Fatou-Bieberbach domains. J E Fornaess, N , Duke Math. J. 65J. E. Fornaess, N. Sibony: Complex Hénon mappings in C 2 and Fatou-Bieberbach domains, Duke Math. J. 65 (1992), pp. 345-380. Hénon mappings in the complex domain. I. The global topology of dynamical space. J H Hubbard, R W Oberste-Vorth, Inst. HautesÉtudes Sci. Publ. Math. No. 79Hubbard, J. H., Oberste-Vorth, R. W.: Hénon mappings in the complex domain. I. The global topology of dynamical space. Inst. HautesÉtudes Sci. Publ. Math. No. 79 (1994), 5--46. Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials. Real and complex dynamical systems. J H Hubbard, R W Oberste-Vorth, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 464Hubbard, J. H.; Oberste-Vorth, R. W.: Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials. Real and complex dynamical systems (Hillerød, 1993), 89-132, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464 S Morosawa, Y Nishimura, M Taniguchi, T Ueda, Holomorphic dynamics. S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda: Holomorphic dynamics, Translated from the 1995 Japanese original and revised by the authors. Cambridge Studies in Advanced Mathematics. 66Cambridge University PressJapanese original and revised by the authors. Cambridge Studies in Advanced Mathematics, 66, Cambridge University Press, Cambridge, (2000). Knowledge City, Sector -81, Mohali, Punjab-140306, India Email address: [email protected], [email protected]. Indian Institute of Science Education and Research MohaliIndian Institute of Science Education and Research Mohali, Knowledge City, Sector -81, Mo- hali, Punjab-140306, India Email address: [email protected], [email protected]
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[ "Risperidone response in patients with schizophrenia drives DNA methylation changes in the immune and neuronal systems", "Risperidone response in patients with schizophrenia drives DNA methylation changes in the immune and neuronal systems" ]
[ "Ph.DAna Lokmer [email protected] \nFondation FondaMental\nCréteilFrance\n", "Ph.DCharanraj Goud Alladi ", "Réjane C1 \nOptimisation Thérapeutique en Neuropsychopharmacologie (OTeN)\nUMR-S 1144\nUniversité de Paris\nINSERM\nParisFrance\n", "Pharm.DTroudet \nUniv Paris Est Créteil\nINSERM\nIMRB, Translational Neuropsychiatry\nCréteilFrance\n\nFondation FondaMental\nCréteilFrance\n", "Delphine D Bacq-Daian \nCentre National de Recherche en Génomique Humaine (CNRGH)\nUniversité Paris-Saclay\nCEA\n91057EvryFrance\n", "Ph.D. dAnne Boland-Auge \nCentre National de Recherche en Génomique Humaine (CNRGH)\nUniversité Paris-Saclay\nCEA\n91057EvryFrance\n", "Violaine Latapie \nUniv Paris Est Créteil\nINSERM\nIMRB, Translational Neuropsychiatry\nCréteilFrance\n\nFondation FondaMental\nCréteilFrance\n", "Ph.DJean-François Deleuze \nCentre National de Recherche en Génomique Humaine (CNRGH)\nUniversité Paris-Saclay\nCEA\n91057EvryFrance\n", "M.D. eRavi Philip Rajkumar \nDepartment of Pharmacology\nJawaharlal Institute of Postgraduate Medical Education and Research\nPuducherry-605006India\n", "Deepak Gopal Shewade ", "Ph.DM F Pharm \nDepartment of Psychiatry\nJawaharlal Institute of Postgraduate Medical Education and Research\nPuducherry-605006India\n\nCentre National de Recherche en Génomique Humaine (CNRGH)\nInstitut de Biologie François Jacob\nCEA\nUniversité Paris-Saclay\nEvryFrance\n", "M.DFrank Bélivier \nFondation FondaMental\nCréteilFrance\n\nOptimisation Thérapeutique en Neuropsychopharmacologie (OTeN)\nUMR-S 1144\nUniversité de Paris\nINSERM\nParisFrance\n\nDépartement de Psychiatrie et de Médecine Addictologique\nHôpitaux Lariboisière-Fernand Widal\nGHU APHP Nord\nParisFrance\n\nParis Est Créteil, Inserm U955, IMRB, Neuropsychiatrie Translationnelle Faculté de Médecine\nUniv\n8 rue du Général Sarrail94010Créteil cedexFrance\n", "Ph.D. cCynthia Marie-Claire \nOptimisation Thérapeutique en Neuropsychopharmacologie (OTeN)\nUMR-S 1144\nUniversité de Paris\nINSERM\nParisFrance\n", "Ph.DStéphane Jamain \nUniv Paris Est Créteil\nINSERM\nIMRB, Translational Neuropsychiatry\nCréteilFrance\n\nFondation FondaMental\nCréteilFrance\n", "DrAna Lokmer " ]
[ "Fondation FondaMental\nCréteilFrance", "Optimisation Thérapeutique en Neuropsychopharmacologie (OTeN)\nUMR-S 1144\nUniversité de Paris\nINSERM\nParisFrance", "Univ Paris Est Créteil\nINSERM\nIMRB, Translational Neuropsychiatry\nCréteilFrance", "Fondation FondaMental\nCréteilFrance", "Centre National de Recherche en Génomique Humaine (CNRGH)\nUniversité Paris-Saclay\nCEA\n91057EvryFrance", "Centre National de Recherche en Génomique Humaine (CNRGH)\nUniversité Paris-Saclay\nCEA\n91057EvryFrance", "Univ Paris Est Créteil\nINSERM\nIMRB, Translational Neuropsychiatry\nCréteilFrance", "Fondation FondaMental\nCréteilFrance", "Centre National de Recherche en Génomique Humaine (CNRGH)\nUniversité Paris-Saclay\nCEA\n91057EvryFrance", "Department of Pharmacology\nJawaharlal Institute of Postgraduate Medical Education and Research\nPuducherry-605006India", "Department of Psychiatry\nJawaharlal Institute of Postgraduate Medical Education and Research\nPuducherry-605006India", "Centre National de Recherche en Génomique Humaine (CNRGH)\nInstitut de Biologie François Jacob\nCEA\nUniversité Paris-Saclay\nEvryFrance", "Fondation FondaMental\nCréteilFrance", "Optimisation Thérapeutique en Neuropsychopharmacologie (OTeN)\nUMR-S 1144\nUniversité de Paris\nINSERM\nParisFrance", "Département de Psychiatrie et de Médecine Addictologique\nHôpitaux Lariboisière-Fernand Widal\nGHU APHP Nord\nParisFrance", "Paris Est Créteil, Inserm U955, IMRB, Neuropsychiatrie Translationnelle Faculté de Médecine\nUniv\n8 rue du Général Sarrail94010Créteil cedexFrance", "Optimisation Thérapeutique en Neuropsychopharmacologie (OTeN)\nUMR-S 1144\nUniversité de Paris\nINSERM\nParisFrance", "Univ Paris Est Créteil\nINSERM\nIMRB, Translational Neuropsychiatry\nCréteilFrance", "Fondation FondaMental\nCréteilFrance" ]
[]
Background:The choice of efficient antipsychotic therapy for schizophrenia relies on a timeconsuming trial-and-error approach, whereas social and economic burden of the disease calls for faster alternatives. Material & methods: In search for predictive biomarkers of antipsychotic response, we analyzed blood methylomes of 28 patients before and four weeks into risperidone therapy. Results: We identified several CpGs exhibiting response-specific temporal dynamics in otherwise temporally stable methylomes and observed noticeable global response-related differences between good and bad responders. These were associated with the genes involved in immunity, neurotransmission and neuronal development. Polymorphisms in many of these genes were previously linked with schizophrenia etiology and antipsychotic response. Conclusion:Antipsychotic response seems to be shaped by both stable and medication-induced methylation differences.Plain Language SummaryThe most common way to treat schizophrenia is antipsychotic medication. However, not all antipsychotics work for all patients. The only way to find a suitable antipsychotic is to prescribe one and wait, sometimes for months, to see if it works. Finding an alternative to this "trial & error" method would help reduce patient suffering and costs for healthcare systems. The idea is to look in the DNA of our blood cells for the specific marks that can change in response to our lifestyle or health condition. These marks could help us predict how the patients will react to the drug. In other words, they can serve as biomarkers of antipsychotic response. So we examined the blood of schizophrenia patients before and four weeks after starting the medication. We found that the patients who did not respond well to the drug had different marks on the genes involved in immune 3 defense and nervous system functioning. Some of these genes also play role in the development of schizophrenia, whereas others can directly affect what happens to the drug in the patient's body.Still, we could not predict how the patients will react with certainty, because we examined only 28 patients. However, we found valuable targets for future research.Tweetable AbstractBlood methylomes of schizophrenia patients suggest the role of immune and neuronal systems in antipsychotic treatment response and reveal potential targets for antipsychotic response biomarker research.Summary Points Trial & error method of choosing an efficient antipsychotic treatment for schizophrenia patients prolongs suffering and places considerable burden on healthcare systems. Blood represents an easily accessible source of potential biomarkers predictive of antipsychotic response. We examined blood methylomes of good and bad responders to risperidone treatment, just before and four weeks after starting the therapy to look for predictive biomarkers of antipsychotic response. Blood methylomes differ slightly between the good and bad responders and are overall temporally stable, but we did observe some response-specific shifts following the treatment. Differentially methylated genes were mostly linked to the genes involved in immunity, neurotransmission and neuronal functioning and development.4 Some of the differentially methylated genes, such as CACNA2D2 or NAV, have been previously implicated in schizophrenia risk. We also observed differential methylation of glutathione S-transferase GSTM1 and MGST1 genes, whose polymorphisms were shown to affect antipsychotic treatment response and their side effects. Although we could not identify definitive antipsychotic response biomarkers in our small cohort, our results suggest potential targets for the future response biomarker research.
10.2217/epi-2023-0017
[ "https://export.arxiv.org/pdf/2304.03131v1.pdf" ]
257,533,546
2304.03131
675442a09aacf0bceada6e4b44c90dfab780b25e
Risperidone response in patients with schizophrenia drives DNA methylation changes in the immune and neuronal systems Ph.DAna Lokmer [email protected] Fondation FondaMental CréteilFrance Ph.DCharanraj Goud Alladi Réjane C1 Optimisation Thérapeutique en Neuropsychopharmacologie (OTeN) UMR-S 1144 Université de Paris INSERM ParisFrance Pharm.DTroudet Univ Paris Est Créteil INSERM IMRB, Translational Neuropsychiatry CréteilFrance Fondation FondaMental CréteilFrance Delphine D Bacq-Daian Centre National de Recherche en Génomique Humaine (CNRGH) Université Paris-Saclay CEA 91057EvryFrance Ph.D. dAnne Boland-Auge Centre National de Recherche en Génomique Humaine (CNRGH) Université Paris-Saclay CEA 91057EvryFrance Violaine Latapie Univ Paris Est Créteil INSERM IMRB, Translational Neuropsychiatry CréteilFrance Fondation FondaMental CréteilFrance Ph.DJean-François Deleuze Centre National de Recherche en Génomique Humaine (CNRGH) Université Paris-Saclay CEA 91057EvryFrance M.D. eRavi Philip Rajkumar Department of Pharmacology Jawaharlal Institute of Postgraduate Medical Education and Research Puducherry-605006India Deepak Gopal Shewade Ph.DM F Pharm Department of Psychiatry Jawaharlal Institute of Postgraduate Medical Education and Research Puducherry-605006India Centre National de Recherche en Génomique Humaine (CNRGH) Institut de Biologie François Jacob CEA Université Paris-Saclay EvryFrance M.DFrank Bélivier Fondation FondaMental CréteilFrance Optimisation Thérapeutique en Neuropsychopharmacologie (OTeN) UMR-S 1144 Université de Paris INSERM ParisFrance Département de Psychiatrie et de Médecine Addictologique Hôpitaux Lariboisière-Fernand Widal GHU APHP Nord ParisFrance Paris Est Créteil, Inserm U955, IMRB, Neuropsychiatrie Translationnelle Faculté de Médecine Univ 8 rue du Général Sarrail94010Créteil cedexFrance Ph.D. cCynthia Marie-Claire Optimisation Thérapeutique en Neuropsychopharmacologie (OTeN) UMR-S 1144 Université de Paris INSERM ParisFrance Ph.DStéphane Jamain Univ Paris Est Créteil INSERM IMRB, Translational Neuropsychiatry CréteilFrance Fondation FondaMental CréteilFrance DrAna Lokmer Risperidone response in patients with schizophrenia drives DNA methylation changes in the immune and neuronal systems 1 Equal contributions. 1 *Corresponding authors: Background:The choice of efficient antipsychotic therapy for schizophrenia relies on a timeconsuming trial-and-error approach, whereas social and economic burden of the disease calls for faster alternatives. Material & methods: In search for predictive biomarkers of antipsychotic response, we analyzed blood methylomes of 28 patients before and four weeks into risperidone therapy. Results: We identified several CpGs exhibiting response-specific temporal dynamics in otherwise temporally stable methylomes and observed noticeable global response-related differences between good and bad responders. These were associated with the genes involved in immunity, neurotransmission and neuronal development. Polymorphisms in many of these genes were previously linked with schizophrenia etiology and antipsychotic response. Conclusion:Antipsychotic response seems to be shaped by both stable and medication-induced methylation differences.Plain Language SummaryThe most common way to treat schizophrenia is antipsychotic medication. However, not all antipsychotics work for all patients. The only way to find a suitable antipsychotic is to prescribe one and wait, sometimes for months, to see if it works. Finding an alternative to this "trial & error" method would help reduce patient suffering and costs for healthcare systems. The idea is to look in the DNA of our blood cells for the specific marks that can change in response to our lifestyle or health condition. These marks could help us predict how the patients will react to the drug. In other words, they can serve as biomarkers of antipsychotic response. So we examined the blood of schizophrenia patients before and four weeks after starting the medication. We found that the patients who did not respond well to the drug had different marks on the genes involved in immune 3 defense and nervous system functioning. Some of these genes also play role in the development of schizophrenia, whereas others can directly affect what happens to the drug in the patient's body.Still, we could not predict how the patients will react with certainty, because we examined only 28 patients. However, we found valuable targets for future research.Tweetable AbstractBlood methylomes of schizophrenia patients suggest the role of immune and neuronal systems in antipsychotic treatment response and reveal potential targets for antipsychotic response biomarker research.Summary Points Trial & error method of choosing an efficient antipsychotic treatment for schizophrenia patients prolongs suffering and places considerable burden on healthcare systems. Blood represents an easily accessible source of potential biomarkers predictive of antipsychotic response. We examined blood methylomes of good and bad responders to risperidone treatment, just before and four weeks after starting the therapy to look for predictive biomarkers of antipsychotic response. Blood methylomes differ slightly between the good and bad responders and are overall temporally stable, but we did observe some response-specific shifts following the treatment. Differentially methylated genes were mostly linked to the genes involved in immunity, neurotransmission and neuronal functioning and development.4 Some of the differentially methylated genes, such as CACNA2D2 or NAV, have been previously implicated in schizophrenia risk. We also observed differential methylation of glutathione S-transferase GSTM1 and MGST1 genes, whose polymorphisms were shown to affect antipsychotic treatment response and their side effects. Although we could not identify definitive antipsychotic response biomarkers in our small cohort, our results suggest potential targets for the future response biomarker research. Introduction Schizophrenia (SCZ) is a heritable psychiatric disorder affecting over 20 million individuals worldwide [1] and ranking among the top 20 causes of disability according to the Global Burden of Disease Studies [2]. Its big impact on health, social and economic systems can be explained by early onset of its severe symptoms, its chronic nature and co-occurrence with various somatic diseases, leading to a life-long productivity loss, mental and physical suffering, and reduced life expectancy [1,3]. Development of efficient treatments for SCZ is therefore of paramount importance for the affected individuals and the society as a whole. Despite the decades of intense research and development of new drugs and treatments following the introduction of the first antipsychotic (AP) medication more than 60 years ago, only a minority of SCZ patients (median of 13.5%) attain durable recovery, i.e. clinical remission and good social functioning for a minimum of two years [4]. The use of antipsychotics is not only associated with severe adverse effects, but also plagued with a varying and unpredictable efficacy in alleviating SCZ symptoms [5,6]. Due to the low predictability of AP treatment success, trial-and-error remains the standard method for therapy adjustment, often taking weeks or months to select a suitable AP and thus prolonging patients' suffering. The need for early and accurate prediction of patient response to medication is further accentuated by the fact that early treatment start is associated with better clinical outcomes [7]. Although the traits such as age at disease onset, ethnicity and baseline symptom severity may help predict the AP response to a certain extent (review in [8]), we recently showed that this prediction was improved by combining clinical data with biological markers [9,10]. Indeed, a considerable individual phenotypic and matching genotypic variability observed in SCZ patients suggests that personalized medicine, including the identification of predictive biomarkers, holds the key to successful SCZ treatment [11]. Most of the research regarding the response to antipsychotics has so far focused on the underlying genetic and genomic variation (reviews in [12,13]). Pharmacogenetic and pharmacogenomic studies identified a number of candidate polymorphisms in the genes involved in dopaminergic and serotonergic system, glutamate signaling [14] as well as various other genes that may serve as predictive biomarkers of antipsychotic response [12]. Interestingly, Ruderfer et al. [15] found an overlap between the SNPs associated with SCZ risk and AP efficacy, not only suggesting the existence of shared mechanisms between SCZ pathogenesis and treatment response, but also demonstrating that the SCZ treatment response is polygenic and complex in nature just as the disease itself. As the treatment outcome is influenced not only by genetics, but also by its interplay with environmental factors, gene expression (GE) profiles may provide more specific information about the AP efficacy than genetic data. Comparisons of GE profiles between healthy controls and patients before and after treatment suggest reversal to "normal" profiles in the treated patients [16,17]. However, there is virtually no overlap between the genes identified in different studies. The reasons for this may include various factors, such as small sample size, omission of important covariates, large individual variation of GE profiles as well as analysis-related technical issues (review in [18]). Similarly, the only two studies [10,19] explicitly trying to predict the treatment outcome from the baseline GE profiles identified different sets of genes as candidate predictive biomarkers, highlighting once more the complex nature of AP response and the role of unknown individual and environmental factors. Similar to GE, epigenetic marks reflect both genetic and environmental influences, while being more temporally stable and thus likely representing a more reliable source of predictive biomarkers [20]. Although there is a considerable amount of research regarding the epigenetic mechanisms involved in SCZ (review in [21]), most of the studies so far have not been designed to differentiate between the epigenetic signatures of SCZ pathology and medication use, with very few specifically addressing the relationship between DNA methylation (DNAm) and AP treatment response heterogeneity (reviews in [22][23][24] In order to improve our understanding of the link between DNAm profiles and variation in antipsychotic treatment response, we analyzed 56 blood methylomes from 28 patients before and four weeks after starting risperidone treatment. All patients were of Tamil origin, excluding the potential confounding effects of mixed ancestry. To further minimize the influence of confounding factors, our analysis was corrected for sex, age and tobacco use. We aimed to identify specific methylation sites that may help predict the AP response, but also to shed light on the functions and processes associated with variation in treatment efficacy and thus contribute to a better understanding of the broad epigenetic mechanisms underlying the risperidone treatment response. Methods The methods are described in detail in the supplementary material. Recruitment, sampling design and treatment response assessment The cohort consists of 28 Tamil speaking adults, diagnosed with schizophrenia (total score at Positive and Negative Syndrome Scale, PANSS ≥30) and newly prescribed risperidone (4-8 mg/day). Patients who were prescribed antipsychotics other than risperidone, enganged in substance abuse and pregnant or lactating women were excluded from the study. Treatment compliance was confirmed by the patients' family members or caregivers. None of the patients received any antipsychotic medication at least four weeks before the baseline visit. All except one were comedicated with at least one additional medicament for sleep or to alleviate side effects. Good treatment response was defined as at least 20% reduction of total PANSS score after four weeks of treatment [32]. Blood samples for the methylome analysis were taken during baseline and follow-up visits. Patients' characteristics summary, including the differences between the good and bad responders can be found in Table 1. Ethical statement Informed consent was obtained from each study participant and the patient's legally acceptable representative (LAR) or a family member. The Institutional Ethics Committee approved the study protocol before the commencement of the study (IEC Project No. JIP/IEC/4/2013/189). Generation and quality control of methylation data DNA was extracted from the blood cells using QIAamp Blood Midi Kit (Qiagen, Germany) and subsequently used for bisulfite conversion with EpiTect Fast 96 Bisulfite Kit (Qiagen, Germany). Genome-wide methylation was assessed using Infinium Methylation EPIC BeadChip [33] on the Centre National de Recherche en Génomique Humaine (CNRGH) automated platform (Illumina Inc., USA). All steps were performed following the manufacturers' protocols. Quality control, preprocessing, statistical analysis and data visualization were all done in R statistical framework [34], version 4.1.3. We first filtered out low quality probes, as they can negatively influence the normalization procedure [35], and checked sample quality [36]. In addition to low quality, probes, we also removed previously identified 44,570 cross-reactive probes (binding to multiple sites in the genome [45,46] ). Finally, to reduce the amount of technical variation within a single array, we performed a three-step (background, dye-bias and design-bias correction) normalization of the data. [50]. Estimation of blood cell type composition and batch effects We estimated the relative abundances of six leukocyte types from the methylation data with the flow.sorted.blood.EPIC v. 1.10.1 package [40]. as the leukocyte composition is an important source of DNAm variability (for review see [41]). As the blood cell types per se were not of primary interest in this study and six additional variables could negatively affect inference, we used the first principal component (PC1, explaining 84.6% of variability, Supplementary Figure 1) of the PCA based on the leukocyte composition data to represent the cell type heterogeneity. Similarly, to avoid overfitting due to a high number of independent variables and to prevent false positives due to a tendency of multilevel factors to appear significant in the statistical analysis, we we used the two "unwanted variation" ordination axes as a proxy for batch effects in the models. The batch effects included a "control matrix", as described in Fortin et al. [42], chip identity, position on the chip and sampling centre (Supplementary Figure 2). Multivariate analysis In order to examine global blood methylation patterns, we performed a PCA on all 56 samples, using variable (SD of β-values distribution < 0.1) autosomal probes only (N = 39,239), to reduce computational burden and the effect of noise [35]. This initial PCA (Supplementary Figure 3) revealed two outlier individuals (P9 and P14) that were excluded from further analysis. Of note, these two individuals had nothing conspicuous in common (they differed by sex and response to treatment from one another, had distinct DNAm profiles according to the PCA and did not differ from other patients either by duration of psychosis or comedication). In both cases, the samples from the same individual grouped together, indicating that the lack of similarity to other samples was probably due to unknown sources of biological variation (e.g. genetics or physiological condition). All the reported results are based on the remaining 52 samples from 26 individuals (19 good and 7 bad responders). To assess the difference in magnitude of medication-induced methylation changes between the good and bad responders, we compared the pairwise Mahalanobis (multivariate) distances between the paired samples by linear regression. In order to extract CpGs significantly correlated (absolute correlation values ≥0.2) with the main axes (PC1 and PC2) of methylome variation and to assess the relationship between the risperidone response and other factors (baseline and follow-up PANSS scores, visit, sex, smoking, alcohol use, blood cell type composition, chip number, position on the chip, patient identity, unwanted variation) and those axes, we calculated the corresponding correlations using dimdesc function from FactoMineR [43]. Gene set enrichment analysis In order to gain insight into the function of genes associated with the CpGs significantly correlated with either positive or negative sides of the two main principal components (PCs), we conducted an enrichment analysis for each of the four groups of CpGs separately (i.e. PC1-, PC1+, PC2-, PC2+, Figure 1D) using the gometh method, adapted to the biases specific for the methylation data [61] . We used only the variable 39,239 probes as a CpG universe to avoid sample source bias [45].We tested for the enrichment of Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways and gene ontology (GO) categories. Both databases contain a large number of categories/pathways to be tested, strongly reducing the likelihood for any of the p-values to remain significant at any of the commonly used significance cutoffs after the multiple testing correction in the small datasets. However, gene-set enrichment analysis is a multi-step procedure and some authors argue that a pvalue adjustment does not correctly account for the family-wise error rate and advise to use raw pvalues and term ranking to select potentially interesting pathways and processes instead [46]. Therefore, we kept the terms/pathways with raw p≤0.01 and additionally filtered the results based on the number of differentially methylated genes and pathway/term size and coverage. In order to simplify the interpretation of the GO enrichment analysis results, we used the GOxploreR v. 1.2.4 package to prioritize GO terms [47] and GOSemSim v. 2.18.1 package [48,49] to group them based on their semantic similarity (0.7 similarity threshold). In addition, we performed a gene-set enrichment analysis of the variable 39,239 autosomal CpGs with all probes on the chip that passed our quality control step (N = 821,287) as the universe in order to identify functional categories that were over-represented in the variable portion of the blood methylome of our cohort. Association between individual CpG methylation levels and risperidone response To analyze the relationship between the risperidone response and methylation at the probe level, , we ran generalized linear mixed models with beta family (glmmTMB v. 1.1.2.3 package [50]) using visit, response (defined either as binary variable or %PANSS improvement) and visit x response interaction as variables of interest. We included leukocyte composition, sex, smoking and two variables describing the unwanted variation as covariates. We removed the models with convergence, multicollinearity, outlier, uniformity and dispersion issues (DHARMa v.0.4.6, [51]). We compared these models to the corresponding null models with a likelihood ratio test (LRT) to obtain a model significance and subsequently adjusted the p-values for multiple testing using the Benjamini-Hochberg procedure. However, due to the exploratory nature of our analysis, we included all the models with raw p-values ≤ 0.05 in the results. Still, we clearly indicate which models remained significant at 0.05 level following the Benjamini-Hochberg (B-H) correction. In addition, we added the information about the experimental evidence of correlation between the blood and brain methylation values of these CpGs from the IMAGE-CpG tool [52]. Finally, we used the top significant CpGs (absolute value of coefficient estimates from glmmTMB ≥0.18 for visit x response interaction and top twenty CpGs for the main effect of response) to try to predict the risperidone response using the stability variable selection method (package mboost v. 2.9-5 [53,54]). Data and code availability All data including R objects and figures that are not included as a supplementary material are available on Figshare. Raw methylation intensity data are deposited in the ArrayExpress repository under the accession number E-MTAB-11921. All scripts necessary to reproduce the presented analysis are available upon request from the authors. Results Global methylation patterns differ between good and bad responders and are more influenced by medication in good responders PCA of variable autosomal probes revealed small, but noticeable differences in blood methylation patterns between good (N = 19) and bad (N = 7) responders at baseline and after four-week risperidone therapy ( Figure 1A and 1B). The methylomes from the same individual were, to a large extent, temporally stable, but we did observe slight changes following the treatment ( Figure 1C). Although not statistically significant, these shifts were more pronounced in good responders, as reflected by higher Mahalanobis distances between the corresponding paired samples (mean ± SD = 0.73 ± 0.524) compared with those from the bad responders (mean ± SD = 0.36 ± 0.244; Anova: Figure 1C). F 1,24 = 3.16, p = 0.088, Global methylation patterns are correlated with blood cell type composition, PANSS score and %PANSS improvement To identify factors that best describe variation along the first two ordination axes, we calculated correlations between these factors and the axes, filtering out those with correlation values <0.2 ( Figure 1D). In addition to the variables describing technical and patient identity related variation, Table 1) and strongly correlated with the blood cell composition PC, probably reflecting a higher proportion of B-cells in good responders both at baseline and follow-up visits (and a marginally significant lower proportion of the CD4+ T-cells at the baseline, Table 1). In a similar manner, we calculated correlations between the PCs and CpG methylation values, keeping again only those with the absolute correlation values ≥0.2 (Supplementary Table 2). Overall, 20,747 (53%) of the variable autosomal variables were significantly correlated with the first two PCs. Specifically, 9,986 and 2,975 CpGs were positively and negatively correlated with the PC1 (representing the groups PC1+ and PC1-, Figure 1D), and 6,326 and 3,660 CpGs were positively and negatively correlated with the PC2 (representing the groups PC2+ and PC2-, Figure 1D), respectively. Some of the genes (e.g. RELN, GRIN2A, GRIN2B, DLG2) associated with the CpGs significantly correlated with the PC2, and thus to a certain extent co-varying with the treatment response (% PANSS improvement or follow-up PANSS) have been previously linked to the brain development, SCZ and other psychiatric disorders (Supplementary Table 2). GO enrichment analysis revealed many variable CpGs associated with the genes involved in immunity and neuron functioning and development GO enrichment analysis of the four tested CpG groups (PC1-, PC1+, PC2-, PC2+, Figure 1D) revealed the enrichment of biological processes (BP) related to immunity, neurogenesis and synaptic signaling, in addition to more general regulatory and metabolic processes (Supplementary Table 3). Specifically, the CpGs with a tendency towards higher methylation values in bad responders and a lower relative abundance of B-cells (i.e. PC1-) were annotated to the genes primarily involved in different aspects of immunity and blood cells differentiation (e.g. positive regulation of inflammatory response, adaptive immune response, establishment of T-cell polarity), and to a lesser extent in synaptic signaling (e.g. excitatory postsynaptic potential, long-term postsynaptic potentiation), presynapse assembly, but also in ERBB signaling and amyloid fibril formation. The processes enriched along the opposite side of the PC1 (i.e. PC1+, the CpGs more methylated in the good responders) included, among other, Fc receptor signaling pathway, endosomal and intracellular transport, as well as terms related to phagocytosis, inflammatory response (interleukin pathway and neutrophil degranulation), JUN kinase activity and neuron projection development (Supplementary Table 3 Table 3). This held largely true for CC, with the enriched terms dominated by neuron-related structures for all except the PC1+ group (e.g. Golgi apparatus subcompartment, AP-molecular functions over-represented in the PC2+ group were related almost exclusively to neuronal signaling (e.g. NMDS glutamate receptor activity), whereas the the PC1-and PC2-groups were associated with more general, higher level terms (e.g. steroid dehydrogenase activity and acting binding, respectively). Finally, we wanted to check which processes and cellular components were over-represented in the variable CpG subset that we used as input for the multivariate analysis compared to the universe containing all the probes present on the EPIC chip. Among other, these processes included: transmission of nerve impulse, transport across blood-brain barrier, neuron projection morphogenesis, regulation of neurotransmitter receptor activity, dopamine receptor signaling pathway, modulation of chemical synaptic transmission myeloid leukocyte mediated immunity and leukocyte degranulation (Supplementary Figure 6). The enrichment of neuron-functioning related processes was further reflected in the enriched cellular components, to a large extent associated with neurons (neuronal dense core vesicle, axon, presynapse, dendrite, neuronal cell body etc.). This suggests that a significant part of epigenetic variation observed in this cohort is related to immunity and nervous system functioning. Good response to medication is associated with hypermethylation of CpGs related to nicotine and morphine addiction KEGG pathways In a corresponding KEGG pathway analysis, we identified several immunity, cancer and infection related pathways (Supplementary Table 4) associated with either pole of the PC1. More particularly, the PC1-group was characterized by the pathways involved in adaptive immune response (e.g. T cell receptor signaling), endometrial and small cell lung cancer and infection (toxoplasmosis and Epstein-Barr virus infection), whereas the PC1+ group was characterized by Yersinia infection and lysosome pathways. In addition, chemokine signaling pathway, involved in inflammatory responses and general functioning of the immune system, was enriched in both PC1-related groups. Some differentially methylated genes from this pathway were detected in both groups, suggesting that the phenotypic variability associated with the PC1 could be partially explained by gene-specific epigenetic mechanisms only. However, most of the genes were detected in either PC1-or PC1+ group, indicating the existence of additional mechanisms acting on different components of these pathways. Regarding the PC2, no pathways were over-represented in the PC2-group, whereas the PC2+ group was characterized by nicotine and morphine addiction pathways ( Supplementary Figure 7). Differentially methylated CpGs related to risperidone response are concentrated in gene bodies and outside of CpG islands Although we examined the relationship between individual CpG methylation values and response to medication for both %PANSS improvement and binary (good/bad) response, we focused on the latter in the rest of the analysis for simplicity. Table 5). The majority of these probes were situated in the gene bodies (64%) and outside (68%) of the CpG islands and shores (i.e. in the "open sea" areas). Those proportions were different from those observed for all the probes on the EPIC chip, with more probes than expected situated outside the CpG islands and in 5'-UTR regions (Fisher's exact test simulated p-value = 0.0005 and 0.03, respectively, Figure 3A and Figure 4A). As a gene can be represented by multiple CpGs on the EPIC chip, we also wanted to determine the proportion of differentially methylated CpGs related to each of the above 193 genes. We found that a large majority of CpGs (97.5% on average, min = 72%, max = 99.9%) annotated to these genes were not differentially methylated between the good and bad responders. Some of the CpGs with the highest estimated methylation difference between the good and bad responders (Supplementary Table 5 Table 2, Supplementary Table 6). In a corresponding analysis, we identified 237 CpGs in 221 genes (or 208 CpGs in 194 genes after B-H correction) with distinct temporal dynamics in good and bad responders (i.e. with a significant response x visit interaction, Supplementary Table 5 Table 3, Supplementary Table 6). However, note that the model included both baseline and follow-up methylation values for those CpGs, and it was not possible to predict the response from the baseline values only. Discussion SCZ patients vary widely in their response to AP treatment and finding a suitable medication in individual cases may take a long time. In this study, we explore blood methylomes of 28 SCZ patients before and after risperidone treatment in search for epigenetic signatures that might help predict the treatment outcome. Although we could not identify any particular CpG whose methylation values would predict response to risperidone treatment, we detected small but noticeable differences in global methylation patterns between good and bad responders and we observed little change between the baseline and follow-up visits for either group. The two groups were in addition characterized by temporally stable, but different blood cell composition, notably by a higher proportion of B lymphocytes in good responders. Finally, the functional analysis of significantly differentially methylated probes in our cohort suggests that the level of DNA methylation of genes involved in immunity, neurotransmission, and neuronal development could play a role in AP response. Previous research has shown that SCZ patients sometimes have higher neutrophil-to-lymphocyte ratio (NLR) compared to healthy controls and revealed its positive correlation with symptom severity as well as NLR decrease in medicated patients (review in [55]). If and how the percentage of B-cells is related to SCZ pathology and treatment response remains unclear, although some studies reported a decrease of the B-cells percentage upon AP treatment (review in [56]). We detected no change of neutrophil relative abundance at the follow-up, nor any correlation with PANSS score (not shown), but we did find marginally lower neutrophil proportion in good responders as well as consistently (on both visits) lower percentage of B-cells in bad responders, suggesting that immune status and inflammation could be important for predicting AP response. However, note that we only have indirect blood cell composition estimates based on methylation data and not absolute counts, which are necessary for accurate description of leukocyte populations. The role of inflammation, stress and infection in SCZ has been recognized for a long time (review in [78,79].). Similarly, some evidence suggests that the levels of certain cytokines may help predict the outcome of SCZ treatment [59,60]. In our cohort, we observed response-specific changes following the treatment for three interleukin receptor-associated genes -IL23R, IL1R1 and IL18RAP1 -as well as IFNL1 cytokine (Supplementary Table 5 Increased titers of Toxoplasma gondii antibodies have been linked to both acute and chronic psychosis [61] and our results suggest that infections might affect the response to AP just as they affect the SCZ etiology, by driving DNA methylation changes. However, it is so far unclear how the methylation shifts observed here translate into gene expression and physiological processes, as the relationship between methylation and gene expression depends on various factors, such as tissue, CpG position and expression level of the gene in question [62,63]. It has been shown that genetic variation affects epigenetic mechanisms in SCZ [64,65]. Although we do not have genetic data to test the relationship between genetic polymorphisms, methylation patterns and AP response explicitly, we detected differences in methylation of CpGs related to the genes previously associated with SCZ treatment. For example, symptom improvement in our study was associated with glutamate receptor signaling pathway and NMDA receptor activity, just as differential methylation of the related genes and SNPs in glutamate receptors have been linked to AP treatment outcome by both candidate gene [14] and genome-wide [66] approach previously. Similarly, we detected differential methylation of MHC-related genes (i.e. HLA-DQ1, HLA-DRB1 and a pseudogene HLA-H in our study (Supplementary Table 5), which also affect response to AP drugs [67]. While the exact genes detected in our study do not correspond to the genes reported previously, the functional overlap suggests that genetic and epigenetic mechanisms involving the same biological processes contribute to the AP treatment outcome. The lack of perfect match in this case could be explained by the choice of method (candidate vs. whole genome approach), as well as other biological (medication, patient characteristics) and technical (e.g. study size) factors. Although polymorphisms within cytochrome P450 gene CYP2D6 have previously been linked with variation in risperidone response (for review see [68]), we found no evidence of differential methylation of any of the nine CpGs associated with this gene on the EPIC Infinium chip. On the other hand, three CpGs in the glutathione S-transferase GSTM1 ( Table 2) were among the most differentially methylated between the good and bad responders, whereas a CpG associated with another glutathione S-transferase MGST1 (Table 3) was among the probes exhibiting the most pronounced response-specific temporal dynamics. Glutathione S-transferases are involved in xenobiotic metabolism and could therefore also directly shape antipsychotic response. Notably, Pinheiro et al. [69] found that polymorphisms in GSTM1 affect the risk of treatment-resistant schizophrenia, whereas polymorphisms in MGST1 have been associated with the side effects (sinus bradycardia) of second-generation antipsychotics in Chinese population [70]. Glutathione Stransferases could therefore also present a potentially interesting target for future studies of APresponse predictive biomarkers. So far, few studies examined epigenetic mechanisms underlying AP response and even fewer focused on methylation profiles of human peripheral tissues, which have potential as a source of clinically relevant biomarkers (reviews in [22,23]). For example, methylation of serotonergic pathway genes has been implicated in SCZ etiology, as well as in response to AP [71] using targeted approach, but we found no evidence for this in our data. Similarly, we did not detect differential methylation of previously examined dopaminergic system genes (ANKK1 and DRD2 [72]), nor any other genes identified by genome-wide approach [25]. On the other hand, we found several differentially methylated procadherin and cadherin genes, corroborating the results of a study of olanzapine-induced changes in rat brains [73]. These discrepancies are likely due to a number of different factors such as small sample size (making it hard to detect presumably small-effect differences contributing to the treatment outcome [15]), known and unknown biological variation including ethnicity [74], heterogeneity of experimental conditions or different APs and their combinations used in these studies [75]. In addition, most studies focused on specific genes or groups of genes [71][72][73], with the whole methylome approach remaining an exception ([25], this study). Finally, apart from our study, only Rukova et al. [25] applied a repeated-measures experimental design, whereas others focused either on differences before or after treatment. Overall, we detected a number of genes whose methylation is potentially relevant to AP response, many of which are involved in neurotransmission, neuronal development and functioning and other processes that have been previously implicated in SCZ etiology. For example, HDAC4 methylation differed between the good and bad responders both on average and by its temporal dynamics, indicating epigenetic mechanisms involved in histone acetylation may influence the treatment outcome as well. On the other hand, we were not able to predict the outcome from the baseline methylation values of any of the genes identified as significantly affecting response in the probewise analysis (Table 2, Supplementary Table 6). However, if we included baseline and followup methylation values -in other words, methylation temporal dynamics -we identified one CpG, annotated to CACNA2D2, encoding the Calcium Voltage-Gated Channel Auxiliary Subunit Alpha2delta 2, as with a possible predictor of AP response ( Table 3, Supplementary Table 6). Calcium voltage-gated channels play role in neurotransmission and neuron plasticity and polymorphisms in their genes have been associated with SCZ risk and pathogenesis [76][77][78]. We identified an additional gene, NAV (neuron navigator 2), whose methylation temporal dynamics was moderately correlated with AP response. [96]. Polymorphisms in NAV2 gene, essential for brain development, affect onset of Alzheimer disease [80] and associated episodic memory phenotypes [81]. To summarize, functions and associations of these genes and other candidates, for example RNF19A (involved in reduced adult neurogenesis [82]), SHANK2 (linked with mania-like behavior [83] and autism [84]), LMF1 (implicated in metabolic disorders [85]) or those significantly indirectly correlated with the symptom improvement only in the multivariate, but not in the probewise analysis (e.g. RELN, DLG2, CUX2), cover various aspects of SCZ etiology and manifestations and support the hypothesis that response to AP treatment is complex and shaped by multiple interacting processes and underlying epigenetic mechanisms. At this point, we would like to address the limitations of our study and some general issues that require further attention in the future. The small size of our cohort (twenty good and eight bad responders, with one biological outlier in each of these groups that were therefore removed from the analysis, Supplementary Figure 3) does not allow for strong conclusions, especially as our results suggest that there is no single (or few) large-effect CpG site predicting the outcome of SCZ treatment. We can assume that the size of the cohort at least partially explains why we were not able to predict the response from the baseline methylation values. Other factors that might have affected our ability to detect conclusive correlations between methylation and AP response include variable duration of psychosis, inflammation status [57], as well as comedication [86] of the patients in our cohort. Furthermore, correlation between blood and brain methylation values for most of the genes in our studie is small or unknown [52]) and even so, the effect of methylation on gene expression level is not straightforward [21,62,87]. Despite these issues, we were able to demonstrate small global differences between the blood methylomes of SCZ patients varying in their response to medication and found that many of the implicated genes and processes have been previously associated with genetic basis of both SCZ pathology and AP response. In addition, monitoring of temporal methylation dynamics of CACNA2D2-associated CpGs after starting medication could enable more rapid switch of therapy where needed. However, in order to improve our search for predictive biomarkers and address the limitations listed above, we call for longitudinal study designs with bigger and well characterized cohorts (inflammatory markers, smoking, sex, age, ethnicity...), including validation sets, preferably with the same treatment and psychosis duration [24]. In addition to longitudinal probing of SCZ patients, the inclusion of healthy controls could contribute to understanding and interpretation of methylation changes observed after treatment and linking them to clinical outcomes. Finally, we need to improve our knowledge about the relations between the epigenomes from different tissues as well as about the relationship between genetic and epigenetic variation in SCZ etiology and AP pharmacodynamics [23]. Conclusion We present here the first (to the best of our knowledge) repeated-measures study aiming at better understanding and prediction of the antipsychotic (AP) treatment response from the blood methylomes of schizophrenia patients with a genome-wide coverage (including both CpG islands and open sea areas). Overall, response heterogeneity seems to be shaped by both stable differences in methylation levels as well as by response-specific shifts following the treatment start. Despite the small sample size, the genetic uniformity and detailed description of our cohort allowed us to identify potential processes (mainly related to immunity and neuronal functioning) involved in the AP response and suggest potential targets, such as CACNA2D2 or glutathione S-transferase MGST1 and GSTM1 genes, for the future biomarker development. We look forward to the future studies with larger cohorts and thus the power to identify additional potential epigenomic biomarkers of antipsychotic response as well as to examine the utility of those identified in our work. Declaration of interest The authors declare they have no competing financial interests or personal relationships that could influence the reported work. Figure 1D) are grouped by semantic similarity at 0.7 cutoff. The size of rectangles is proportional to the significance of the corresponding term. Sets of semantically similar terms are divided from each other by thick border lines. Table 3 Table 3. CpGs with the highest estimated differences in temporal dynamics between the good and bad responders. The CpGs are ordered by the absolute value of the coefficient estimate (log odds ratio). Raw p-values and adjusted pvalues represent the significance of the model including response, visit, their interaction and the covariates compared with the null model by Likelihood Ratio Test (LRT). Last column shows selection probabilities for the prediction of risperidone response for all selected CpGs or the maximum selection probability in case none of the CpGs was selected. we found a strong negative correlation of the first principal component (PC1) with the blood cell composition PC (cor = -0.95, p<10 -6 ) and a moderate positive correlation with the baseline PANSS score (cor = 0.37, p = 0.008, Supplementary Table 1). On the other hand, the follow-up PANSS score was negatively correlated with the PC2 (cor = -0.33, p = 0.018, Supplementary Table 1) and thus the %PANSS improvement was correlated with it positively (cor = 0.29, p = 0.04). The PC1 partially separated good and bad responders (cor = 0.15, p = 0.05, Supplementary ,Figure 2). Whereas the processes enriched along the PC1 were mainly related to immunity, those enriched along the PC2 were dominated by neuron functioning and development. Specifically, the CpGs with higher methylation values in the patients with a higher follow-up PANSS score (PC2-), were associated with the genes involved in synapse organization and functioning (e.g. postsynaptic membrane organization, neurotransmitter-gated ion channel clustering, regulation of synapse vesicle clustering), axon development (e.g. axonal fasciculation) and signaling pathways involved in development, carcinogenesis and neurogenesis (positive regulation of Notch signaling pathway, positive regulation of Wnt pathway). In addition, this group was characterized by the enrichment of processes related to gastric motility, circadian clock entrainment and cardiac muscle development(Figure 2). On the other hand, the processes associated with symptome improvement (PC2+) encompassed cognition and synaptic signaling (e.g. ionotropic glutamate receptor signaling pathway, regulation of N-methyl-D-aspartate (NMDA) receptor activity, excitatory chemical synaptic transmission) and some regulatory (e.g. positive regulation of renal sodium excretion), metabolic and developmental processes (e.g. blood vessel maturation).We further wanted to know if the enriched biological processes were reflected in other gene ontology categories, namely in the cellular components (CC) and molecular functions (MF, Supplementary Our choice was supported by a strong correlation of the coefficient estimates for both definitions of response in the corresponding models (Spearman's rho between the %PANSS improvement and binary response for the response main effect = 0.78 [0.76, 0.79] and response x visit interaction = 0.76 [0.74, 0.78]). Probewise analysis of CpGs from the genes associated with the enriched GO BP terms or KEGG pathways revealed 216 CpGs assigned to 193 unique genes (or 194 CpGs in 178 genes after BH correction) that were on average significantly differentially methylated between the good and bad responders regardless of the visit (i.e. response main effect, Supplementary and Supplementary Figure 8 , 8Table 2) were characterized by a higher methylation in good responders (e.g. those associated with the genes COL13A1, SORCS2, CASZ1, HDAC4 and HTT), whereas others had higher methylation values in bad responders (e.g.those annotated to the RB1, AKAP6, PKHD1. DISP3, ABLIM2, SPRED2, BMP7 or GSTM1 genes). Some of these genes are involved in neuron differentiation (DISP3, ABLIM2), while others have been linked to neuromuscular disorders (COL13A1), Huntington disease (HTT, SORCS2), schizophrenia (AKAP6) and drug detoxification (GSTM1). However, none of these or any other of the top 20 differentially methylated CpGs were kept as a reliable predictor of the risperidone response by the stability variable selection procedure (maximum selection probability 0.46 for a COL13A1-related CpG for baseline methylation values model and 0.52 for CASZ1 CpG for both visits model, and Supplementary Figure 9). Again, the majority of these probes were situated more often than expected by chance in the open sea areas (64%, Fisher's exact test p-value = 0.0005, Figure 3B-C) and concentrated in the gene bodies (69%, Fisher's exact test p-value = 0.001, Figure 4B-C) . Once more, these differentially methylated CpGs represented only a minority of the total CpGs on the chip associated with their corresponding genes (2.5% on average, min = 0.13%, max = 33%). Only four of these CpGs, annotated to the NAV2, MGST1, BPGM and CACNA2D2 genes, were represented by good quality models with the absolute values of glmmTMB coefficient estimates ≥0.18. Each of those CpGs exhibited a specific pattern of change. Methylation of the NAV2-associated CpG, a gene involved in cellular growth and migration, and of CACNA2D2-related CpG, involved in opioid and gabapentin metabolism, decreased in bad responders and increased slightly in good responders following the treatment. On the other hand, the methylation of the CpG associated with MGST1, a detoxification enzyme, increased in bad responders and decreased in the good ones, converging at a similar level at the follow-up visit for both groups. The remaining CpG was associated with BPGM (a gene influencing erythrocyte oxygen affinity) was more strongly methylated in good responders at the baseline, with the values dropping to the stable bad-responder levels at the follow-up. Stability selection of the variables potentially predictive for the treatment response identified the CACNA2D2-associated CpG as a good candidate (0.82 selection probability), followed by NAV2 (0.52. ), and enrichment of chemokine signaling, Th1 and Th2 cell differentiation and several cancer and infection (including toxoplasmosis) pathways related genes (Supplementary Figure 7, Supplementary Table 4). in schizophrenia: Progress and future directions. Prog. Neuropsychopharmacol. Biol. Psychiatry. 81, 38-49 (2018). ** Thorough review of the progress in understanding of the epigenetic mechanisms of antipsychotic response. 24. Swathy B, Banerjee M. Understanding epigenetics of schizophrenia in the backdrop of its antipsychotic drug therapy. Epigenomics. 9(5), 721-736 (2017). * A perspective and recommendations in order to disentangle epigenetic mechanisms behind the schizophrenia pathogenesis and antipsychotic treatment response. 25. Rukova B, Staneva R, Hadjidekova S, Stamenov G, Milanova V, Toncheva D. Whole genome methylation analyses of schizophrenia patients before and after treatment. Biotechnol. Biotechnol. Equip. 28(3), 518-524 (2014). 26. Galanter JM, Gignoux CR, Oh SS, et al. Differential methylation between ethnic sub-groups reflects the effect of genetic ancestry and environmental exposures. eLife. 6, e20532 (2017). 27. Horvath S, Raj K. DNA methylation-based biomarkers and the epigenetic clock theory of ageing. Nat. Rev. Genet. 19(6), 371-384 (2018). 28. Lieberman JA, Stroup TS, McEvoy JP, et al. Effectiveness of Antipsychotic Drugs in Patients with Chronic Schizophrenia. N. Engl. J. Med. 353(12), 1209-1223 (2005). 29. Zeilinger S, Kühnel B, Klopp N, et al. Tobacco Smoking Leads to Extensive Genome-Wide Changes in DNA Methylation. PLOS ONE. 8(5), e63812 (2013). Figure 1 . 1PCA of within-normalized methylation values of variable autosomal probes without outlier individuals. The panels show individual samples at the A) baseline, B) follow-up and C) both visits together. D) The same PCA as in C), but showing factors with the correlation of at least 0.2 with any of the two depicted PCs. PC-correlated CpGs, PANSS score components and unwanted variation variables are omitted for simplicity (complete results can be found in the Supplementary Table 2 & 3). On A) and B), big plotting characters denote centroids, the ellipses the 95% confidence regions for centroids locations. On C), the lines connect the baseline (dot) and follow-up (triangle) samples from the same individual. On D), the ellipses show app. areas that contain variables and CpGs significantly positively correlated with the corresponding side of the axis and therefore belonging to one of the four groups defined in the Results (PC1-, PC1+, PC2-, PC2+). Figure 2 . 2Treemaps of prioritized GO BP terms. The GO terms enriched in A) PC1-, B) PC1+, C) PC2-and D) PC2+ groups (s. Figure 3 . 3Pie charts showing the distribution of probes in relation to CpG islands. CpGs significant for the A) main effect of response, B) response x visit interaction and C) all CpGs on the EPIC chip. Figure 4 . 4Upset plots showing the distribution of probe positions in the genes. CpGs significant for the A) main effect of response, B) response x visit interaction and C) all CpGs on the EPIC chip. ). A single study up to date using a repeated measures design and a genome-wide approach to search for CpG islands predictive of AP response remains that of Rukova et al. [25]. Rukova et al. identified nine differentially methylated regions (DMRs) that distinguished good from bad responders at the baseline, but only in men. This example highlights the need for detailed patient description in the quest for AP response predictive biomarkers and this holds especially true for traits with a known influence on SCZ risk, antipsychotic response and DNAm profiles, such as age, sex or tobacco use [26-31]. Table_1 Table 1 . Table_11Cohort description including the differences between good and bad responders. Means are shown with corresponding standard deviations.Bad responders (N=8) Good responders (N=20) Test Stat p-value Effect size % males 75 50 Fisher's exact test 0.40 Age (years) 34.7 ± 12.31 32.2 ± 7.16 Welch Two Sample t-test t = 0.55, df = 8.96 0.60 Weight (kg) 58.9 ± 15.91 55.7 ± 14.89 Welch Two Sample t-test t = 0.49, df = 12.22 0.64 Past or current alcohol user (%) 12.5 20 Fisher's exact test 1 Past or current smoker (%) 25 30 Fisher's exact test 0.28 Duration of psychosis (months) 19.1 ± 22.51 35.9 ± 30.25 Welch Two Sample t-test t = -1.61, df = 17.42 0.12 Total PANSS score at baseline 89.5 ± 17.48 93.05 ± 12.62 Welch Two Sample t-test t = -0.52, df = 10.06 0.61 adaptor complex, Supplementary Figure 4), with the corresponding enriched molecular functions (e.g. MAPKKK activity, ubiquitin protein ligase activity, Supplementary Figure 5). 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[ "https://github.com/immunomethylomics/FlowSorted.Blood.EPIC." ]
[ "Virtual photons to fourth order CHPT with nucleons #1", "Virtual photons to fourth order CHPT with nucleons #1" ]
[ "Guido Müller [email protected] \nInstitut für Theoretische Physik Boltzmanngasse 5\nUniversität Wien\nA-1090WienAustria\n" ]
[ "Institut für Theoretische Physik Boltzmanngasse 5\nUniversität Wien\nA-1090WienAustria" ]
[]
Baryon chiral perturbation theory offers another possibility of investigating isospin violation. As first stressed by Weinberg [1], reactions involving nucleons and neutral pions can lead to gross violations of isospin, e.g. in the scattering length difference a(π 0 p) − a(π 0 n) he predicted an effect of the order of 30%. This is because chiral symmetry and isospin breaking appear at the same order and the leading isospin symmetric terms involving neutral pions are suppressed due to chiral symmetry. It is, however, known that precise and complete one loop calculations in the baryon sector should be carried out to fourth order since it has also been shown that in many cases one loop graphs with exactly one dimension two insertion are fairly large. Most calculations in baryon CHPT are performed in heavy baryon chiral perturbation theory (HBCHPT)[5,6]. This is based on the observation that a straightforward extension of CHPT with baryons treated fully relativistically leads to a considerable complication in the power counting since only nucleon three-momenta are small compared to typical hadronic scales, as discussed in detail in ref.[7]. However, one has to be careful with strict non-relativistic expansions since in some cases they can conflict structures from analyticity, as discussed e.g. in ref.[8]. Therefore, in ref.[9], a novel scheme was proposed which is based on the relativistic formulation but circumvents the power counting problems (to one loop) by a clever separation of the loop integrals into IR singular and regular parts. In this formulation, all analytic constraints are fulfilled by construction.The starting point of our approach is to construct the most general chiral invariant Lagrangian built from pions, nucleons and external scalar, pseudoscalar, vector, axial-vector sources and virtual photons, parametrized in terms of the vector field A µ (x). The Goldstone bosons are collected in a 2×2 matrix-valued field U(x) = u 2 (x). The nucleons are described by structureless #1
null
[ "https://arxiv.org/pdf/hep-ph/9908470v1.pdf" ]
15,590,488
hep-ph/9908470
3bbddb2133b8bbbf231fa507de5f1a9d43152a9d
Virtual photons to fourth order CHPT with nucleons #1 26 Aug 1999 Guido Müller [email protected] Institut für Theoretische Physik Boltzmanngasse 5 Universität Wien A-1090WienAustria Virtual photons to fourth order CHPT with nucleons #1 26 Aug 1999 Baryon chiral perturbation theory offers another possibility of investigating isospin violation. As first stressed by Weinberg [1], reactions involving nucleons and neutral pions can lead to gross violations of isospin, e.g. in the scattering length difference a(π 0 p) − a(π 0 n) he predicted an effect of the order of 30%. This is because chiral symmetry and isospin breaking appear at the same order and the leading isospin symmetric terms involving neutral pions are suppressed due to chiral symmetry. It is, however, known that precise and complete one loop calculations in the baryon sector should be carried out to fourth order since it has also been shown that in many cases one loop graphs with exactly one dimension two insertion are fairly large. Most calculations in baryon CHPT are performed in heavy baryon chiral perturbation theory (HBCHPT)[5,6]. This is based on the observation that a straightforward extension of CHPT with baryons treated fully relativistically leads to a considerable complication in the power counting since only nucleon three-momenta are small compared to typical hadronic scales, as discussed in detail in ref.[7]. However, one has to be careful with strict non-relativistic expansions since in some cases they can conflict structures from analyticity, as discussed e.g. in ref.[8]. Therefore, in ref.[9], a novel scheme was proposed which is based on the relativistic formulation but circumvents the power counting problems (to one loop) by a clever separation of the loop integrals into IR singular and regular parts. In this formulation, all analytic constraints are fulfilled by construction.The starting point of our approach is to construct the most general chiral invariant Lagrangian built from pions, nucleons and external scalar, pseudoscalar, vector, axial-vector sources and virtual photons, parametrized in terms of the vector field A µ (x). The Goldstone bosons are collected in a 2×2 matrix-valued field U(x) = u 2 (x). The nucleons are described by structureless #1 relativistic spin- 1 2 particles, the spinors being denoted by Ψ(x) in the relativistic case or by the socalled light component N(x) in the heavy fermion formulation. The effective field theory admits a low energy expansion, i.e. the corresponding effective Lagrangian can be written as L eff = L (2) ππ + L (4) ππ + L (1) πN + L (2) πN + L (3) πN + L (4) πN + . . . ,(1) where the ellipsis denotes terms of higher order not considered here. For the explicit form of the meson Lagrangian and the dimension one and two pion-nucleon terms, we refer to ref. [11]. More precisely, in the pion-nucleon sector, the inclusion of the virtual photons modifies the leading term of dimension one and leads to new local (contact) terms starting at second order [2]. In particular, since the electric charge related to the virtual photons always appears quadratic, the following pattern for the terms in the electromagnetic effective Lagrangian emerges. At second order, we can only have terms of order e 2 , at third order e 2 q and at fourth order e 2 q 2 or e 4 (besides the standard strong terms).The inclusion of the virtual photons proceeds with, Q ± = 1 2 u Q u † ± u † Q u ,(2) which under chiral SU(2) L ×SU(2) R symmetry transform as any matrix-valued matter field (Q defines the charge matrix). In particular, to lowest order one finds (in the relativistic and the heavy fermion formulation) L (1) πN =Ψ iγ µ ·∇ µ − m + 1 2 g A γ µ γ 5 ·ũ µ Ψ =N iv ·∇ + g A S ·ũ N + O( 1 m ) ,(3)with∇ µ = ∇ µ − i Q + A µ ,ũ µ = u µ − 2 Q − A µ ,(4) and #3 Ψ(x) = exp{imv · x} (N(x) + h(x)) . Furthermore, v µ denotes the nucleons' four-velocity, S µ the covariant spin-vectorà la Pauli-Lubanski and g A the axial-vector coupling constant. These virtual photon effects can only come in via loop diagrams since by definition a virtual photon can not be an asymptotic state. At second order, local contact terms with finite low-energy constants (LECs) appear. We call these LECs f i for the heavy baryon approach and f ′ i in the relativistic Lagrangian. As stated before, the em Lagrangian is given entirely in terms of squares of Q ± (and their traceless companions), L (2) πN,em = 3 i=1 F 2 π f ′ iΨ O (2) i Ψ = 3 i=1 F 2 π f iN O (2) i N ,(6) with the O 1 = Q 2 + −Q 2 − , O(2)2 = Q + Q + , O(2)3 = Q 2 + +Q 2 − .(2) #3 We do not spell out the details of how to construct the heavy nucleon EFT from its relativistic counterpart but refer the reader to the extensive review [11]. Notice furthermore that only the second term in eq.(6) has an isovector piece and contributes to the neutron-proton mass difference [2]. The factor F 2 π in eq.(6) ensures that the em LECs have the same dimension as the corresponding strong dimension two LECs. From the third order calculation of the proton-neutron mass difference [2] one deduces the value for f 2 , f 2 = −(0.45 ± 0.19) GeV −1 . The Lagrangian to third order takes the form L (3) πN,em = 12 i=1 F 2 π g ′ iΨ O (3) i Ψ = 12 i=1 F 2 π g iN O (3) i N ,(8)with the O (3) i monomials in the fields of dimension three [2][3] , in their relativistic form and the heavy baryon counterparts. Again, for the g i to have the same mass dimension as the d i of the strong sector defined in ref. [10], we have multiplied them with a factor of F 2 π . Thus the g i (g ′ i ) scales as [mass −2 ]. So the complete fourth order pion-nucleon Lagrangian with virtual photons is given by [ 3] L (4) πN,em = 5 i=1 F 4 π h ′ iΨ O (e 4 ) i Ψ + 90 i=6 F 2 π h ′ iΨ O (e 2 p 2 ) i Ψ (9) with the O (4) i monomials in the fields of dimension four [3]. To be consistent with the scaling properties of the dimension two and three LECs, the h i are multiplied with powers of F 2 π such that the first five LECs take dimension [mass −3 ] while the others are of dimension [mass −1 ]. The scalar form factor The scalar form factor of the nucleon is defined via N(p ′ )| m uū u + m dd d |N(p) =ū(p ′ )u(p) σ(t) , t = (p ′ − p) 2 ,(10) for a nucleon state |N(p) of four-momentum p. At t = 0, which gives the much discussed pionnucleon σ-term, one can relate this matrix element to the so-called strangeness content of the nucleon. A direct determination of the σ-term is not possible, but rather one extends pion-nucleon scattering data into the unphysical region and determines σ πN (t = 2M 2 π ), i.e. at the so-called Cheng-Dashen point. The relation to the σ-term is given by the low-energy theorem of Brown, Peccei and Pardee [13], σ πN (2M 2 π ) = σ πN (0) + ∆σ πN + ∆R (11) where ∆σ πN parametrizes the t-dependence of the sigma-term whereas ∆R is a remainder not fixed by chiral symmetry. The most systematic determination of ∆σ πN has been given in ref. [14], ∆σ πN = (15 ± 1) MeV. The remainder ∆R has been bounded in ref. [15], ∆R < 2 MeV. It was shown that the third order effects can shift the proton σ-term by about 8% and have a smaller influence on the shift to the Cheng-Dashen (CD) point [2]. In [3] we worked out explicitly the isospin violating corrections to this shift to fourth order. This is motivated by the fact that in the difference most of the counterterm contributions drop out, more precisely, only momentumdependent contact terms can contribute to the shift. Such terms only appear at fourth order since due to parity one needs two derivatives and any quark mass or em insertion accounts for at least two orders [3]. It can be decomposed as σ (4) πN (t) = σ (4),IC πN (t) + σ (4),IV πN (t) .(12) The isospin-conserving strong terms have already been evaluated in ref. [15]. Here, we concentrate on the em corrections ∼ 1 (in isospin space) and all terms ∼ τ 3 . We have eye graphs and tadpoles with insertions ∼ f 2 , c 5 . These can be evaluated straightforwardly. Because of the tiny coefficients appearing in the evaluation of the loop contributions, these are only fractions of an MeV, ∆σ 4,IV,loop πN = −0.05 MeV and can thus be completely neglected. For the counterterm contributions, setting all appearing LECs on the values obtained from dimensional analysis as explained [3], one finds a total contribution ∆σ 4,IV,ct πN = ±0.01 MeV. For the IC em terms, we find (setting again f 1,3 = ±1/4π) a completely negligible loop contribution (less than 0.01 MeV) and the counterterms give ±0.7 MeV for the LECs estimated from dimensional analysis. Note, however, that if the numerical factors f 1,3 /(4π) are somewhat bigger than one, one could easily have a shift of ±2 MeV, which is a substantial electromagnetic effect. Neutral pion scattering off nucleons As pointed out long time ago by Weinberg [1], the difference in the S-wave scattering lengths for neutral pions off nucleons is sensitive to the light quark mass difference, a(π 0 p) − a(π 0 n) = 1 4π 1 1 + M π + /m p −4B(m u − m d )c 5 F 2 π + O(q 3 ) = 1 4π 1 1 + M π + /m p ∆ 2 (M π 0 ) + O(q 3 ) .(13) It was shown in ref. [2] by an explicit calculation that to third order there are no corrections to this formula. This is based on the fact that the electromagnetic Lagrangian can not contribute at this order since the charge matrix has to appear quadratic and never two additional pions can appear. However, at next order one can of course have loop graphs with one dimension two insertion and additional em counterterms. To obtain the first correction to Weinberg's prediction, eq.(13), one thus has to compute the fourth order corrections [3]. These are due to strong dimension two insertions ∼ c 5 and em insertions ∼ f 2 . For the difference a(π 0 p)−a(π 0 n) we only have to consider the operators ∼ τ 3 [3] Consider first the loop contributions. Since we can not fix the counterterms from data, we are left with a spurious scale dependence which reflects the theoretical uncertainty at this order. For λ = {0.5, 0.77, 1.0} GeV we find ∆ str 2 = {−7.1, 0.9, 5.7} · 10 −2 , ∆ em 2 = {11.5, 12.0, 12.3} · 10 −2 . The counterterms are estimated based on dimensional analysis at the scale λ = M ρ and give a contribution of about −0.3 · 10 −2 . Even if the LECs would be a factor of ten larger than assumed, the counterterm contribution would not exceed ±3%. Altogether, the correction to Weinberg's prediction, eq.(13), are in the range of 4 to 18 percent, i.e. fairly small. Finally, we wish to mention that in ref. [16] isospin-violation for neutral pion photoproduction off nucleons was discussed which allows one to eventually measure directly the very small π 0 p scattering length by use of the final-state theorem. Summary We have developed the whole mechanism to calculate isospin violation effects in the framework of CHPT. In order to get the correct size of isospin violation one has to include all non-isospin symmetric sources like the electromagnetic interaction, the quark mass difference (π 0 − η mixing), explicit photon loops. In future calculations one has to pin down the two LECs of the second order em Lagrangian f 1,3 which are not well known and are estimated only by dimensional arguments. AcknowledgmentsI am grateful to Ulf-G. Meißner for enjoyable collaboration. . S Weinberg, Trans. N.Y. Acad. of Sci. 38185S. Weinberg, Trans. N.Y. Acad. of Sci. 38 (1977) 185. . - G Ulf, S Meißner, Steininger, Phys. Lett. 419403Ulf-G. Meißner and S. Steininger, Phys. Lett. B419 (1998) 403. . - G Ulf, G Meißner, Müller, Nucl.Phys. 556in pressUlf-G. Meißner and G. Müller, Nucl.Phys. B556 (1998) in press. . N Fettes, - G Ulf, S Meißner, Steininger, hep-ph/9811366Phys. Lett. B. in pressN. Fettes, Ulf-G. Meißner and S. Steininger, hep-ph/9811366, Phys. Lett. B (in press). . E Jenkins, A V Manohar, Phys. Lett. 255558E. Jenkins and A.V. Manohar, Phys. Lett. B255 (1991) 558. . V Bernard, N Kaiser, J Kambor, -G Ulf, Meißner, Nucl. Phys. 388315V. Bernard, N. Kaiser, J. Kambor and Ulf-G. Meißner, Nucl. Phys. B388 (1992) 315. . J Gasser, M E Sainio, A Švarc, Nucl. Phys. 307779J. Gasser, M.E. Sainio and A.Švarc, Nucl. Phys. B307 (1988) 779. . V Bernard, N Kaiser, -G Ulf, Meißner, Nucl. Phys. 611429V. Bernard, N. Kaiser and Ulf-G. Meißner, Nucl. Phys. A611 (1996) 429. . T Becher, H Leutwyler, hep-ph/9901384T. Becher and H. Leutwyler, hep-ph/9901384. . N Fettes, - G Ulf, S Meißner, Steininger, Nucl. Phys. 640199N. Fettes, Ulf-G. Meißner and S. Steininger, Nucl. Phys. A640 (1998) 199. . V Bernard, N Kaiser, -G Ulf, Meißner, Int. J. Mod. Phys. 4193V. Bernard, N. Kaiser and Ulf-G. Meißner, Int. J. Mod. Phys. E4 (1995) 193. . - G Ulf, G Meißner, S Müller, Steininger, hep-ph/9809446Ann. Phys. (NY). in pressUlf-G. Meißner, G. Müller and S. Steininger, hep-ph/9809446, Ann. Phys. (NY), in press. . L S Brown, W J Pardee, R D Peccei, Phys. Rev. 42801L.S. Brown, W.J. Pardee and R.D. Peccei, Phys. Rev. D4 (1971) 2801. . J H Gasser, M E Leutwyler, Sainio, Phys. Lett. 253252J. Gasser. H. Leutwyler and M.E. Sainio, Phys. Lett. B253 (1991) 252. . V Bernard, N Kaiser, -G Ulf, Meißner, Phys. Lett. 389144V. Bernard, N. Kaiser and Ulf-G. Meißner, Phys. Lett. B389 (1996) 144. . A M Bernstein, Phys. Lett. 44220A.M. Bernstein, Phys. Lett. B442 (1998) 20.
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[ "COMPARISON OF ALGORITHMS TO CALCULATE QUADRATIC IRREGULARITY OF PRIME NUMBERS", "COMPARISON OF ALGORITHMS TO CALCULATE QUADRATIC IRREGULARITY OF PRIME NUMBERS" ]
[ "Joshua Holden " ]
[]
[]
In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k 0 , using the values of the zeta function ζ k 0 at negative integers as our "higher Bernoulli numbers". In the case where k 0 is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of k 0 -irregularity (more generally, "quadratic irregularity") of a prime number.
10.1090/s0025-5718-01-01341-2
[ "https://arxiv.org/pdf/math/0010286v1.pdf" ]
2,829,076
math/0010286
da8437fd85b83a6e09b1b408d1401b0b9eaff3e4
COMPARISON OF ALGORITHMS TO CALCULATE QUADRATIC IRREGULARITY OF PRIME NUMBERS 29 Oct 2000 Joshua Holden COMPARISON OF ALGORITHMS TO CALCULATE QUADRATIC IRREGULARITY OF PRIME NUMBERS 29 Oct 2000 In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k 0 , using the values of the zeta function ζ k 0 at negative integers as our "higher Bernoulli numbers". In the case where k 0 is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of k 0 -irregularity (more generally, "quadratic irregularity") of a prime number. Definitions Let k 0 be a totally real number field, and let p be an odd prime. Let k 1 = k 0 (ζ p ), where ζ p n will denote a primitive p n -th root of unity. Let ∆ = Gal(k 1 /k 0 ), and let δ = |∆|. Let p e be the largest power of p such that ζ p e ∈ k 0 (ζ p ). Definition 1. Let ζ k0 be the zeta function for k 0 . We say that p is k 0 -regular if p is relatively prime to ζ k0 (1 − 2m) for all integers m such that 2 ≤ 2m ≤ δ − 2 and also p is relatively prime to p e ζ k0 (1 − δ). The number of such zeta-values that are divisible by p will be the index of k 0 -irregularity of p. According to a well-known theorem of Kummer, p divides the order of the class group of Q(ζ p ) if and only if p divides the numerator of a Bernoulli number B m for some even m such that 2 ≤ m ≤ p − 3. Such primes are called irregular; the others are called regular. In the setting we have described above, the author proved in his thesis ( [7], see also [8]), building on work of Greenberg and Kudo, that under a certain technical condition Kummer's criterion can be extended to give information about whether p divides the class group of k 0 (ζ p ). To be exact, let k + 1 denote the maximal real subfield of k 1 , which is equal to k 0 (ζ p + ζ −1 p ). Let h(k 1 ) denote the class number of k 1 and h + (k 1 ) denote the class number of k + 1 . It is known that h + (k 1 ) | h(k 1 ); we let the relative class number h − (k 1 ) be the quotient. Theorem 1 (Greenberg, Holden). Assume that no prime of the field k + 1 lying over p splits in k 1 . Then p divides h − (k 1 ) if and only if p is not k 0 -regular. As an application, we note that one common way of constructing public-key cryptographic systems is to utilize the problem of finding a discrete logarithm in 1991 Mathematics Subject Classification. Primary 11Y40, 11Y60, 11Y16, 11B68; Secondary 11R42, 11R29, 94A60, 11R18. Key words and phrases. Bernoulli numbers, Bernoulli polynomials, irregular primes, zeta functions, quadratic extensions, cyclotomic extensions, class groups, cryptography. To appear in Mathematics of Computation. 1 some abelian group. In order to make sure that the discrete logarithm problem is computationally hard, one needs to know something about the structure of the group involved, e.g. that it is divisible by a large prime. Theorem 1 shows that if p is a large k 0 -irregular prime and the conditions of the theorem are met, then the class group of k 0 (ζ p ) may be especially suitable for cryptography. (One should see [3] for more on the use of class groups in cryptography.) For the case we consider, k 0 will be a real quadratic field Q( √ D), with D a positive fundamental discriminant. For such a k 0 , we will say that primes are Dregular or have given index of D-irregularity, and we will let the zeta function ζ k0 be also denoted by ζ D . (More generally, we may refer to the concept as "quadratic irregularity".) In this case δ will be equal to p − 1 unless D = p, in which case δ = (p − 1)/2. Also, e is always equal to 1 when p does not divide the order of k 0 over Q, which is true in this case since p is odd. For the condition in Theorem 1 that no prime of the field k + 1 lying over p splits in k 1 to be satisfied it is sufficient that p should not divide D, and we should also note that since p does not divide the degree of k 0 = Q( √ D) over Q, a theorem of Leopoldt shows that p divides h(k 1 ) if and only if p divides h − (k 1 ). In general, we will consider three cost models for the time of multiplication: first using naive multiplication (O(tt ′ )), second using Schönhage-Strassen fast multiplication or a similar method (O(t lg(t ′ ) O(1) ), and third using a model where multiplication (or addition) takes constant time regardless of the size of the factors (O(1)). We do not expect constant time multiplication to occur asymptotically in the real world, but it can provide useful estimates in situations where the size of the numbers involved is small compared to the word size of the actual computer in question. (In these running time bounds, t is the number of bits in the larger multiplicand and t ′ the number of bits in the smaller.) First formula Siegel's first formula to compute ζ D (1 − 2m) for m ≥ 1 an integer is analogous to the formula ζ(1 − 2m) = −B 2m /(2m). Using elementary methods, Siegel showed that similarly ζ D (1 − 2m) = B 2m 4m 2 D 2m−1 D j=1 χ(j)B 2m (j/D).(1) Here χ(j) = D j , the Kronecker symbol, and B 2m (j/D) indicates the 2m-th Bernoulli polynomial evaluated at the fraction j/D. The Bernoulli polynomial B r (x) can be computed from the Bernoulli numbers as B r (x) = r s=0 r s B r−s x s . It is not difficult to estimate the sizes of the numbers involved. We will assume throughout that B m , 1 ≤ m ≤ M , are precomputed over the common denominator of the final result, and stored in this fashion each has size O(m(lg m + lg D)) bits for a total table size of O(M 2 (lg M + lg D)) bits. (The precomputation does not in fact add to the asymptotic running time.) The rational numbers B 2m (j/D) can also be stored in O(m(lg m + lg D)) bits, as can the total. See [6] for more details. However we can do better than this, using a cross between Horner's method of evaluating polynomials and an algorithm used by Brent to calculate Bernoulli numbers, as was previously discussed by the author in [6]. This method gives a total time of O(M 3 (lg M + lg a + lg b) O(1) ) using either constant or fast multiplication, and time O(M 2 ) using constant time multiplication. Using either of these algorithms to compute ζ D (1 − 2m), 2 ≤ 2m ≤ M , Second formula Siegel's second formula is, as I said, derived from the theory of modular forms. In general, for k 0 a totally real number field as above, it says that ζ k0 (1 − 2m) = −2 n c −1 2mn r l=1 c 2mn,l s k0 l (2m), where n = [k 0 : Q], c 2mn = c 2mn,0 and c 2mn,l are rational integers depending only on 2mn and l (given by explicit formulas which we will discuss), r = ⌊mn/6⌋ if 2mn ≡ 2 modulo 12 ⌊mn/6⌋ + 1 otherwise, and s k0 l is a sum over norms of ideals in the ring of integers of k 0 , namely s k0 l (2m) = ν∈(d) −1 , ν≫0, tr(ν)=l σ 2m−1 ((ν)d), where σ 2m−1 (A) = B|A N (B) 2m−1 is a generalization of the usual sum of powers function and d is the different of k 0 . In the quadratic case this all becomes much easier: ζ D (1 − 2m) = −4c −1 4m r l=1 c 4m,l s D l (2m), r = ⌊m/3⌋ + 1,(2)s D l (2m) = ν∈( √ D) −1 , ν≫0, tr(ν)=l σ 2m−1 ((ν √ D)), and s D l (2m) can also be expressed in terms of a purely arithmetic function e 2m−1 (n), as follows: s k0 l (2m) = j|l χ D (j)j 2m−1 e 2m−1 ((l/j) 2 D) and e 2m−1 (n) = x 2 ≡n (mod 4) |x|≤ √ n σ 2m−1 n − x 2 4 where σ 2m−1 (n) = d|n d 2m−1 is the usual sum-of-powers function. (See [10], [11], [4] and [5] for more detailed descriptions of these formulas, and for their derivations.) The coefficients c 4m,l are most easily expressed as the coefficients of a certain power series, and can be computed as needed without adding to the asymptotic running time. We will give explicit formulas for the power series and discuss its computation in Section 4. It is not hard to prove that in this form c 4m,l is of size O(m). However, it is more efficient to rearrange the terms of the formula as follows: ζ D (1 − 2m) = −4c −1 4m r l=1 c 4m,l s k0 l (2m) = −4c −1 4m r l=1 c 4m,l j|l χ D (j)j 2m−1 e 2m−1 ((l/j) 2 D) = −4c −1 4m r k=1   ⌊r/k⌋ j=1 χ D (j)j 2m−1 c 4m,jk   e 2m−1 (k 2 D)(3) This rearrangement of the formula requires fewer calls to compute e 2m−1 by a factor of lg m. Using this version of the formula, the time necessary to compute all ζ D (1 − 2m), 2 ≤ 2m ≤ M , using naive multiplication is O(M 3 √ D L(M ) O(1) L(D) O(1) + M 5 √ D(lg M + lg D) O(1) ). This is much worse than the best algorithm based on (1) in terms of M , but it is better in terms of D. Also, except for one final division by c −1 4m , all of the arithmetic in this formula deals only with rational integers; unlike the previous formulas. Note that the first term comes from the factoring process, while the second term comes from multiplications. It should be noted that the asymptotic running time of this algorithm is greatly improved by using Schönhage-Strassen fast multiplication or constant time multiplication, in which cases the second term becomes smaller than the first and the running time becomes O(M 3 √ D L(M ) O(1) L(D) O(1) ) This is still worse than using (1) in terms of M , but only by a subexponential factor. It should also be noted that (2) and (3) Since the exponent 2m − 1 used in the σ 2m−1 (n) function changes as m does, we can dispose of the table when we change m. The table that we need to keep requires at most O(M 3 D(lg M + lg D)) bits of storage, which could be a significant barrier. More efficient storage of the important information may be valuable here; we will discuss this somewhat more in Section 5. Computing the numbers c 4m,l The integers c 4m,l are defined as follows. Let G k = 1 − 2k B k ∞ n=1 σ k−1 (n) q n be the (normalized) Eisenstein series of order k for k = 6, 10, and 14. (For the general c 2mn,l one also needs k = 0, 4, and 8.) Let ∆ = q ∞ n=1 (1 − q n ) 24 = q ∞ n=0 (−1) n (2n + 1) q n(n+1)/2 8 . be the discriminant series. Let r = ⌊m/3⌋ + 1 as before, and let T 4m = G 12r−4m+2 ∆ −r = ∞ n=−r c 4m,−n q n . Then c 4m = c 4m,0 , and the other c 4m,l for 1 ≤ l ≤ r can also be read off as coefficients of T 4m . Luckily, the expression 12r − 4m + 2 only takes on the values 6, 10, and 14. (In the more general case we can define T 2mn similarly; the expression 12r − 2mn + 2 can take on the values 0, 4, and 8 in addition to those above.) The best algorithm known to the author for calculating these coefficients goes roughly as follows. At the start of the computations for ζ D (1 − 2m), 2 ≤ 2m ≤ M , calculate G 6 , G 10 , G 14 , and ∆ −1 with the maximum number of coefficients necessary (about M/12). Instead of trying to compute all of the needed series ∆ −r at once, we calculate it as a running product which only needs to be updated when r changes. Then, whenever m changes, we multiply truncated versions G 12r−4m+2 and ∆ −r (with about m/6 coefficients each) to find the required coefficients of T 4m . The series ∆ −1 can also be expressed as ∆ −1 = q ∞ n=1 (1 − q n ) −24 = q ∞ n=1 1 (1 − q n ) 24 = q ∞ n=0 p(n) q n 24 where p(n) takes on integer values and is well-known as the partition function from additive number theory. Hardy and Ramanujan proved an asymptotic expression for p(n) which shows that lg p(n) is of order Table 1 presents the various algorithms, for comparison. We present the asymptotic order of the running time to compute ζ D (1 − 2m), 2 ≤ 2m ≤ M , for a given D, using the three methods of multiplication discussed earlier. Summary and future work The factor of lg M in the times for algorithms based on (2) has been included to emphasize that these algorithms are slower than those based on (3), even though the factor of lg M could be absorbed into that of L(M ) O(1) . Multiplication Time order (1) naive We also provide, in Tables 2 and 3, tables of actual timings for some of the algorithms, using naive multiplication. The times were measured on a Sun SPARC Ultra-1 computer using the GP-Pari interpreted language. (See [2].) The data computed by these programs will be analyzed in a future paper. Table 2 measures the time to compute ζ D (1 − 2m), 2 ≤ 2m ≤ M , for a given D, in hours, minutes, and seconds. The number in parentheses indicates the size of the stack used, in bytes. These numbers are only a very rough guide to the actual amount of memory used. Table 3 measures the time to compute ζ d (1 − 2m), 2 ≤ 2m ≤ M , 5 ≤ d ≤ D. We use the algorithm based on (3), both with and without keeping a table of σ 2m−1 (n) as described earlier. The units and stack size numbers should be interpreted as in Table 2. As one can see, memory usage for algorithms based on (3) with a table of values σ 2m−1 (n) goes up quite quickly, and even so a lot of redundant work is being done. For one thing, the same numbers n will have to be factored repeatedly for different values of 2m − 1, but they will lead to different values of σ 2m−1 (n) so the actual factorization would need to be stored and not just a function value. This would be even more memory intensive. Carl Pomerance has suggested some approaches to this, including using a cache rather than a complete table, and storing only the largest prime factor rather than a complete factorization. The question of a cache leads naturally to the question of which numbers will appear as integer values of (k D − x 2 )/4 as k and D vary, and how often. Henri Cohen, in [5], gives some variations on Siegel's formula which could cut down on the number of times σ 2m−1 (n) needs to be computed. Algorithms based on these might provide a linear speedup over the algorithms presented here, but the asymptotic behavior would probably be the same. M 4 D(lg M + lg D) O(1) (1) fast M 3 D(lg M + lg D) O(1) (1) constant M 2 D(lg M + lg D) O(1) (1) from [6] naive M 3 D(lg D + lg M ) O(1) (1) from [6] fast M 3 D(lg D + lg M ) O(1) (1) from [6] constant M 2 D(lg M + lg D) O(1) (2) naive M 3 √ D L(M ) O(1) L(D) O(1) lg M +M 5 √ D(lg M + lg D) O(1) (2) fast M 3 √ D L(M ) O(1) L(D) O(1) lg M (2) constant M 3 √ D L(M ) O(1) L(D) O(1) lg M (3) naive M 3 √ D L(M ) O(1) L(D) O(1) +M 5 √ D(lg M + lg D) O(1) (3) fast M 3 √ D L(M ) O(1) L(D) O(1) (3) constant M 3 √ D L(M ) O(1) L(D) O(1) The anonymous reviewer has suggested that some of the arithmetic could possibly be speeded up by the use of modular techniques and the Chinese remainder theorem. This certainly deserves more consideration. Another prospect for future work is the analysis of "first-hit" versions of these algorithms, namely determining how long we should expect to search before finding, say, the first D-irregular prime larger than a certain bound for D in a given range. This might be particularly useful for cryptographic applications, in which we would be explicitly looking for a class group (or a small number of them) with a hard discrete logarithm problem. This will be explained in more detail in [9]. A first attempt at an algorithm based on(1) might compute B 0 (α), . . . , B M (α) naively from the formula. The time taken for this would be dominated by the powerings. For α = a/b some rational number, the total time with naive multiplication would be O(M 4 (lg M + lg a + lg b) O(1) ). Using fast multiplication instead of naive multiplication would improve this to O(M 3 (lg M + lg a + lg b) O(1) ), while with constant time multiplication we need only time O(M 2 lg M ) regardless of a and b. is then relatively straightforward. Note that the Kronecker symbol χ(j) can be computed in time O(lg 2 D). The slower version of the algorithm has time O(M 4 D(lg M + lg D) O(1) ) with naive multiplication, O(M 3 D(lg M + lg D) O(1) ) with fast multiplication, and O(M 2 D(lg M +lg D) O(1) ) with constant time multiplication. The faster version runs in time O(M 3 D(lg D + lg M ) O(1) ) with either naive or fast multiplication (the O(1) factor is different, of course) and again in time O(M 2 D(lg M + lg D) O(1) ) with constant time multiplication. The running time for calculating the function e 2m−1 (n) is complicated by the need for factoring; we use here an estimate based on the elliptic curve factoring method (we would expect something very similar with any of the other standard subexponential methods) to get an expected running time involving the function L(x) = e √ log x log log x . Given this, we get an expected running time to compute e 2m−1 (n) of O( √ n L(n) 1+o(1) + (2m − 1) 2 √ n lg 2 n) using naive multiplication. If we now applied (2) as written to compute all ζ D (1 − 2m), 2 ≤ 2m ≤ M , we would get a running time of O(M 3 √ D L(M ) O(1) L(D) O(1) lg M + M 5 √ D lg M (lg M + lg D) O(1) ), again using naive multiplication. also present opportunities for time savings when computing zeta-values for multiple D in the same range of M , at a sacrifice of memory space. The controlling factor in the speed of the algorithm is the number of times that σ 2m−1 (n) must be calculated. Note that in computing allζ d (1 − 2m), 5 ≤ d ≤ D,there can only be O(m 2 D) different values of n. However, following the algorithm strictly, we would ordinarily make O(m 2 D 3/2 ) calls to the subroutine that calculates this function. Thus if we compute all ζ d (1 − 2m), 5 ≤ d ≤ D, storing values of σ 2m−1 (n) as we compute them, and then repeat this process for each m in the range 2 ≤ 2m ≤ M , the running time should be O(D L(D) O(1) ) in terms of D, rather than O(D 3/2 L(D) O(1) ) as one would obtain following the algorithm strictly. This compares very favorably with the time of O(D 2 ) in terms of D which holds for algorithms using (1). √ n. (See, for example, Chapter 14 of [1].) From this it is easy to show that the coefficients of ∆ −r take at most O(M 1.5 ) bits of storage each, as do the coefficients of G 12r−4m+2 and T 4m . Thus the storage required for all of the computation of c 4m,l necessary for a fixed m is O(M 2.5 ). The resulting table, which is of course independent of d, can be used to compute ζ d (1 − 2m) for any range of d and then disposed of when m is changed. As far as the time for this algorithm is concerned, computing the tables necessary for all ζ d (1 − 2m), 2 ≤ 2m ≤ M , 5 ≤ d ≤ D, takes time O(M 5 ) with naive multiplication of integers and also of polynomials. By using FFT methods to multiply polynomials the time can be reduced to O(M 3.5 lg M lg lg M ) with fast multiplication of integers and O(M 2 lg M ) with constant time multiplication of integers. (The use of FFT methods here was suggested by A.O.L. Atkin and Will Galway.) This is within our previously established time bounds in the naive and constant time cases; however in the fast multiplication case it could add to the total asymptotic time in terms of M , which for the previous parts of the algorithm was established as O(M 3 L(M ) O(1) ) in terms of M . On the other hand it should be noted that these calculations only need to be done once per value of m no matter how many values of d one is examining. Also the constant involved in the O(M 3.5 lg M lg lg M ) seems to be quite good in practice compared to that in the O(M 3 L(M ) O(1) ). Table 1 . 1Comparison of algorithms for ζ D (1 − 2m), 2 ≤ 2m ≤ MEquation used Table 2 . 2Calculating ζ D (1 − 2m), 2 ≤ 2m ≤ MD M time (1) from [6] (stacksize) time (3) (stacksize) 5 100 3.155 (10M) .838 (10M) 101 100 55.561 (10M) 3.758 (10M) 501 100 4:50.282 (10M) 12.480 (10M) 1001 100 10:52.411 (10M) 20.670 (10M) 5001 100 48:27.107 (12M) 1:43.548 (10M) 5 500 2:05.670 (4M) 8:11.603 (4M) 101 500 42:38.612 (4M) 1:18:26.615 (4M) 501 500 3:48:18.615 (8M) 4:45:20.438 (4M) 1001 500 7:49:56.048 (12M) 7:49:00.783 (4M) 5 1000 10:05.903 (4M) 3:55:46.908 (4M) 5 2000 1:14:01.992 (16M) 118:17:46.020 (4M) Table 3 . 3Calculating ζ d (1 − 2m), 2 ≤ 2m ≤ M , 5 ≤ d ≤ DD M time (3) (stacksize) time (3) with table (stacksize) 100 100 1:19.638 (4M) 1:27.983 (4M) 500 100 18:34.377 (4M) 13:57.166 (8M) 1000 100 1:03:50.464 (4M) 37:52.552 (12M) 5000 100 26:39:46.109 (4M) 7:10:44.870 (64M) AcknowledgementsThe author would like to thank A.O.L. Atkin, Will Galway, and the members of the NMBRTHY electronic mailing list for their very helpful suggestions, and Robert Harley for generating some of the precomputed tables used in the computations reported in this paper. He would also like to especially thank Johannes Buchmann, Henri Cohen, and Carl Pomerance for suggestions and encouragement, and Gary Walsh for his encouraging remarks on an early version of this paper. Finally, he would like to thank the anonymous reviewer for encouragement and several helpful suggestions. Introduction to analytic number theory, Undergraduate Texts in Mathematics. Tom M Apostol, Springer-VerlagTom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathe- matics, Springer-Verlag, 1976. User's guide to PARI-GP, Laboratoire A2X. C Batut, K Belabas, D Bernardi, H Cohen, M Olivier, 2.0.9Université Bordeaux IC. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, User's guide to PARI-GP, Laboratoire A2X, Université Bordeaux I, version 2.0.9 ed., May 13, 1998, <http://hasse.mathematik.tu-muenchen.de/ntsw/pari/Welcome.html>, <ftp://megrez.math.u-bordeaux.fr>. A one way function based on ideal arithmetic in number fields. Johannes Buchmann, Sachar Paulus, Advances in cryptology-CRYPTO '97. Burton S. Kaliski, JrSpringer-Verlag1294Johannes Buchmann and Sachar Paulus, A one way function based on ideal arithmetic in number fields, Advances in cryptology-CRYPTO '97 (Burton S. Kaliski, Jr, ed.), Lecture Notes in Computer Science, vol. 1294, Springer-Verlag, 1997, pp. 385-394. Sums involving the values at negative integers of L-functions of quadratic characters. Henri Cohen, Math. Ann. 217Henri Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), 271-285. Variations sur un thème de Siegel et Hecke, Acta Arith. 30, Variations sur un thème de Siegel et Hecke, Acta Arith. 30 (1976), 63-93. Irregularity of prime numbers over real quadratic fields. Joshua Holden, Algorithmic Number Theory: Third International Symposium; Proceedings. J. P. BuhlerSpringer-Verlag1423Springer Lecture Notes in Computer ScienceJoshua Holden, Irregularity of prime numbers over real quadratic fields, Algorithmic Number Theory: Third International Symposium; Proceedings (J. P. Buhler, ed.), Springer Lecture Notes in Computer Science, vol. 1423, Springer-Verlag, 1998, pp. 454-462. On the Fontaine-Mazur conjecture for number fields and an analogue for function fields. Brown UniversityPh.D. thesis, On the Fontaine-Mazur conjecture for number fields and an analogue for function fields, Ph.D. thesis, Brown University, 1998. On the Fontaine-Mazur Conjecture for number fields and an analogue for function fields. J. Number Theory. 81, On the Fontaine-Mazur Conjecture for number fields and an analogue for function fields, J. Number Theory 81 (2000), 16-47. First-hit analysis of algorithms for computing quadratic irregularity, (In preparation). , First-hit analysis of algorithms for computing quadratic irregularity, (In prepara- tion). Bernoullische Polynome und quadratische Zahlkörper. Carl Ludwig Siegel, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II. 2Carl Ludwig Siegel, Bernoullische Polynome und quadratische Zahlkörper, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 2 (1968), 7-38. On the values at negative integers of the zeta-function of a real quadratic field. Don Zagier, Enseign. Math. 2Don Zagier, On the values at negative integers of the zeta-function of a real quadratic field, Enseign. Math. (2) 22 (1976), 55-95.
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[ "The generation of helical magnetic field in a viable scenario of Inflationary Magnetogenesis", "The generation of helical magnetic field in a viable scenario of Inflationary Magnetogenesis" ]
[ "Ramkishor Sharma \nDepartment of Physics & Astrophysics\nUniversity of Delhi\n110007New DelhiIndia.\n", "Kandaswamy Subramanian \nIUCAA\nPost Bag 4\n\nPune University Campus\n411007Ganeshkhind, PuneIndia\n", "T R Seshadri \nDepartment of Physics & Astrophysics\nUniversity of Delhi\n110007New DelhiIndia.\n" ]
[ "Department of Physics & Astrophysics\nUniversity of Delhi\n110007New DelhiIndia.", "IUCAA\nPost Bag 4", "Pune University Campus\n411007Ganeshkhind, PuneIndia", "Department of Physics & Astrophysics\nUniversity of Delhi\n110007New DelhiIndia." ]
[]
We study the generation of helical magnetic fields in a model of inflationary magnetogenesis which is free from the strong coupling and back-reaction problems. To generate helical magnetic fields, we add an f 2F µν Fµν term to the lagrangian of Ratra model. The strong coupling and back-reaction problems are avoided if we take a particular behaviour of coupling function f , in which f increases during inflation and decreases post inflation to reheating. The generated magnetic field is fully helical and has a blue spectrum, dρB/d ln k ∝ k 4 . This spectrum is obtained when coupling function f ∝ a 2 during inflation. The scale of reheating in our model has to be lower than 4000 GeV to avoid back-reaction post inflation. The generated magnetic field spectrum satisfies the γ-ray bound for all the possible scales of reheating. The comoving magnetic field strength and its correlation length are ∼ 4 × 10 −11 G and 70 kpc respectively, if reheating takes place at 100 GeV. For reheating at the QCD scales of 150 MeV, the field strength increases to ∼ nano gauss, with coherence scale of 0.6 Mpc.
10.1103/physrevd.97.083503
[ "https://arxiv.org/pdf/1802.04847v1.pdf" ]
119,235,694
1802.04847
536bd948bff5abc2d646d1bb6a85a6e2c7ca45b2
The generation of helical magnetic field in a viable scenario of Inflationary Magnetogenesis 13 Feb 2018 Ramkishor Sharma Department of Physics & Astrophysics University of Delhi 110007New DelhiIndia. Kandaswamy Subramanian IUCAA Post Bag 4 Pune University Campus 411007Ganeshkhind, PuneIndia T R Seshadri Department of Physics & Astrophysics University of Delhi 110007New DelhiIndia. The generation of helical magnetic field in a viable scenario of Inflationary Magnetogenesis 13 Feb 2018 We study the generation of helical magnetic fields in a model of inflationary magnetogenesis which is free from the strong coupling and back-reaction problems. To generate helical magnetic fields, we add an f 2F µν Fµν term to the lagrangian of Ratra model. The strong coupling and back-reaction problems are avoided if we take a particular behaviour of coupling function f , in which f increases during inflation and decreases post inflation to reheating. The generated magnetic field is fully helical and has a blue spectrum, dρB/d ln k ∝ k 4 . This spectrum is obtained when coupling function f ∝ a 2 during inflation. The scale of reheating in our model has to be lower than 4000 GeV to avoid back-reaction post inflation. The generated magnetic field spectrum satisfies the γ-ray bound for all the possible scales of reheating. The comoving magnetic field strength and its correlation length are ∼ 4 × 10 −11 G and 70 kpc respectively, if reheating takes place at 100 GeV. For reheating at the QCD scales of 150 MeV, the field strength increases to ∼ nano gauss, with coherence scale of 0.6 Mpc. I. INTRODUCTION Cosmic magnetic fields have been detected from planetery scales [1] to galaxy clusters scales [2][3][4]. γ-ray observations of Blazars suggest their presence in the voids as well [5,6]. However the origin of these fields do not have a fully satisfactory explanation. Astrophysical scenarios [7][8][9][10][11] for generating these fields involve battery effects to create a seed field which is later amplified to the observed strength by the dynamo action [12][13][14][15]. However, the presence of coherent magnetic fields in void regions favor primordial scanarios of generation. Several possible scenarios for this have been suggested in literature (for reviews, see [38][39][40][41][42]). Inflationary magnetogenesis is one of the possible scenarios to generate fields that are coherent over large scales. However the generation of magnetic field during inflation in the standard physics is not possible due to the conformal invariance of the electromagnetic field [43]. A breaking of conformal invariance is necessary. This has been done in many models by taking a time dependent function coupled with the kinetic energy term of the electromagnetic (EM) field. This time dependent coupling term can arise through the coupling between inflaton field and EM field (Ratra model) [18] or by taking a non-minimal coupling of the EM field to the gravitational field [16]. Although Ratra model generates sufficient strength of the magnetic field, it potentially suffers from strong coupling and back-reaction problems [44]. These problems have been resolved by further modification of the Ratra model [29,31,45]. In Ref. [45], to resolve these problems f is assumed to increase during inflation and decrease back to its initial value post inflation. Firstly this behaviour of f circumvents the strong coupling problem. Moreover, for a small enough inflationary and reheating scales, the model also does not suffer from the backreaction problem. Indeed demanding no back-reaction post inflation, bounds on the inflationary scale and reheating scale has been obtained. The generated magnetic field strength in this model can also explain the magnetic field strength suggested by γ-ray observation, below a certain reheating scale. In the model discussed above, the generated magnetic field is of non-helical nature. In this paper we look at the possible generation of helical magnetic fields. The nonlinear evolution of magnetic field in helical case differs from non-helical one. Due to the helicity conservation, the magnetic field strength decreases at a slower rate and the correlation length increases at a higher rate compared to the non-helical case. Helicity conservation gives us a more optimistic evolution of the magnetic field and this can relax the bound on the reheating scale given in [45] further. Moreover, it has been claimed in the literature that gamma ray observations of the Blazars indicate the presence of helical magnetic field in intergalactic medium [46,47]. To generate helical magnetic field, we add a parity breaking term in the Electromagnetic action. We also compare our generated magnetic field strength with the constraints from γ-ray observation. This paper is organised as follows. In section II we discuss the generation of helical magnetic field during inflation. Section III discusses the generation of the field post inflation to reheating and we obtain a relation between inflationary scale and reheating scale by demanding no back-reaction post inflation. Section IV incorporates the nonlinear evolution of magnetic field and its correlation length after generation. Results of our model are also given there. In section V, we compare our results with the γ-ray observation. Our conclusions are presented in VI. II. GENERATION OF HELICAL MAGNETIC FIELD DURING INFLATION We start with the action for the electromagnetic field in which the conformal invariance is explicitly broken by introducing a time dependent function f 2 multiplying F µν F µν in the action. To generate helical magnetic field, we also add a f 2 F µνF µν in the action. Thus we take the action to be of the form, S = − √ −gd 4 x[ f 2 (φ) 16π F µν F µν + F µνF µν + j µ A µ ] − √ −gd 4 x 1 2 ∂ ν φ∂ ν φ + V (φ)(1) Here F µν = ∂ µ A ν − ∂ ν A µ andF µν = (1/2)ǫ µναβ F αβ , where A µ is the EM 4-potential and ǫ µναβ is fully antisymmetric tensor defined as ǫ µναβ = 1/ √ −g η µναβ . η µναβ is levi-civita symbol whose values are ±1 only. The term j µ A µ represents the interaction where j µ is the four current density. Remaining part of the action incorporates the evolution of the inflaton field. In this paper we adopt the following nomenclature. Greek indices µ, ν.... are for space-time coordinates and Roman indices i, j, k.... are for spatial coordinates. Our metric convention is g µν = diag(−, +, +, +). For further analysis we assume negligible free charge density during inflation. Hence, we neglect the interaction term. To obtain the equation of motion of 4-potential A µ we vary the action with respect to the A µ . [f 2 F µν +F µν ] ;ν = 0 1 √ −g ∂ ∂x ν √ −gf 2 (φ) g µα g νβ F αβ + 1 √ −g η µναβ F αβ = 0(2) Varying the action with respect to scalar field we obtain the equation governing the evolution of the scalar field as 1 √ −g ∂ ∂x ν √ −gg µν ∂ µ φ − dV dφ = f 2 df dφ F µν F µν + F µνF µν(3) Here EM field is assumed to be a test field. Hence, it will not affect the geometry of spacetime. In the inflationary era the universe is dominated by scalar field φ which is a time dependent homogeneous field. We work in a frame work of homogeneous and isotropic universe, described by the FRW line element. ds 2 = −dt 2 + a 2 (t)[dx 2 + dy 2 + dz 2 ] = a 2 (η)[−dη 2 + dx 2 + dy 2 + dz 2 ](4) In this new coordinate system (η, x, y, z), we can define fundamental observers with four velocity ( 1 a , 0, 0, 0). It is convenient to work in Coulomb gauge, ∂ j A j = 0 , A 0 = 0 For µ = i, Eq. (2) is A ′′ i + 2 f ′ f (A ′ i + η ijk ∂ j A k ) − a 2 ∂ j ∂ j A i = 0 (5) Here prime(') denotes derivative with respect to η and ∂ j is defined as ∂ j ≡ g jk ∂ k = a −2 δ jk ∂ k . η ijk represents the three dimensional levi-civita symbol. To quantize the EM field, we calculate the conjugate momentum (Π i )of A i field, promote these fields to operators and impose the canonical quantization condition, Π i = δL δȦ i ,(6)[A i , Π j ] = i d 3 k 2π 3 e k·( x− y) P i j (k).(7) Here P i j = δ i j − δ jm (k i k m /k 2 ) is used to ensure the coulomb gauge condition. We fourier transform A i using the momentum space operators (b λ and b † λ ). A i ( x, η) = √ 4π d 3 k 2π 3 λ=1,2 ǫ i λ A λ (k, η)b λ ( k)e i k· x + A * λ (k, η)b † λ ( k)e −i k· x .(8) Here ǫ i λ represents the polarisation vector which can be written in terms of the 3-dimensional orthonormal unit vectors as ǫ i λ =ǫ i λ a , ǫ i 3 =k a(9) Hereǫ i λ are unit 3-vectors, orthogonal tok and each other. It is useful to define a new variableĀ λ as, A λ = aA λ (k, η). Substituting Eq.(8) in Eq. (5). λ b λ ǫ iλ Ā ′′ λ + 2 f ′ fĀ ′ λ + k 2Ā λ + 2f ′ f η ijmǫmλ k jĀλ = 0 (10) To simplify this equation further let's choose a different set of basis vector defined asǫ + = (ǫ 1 + iǫ 2 )/2 and ǫ − = (ǫ 1 − iǫ 2 )/2. In terms of these new basis vectors, λĀ λǫλ b λ =Ā +ǫ+ b + +Ā −ǫ− b − . This set of basis vectors are known as the helicity basis. Then Eq.(10) reduces toĀ ′′ h + 2 f ′ f Ā ′ h + hkĀ h + k 2Ā h = 0(11) Here h = ±1 represents the helicity sign. The equation of motion in terms of A h = fĀ h (k, η) turns out to be, A ′′ h (k, η) + k 2 − f ′′ f + 2hk f ′ f A h (k, η) = 0(12) Before we solve the above equation for a particular f (φ), it will be of interest to calculate the energy density of the EM field. To do this we calculate the energy momentum tensor of the EM field. T µν ≡ − 2 √ −g δ √ −gL δg µν = f 2 4π g αβ F µα F νβ − g µν F αβ F αβ 4(13) We define the electric and magnetic field four vectors as B µ ≡F µν u ν and E µ ≡ F µν u ν . For the observer u µ = (1/a, 0, 0, 0), the time component of these vectors are zero. The spatial components are given by B i = (1/a)η ijk δ jm δ kn ∂ m A n and E i = −(1/a)∂ η A i . Then the EM energy densities of ground state measured by the observer with 4-velocity u µ = (1/a, 0, 0, 0) are given by, ρ B = 0|T B µν u µ u ν |0 and ρ E = 0|T E µν u µ u ν |0 ,(14) where we have separated the total energy density into the magnetic part and the electric part. We express these parts in terms of A i as, ρ B = 0| f 2 8π ∂ i A n ∂ j A l g ij g nl − g il g nj |0 (15) = 0| f 2 8π B i B i |0 and ρ E = 0| f 2 8π A ′ i A ′ j g ij |0 (16) = 0| f 2 8π E i E i |0 After substituting A i from Eq.(8) into Eq.(16) and Eq. (17), and using the helicity basis, we reduce the energy densities in terms of A + and A − . ρ B ≡ dk k dρ B (k, η) d ln k = dk k 1 (2π) 2 k 5 a 4 |A + (k, η)| 2 + |A − (k, η)| 2 (17) ρ E ≡ dk k dρ E (k, η) d ln k = dk k f 2 (2π) 2 k 3 a 4 A + (k, η) f ′ 2 + A − (k, η) f ′ 2(18) In deriving the above expressions we have used the following properties, b h |0 = 0 0|b h ( k)b † h ′ ( k ′ )|0 = (2π) 3 δ hh ′ δ 3 ( k − k ′ ) In our analysis, we work with the spectral energy densities of magnetic and electric fields given by (dρ B (k, η)/d ln k) and (dρ E (k, η)/d ln k) respectively. These spectral densities represent the energy contained in logarithmic interval in k-space. We now turn to calculation of the magnetic and electric spectral energy densities for a particular type of coupling function. Let us assume the form of the coupling function to be a simple power law as, f 1 (a) = f i a a i α .(19) We assume that during inflation f (φ) has a form such that f evolves with a as given in (19). Here a i represents the value of scale factor at the beginning of inflation and α is a real constant. By assuming the background to be purely de-sitter during inflation (a ∝ η −1 ), we get the following f ′′ 1 f 1 = α(α + 1) η 2 .(20) For this form of coupling function Eq.(12) reduces to A ′′ h (k, η) + k 2 − α(α + 1) η 2 − 2hk α η A h (k, η) = 0. (21) To solve Eq.(21), we rewrite it by defining some new variables, µ 2 ≡ α(α + 1) + 1 4 , κ ≡ iαh, z ≡ 2ikη. Then the equation takes the form, ∂ 2 A h (k, η) ∂z 2 + 1 z 2 1 4 − µ 2 + κ z − 1 4 A h (k, η) = 0.(22) The solutions of this equation are Whittaker functions [48], A h = c 1 W κ,µ (z) + c 2 W −κ,µ (−z).(23) To determine the coefficients c 1 and c 2 , we have matched the solution with the bunch davies vacuum in the sub horizon limit | − kη| >> 1, A h = e −ikη √ 2k . After matching we get, A h = e iπκ 2 √ 2k W κ,µ (z) = e −hπα 2 √ 2k W iαh,α+ 1 2 (2ikη) At the end of the inflation, all the modes of cosmological interest will be outside the horizon. To get the spectral magnetic energy density of these modes, we need to write the above expression in the super horizon limit. A h = C h √ 2k (−kη) −α + h(−kη) −α+1 + D h √ 2k (−kη) 1+α − hα 1 + α (−kη) 2+α(24) Here C h = e −hπα 2 (−2i) −α Γ(1 + 2α) Γ(1 + α + ihα) , D h = e −hπα 2 (−2i) 1+α Γ(−1 − 2α) Γ(−α − ihα) .(25) Substituting Eq.(24) into Eq.(17) the spectral magnetic energy density comes out to be, dρ B d ln k = 1 8π 2 H 4 f |C + | 2 + |C − | 2 (−kη) −2α+4 + |D + | 2 + |D − | 2 (−kη) 2α+6 .(26) Similarly the spectral electric energy density is given by, dρ E d ln k = 1 8π 2 H 4 f |C + | 2 + |C − | 2 (−kη) −2α+4 + |D + | 2 + |D − | 2 (1 + 2α) 2 (−kη) 2α+4 . (27) For the expressions in Eq. (26) and Eq. (27), we have only kept the dominant terms. The reason of two branches in the above expressions is that α can be positive or negative. In the magnetic field spectrum the first branch dominates when α > 1/2 and the other branch dominates when α < 1/2. For the electric field spectrum the first branch dominates when α > −1/2 and the other branch dominates when α < −1/2. There are two possible value of α (namely, α = 2, −3) for which the magnetic field spectrum is scale invariant. For the case α = −3, when the magnetic field is scale invariant the electric field spectrum diverges as (−kη) −2 . This implies that the electric field energy density may overshoot the inflaton energy density during inflation in this case and our assumption of EM field being a test field would no longer be valid. This problem is known as the back-reaction problem [44]. However, for the case α = 2, both the magnetic and electric field spectrum are scale invariant, and this avoids the back reaction problem. For this case the coupling function f is proportional to a 2 , which means that f will be very large at the end of inflation compared to its initial value. If we assume that f becomes a constant at the end of inflation, then the effective EM charge e f = 1/f 2 . Suppose we demand that at the end of inflation e f should have the observed value, then it will be very large at the beginning of inflation. Due to this large value of e f , our perturbative analysis of field theory would no longer be valid. This problem is known as strong coupling problem [44]. On the other hand, if e f has the observed value at the beginning of inflation, it will have a very small value at the end of inflation, avoiding the strong coupling problem. This case is further explored in the next section. The branch α = 2 is also preferred, as we discussed in detail in Ref. [45], because it evades the constraints imposed by the possibility of increased conductivity due to the Schwinger effect [31]. The magnetic energy spectrum at the end of inflation for α = 2, dρ B d ln k ≈ 9 e 4π 320π 3 H 4 f(28) We note that this value is larger than the non-helical case by a factor of (e 4π /80π) ≈ 10 3 . III. EVOLUTION AFTER INFLATION As discussed in the last section, there are two possible scenarios for obtaining a scale invariant magnetic spectrum. The case in which f is increasing during inflation avoids the back-reaction problem. Moreover if we assume that f begins with a value of unity at the onset of inflation and increases during the inflationary phase, there will be no strong coupling problem. At end of inflation, however, the EM field will be very weakly coupled to the charged particles as f is much larger than unity. We address this issue in [45] by postulating that from the end of inflation onwards f decreases and attains a value of unity at reheating and remains unity thereafter. This ensures that EM action is again in the standard conformally invariant form after the reheating era. Thus the deviation from the standard form is onset of inflation to the reheating. We assume the universe to be matter dominated from the end of inflation to reheating. We consider α > 1/2 for further analysis. In this era the evolution of the scale factor is as follows, a = a 3 f H 2 f 4 η + 3 a f H f 2 and the coupling function f is assumed to evolve as f ∝ a a f −β Here a f is the value of scale factor at the end of inflation. We calculate the constant of proportionality by demanding the continuity of f at the end of inflation. f 2 = f i a f a i α a a f −β To estimate the EM energy densities in this era, we need to solve Eq.(21) for this new coupling function. Solution is given by, A 2h = d 1 M 2iβh,−(2β+ 1 2 ) (2ikζ) + d 2 M 2iβh,2β+ 1 2 (2ikζ) (29) Here ζ = η + 3/(a f H f ) and M 2iβh,−(2β+ 1 2 ) (2ikζ) represents the second kind of whittaker function [48]. To calculate d 1 and d 2 , we need the above expression in the super horizon limit. In this limit Eq.(29) becomes, A 2h = d 1 (2i) −2β (kζ) −2β − h(kζ) −2β+1 + d 2 (2i) 2β+1 (kζ) 2β+1 + 2hβ 1 + 2β (kζ) 2β+2 and, A 2h = A 2h f 2 = k H f −2β d 3 (1 − h(kζ)) + d 4 (kζ) 4β+1 + 2hβ 1 + 2β (kζ) 4β+2(30) Here d 3 and d 4 are two new constants. They can be expressed in terms of d 1 and d 2 . We demand that at the end of the inflation bothĀ h ,Ā 2h and their derivatives have to be matched. After matching we get, d 3 = C h √ 2k k H f −α+2β 1 + 3h k a f H f and d 4 = C h √ 2k k H f −α+2β 3h 2 2(4β + 1) 2k a f H f −4β+1 1 − 2hβ 1 + 2β 4β + 2 4β + 1 2k a f H f −1 In d 3 and d 4 expressions, we only take the contribution of dominant terms. Energy densities after inflation evolve as dρ B d ln k = C 1 8π 2 k 4 a 4 f 2 2 (a) k H f −2α 1 + 9 4(4β + 1) 2 2k a f H f −8β+2 2k aH 8β+2 + 3h 2 (4β + 1) 2k a f H f −4β+1 2k aH 4β+1 dρ E d ln k = 1 8π 2 k 4 a 4 f 2 2 (a) k H f −2α C 1 + 9 4 C 1 2k a f H f −8β+2 2k aH 8β + 3C 2 2k a f H f −4β+1 2k aH 4β Here C 1 = |C + | 2 + |C − | 2 and C 2 = |C + | 2 − |C − | 2 . At the end of inflation, the first term inside the bracket in the expressions of dρ B /d ln k dominates for all the modes outside the horizon and gives a scale invariant magnetic field spectrum for α = 2. As f decreases post inflation, this term also decreases and becomes very small at reheating. Although the second and third term are very small compared to the first term for the mode k i = a i H f at the end of inflation, the contribution from these terms compared to the first term increases as f decreases post inflation. Consequently, the second and third term overshoot the first term before reheating. The second term is 36/((4β + 1) 2 a 4 f ) × (a r /a f ) 4β+1 times larger than the first term and 6/((4β +1)a 2 f )×(a r /a f ) 2β+1/2 times larger than the third term at reheating for the mode which exit the horizon at the beginning of inflation (k i = a i H f ). It is even larger for all other modes of interest. After taking the contribution of the dominant term at reheating in the above expressions, we get the following expression for EM energy densities at reheating, dρ B d ln k r = 9 32π 2 k 4 a 4 r f 2 2 (a r ) k H f −2α C 1 1 (4β + 1) 2 2k a f H f −8β+2 2k a r H r 8β+2 (31) dρ E d ln k r = 9 32π 2 k 4 a 4 r f 2 2 (a r ) k H f −2α C 1 2k a f H f −8β+2 2k a r H r 8β(32) Here a r and H r are the scale factor and Hubble parameter at reheating respectively. Fig. (1) shows the evolution of EM and the inflaton energy densities with the scale factor, both during and after inflation. From Fig.(1) one can see that EM energy densities are increasing after inflation. It is necessary that EM energy density does not overshoot the energy density of the universe before the coupling function f reaches to its pre-inflationary value. Since EM energy density has a monotonically increasing behaviour, if ρ E + ρ B < ρ φ is satisfied at reheating, it will be valid throughout the post inflationary era prior to reheating. The total EM energy density at reheating is, After substituting k r = a r H r = a f H f e −Nr/2 in Eq. (33) and using the definition of N and N r , we get ρ E + ρ B r = kr aiH f d ln k dρ E (k, η) d ln k r + dρ B (k, η) d ln k r = kr aiH f d ln k 9 32π 2 k 4 a 4 r f 2 2 (a r )C 1 k H f −2α 2k a f H f −8β+2 2k a r H r 8β 1 + 1 (4β + 1) 2 2k a r H r 2 = 9 32π 2 f 2 2 (a r ) a 4 r C 1 k r H f −2α 2k r a f H f −8β+2ρ E + ρ B r = C 3 H 4 f e α(2N +Nr)−7Nr(34) Here C 3 = 9C 1 8π 2 1 6 − 2α + 4 (8 − 2α)(4β + 1) 2 . In the above expression, we also use β = αN/N r which is obtained by demanding f (a r ) = 1. In order that ρ E + ρ B | r < ρ φ | r , we require 2α(N + N r ) − (7 + α)N r < ln π 2 g r 30C 3 − 4 ln H f T r .(35) Here we use ρ φ | r = g r (π 2 /30)T 4 r where T r and g r represent reheating temperature and relativistic degree of freedom, respectively, at reheating. We have several variables in the above expression but they are all not independent. To reduce the expression in terms of the minimum number of variables (independent variables), we use the following constraint. From the isotropy of the cosmic microwave background radiation, we find the following relation. N + N r > 66.9 − ln T r H f − 1 3 ln g r g 0 .(36) Here, g 0 is the relativistic degree of freedom in the universe at present. The above relation has been derived from the fact that the present observable universe has to be inside the Hubble radius at the beginning of inflation. In the above expression, we also assume a radiation dominated era from reheating till today. By the assumption of matter dominance post inflation to reheating, N r can be written in terms of H f and T r . N r = 1 3 ln ρ φ | inf ρ φ | r = 1 3 ln 90H 2 f 8πGπ 2 g r T 4 r(37)+ 134α + (2α + 4) ln H f T r − 4(7 + α) 3 ln   4 3H 2 f 8πG 1 T r   < 0.(38) If reheating temperature and the scale of inflation satisfy the above bound, there will not be any back-reaction and strong coupling problem in our prescribed model till reheating. For further analysis, we assume a particular value of α and calculate the possible inflationary scales (H f ) for different reheating scales (T r ) using the bound in Eq. (38). For these T r and H f , we calculate N and N r . We have also calculated correlation length of magnetic field and its strength at this scale at reheating using the following expressions, L c = a r kr 0 2π k dρB (k,η) d ln k d ln k kr 0 dρB (k,η) d ln k d ln k B[L c ] = 8π dρ B (k, η) d ln k k= 2πar Lc .(39) In Fig.(2), we plot the evolution of the |A h |/a 2 , (which appears in Eq. (17) for ρ B ), with scale factor for different helcity modes from the beginning of inflation to the epoch of reheating. The black solid curve and red dotted curve shows the evolution of the negative helicity and positive helicity mode, respectively. Blue dot-dashed curve shows the evolution of |A h |/a 2 if the parity breaking term is not present in the action (non-helical case) and for future purpose we name this mode as zero helicty mode. If f had been a constant equal to 1, red solid curve would have represent the evolution. Black dotted curve shows the |A h |/a 2 for Bunch Davies vacuum. In the left panel, first vertical line is for the epoch of horizon crossing during inflation and second vertical line is for the end of inflation. It is evident from the figure that negative helicity mode has larger value than the zero helicty mode and the positive helicity mode. This means that in the helical case, magnetic energy density is larger compared to the non-helical case at the end of inflation and it is almost fully helical because the strength of positive helicity mode is negligible compared to negative helicity mode. It is also evident from this panel that without the coupling IV. EVOLUTION OF MAGNETIC FIELD AFTER REHEATING To determine the magnetic field strength and its correlation length at present, we need to evolve the magnetic field from the epoch of reheating to today. As we have seen in the last section, our generated magnetic field has a blue k 4 spectra on super horizon scales at the time of generation. After reheating, the universe is dominated by radiation and in radiation dominance Hubble radius increases faster than the wavelength of a mode. Due to this modes start to re-enter the horizon. As the Alfvén crossing time for a mode becomes smaller than the comoving Hubble time, non-linear effects due to the magnetic field coupling to the plasma come into picture. If we consider only the flux frozen evolution of the magnetic field B ∝ 1/a 2 , then magnetic field strength and its correlation length at present are given by the following expressions. L c0 =L c a 0 a r , B 0 [L c0 ] =B[L c ] a 0 a r −2 .(40) However if we incorporate the non-linear effects and the consequent turbulent decay of the magnetic field, its strength and correlation scale have different scaling behaviour. Because of magnetic helicity conservation, inverse cascade takes place and magnetic energy transfers from smaller length scales to larger scales. This phenomenon has been discussed in [42,49,50] and also confirmed by the numerical simulations [49][50][51][52][53][54]. After using the results discussed in [42,49,50], we get the following scaling laws for the correlation scale L N L c0 and the strength of the field at this scale B N L 0 [L N L c0 ]. L N L c0 =L c0 a m a r 2/3 , B N L 0 [L N L c0 ] =B 0 [L c0 ] a m a r −1/3 .(41) Here a m represents the scale factor at the matter radiation equility. As discussed in [45] there is no significant change in the comoving coherence length and field strength in the matter dominated era after a m . Thus Eq. (41) gives reasonable estimates of the present day comoving field strength and corerealtion length. To estimate the maximum possible value of the magnetic field at different reheating scale, firstly we take the lowest possible scale of reheating (5MeV) allowed by the Big Bang Nucleosynthesis bound [55]. We have also considered reheating scales around QCD phase transition, Electro-Weak phase transition and at 1000 GeV. For each of these reheating temperatures we calculate the bound on the scale of inflation using Eq. (35). Further we calculate the magnetic field strength and its correlation length both assuming frozen field evolution Eq.(40) and with turbulent decay using Eq. (41). The results are given in Table I. The bound obtained in Eq. (38) suggests that as we increase the reheating scale (T r ), inflationary scale (H f ) decreases. Since reheating occurs after end of inflation, the above behaviour suggests that the highest possible reheating scale is ≈ 4000 GeV for α = 2. We consider several reheating scales below this highest possible reheating scale. If we consider reheating at 1000 GeV, the maximum possible magnetic field strength is 7.1 × 10 −12 G and its correlation length is 0.03 Mpc. We also calculate magnetic field strength and its correlation length for T r = 100 GeV, T r = 150 MeV and T r = 5 MeV and the results are shown in Table I. It is evident from the table that as we decrease the reheating temperature, the maximum possible magnetic field strength as well as its correlation length increases. Specifically, we have B N L 0 ∼ 3.9 × 10 −11 G, ∼ 9.9 × 10 −10 G, 6.4 × 10 −9 G for respectively T r = 100 GeV, T r = 150 MeV and T r = 5 MeV and L N L c0 = 0.07 Mpc, 0.6 Mpc and 1.6 Mpc respectively for the same reheating temperatures. In the above estimates, we have assumed that the EM energy density reaches a value equal to the energy density in the inflaton field at reheating. Suppose (ρ E + ρ B )| r = ǫρ φ | r , then for a particular T r , the above estimated magnetic field strength will be decreased by a factor of √ ǫ but the correlation scale will remain the same. In this case, the maximum allowed value of reheating temperature is also decreased by a modest amount. For example with ǫ = 10 −4 , the maximum T r becomes ≈ 1300 GeV. V. γ-RAY CONSTRAINTS We now consider the constraints from the γ-ray observation. Non detection of GeV photons in Blazars observation by the Fermi telescope puts a lower bound on the magnetic field strength [5,6]. Emitted photons of TeV energy from blazars interacts with the Extragalactic background light and gives rise to pair production. These produced particles interacts with the Cosmic Microwave Background and generate the GeV energy photons by inverse compton scattering. If a magnetic field is present in the intergalactic medium, it can affect the trajectories of these charge particle and observed surface brightness of GeV emission due to inverse compton scattering can be suppressed. Further, If we consider that size of secondary emitting region is larger than the point spread function of the telescope then this gives a lower bound of B ∼ 10 −15 G strength at the coherence scale ≥ 1 Mpc. If the flux suppression mechanism is due to the time delay of secondary emission then one gets a lower bound of B ∼ 10 −17 G at the same coherence scale. Below 1 Mpc as the coherence length decreases the lower bound on the magnetic field strength increases as L −1/2 c in both the cases. Here L c is the comoving correlation length of the magnetic field. In Fig.(3), we have plotted the maximum possible magnetic field strength generated in our model for different reheating scales as a dashed red curve and the two bounds on the magnetic field strength obtained from the nondetection of γ-rays from blazars as respectively a solid black curve and as a dotdashed blue curve for the two different mechanism, respectively. In the left panel we have only considered the flux frozen evolution of the magnetic field after generation and estimated magnetic field strength and its correlation length. In right panel we have estimated the magnetic field strength and its correlation length by incorporating the non-linear evolution. It is evident from the figure that the generated magnetic field strength satisfies the γ-ray observation for all possible reheating scales in our model. The shaded region in the figure represents the allowed values of magnetic fields from γ-ray observation. In Fig.(3), we have shown the constraints only for a certain range of the reheating temperature. BBN gives us a lower bound of 5 MeV for the scale of reheating. The reason for the upper bound is discussed in the previous section. For α = 2 case, it is at 4000 GeV. We see from Fig.(3) that all our allowed models with T r < 4000 GeV, lead to magnetic fields well above the lower bound required by the γ-ray observations. ) that can be generated by taking the nonlinear evolution of helical magnetic fields. The shaded region in both the figures represents all the allowed magnetic field strengths from γ-ray constraints. VI. CONCLUSION We have studied here the generation of helical magnetic field during inflation. Generation of magnetic field within standard physics during inflation is not possible because of the conformal invariance of EM field. To generate magnetic field during inflation, we have adopted the Ratra model in which a coupling between EM field and inflaton field is assumed. We have added a parity violating term with the same coupling in our action to generate a helical magnetic field. However this model has the well known strong coupling and back-reaction problems during inflation. We have described these problems and attempted to resolve them by adopting a particular behaviour of coupling function f . In our model f starts with a value of unity and increases during inflation so that there is no strong coupling and back-reaction problems. After inflation and before reheating it decreases such that it attains the pre-inflationary value at reheating to match with the observed coupling constant between charged fields and EM field. By demanding that there is no back-reaction of the generated fields post inflation, we get a bound on inflationary and reheating scales. For this type of evolution, the magnetic field spectrum at reheating is blue and can not be shallower than dρ b /d ln k ∝ k 4 spectra. This spectra is obtained when f ∝ a 2 during inflation. We have discussed this case in detail and estimate the magnetic field energy density and its correlation length for different reheating scales. If reheating happens at 100 GeV then the comoving magnetic field strength is 3.4×10 −7 G and its correlation length is 8.8×10 −10 Mpc if we only consider the flux frozen evolution. Magnetic field strength and correlation length change to 3.9×10 −11 G and 0.07 Mpc if we incorporate the non-linear evolution whereby the helical field decays due to the generated magnetohydrodynamics turbulence, conserving magnetic helicity. The generated magnetic field is almost fully helical. The generated magnetic field strength at the end of inflation is larger compared to non-helical case considered in Ref. [45]. Moreover the maximum possible reheating scale in the helical case is ≈ 4 × 10 3 GeV which was ≈ 10 4 GeV in the non-helcial case. We have also shown that the generated magnetic field in our model satisfies the γ-ray constraints for all the allowed reheating scales. The behaviour of coupling function which we have adopted, could be obtained in hybrid inflationary scenarios [56]. In hybrid inflation, one has two scalar fields with one dominating during inflation and providing the necessary condition for inflation and other field ends the inflation. We can consider our coupling function as a function of both the fields such that when first field rolls down the potential during inflation, f increases and f decreases when other field evolves. To summarize, we have suggested a viable scenario for inflationary generation of helical magnetic fields, which does not suffer from the back reaction or strong coupling problems. In our model the generated field is almost fully helical. As we increase the reheating scale both the magentic field strength and its correlation length decrease. However they satisfy the γ-ray constraints for all the allowed values of reheating scales. The generated magnetic field strength and its correlation length are larger for the helical case compared to the non-helical case. For a re-heating scale at 100 GeV the magnetic field strength and its correlation length were 6.8 × 10 −13 G and 7.3 × 10 −4 Mpc for the non-helical case [45] but 3.9 × 10 −11 G and 0.07 Mpc for the helical case. Cosmic Microwave Background and structure formation have mainly focused on nearly scale invariant spectra [39,42,[57][58][59][60][61][62][63][64]. It would be of interest to revisit these effects for the blue spectra (see for example [65]) predicted by our consistent models of inflationary magnetogenesis. ACKNOWLEDGMENTS RS and TRS acknowledge the facilities at IUCAA Resource Center, University of Delhi as well as the hospitality and resources provided by IUCAA, Pune where part of this work has been done. RS acknowledges CSIR, India for the financial support through grant 09/045(1343)/2014-EMR-I. TRS acknowledges SERB for the project grant EMR/2016/002286. FIG. 1 : 1analysis we have used two new variable de-fined as, In this figure we have taken α = 2 and T r = 100 GeV. It shows the evolution of ρ φ , ρ E and ρ B with scale factor. First vertical bold black lines is for the value of a f and second is for the value of a r . This figure shows that the energy of EM field does not overshoot the energy of the scalar field φ which decides the background geometry if the scale of inflation and reheating satisfies the bound in Eq.(38). FIG. 2 : 2In this figure we have plotted the |A h |/a 2 vs scale factor (a). Here we have assumed α = 2, T r = 100GeV and k = 10 5 H f . In the left panel, we have shown that how vector potential evolves for positive helicity, negative helicity, zero helicity and for the case of constant coupling (f = 1) during inflation. In the second plot, we have shown the evolution of the same modes post inflation to reheating.between inflaton and EM field, the strength of the magnetic field is very small for the modes which have crossed the horizon much earlier than end of the inflation. The right panel ofFig.(2) also follows the same colour coding and it shows the post inflationary evolution till reheating. In this panel, first vertical line is for epoch of end of inflation and the second vertical line is for the epoch of reheating. This panel shows that the strength of the mode decreases post inflation. Subsequently there is a transition and the mode starts to increase till reheating. The reason for the transition is as follows. The branch which dominated during inflation leads to both an initially dominant decaying mode and a subdominant growing mode after the transition to the matter dominated era post inflation. The initially dominant mode decreases as f decreases, while the initially subdominant one increases with time. In further evolution, naturally there is a point where both the branch crosses each other, this point is the transition point in the right panel. Subsequent growth of the field continues till the reheating epoch (indicated by the second vertical line). Finally on reheating the electric field gets damped by the increased plasma conductivity. The magnetic field evolves further as discussed below. FIG. 3 : 3We have assumed α = 2 in plotting these figures. The black curve and dotdashed blue curve in both the figures correspond to the lower bound on observed magnetic field strength constrained by the gamma ray observations for two different mechanism. These bounds are estimated at the correlation length of the generated magnetic field. The red dashed curve in the left panel represents the maximum magnetic field strength (B 0 [L c0 ]) that can be generated in our model by taking flux freezing evolution. 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[]
[ "A Grammar of Hypotheses for Visualization, Data, and Analysis", "A Grammar of Hypotheses for Visualization, Data, and Analysis" ]
[ "Ashley Suh ", "Ab Mosca ", "Eugene Wu ", "Remco Chang H D H V H A " ]
[]
[]
We present a grammar for expressing hypotheses in visual data analysis to formalize the previously abstract notion of "analysis tasks." Through the lens of our grammar, we lay the groundwork for how a user's data analysis questions can be operationalized and automated as a set of hypotheses (a hypothesis space). We demonstrate that our grammar-based approach for analysis tasks can provide a systematic method towards unifying three disparate spaces in visualization research: the hypotheses a dataset can express (a data hypothesis space), the hypotheses a user would like to refine or verify through analysis (an analysis hypothesis space), and the hypotheses a visualization design is capable of supporting (a visualization hypothesis space). We illustrate how the formalization of these three spaces can inform future research in visualization evaluation, knowledge elicitation, analytic provenance, and visualization recommendation by using a shared language for hypotheses. Finally, we compare our proposed grammar-based approach with existing visual analysis models and discuss the potential of a new hypothesis-driven theory of visual analytics.
null
[ "https://export.arxiv.org/pdf/2204.14267v2.pdf" ]
257,921,939
2204.14267
05b4010981e21b4bde2c6ed95881b638202c485a
A Grammar of Hypotheses for Visualization, Data, and Analysis Ashley Suh Ab Mosca Eugene Wu Remco Chang H D H V H A A Grammar of Hypotheses for Visualization, Data, and Analysis All of the user's hypotheses can be answered from the data & visualization H H D H V H A Some of the user's hypotheses can be answered from the data & visualization H H D H V H A None of the user's hypotheses can be answered from the data & visualization H H D H V H H A H V H H V H A H H A H V H H A H D H Inappropriate Visualization We present a grammar for expressing hypotheses in visual data analysis to formalize the previously abstract notion of "analysis tasks." Through the lens of our grammar, we lay the groundwork for how a user's data analysis questions can be operationalized and automated as a set of hypotheses (a hypothesis space). We demonstrate that our grammar-based approach for analysis tasks can provide a systematic method towards unifying three disparate spaces in visualization research: the hypotheses a dataset can express (a data hypothesis space), the hypotheses a user would like to refine or verify through analysis (an analysis hypothesis space), and the hypotheses a visualization design is capable of supporting (a visualization hypothesis space). We illustrate how the formalization of these three spaces can inform future research in visualization evaluation, knowledge elicitation, analytic provenance, and visualization recommendation by using a shared language for hypotheses. Finally, we compare our proposed grammar-based approach with existing visual analysis models and discuss the potential of a new hypothesis-driven theory of visual analytics. The relationships between three hypothesis spaces, formalized using our proposed hypothesis grammar. A visualization hypothesis space (H V ) expresses the set of hypotheses a visualization can evaluate, a data hypothesis space (H D ) expresses the set of hypotheses a dataset can evaluate, and an analysis hypothesis space (H A ) expresses the set of hypotheses a user wants to evaluate through analysis. The intersection of these spaces can help us understand how the hypotheses a user would like to answer through analysis can either be fully answered by data and visualization, partially answered, or not answerable at all. INTRODUCTION Visual analytics has long been hallmarked for its ability to support users through complex analysis tasks, such as training and debugging machine learning models [72], hypothesis testing [41], knowledge generation [73], and decision-making [24]. Despite this ubiquity, many contributed visual analysis systems are mechanistically agnostic to the user's analysis tasks, and ultimately their goals. The challenge of defining and operationalizing "tasks" in visual analysis has been an open problem for more than 40 years, with the first mention of the problem dating back to J. Mackinlay 1986 [54]. Since then, numerous papers have attempted to address this challenge [3,7,13,23,26,38,52,58,75,78,79]. Despite these efforts, the definition of "task" in visual analysis remains abstract. Originally used to describe data retrieval operations such as Lookup Value, or Compare Values, tasks have grown to include statistical and machine learning operations [34] (e.g., identify clusters, discover outliers) and social interactions (e.g., Present Results or Enjoy a Visualization [7]). While this richness allows us to describe the plethora of ways in which tasks proliferate a user's interactions with visualization, these broad and diverse definitions of tasks make it increasingly difficult to operationalize them for visual analytics. An operationalizable definition of "tasks" in visualization can have a profound impact on visualization research. In addition to its application to automatic visualization recommendation [54], a composable definition of tasks can lead to potential breakthroughs in visualization design, evaluation, analysis provenance, natural language analytic tools, as well as knowledge and belief elicitation,. Conversely, without this definition, our community will continue to see a disconnect between empirical research on visualization tasks, and the practical implementation of that research in visual analytic tools. In this paper, we propose a hypothesis grammar that builds a basis for operationalizing tasks in visual data analysis. Our grammar is grounded in the concept of scientific hypotheses, which offer a practical and welldefined starting point for concretizing the relationship between analysis tasks, data, and visualization. Through this formalism, we are able to describe and analyze three spaces (or sets) that can be expressed with our grammar: the hypotheses a dataset can evaluate (a data hypothesis space), the hypotheses a user would like to evaluate through analysis (an analysis hypothesis space), and the hypotheses a visualization design can evaluate (a visualization hypothesis space). The development and exploration of the hypothesis grammar takes a step towards making tasks operational, allowing us to unify three areas (data, analysis tasks, visual encodings) that are often researched in isolation. The unification of these spaces enables a new methodology for the visualization community to reason about concepts that had not been possible or easy to do previously, including a hypothesis-driven approach to analysis. For example, during analysis, a user could express their questions using our grammar, then examine whether these questions are answerable with the available data and the visualization tool. Concepts such as "exploration," "insight," and "knowledge verification" can be captured through the relationship and intersection of these hypothesis spaces (e.g., gaining new insights might "shrink" the analysis hypothesis space by "removing" hypotheses that are irrelevant to the user). Visualizations can be formalized, not by their data encodings, but in terms of the hypotheses that the visualizations can be used to verify, thereby enabling the possibility of a novel hypothesis-based theory of visualization. In the following sections, we elaborate on the merits of modeling analysis tasks using a grammar-based approach for scientific hypotheses. We then introduce our proposed grammar, and demonstrate how visualization experts can use it to reason about the interconnection of data, analysis tasks, and visual encodings. We discuss the formalization of three hypothesis spaces generated with our grammar, and how they can spur new directions in visualization research. Finally, we provide a demonstration of our grammar through an analysis case study, and conclude with a direct comparison to existing work in this area. To summarize, we make the following contributions in this paper: • We introduce a Grammar of Hypotheses to operationalize analysis tasks with respect to a user's questions, data, and visualization. • We use the grammar to formalize three "hypothesis spaces": an analysis hypothesis space, a data hypothesis space, and a visualization hypothesis space. • We present case studies to demonstrate how the unification of these three spaces can inform future visualization research. • We provide a comparison to existing visual analysis theories to illustrate how our hypothesis grammar augments current practices, suggesting a new exploratory hypothesis analysis methodology. BACKGROUND & MOTIVATION The concept of a "hypothesis" has varying implications depending on the context and community in which it is used. In this section, we provide a brief background on the use of hypotheses and grammars in the context of the visualization and science communities. From this background, we motivate why a grammar-based approach -specifically, a grammar of hypotheses -is an advantageous method to begin operationalizing visual analysis tasks. Hypotheses Scientific Hypotheses In the science community, the precise criteria for a scientific hypothesis has been well-studied [71,83] and is rigorously defined as a proposition (or set of propositions) proposed as a tentative explanation for an observed situation or phenomenon [19,30]. A valid scientific hypothesis must be both testable and falsifiable [70]. Researchers have proposed a number of requirements for evaluating the quality of a scientific hypothesis [63,87]. For example, work by Quinn and George [71] states that a "good" hypothesis must have the following properties: • it makes sense; • it is empirical, a (partial) scientific relation; • it is adequate, a scientific relation between at least two variables; • it is precise-a qualified and/or quantified relation; • it states a test, an explicit statement of a test. Notably, research in this area distinguishes between scientific hypotheses and statistical hypotheses [39,85]. Scholars have argued that the two are distinct due to statistical hypotheses' use of confidence thresholds (e.g., p-values) and the reliance on null and alternate hypotheses -both of which are inappropriate for scientific hypotheses [6]. In this work, we base our grammar on the definition and criteria for scientific hypotheses. This provides (1) a formal definition on which a grammar can be based, and (2) structure for thinking about hypothesis generation and refinement in visual data analysis. Significance of Hypotheses in Visual Analytics A central goal of visual analytics is supporting users through forming, refining, and validating hypotheses [40,41]. In the Sensemaking process [69], the iterative refinement of hypotheses is an integral step before the presentation of analysis findings. In the Knowledge Generation Model [73], a hypothesis is generated from knowledge and insight, and is the driving motivation behind a user's (inter)actions with a visual analytics system. Pike et al. [68] write, "[in] constructing hypotheses or 'abductions', the analyst is engaged in exploration of the data space and the formation of mental models to explain observations." These theoretical models reflect how the visual analytics community considers the concept of hypotheses integral to effective design. However, while many visual analytic models discuss the importance of supporting hypothesis generation and verification, there are few operational means of both supporting and assessing hypotheses in visual data analysis. The grammar proposed in this paper allows us to combine the ethos of visual analytics hypotheses with the rigor and precision of scientific hypotheses, resulting in a more concrete way to model and reason about sensemaking and visual analysis. We discuss the relationship of our grammar with existing theories in Section 8. Grammars A grammar can disambiguate the subjective view of a user's task, allowing for new designs of visualization systems to reason and operate over both data and the user's analysis goals. We discuss how grammars have been used with success in the science and visualization communities. Use of Grammars in Science Grammars for representing the syntax of hypotheses have been used previously for science education [31,63]. A large focus is on building tools (on top of a grammar) to help students build expressions from variables (i.e. data attributes) and relations (e.g., 'increases' or 'positively correlated with') to create a hypothesis that meets the criteria discussed in Section 2.1.1 (e.g., if A increases, then B increases). For example, Kroeze et al. [51] use a context-free grammar (CFG) to automatically parse and give (semantic) feedback on a student's hypotheses by using the parsing tree. A subset of this CFG (in the domain context of electrical circuits) is: HYPOTHESIS -> if ACTION then ACTION ACTION -> VAR MODIFIER VAR -> voltage | brightness MODIFIER -> increases | decreases An example hypothesis parsable from this CFG includes "if voltage increases, then brightness increases." In the development of our hypothesis grammar, we drew inspiration from many works in science education [22,87]. We present our own version of a hypothesis grammar, but applied more broadly to analysis tasks within the context of visualization and visual analysis, in Section 3. Grammars in Visualization & Visual Analytics A core theory in visualization today is the concept of a graphical grammar, which decomposes a data visualization into orthogonal components that are easy to learn and automatically analyze, implement, and reason about. The use of grammars in visualization was popularized by the development of the Grammar of Graphics by Leland Wilkinson [90] in 1999. The Grammar of Graphics provides a set of rules such that a visualization is constructed by mapping data attributes to visual properties, e.g., position, size, color, and shape. The Grammar of Graphics has been used as the basis for several popular visualization tools, such as R's ggplot2 [88], and has inspired the development of many other visualization grammars [53,57,67,77,91]. The use of grammars has enabled the development of powerful and flexible visualization tools that can be easily customized to meet the needs of diverse users and applications. Grammars have also provided a new means to reason about visualization design and theory. By building on top of the Grammar of Graphics, Vega-Lite [76] elevated interaction as a first-class citizen to design, allowing the specification of visualizations and dashboards to include interactivity. Systems like Voyager 2 [94] and Draco [62] similarly leverage the flexibility and robustness of grammars to enhance visualization recommendation. Grammars have helped the visualization research community take great strides in supporting users through the construction of custom visualizations with concise, intuitive, structured, and consistent languages. They have also facilitated new theories to reason over visual encodings and interaction techniques. However, we observe that visualization grammars used today are still agnostic to a user's analysis task -despite the fundamental principle that supporting a user's task is essential to good visualization design [79]. Moreover, understanding which visualizations excel at answering questions beyond simple analytic tasks [3,74], or evaluating hypotheses (and to what capacity) is difficult to reason about. In Section 5, we illustrate how a hypothesis grammar can help bridge these gaps. Why a Grammar of Hypotheses? There is a plethora of domain-specific languages related to data manipulation (and data visualization), ranging from database query languages such as SQL [12] and DataLog [1], to recent business intelligence languages like Malloy [56]. These languages are expressive and practically useful. However, from the perspective of visualization research, their expressivity and generalizability make them difficult to critically think through the relationships between data, tasks, and visualization. Our proposed hypothesis grammar is designed with the opposite goals in mind. We do not intend for the grammar to be implemented as a practical programming language at this moment. Instead, it is deliberately made narrow to reason about visualization, visual data analysis, and operational tasks. We consider the simplicity of the grammar a desirable property, as it provides a minimum viable product to illustrate how we can resolve a subset of complex problems that visualization researchers have been grappling with for decades [54]. It is relevant to note that the design of our proposed hypothesis grammar is inspired by SQL. In fact, the hypothesis grammar can be seen as a very limited subset of SQL where every statement in the grammar can be trivially converted to and evaluated with SQL. As such, all examples shown in this paper can be executed by a relational database system with minimum modification. We discuss these implications, and considerations for other languages that could be used, in Section 9. HYPOTHESIS GRAMMAR Before discussing the implications of a grammar-based approach for tasks, we first need to establish a baseline with our hypothesis grammar. We start with an illustrative example, showing how the hypotheses discussed in Section 2 can be formed from a simple tabular dataset and toy grammar. We then walk through a generalizable hypothesis grammar that is capable of expressing hypotheses from tabular datasets. We present this grammar in the most simple representations as possible, and leave nuances and complexities for supplemental material. Illustrative Hypothesis Grammar Before introducing the full hypothesis grammar, we begin with small, illustrative examples. The complete grammar is presented in Section 3.2. Definition Our simple grammar is based on the definition of a hypothesis, specifically, that a hypothesis is "a qualifiable relation between at least two variables." The goal is to design a grammar that can express relations such as: average(sales)>average(cost) or average(cost)>10. Based on this definition, we define a hypothesis as: In this grammar, a hypothesis (hyp) is defined with expressions (expr). An expression can be a data attribute (attr) such as sales, a constant (in this case, a number), or a function (func) over another expression. Data attributes (attr) are currently unresolved in this grammar, which we will resolve later in Section 4. Since the evaluation of a hypothesis results in a binary true or false, the operator (op) is limited to binary relations (such as >, <, =, etc.). The list of functions for a hypothesis grammar needs to be preregistered, similar to registering a user-defined function in a SQL database. For simplicity, we assume that the list of functions includes the typical aggregation (such as AVG, SUM, MIN, etc.) and analytic functions (such as CORR for correlation, STDDEV for standard deviation, etc.) that are commonly supported by SQL databases. Lastly, we introduce the notion of a predicate (pred), which functions similarly to a WHERE clause in SQL queries to filter data. For example, a predicate can express [year=2023] to filter data by the year 2023. Examples We provide examples of the use of this grammar to express hypotheses: • "Is my company profitable?" can translate to the hypothesis "the total amount of sales is higher than cost and expenditure." This is formulated as: SUM(sales)> ADD(SUM(cost), SUM(expenditure)). • "If the temperature of an object goes up, its volume will go up" is a scientific hypothesis. It can be expressed as "there is a positive correlation between temperature and volume." Let CORR compute the correlation between two variables. We can reformulate as: CORR(temperature, volume)>0.9. • "Car horsepower (hp) since 2000 follows a normal distribution." Let KS-NORMAL be the Kolmogorov Smirnov test with alpha value 0.05 to evaluate if a variable follows a normal distribution. This hypothesis can be formulated as: KS-NORMAL(hp)>0.05 [year>2000] Beyond these examples, we also demonstrate in Section 6 how the hypothesis grammar can be applied to analyze the VAST 2017 Challenge Mini-Challenge 1 (the Boonsong Kekagul Nature Reserve challenge). Complete Hypothesis Grammar Our complete hypothesis grammar (in PEG notation [29]) is shown in Table 1. Other notations could be used, we discuss in Section 9. The complete grammar (which we refer to as our base grammar) is similar to the illustrative grammar, but with four minor changes: A statement, which evaluates to either true or false, describing the relationship between at least one expression and one variable expr func "(" (expr ("," expr)? )? ")" | var A nullary, unary, binary expression, or variable. A condition expression that evaluates to true or false for a given var func str • (hyp & The name of a function with explicit input and output types from a predefined library or table. op = | < | > | ... A boolean operation. attr str An attribute reference. Grounded to attributes in dataset. const str | number | .. | array Constants are specified as part of the hypothesis, and may be any scalar, or array that isn't derived from the database. Table 1: Breaking down each term in our proposed hypothesis grammar (referred as the Base Grammar), as described in Section 3. • (func(expr, expr,...)): The number of parameters to a function is now unlimited. This allows for the following expression func(expr 1 , expr 2 , expr 3 ,...) where each expr i is a parameter of to the function. We note that the immediate, widespread implementation of the grammar was not the primary focus when we designed our hypothesis grammar. Instead, we use the hypothesis grammar as a way to illustrate how a grammar based on hypotheses can be used to reason about properties and concepts in visualization research. In the section below, we describe how the grammar can formalize and unify the concepts of data, analysis tasks, and visualization. HYPOTHESIS SPACES We first observe that the hypothesis grammar can be used as a set of production rules to generate valid hypotheses -that is, statements that are acceptable (parsable) by the hypothesis grammar. Conversely, we can consider all possible hypothesis statements that are acceptable by the hypothesis grammar as a set, which we refer to as a hypothesis space. We use this notion to describe the hypotheses spaces of a dataset (a data hypothesis space), a visualization (a visualization hypothesis space), and an analyst's analysis tasks (an analysis hypothesis space). We illustrate each of these spaces using PEG notation, while omitting the quotations for simplicity. Table 1 shows all full marks. Full Hypothesis Space (H) The full (infinite) hypothesis space is denoted as H and represents all possible hypotheses that can be generated. Formally, it is defined as all statements that can be parsed by the base grammar (see Table 1). Data Hypothesis Space (H D ) H D A data hypothesis space is denoted as H D . The space is defined by 'grounding' the base grammar to a dataset D. Informally, we say that H D describes all hypotheses that can be evaluated or verified using the dataset D, therefore, H D is a strict subset of H. To illustrate this, consider a simple dataset (shown to the right) with two numerical attributes, cost and price. As shown below, we can extend the base grammar by grounding it to this data. The highlighted text in red represents changes to the original grammar in Section 3.2. Although the change seems minimal, the implication of grounding the data to the grammar is significant. Assuming a finite set of operators and functions, this grammar describes all possible hypotheses that we can now pose with this simple two-attribute dataset. We note that the number of hypotheses is not finite due to the recursive expansions of the expr rule and the infinite possible resolutions of const. The grammar can be made finite by applying restrictions on the depth of the recursion and binning the possible values. Given this grounded grammar, we can now define a set of verifiable hypotheses (acceptable statements) based on whether the grammar and dataset are able to parse each statement. Analysis Hypothesis Space (H A ) H A An analysis hypothesis space is denoted as H A . This space represents the set of analysis questions (expressed as hypotheses) that a user wants to evaluate, independent of the data that is available to them, or the consideration of what visualization to use. To illustrate H A , consider the previous grammar used for H D . If a user thinks that interest-rate, the year, and stock-price might be correlated with cost, she might add them along with CORR to the previous grammar as follows: This grammar represents, in the form of hypotheses, the analysis questions that a user might have. It is important to note that the attributes interest-rate, year, and stock-prices are not in the dataset, and simply represent attributes that the user is interested in. Similarly, the attribute price has been removed from the grammar because it is not an attribute that the user cares about in this analysis. In this regard, the analysis hypothesis space can be created independently of the data. In Section 6, we use the 2017 VAST Challenge to demonstrate how realistic analysis tasks can be formalized as H A . Visualization Hypothesis Space (H V ) H V The This grammar (and the hypothesis space) is specific to this instance of V b , and will change if its design specification or dataset changes. For example, if V b now renders 100 bars, evaluating if the barchart exhibits a normal distribution may be a valid hypothesis. Conversely, it may no longer be valid to compare the heights of bar 5 with bar 95 because perceptually this becomes much harder to do, especially if the width of the visualization is large [84]. Visualization Hypothesis Space and Graphical Perception: Finding the limitations of human perception and cognition with different visual encodings is still an active research area in the visualization community. There is, at this time, no definitive mapping between all visualization designs and their task effectiveness. However, research has begun to propose equivalent design spaces (e.g., [21,32,66,75,96]). Recent work such as Saket et al. [74], Kim et al. [44], and Mortiz et al. [62] have started to empirically evaluate and model the effectiveness of visualizations for analytics tasks. Our grammar for describing the hypothesis space of a visualization currently cannot account for these perceptual models in a generalized way. However, we will further discuss the implications of quantifying and evaluating visualization hypothesis spaces in Section 9. Definitions of Hypothesis Spaces To formalize the three hypothesis spaces and apply them to reason about the implications of the spaces, we first define some operations and properties of the hypothesis spaces: Definition 1 (A Hypothesis). A hypothesis statement (h) is said to be in a hypothesis space H 1 (i.e., h ∈ H 1 ) if h can be parsed by the grammar associated with the hypothesis space. Definition 2 (Size of a Hypothesis Space). The size of a hypothesis space H 1 is denoted as |H 1 |, representing the number of statements that can be parsed by the grammar associated with the hypothesis space. IMPLICATIONS OF THE HYPOTHESIS GRAMMAR In the visualization community, grammars have facilitated the formalisms for expressing visual encodings [90], user interactions [76], and visualization recommendations [62,94]. We believe our proposed grammar can similarly help researchers reason about design, analysis, and tasks in terms of the hypotheses they support. Below, we discuss the implications of the relationship between the three hypothesis spaces, how they can be used to model a visual analysis process, and how the grammar can facilitate the formalization of analysis tasks. Unifying the Three Spaces Our proposed hypothesis grammar allows us to express data, visualization, and analysis using the same language. By doing so, we can consider the relationships produced by the unification of the hypothesis spaces H D , H V , and H A -as well as their implications to the roles of data, visualization, and users in visual analytics. Figure 1 provides an illustration of some interesting relationships between these spaces: HH A HV Under-Powered Visualization. If H V ⊂ H A but |H A | is significantly larger than |H V | (i.e., |H A | |H V |) , the visualization is too limited for the user's analysis needs, and a more robust visualization solution or tool is necessary. H V HA Over-Powered Visualization. Conversely, if H A ⊂ H V but |H V | is significantly larger than |H A | (i.e., |H V | |H A |) , the visualization is likely too complex for the user's needs, or can encode broad data and questions (e.g., dashboards). Partial Grammar Specification to Refine Analysis In Section 4, we demonstrated how a grammar can be used to represent a user's analysis tasks, forming an analysis hypothesis space (H A ). Our grammar affords us the benefit of using non-terminals that can be resolved, e.g., with unification [81] (similar to the concept of wildcards [92]). As a result, users can partially specify their analysis goals or tasks through the use of our grammar. Visualization tools like Voyager 2 [94] have shown the success of using wildcards with a grammar to aid in visualization design and recommendation. We posit that similar methodologies with our hypothesis grammar can be used for analysis task and hypothesis refinement. Consider the following grammar (based on Section 4.3's example): In this example, the user has a narrow analysis task to find out what other attributes (attr) might correlate with the attribute cost. The user is only interested in items that have a price above some threshold; however, at this point the user does not have a definitive sense of what that threshold is (const2). The above grammar forms a hypothesis space that is smaller than the original one shown in Section 4.3. We can consider the process of adding specificity to the analysis hypothesis grammar (thereby reducing the size of it) as an user's refinement of their analysis tasks towards resolving a goal. When the size of the analysis hypothesis space reaches one (or a similarly small number), the user should then be able to evaluate the hypotheses analytically [41]. Until then, the user interacts with the visual analysis system to explore H V , grounded to their data. During this process, the user is exposed to analysis tasks (in our formalism: the hypotheses supported by their visualization and data) of which they may not have been previously aware. In this way, our grammar is able to model vagueness in analysis goals (e.g. the unknown const2) and refinement as that vagueness is replaced with new knowledge (e.g. setting const2 to a reasonable threshold). Beyond Refinement to Incorporate Insight In the above illustration, we focused on how partial grammar specifications can be used to clarify H A . As the user obtains a better understanding of their analysis goals (e.g., resolving const2 to a real value, from the previous example), we show that H A similarly becomes more refined. This formulation follows similar analysis models [41,69,73,86] that describe how high-level analysis goals are refined through exploration until confirmatory analyses can occur. However, in practice, conducting analysis is rarely a linear process in which H A begins at one state (and thus one size) and continuously shrinks until H A becomes extremely small. Instead, it is more likely that H A starts at an initial state, represented as a grammar referred to as H A 0 , and moves across analysis states (H A 0 , H A 1 , ..., H A i ) with corresponding hypotheses spaces that "move," expand, and shrink in size at any point. This might happen because the user accesses new data (H D grows), or because the user gains new insights into the data and wants to evaluate different hypotheses than originally (e.g., H A 5 ∩ H A 0 ). Understanding how users acquire new insights, or develop new analysis questions, has largely been studied with open-ended protocols qualitatively [65]. With a grammar to formally express H A , we can now examine and enumerate the relationships between H A , H D , and H V . For example, does H A change when certain visualizations are recommended [94], or with in-situ data augmentation [10]? We discuss these questions as visualization research opportunities in Section 7. Formalizing Analysis Tasks in Visualization The ability to formalize tasks can provide practical benefits to visualization design and automation [24,28]. Towards this goal, researchers have proposed a number of task taxonomies [7,23,43,44,78]. While taxonomies are indeed useful for grouping tasks into logical categories, they do not provide formal definitions for the tasks [60]. For example, the Characterize Distribution task has appeared in a number of low-level task taxonomies [3,11,14,38,97], and while semantically informative, describing a user's task as "characterize distribution" does not provide a definition, criteria, or characterization for the distribution. We demonstrate that our grammar enables the formalization of analysis tasks by providing explicit definitions. We also illustrate that, in some cases, the process of the formalization can reveal ambiguities in a task or taxonomy. For example, we can define the task "characterize distribution" using our grammar as: hyp :-func (attr) < theta func :-fit _ Gaussian | fit _ Linear | fit _ Beta Where the function func fits data to different distributions (i.e., Gaussian, Linear, Beta) and returns the loss (e.g., mean squared error). The hypothesis is evaluated by checking that the resulting loss is below a threshold (theta). We argue that our grammar to describe "characterize distribution" is more precise than current versions in taxonomies, as it provides the specific types of distributions that a user might consider. This grammar also makes explicit the characterization criteria, which is that the loss has to be below some threshold theta. Applying our grammar to define other low-level tasks can reveal potential ambiguity in a taxonomy. For example, the low-level task Find Anomalies appears in many visualization task taxonomies, but how to define an anomaly left up to interpretation. In this case, we are unable to provide a general definition of "find anomaly" using our grammar. Instead, any attempt to define the "find anomaly" task results in defining what an anomaly is. For example: hyp :-attr1 [id = const1] ∈ [ min, max ]) min :-(avg(attr1) -const2 * stdev(attr1)) max :-(avg(attr1) + const2 * stdev(attr1)) const1 :-number This grammar defines an anomaly as any value that is outside some number (const2) of standard deviations away from the average. Although the exact distance is not specified (const2), the grammar defines that anomalies are distance-based and the distance is determined using class average and standard deviation. We propose that the use of the grammar is more useful than current approaches, as it provides more specificity than the categorical name "find anomalies." However, as there is no generalizable way to describe the task, we find that the task itself is ambiguous. We provide a full illustration of how our grammar maps to low-level analytic tasks in the supplemental material. DEMONSTRATION OF THE GRAMMAR To show a practical implementation of our proposed grammar, we use the 2017 VAST Challenge to illustrate the process of expressing sets of analytic questions as a hypothesis space. Finally, we discuss the learning outcomes of this demonstration. 2017 VAST Challenge MC1 The VAST Challenges have a rich history in the visual analytics community [20]. Each challenge is designed to simulate complex, open-ended problems that can be solved through careful data and visual analysis. For this case study, we examine the first mini-challenge (MC1) of the 2017 VAST Challenge. For brevity, we shorten the challenge description and data attribute values in our text. The full description of the challenge can be found online 1 . Problem statement: The population of the Rose-Crested Blue Pipit (a local bird species) is decreasing within a nature preserve. Provide hypotheses to explain any potentially odd behaviors in vehicle activity that might be causing this decrease. Dataset: The challenge includes two datasets: a map of campsites, ranger stations, gates, etc. throughout the preserve, and a sensor log of the vehicular activity that has the schema: Timestamp:datetime, Car -id:str, Car-type:str, Gate-name:str. The metadata as well as a subset of data are: • Timestamp: date and time a sensor reading was taken • Car-id: unique car ID • Car-type: vehicle type (e.g., car, ranger) belonging to a given car-id • Gate-name: name of the sensor (e.g., gate) taking the reading. Gates can be entrances, general gates, ranger areas, or camping sites Expressing an Analysis Hypothesis Space To express an analysis space for the challenge, we first decompose a set of analysis questions and represent them as a hypothesis grammar. We synthesized these questions based on official Challenge submissions found online [5,8,9,98]. For simplicity, our grammar assumes that data transformations (joining tables, extraction, cleaning, etc.) have already been applied. Some of the following examples require deriving new data (e.g., computing duration from two consecutive timestamps) which can be informed through the construction of the grammar. We separately describe the transformations needed for those hypotheses. This grammar lets us evaluate whether some park visitor are not passing through an exit gate once entering. To do so, the grammar counts the number of check-ins at any entrance gate and checks that it is odd (based on the count modulo 2). '~' tests Gate-name against the regular expression for strings that start (^) with "entrance". The non-terminal const1 expresses the space of all hypotheses where a unique Car-id passed the entrance gates an odd number of times. This grammar evaluates if non-ranger cars are entering ranger areas, that is, whether vehicles are (illegally) checking in through ranger gates. This grammar evaluates whether the average car speed in the park is highest between midnight and 7am. To do so, we first pre-compute Speed:float as the distance between two consecutive gate check-ins of the same car, then divide by the time difference. This grammar evaluates whether the distribution of gate check-ins at night follows the same distribution during the day. To evaluate these hypotheses, we must first pre-compute a distribution of the number of gate check-ins per hour in the evenings (DistrEvening) and day time (DistrDay). It then uses KL divergence to confirm that the difference between the two distributions is less than a threshold. This grammar evaluates whether the duration of a visitor's stay in the park is inversely correlated with nesting activity. We need to precompute the average visitor stay length per week or month, and the average amount of nesting over the same time interval. Although the average visitor stay is computable, there is Insufficient Data (H A ⊂H D ) to compute nesting activity, and thus we cannot verify the hypothesis. Complete Analysis Hypothesis Space. Ignoring H6, the resultant analysis hypothesis space H A expresses the above hypotheses, as well as others relevant for addressing the challenge's problem statement: Takeaways One of the values of using a grammar to express an analysis hypothesis space is that it informs us of whether the current data can appropriately evaluate all statements (i.e. the analysis hypothesis space intersects fully with the data hypothesis space). In Q6, we saw that the hypothesis grammar generates statements that cannot be verified, and therefore has inadequate overlap with the challenge's data hypothesis space. Furthermore, it is interesting to note that the original data's hypothesis space H D has little overlap with the analysis hypothesis space H A , as is evident by the numerous attr values not present in the original data. This is in line with the motivation of the VAST Challenge: simple analyses with the raw data will not solve the problem statement. It is through the formation of hypotheses, and the respective data transformations, that makes it possible to solve the challenge. By expressing the first mini-challenge from the 2017 VAST Challenge as a grammar, it becomes clear what data transformations are necessary so that H A ⊂ H D . We believe this procedure of forming an analysis hypothesis space, then curating the data appropriately, can aid in traditional exploratory pactices. Similarly, Q6 makes clear that additional data needs to be collected before it can be answered, or in the worst case, that the question cannot be answered for the challenge.. In summary, by "pre-registering" an analysis hypothesis space, an analyst can know ahead of time (1) the data transformations or feature engineering she will need to perform, and (2) what visualizations can potentially be used to visually evaluate her hypotheses. In addition to informing whether her analysis space overlaps with the data space, H A can also act as a proxy for an initial "design requirements gathering" before designing a visualization system. RESEARCH OPPORTUNITIES Along with the implications of our hypothesis grammar presented in Section 5, we envision multiple research opportunities that can incorporate our grammar such as expressing analyses, recommending and evaluating visualizations, as well as supporting refinement. Hypothesis-Based Visualization Recommendation Related to the work proposed by Mackinlay [54], a grammar for tasks can automate visualization recommendation. Research has looked into task-based recommendations of simple visualizations and simple analytic tasks. For example, Saket et al. trained Kopol, a visualization recommender that suggests visualizations to a user based on a given task and data type [74]. Similarly, Shen et al. proposed a task-based visualization recommendation system [80], which maps a collection of analytic tasks (e.g., 'correlate,' 'find extrema,' 'cluster') to generate and rank candidate visualizations, then outputs the results to a user. A benefit of these recommenders is the reduced effort and expertise required of users to design task-appropriate visualizations. If a user's analysis task is simple enough, we are now able to provide qualified visualizations to satisfy those tasks. This is a significant step forward from custom, manual visualization creation. Complementing these efforts, we believe the opportunity to recommend visualizations based on H A and H D can be equally beneficial. To illustrate this, consider Figure 2. Given a user's dataset and the hypotheses it can express (H D ), visualizations can be recommended such that their spaces overlap the most appropriately with H D (e.g., H V 3 > H V 2 > H V 1), as seen in Figure 2a. Similarly, visualizations can be recommended based on which of their hypotheses spaces H V overlaps the most with H A , which may result in different recommendations (e.g., (H V 1 > H V 3 > H V 2), shown in Figure 2b. This opens new directions in ongoing efforts to understand "taskappropriate" visualizations. Using our proposed grammar-based approach, researchers can begin to study how to automatically express a user's analysis goals (H A ), a candidate visualization's analysis capabilities (H V ), the data (H D ) best mapped to visualizations, and how to use their intersections to inform recommendations. Level Criterion 0 No explanation provided. A nonsense statement, question, observation, or single inference about a single event, person, or object. 1 An inappropriate (e.g., irrational, not falsifiable) explanation. 2 Partial appropriate explanation (e.g., incomplete). 3 Appropriate explanation relating at least two variables in general or nonspecific terms. 4 Precise explanation with qualification of the variables. A specific relationship is provided. This is a hypothesis. 5 In addition to level 4, a test is given. 6 An explicit statement of a (precise) test of a hypothesis is given. Visualization Evaluation Similar to the topic of hypothesis-based recommendations for visualizations is the opportunity to formally evaluate visualizations on their ability to support a grammar of hypotheses. We observe two directions that this research opportunity can take. First, we are now able to evaluate a visualization's efficacy in both expressing and verifying hypotheses. As we showed in Section 4.4, a visual encoding or visualization (e.g., a bar chart) is capable of expressing a set of hypotheses (bar1 > bar2, etc.) -these hypotheses span the visualization's hypothesis space. Just as some visualizations are better suited for visualizing certain data types [54], or supporting certain simple analytic tasks [74], there will be visualizations better suited for expressing and verifying certain hypotheses. For example, to verify hypotheses that pose part-to-whole relationships, a pie chart will be preferable to a line chart. Experiments to quantify efficacy of expressing and verifying hypotheses across different visualizations is a rich and open area for future research. Second, we can evaluate a visualization system by its ability to help users navigate and refine their hypothesis space (H A ). Theoretical models for how users learn from visualization tools, e.g., through knowledge generation [73], the data-frame theory [48], or sensemaking [69] have spurred new frameworks for testing the efficacy of interactive visualization tools. In particular, in terms of these tools' abilities to help users generate and validate knowledge, as well as fit data into a frame (and vice versa), and make sense of data, respectively. Similar in spirit to these models, by tracking how an analysis hypothesis space changes through the use of a tool (e.g., via exploration, as discussed in Section 5.2), we can similarly understand how systems support users through searching, refining, and expanding hypothesis spaces. Guiding Users in Analysis with Grammars & NLP Another area in which our grammar presents research opportunities is guiding users in analysis. Specifically, helping users refine broad, vague analysis goals into executable hypotheses. Users often begin an analysis session with a vague notion of their overarching analysis goal or task. In these cases, modeling analysis tasks as hypotheses allows us to asses and refine under-developed hypotheses into well-defined, testable hypotheses. One method to do so is through the use of Hypothesis Quality Scales, such as Sunal and Haas' [83], shown in Table 2. This quality scale is used to provide rankings for a student's hypothesis such that they can be refined and improved upon. With a hypothesis quality scale, vague or under-specified analysis tasks can be thought of as underdeveloped scientific hypotheses. Scales such as Sunal and Haas' provide the mechanism for adding clarity to hypotheses and analysis tasks. Another possible approach to refining analysis goals is by combining our hypothesis grammar with advancements in NLP and natural language interfaces (NLIs) [4,61,64]. These interfaces can help identify areas of ambiguity in a hypothesis that are apt for refinement. For example, in FlowNL [35], users can query a flow visualization system in plain English to generate relevant graphical views (e.g., "where are hurricanes occurring?"). When their natural language parser does not recognize the user's query, the system will ask the user to clarify a particular word or concept ("what does 'hurricane' represent in the data?") -then stores the user's definition to be reused throughout the user's analysis session. Similar work could be done with the integration of a hypothesis grammar. For example, if the user wants to pose partially-specified hypotheses, an NLI could guide users to refine their analysis questions, goals, and/or hypothesis. FROM EXPLORATORY TO HYPOTHESIS-DRIVEN ANALYSIS The work presented in this paper focuses on how a grammar-based approach for hypotheses can be used to operationalize analysis tasks. This approach offers the benefit of contributing towards the growing body of work in belief elicitation [55] and "hypothesis-first" analysis [18,46]. In visual belief elicitation, a user is asked to first convey their beliefs, assumptions, or predictions about data shown on a visualization. A popular example is the New York Times' "You Draw It" series [2]users are given a blank (or partially complete) line chart, and prompted to draw their prediction for a given question, e.g., "how does a family's income predict their children's college chances?" Empirical research into the effects of visual belief elicitation [50] and "hypothesis-first" analysis shows promise. Users are more engaged with visualizations [33], conduct more deliberate analyses [49], discover fewer spurious and false discoveries [99,100] than with exploratory data analysis [86], and have more accurate recall of data [37,45]. Consequently, a multitude of conceptual frameworks, cognitive models, and design spaces have been proposed to describe the processes of hypothesis-and belief-first analysis [15,46,55]. Hullman and Gelman [36] emphasize the need for theoretical models to unite and complement traditional exploratory practices [41,86] and visualization tools [93,94]. A hypothesis-driven model for visual analytics can contribute towards addressing this gap. As Hullman and Gelman similarly advocate, we believe hypothesis-first interfaces could mitigate analysis risks, aid in interpreting uncertainty, encourage the ability to infer correct conclusions about data, and incorporate the many empirical findings discussed above [49,50,99]. In terms of advancements to visualization theory, a hypothesis-driven approach (which we will refer to as HDA) would be able to support new considerations for the processes of knowledge generation and verification [42,73]. Synthesis / Induction Hypothesis / Deduction The Cycle of Knowledge The figure to the left illustrates how the scientific community considers the cycle of knowledge [42]. Commonly, visualization and data analysis practices follow the principle of exploratory data analysis (EDA) [86], or a "data-first" approach, where an analyst begins analysis by first analyzing the data (illustrated as Synthesis/Induction in the figure). An HDA path would work in reverse, promoting a "hypothesis-first" principle [15,36,47], where analysis begins with an analyst posing a priori hypotheses -similar to the scientific method (illustrated as Hypothesis/Deduction in the figure). We posit that these processes should work in tandem: a data-first approach should result in meaningful hypotheses and insights, and a hypothesis-first approach should result in hypotheses that are validated with data. Given the multitude of observed benefits for 'hypothesis-first' analysis [15,33,37,45,49,50,55,99,100], one may wonder why HDA is not an established, fundamental methodology to visual analytics. Arguments have been made that users do not (and should not) necessarily start analysis with any a priori hypotheses formed [95]. However, we find ourselves in agreement with Felin et al. [27], among others [36,47,59], who argue that "there is no such thing as hypothesisfree data exploration," and that "even informal hunches or conjectures are types of proto-hypothesis." Another hurdle to HDA in visual analytics is the seemingly difficult integration of a user's complex beliefs for data. Currently, there is no operational mechanism to implement well-defined and partiallyspecified hypotheses. Despite the conceptual frameworks and design spaces [15,46,55] that have been contributed, a practical gap still exists in the formalization of "hypothesis-first" visual analytic systems and workflows [36]. We posit that our grammar for hypotheses (as well as its ability to support 'unknowns', Section 5) can provide a baseline for tools to seamlessly integrate a user's hypotheses. Unlike previous research, where hypothesis-first analysis is conducted and studied in "Wizard of Oz" experiments (e.g., [16,50]), we offer an operationalizable method to realize these processes. For now, the exact relationship and boundaries between EDA and HDA are unclear. Perhaps hypothesis-driven analysis should be a concrete component of EDA, formalizing a boundary in which users step from "exploratory" to "confirmatory" analysis phases [86]. In this representation, HDA would provide a coupling of the processes for EDA and CDA. Regardless, HDA offers a realistic way to conceptualize the 'hypothesis-first' analysis process, and our grammar for hypotheses offers a concrete step towards operationalizing systems supportive of it. LIMITATIONS In this paper, we advocate for operationalizing analysis tasks to disambiguate their use in visualization. We pose one possible grammar-based solution using the concept of scientific hypotheses. Below, we provide considerations of the limitations and unknowns for our proposed hypothesis grammar to make tasks operational. Is this Grammar Enough? A formal, universal definition of "task" currently does not exist in visualization literature. Numerous domain-specific tasks have been proposed (e.g., [26,43,52,58,75]), as have tasks specific to particular applications or usage contexts (e.g., [13,23,38,79]). Consequently, the concept of "tasks" in visualization and visual data analysis spans a considerable number of actions, procedures, and needs. A single hypothesis grammar cannot possibly operationalize all definitions of "tasks" in visualization. Our grammar is constrained to expressing analysis questions (as testable relationships between data variables) in the space of (tabular) data, and (mostly static) visual encodings. For example, our grammar can not encode tasks like "Enjoy [7]," as these tasks are not analytical questions in nature. With this said, we believe a grammar-based approach to make analysis tasks operational is paramount. Whether that approach is using our hypothesis grammar or a future 'task-based' grammar, it will provide new methods to push visualization design and theory forward using a new shared language. We envision many grammarbased approaches for tasks in the future, and hope that our hypothesis grammar provides a baseline to improve and reflect upon. Boundedness of the grammar Our grammar is not capable of expressing every possible hypothesis that could be formulated with the English language. Hypotheses are a complex subset of natural language and have been an ongoing object of study. Previous grammars for hypotheses have been contributed in literature [17,87], but are typically limited to a specific domain application (e.g., electrical circuits [51]). Similarly, our hypothesis grammar is restricted to performing analysis and posing questions and statements about data. Assumptions for the data We assume the data is relational and can be accessed from a database (or a table). We also assume the data has been transformed, cleaned, and is not missing any values required by the grammar. This is obviously not always the case with "real-world" analysis. The existence of data transformation languages such as SQL and non-relational query languages suggest that extending the grammar to support data pre-processing is possible. Resulting trade-offs between grammar simplicity and expressiveness must be carefully balanced. Qualifying human judgement Our hypothesis grammar does not take into account uncertain criteria for human judgement. For example, the statement "action movies are better than comedies" could possibly be formed as a hypothesis, but it is subjective. Our grammar does not take these hypotheses into consideration but instead focuses on quantifiable true or false statements that can be posed about data, driven by the notion of scientific hypotheses. Visualization's ability to verify hypotheses We postulate that visualizations can confirm or reject some subset of hypotheses given H V , as illustrated by Wickham et al. with their LineUp technique [89]. However, currently we cannot enumerate all hypotheses that a visualization can be used to answer. There are many open-ended questions that our grammar therefore introduces. If a single visualization is capable of verifying a particular hypothesis, what is the translation to visualization tools in which many visualizations can be shown or suggested to users? Are there some visualizations that are more indicative of supporting a hypothesis than others? Prior work has sought to understand the effectiveness of a visualization in supporting tasks [74]. In future work we aim to investigate if the same can be done for hypothesis spaces. Extensions to other concepts in visual analysis In this work, we demonstrate applications of our grammar with realworld analytic challenges. However, there are other important concepts of data analysis and visualization that we did not have space to address. For example, how uncertainty affects data understanding [82], data augmentation [10], perceptual models and experiments [25], and so on. We hope that our formalism for hypotheses in visualization can provide the springboard for research in all of these areas. CONCLUSION This paper presented a grammar-based approach for operationalizing analysis tasks in visualization. The grammar draws from the science and education literature, and provides a formal language to express (scientific) hypotheses over datasets. We showed how sets of hypotheses (parsable by our grammar) make up a hypothesis space, and can be used to represent the hypotheses a specific dataset can evaluate (data hypothesis space), the hypotheses a visualization can evaluate (visualization hypothesis space), and the hypotheses a user wants to evaluate through analysis (analysis hypothesis space). We illustrated how the use of our grammar can unify a user's analysis goals, the available data, and the capabilities of a visualization system. We demonstrated how it can allow partial specification, and how it can formalize analysis tasks commonly used in visualization research. We provided a case study on the use of our grammar with the 2017 VAST Challenge, which helped identify necessary data transformations, as well as questions that were unanswerable by the data. Finally, we discussed the practical benefits of our hypothesis grammar to aid in future research directions, including a new hypothesis-driven approach (HDA) for visual analytics. Overall, we believe a grammar to articulate tasks will promote better analysis clarity, help evaluate visualization systems, and support hypothesis-based reasoning in visual analytics. Fig. 1: The relationships between three hypothesis spaces, formalized using our proposed hypothesis grammar. A visualization hypothesis space (H V ) expresses the set of hypotheses a visualization can evaluate, a data hypothesis space (H D ) expresses the set of hypotheses a dataset can evaluate, and an analysis hypothesis space (H A ) expresses the set of hypotheses a user wants to evaluate through analysis. The intersection of these spaces can help us understand how the hypotheses a user would like to answer through analysis can either be fully answered by data and visualization, partially answered, or not answerable at all. hyp :-expr op expr [pred] expr :-const | attr | func(expr) | func(expr, expr) op :-> | < | = | ... func :-AVG | MAX | MIN | ... pred :-attr op const const :-number hyp :-expr op expr ([pred]) (& hyp)? expr :-func ((expr (, expr)?)? ) | var var :-attr ([pred])? | const pred :-var op const (& pred)? func :-AVG | MAX | MIN | ... op :-> | < | = | ... const :-number attr :-cost | price hyp :-expr op expr ([pred]) (& hyp)? expr :-func ((expr (, expr)?)?) | var var :-attr ([pred])? | const pred :-attr op const (& pred)? func :-AVG | MAX | MIN | CORR | ... op :-> | < | = | ... const :-number attr := cost | interest-rate | year | stock-price Analytics Solutions. A visual analytics solution can be considered successful if the intersection between H D , H V , and H A is not null (i.e., |H D ∩ H V ∩ H A | > 0). An ideal solution is when a user's questions can be fully answered by the data and the visualization tool such that H A ⊂ H D , H A ⊂ H V , and |H D ∩ H V | > 0. Inappropriate Visualization. The use of an inappropriate visualization occurs when H V ∩ H D (or |H V ∩ H D | is very small) where a visualization or visual encoding cannot support the hypotheses about the data. Ineffective Visualization. The use of an ineffective visualization occurs when H V ∩ H A (or |H V ∩ H A | is very small), that is, a visualization or visual tool cannot support a user's analysis questions. . In the case where H A ⊂ H D , not all of a user's analysis questions can be answered with the given data. This might require acquiring additional data, or performing new data transformations. hyp:-CORR(cost, attr) op const1 [price > const2] attr:-interest-rate | year | stock-price op :-= | < | > | ... const1 :-number const2 := number Q1 : Q1Are park visitors always exiting the park after they enter? hyp :-(mod(count(), 2) = 1)[pred] pred :-[Gate-name~"^entrance" & Car-id=const1] const1 :-number Q3 : Q3Is the park more popular during Pipit mating season? hyp :-count()[Season="Spring"] > count()[Season=const1] const1 :-"Winter" | "Fall" | "Summer" This grammar evaluates whether there are more vehicles visiting the park in the Spring than in other seasons. Season:string can be computed based on the month in Timestamp. Q4: Are cars speeding when rangers are off duty? hyp :-avg(Speed)[Hour∈[0,7)] > avg(Speed)[Hour ∈[0,7)] Q5 : Q5Are there any outliers in vehicle nightly activity? hyp :-KL(DistrEvening, DistrDay) < 10 Q6 : Q6Is the length of visitor stay affecting nesting? hyp :-Corr(StayLengths, NestingAmount) < -0.75 hyp :-expr op expr ([ pred ]) (& hyp)? expr :-func ((expr (, expr)?)? ) | var var :-attr [pred] | const pred :-attr op const func :-! | % | avg | count | Corr | KL op :-= | < | > | != | ∈ | ∈ attr :-Car-id | Car-type | Gate-name | Distance | Duration | Speed | Time | Hour | Check-in | Check-out | Season | Restricted-area | DistrDay | DistrEvening | StayLengths const :-number | str | datetime | boolean between H A and H V . Fig. 2 : 2Using the overlap of hypothesis spaces (Section 4) to inform visualization recommendations. We describe in detail in Section 7.1. 's proposed Presentation Tool (APT) in • Ashley Suh and Remco Chang are with Tufts University. E-mail: {ashley.suh, remco.chang}@tufts.edu • Ab Mosca is with Northeastern University. E-mail: [email protected] • Eugene Wu is with Columbia University. E-mail: [email protected]. hyp): A hypothesis is now recursively defined such that it can contain multiple sub-hypothesis. In other words, a hypothesis can now be written as hyp :-H 1 & H 2 & H 3 & ... where each of the hypothesis H i is separately defined as expr op expr.• (var): Each expression (expr) in a hypothesis (hyp) can have its own predicate. This allows independent filtering for the two ex- pressions. For example, AVG(price)[year<2000] < AVG(price) [year>=2000] is now possible. This now introduces a new symbol called var where var:-attr [pred]. • (pred & pred): Similarly, a predicate can be recursively defined, and can be written as pred :-P 1 & P 2 & P 3 & ... where each of the predicates P i is separately defined as var op const. Term Production Rule Semantic Definition hyp expr op expr ("["pred"]")? ("&" hyp)? visualization hypothesis space, denoted H V , represents all hypotheses that a visualization V is able to evaluate. Similar to how H A can be created without the consideration of the data, here we consider H V independently of any dataset as well. To illustrate the general concept of H V , we provide examples of visualization hypothesis spaces below. Hypothesis Space of a Barchart with 3 Bars: Consider V b , a static barchart graphic that shows three bars (shown left).When V b is rendered as an image, the image has specific property values for the canvas (width, height, border size, spacing, etc.) and contains three bars that have positional attributes, such as their position along the x-axis in pixels (xpos) and their height in pixels (pixelheight). For now, let's only focus on these two attributes. V b can be used to evaluate a few different hypotheses. For example, whether bar 1 is taller than bar 2 (pixelheight[id=1] > pixelheight [id=2]) or if the three bars show an increasing order in height (fit( pixelheight, xpos)> const). Given the limited hypotheses expressible by V b , the hypothesis space should be relatively small and can be expressed as:hyp :-hyp1 | hyp2 hyp1 :-pixelheight[id= num1] op pixelheight[id= num2 & num1 = num2]hyp2 :-fit(pixelheight, xpos) > const attr :-pixelheight | xpos op :-= | < | > | ...pixelheight xpos 1 2 3 num1 :-number num2 :-number const :-number Definition 3 (Containment). Given two hypothesis spaces H 1 and H 2 , if H 1 ⊂ H 2 then |H 1 | < |H 2 |. Further, any solution that can be used to answer hypotheses in H 2 can be used to answer H 1 . Definition 4 (Intersection). Given two hypothesis spaces H 1 and H 2 , their intersection H 1 ∩ H 2 represents hypotheses that can be parsed by both the grammar of H 1 and the grammar for H 2 . 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[ "Application of the Variational Autoencoder to Detect the Critical Points of the Anisotropic Ising Model", "Application of the Variational Autoencoder to Detect the Critical Points of the Anisotropic Ising Model" ]
[ "Anshumitra Baul \nDepartment of Physics and Astronomy\nDepartment of Physics and Astronomy\nLawrence Berkeley National Laboratory\nLouisiana State University\n1 Cyclotron Rd70803, 94720Baton Rouge, BerkeleyLouisiana, CaliforniaUSA, USA\n", "Nicholas Walker \nand Center for Computation and Technology\nLouisiana State University\n70803Baton RougeLouisianaUSA\n", "Juana Moreno \nLouisiana State University\n70803Baton RougeLAUSA\n", "Ka-Ming Tam \nLouisiana State University\n70803Baton RougeLAUSA\n" ]
[ "Department of Physics and Astronomy\nDepartment of Physics and Astronomy\nLawrence Berkeley National Laboratory\nLouisiana State University\n1 Cyclotron Rd70803, 94720Baton Rouge, BerkeleyLouisiana, CaliforniaUSA, USA", "and Center for Computation and Technology\nLouisiana State University\n70803Baton RougeLouisianaUSA", "Louisiana State University\n70803Baton RougeLAUSA", "Louisiana State University\n70803Baton RougeLAUSA" ]
[]
We generalize the previous study on the application of variational autoencoders to the twodimensional Ising model to a system with anisotropy. Due to the self-duality property of the system, the critical points can be located exactly for the entire range of anisotropic coupling. This presents an excellent test bed for the validity of using a variational autoencoder to characterize an anisotropic classical model. We reproduce the phase diagram for a wide range of anisotropic couplings and temperatures via a variational autoencoder without the explicit construction of an order parameter. Considering that the partition function of (d + 1)-dimensional anisotropic models can be mapped to that of the d-dimensional quantum spin models, the present study provides numerical evidence that a variational autoencoder can be applied to analyze quantum systems via Quantum Monte Carlo. arXiv:2111.01112v1 [cond-mat.dis-nn] 1 Nov 2021
10.1103/physreve.107.045301
[ "https://arxiv.org/pdf/2111.01112v1.pdf" ]
240,354,780
2111.01112
6d515aacaef5f2d9fde6eef3579e0b6f779ba7e3
Application of the Variational Autoencoder to Detect the Critical Points of the Anisotropic Ising Model Anshumitra Baul Department of Physics and Astronomy Department of Physics and Astronomy Lawrence Berkeley National Laboratory Louisiana State University 1 Cyclotron Rd70803, 94720Baton Rouge, BerkeleyLouisiana, CaliforniaUSA, USA Nicholas Walker and Center for Computation and Technology Louisiana State University 70803Baton RougeLouisianaUSA Juana Moreno Louisiana State University 70803Baton RougeLAUSA Ka-Ming Tam Louisiana State University 70803Baton RougeLAUSA Application of the Variational Autoencoder to Detect the Critical Points of the Anisotropic Ising Model (Dated: November 2, 2021) We generalize the previous study on the application of variational autoencoders to the twodimensional Ising model to a system with anisotropy. Due to the self-duality property of the system, the critical points can be located exactly for the entire range of anisotropic coupling. This presents an excellent test bed for the validity of using a variational autoencoder to characterize an anisotropic classical model. We reproduce the phase diagram for a wide range of anisotropic couplings and temperatures via a variational autoencoder without the explicit construction of an order parameter. Considering that the partition function of (d + 1)-dimensional anisotropic models can be mapped to that of the d-dimensional quantum spin models, the present study provides numerical evidence that a variational autoencoder can be applied to analyze quantum systems via Quantum Monte Carlo. arXiv:2111.01112v1 [cond-mat.dis-nn] 1 Nov 2021 I. INTRODUCTION Machine learning (ML) has become an indispensable tool to reach the boundaries of scientific understanding in the age of big data. An overflow of information is being analyzed using ML to quantify patterns in a large variety of fields from social networking, object and image recognition, advertising, finance, engineering, medicine, biological physics, and astrophysics among others [1]. Machine learning is a data modeling approach which employs algorithms which favor strategies driven by statistical analysis and based on pattern extraction and inference. ML algorithms, such as deep learning, provide new advances to understand physical data. Opportunities for scientific investigations are being devised particularly in numerical studies, which naturally involve large data sets and complex systems, where explicit algorithms are often challenging. A concerted effort to address large data problems in statistical mechanics and many-body physics using the ML approach is emerging [2][3][4][5][6][7][8][9]. The foundation of ML is deeply connected with statistical physics, and hence is fruitful to combine ML techniques with numerical methods which involve the prediction of phase transition regions. Scaling and renormalization are the core principles to understand macroscopic phenomena from microscopic properties. The way forward for machines to learn from large data sets would incorporate conceptually similar principles [9,10]. Changes in the macroscopic properties of a physical system occur at phase transitions, which often involve a symmetry breaking process [11]. The theory of phase transitions and symmetry breaking was formulated by Landau as a phenomenological model, and was later devised from microscopic principles using the renormaliza-tion group [12]. Phases can be identified by an order parameter, which is zero in the disordered phase and finite in the ordered phase. The order parameter is determined by symmetry considerations of the underlying Hamiltonian. There are states of matter where the order parameter can only be defined in a complicated nonlocal way. These systems include topological states such as quantum spin liquids [13]. A major goal of the ML approach in complex statistical mechanics models or in strongly correlated systems is to detect the phase transitions from the data itself without explicitly constructing any order parameter [2]. The development artificial neural networks to detect phase transitions is a major advance in the area of ML applications in physics. In earlier works, artificial neural networks were based on supervised learning algorithms [2,9]. Labeled data is used to train the supervised learning algorithm, from which the algorithm learns to assign labels to the input data points [14,15]. Apart from supervised learning, another major type of ML is unsupervised learning for which the input data has no label. Conventional unsupervised learning algorithms, such as principal component analysis [16], find structure in unlabeled data without involving any artificial neural network. Here, the data is classified into clusters and labels can then be assigned to the data points accordingly [16]. Autoencoder is a new direction to utilize artificial neural networks in unsupervised machine learning. The very first versions of the autoencoder were being used for dimensional reduction of data before feeding its output into other ML algorithms [17,18]. It is created by encoding an artificial neural network, which outputs a latent representation of the given data, and is utilized as the input of decoding neural network that tries to accurately reconstruct the input data from the latent representation [19,20]. A major shortcoming of autoencoder is the possibility of sharp changes in the latent representation with respect to small differences in the input data. Ideally, the latent representation should be a smooth function of the input data. The variational autoencoder (VAE) represents the latent representation in terms of probability distributions instead of a fixed set of numbers [21,22]. This probabilistic latent representation allows a smooth latent representation. Since 2013, VAEs have been developed to become one of the most successful unsupervised learning algorithms [21]. Promising results are being shown in both encoding and reconstructing data [21][22][23]. VAEs are being successfully applied recently to detect phase transitions in classical spin models [23][24][25]. The input data sets are given by Monte Carlo method and then unsupervised machine learning, such as the VAE is used for deciphering and distinguishing different physics in the input data sets. After successfully applying to classical models, a natural question arises, whether such an approach remains viable for quantum models. In particular, can a VAE be viable to distinguish different quantum phases and transition regions based on the obtained data from quantum Monte Carlo. Recently, various models from statistical mechanics, in particular the Ising model and the Potts model have been investigated [26][27][28]. In this work, we investigate a rather simple quantum model, one-dimensional transverse field Ising model(TFIM), to address the capability and the limitations of the autoencoder. Given that, the critical line of the model can be calculated analytically due to self-dual property [29][30][31]. The TFIM is an excellent test bed to address the various aspects of VAEs in quantum models. This paper is organized as follows. In section II, we briefly describe the transverse field Ising model (TFIM) and the Suzuki-Trotter formulation by mapping it to the anisotropic Ising model. In section III, the Monte Carlo method and the VAE are presented. The results from the VAE are described in the section IV. We conclude and discuss the implication and possible future applications of the method developed in this study in Section V. The self-duality of the anisotropic two-dimensional Ising model and the detail of the VAE are discussed in the appendix. II. TRANSVERSE FIELD ISING MODEL A. Model We consider an Ising model with transverse field [32][33][34]. The Hamiltonian is given as H = − <i,j> J ij σ z i σ z j − Γ i σ x i ,(1) where σ α (α = x, y, z) are the Pauli matrices, which obey the commutation relation, [σ α i , σ β j ] = 2ιδ ij αβγ σ γ i . J ij is the coupling between the spins at sites i and j. Only nearest neighbors coupling is considered in this study. Γ is the transverse field applied in the x-direction. ι is an imaginary number, and σ z has the eigenvalues as ±1. Their eigenvectors are symbolically denoted by | ↑>. and | ↓>, that is | ↑>= 1 0(2) and | ↓>= 0 1 .(3) The order parameter is given by the average magnetization m = i < σ z i > /N (N being the total number of sites) which characterizes the phase transition between paramagnetic and ferromagnetic phases. Without specification, we consider only the one dimensional TFIM with coupling limited to the nearest neighbors. B. Suzuki-Trotter Formalism Instead of working with the quantum spins directly, we follow the standard procedure of mapping a d-dimensional quantum Hamiltonian into a d + 1-dimensional effective classical Hamiltonian by the Suzuki-Trotter transformation [35][36][37]. We apply this to the TFIM system. We first define the longitudinal spin coupling terms and the transverse filed terms as follow H 0 ≡ − <i,j> J ij σ z i σ z j .(4)V ≡ = −Γ i=1 σ x i . H = V + H 0 . The partition function of H reads Z = T r(e −β(H0+V ) ),(5) β is the inverse temperature. The Trotter formula gives, exp(A 1 + A 2 ) = lim M →∞ [exp(A 1 /M )exp(A 2 /M )] M (6) when [A 1 , A 2 ] = 0. Using the trotter formula we have Z = i lim M →∞ s i |[exp(−βH 0 /M )exp(−βV /M )] M |s i ,(7) where s i is the i-th spin configuration of the whole system, and the above summation runs over all the 2 N possible configurations denoted by i. M identifies the number of identity operators. The identity operator formed from the spin operators is given as, I= 2 N i |s i,k s i,k |(8)Z = lim M →∞ T r M k=1 σ i,k , ..., σ N,k | (9) exp(− βH 0 M )exp(− βV M )|σ i,k+1 , ...σ N,k+1(10) Applying the periodic boundary conditions implies σ N +1,p = σ 1,p . After evaluating the expression of the partition function, Z = C N M 2 T r σ (−βH ef f [{σ]}),(11) where C = 1 2 sinh 2βΓ M and the effective classical Hamiltonian H ef f ({σ}) = N <i,j> M k=1 [ −J ij M σ i,k σ j,k (12) − δ ij (2β) ln (coth βΓ M )σ i,k σ i,k+1 ]. σ i,k involved are the eigenvalues of σ z and hence there is no non-commuting part in H ef f . The effective Hamiltonian shows the system of spins in the d + 1-dimensional lattice and one extra label k for each spin variable. Each single quantum spin variable σ i in the original Hamiltonian, now we have an array of M number of classical spins σ i,k . Hence, the partition function of the 1D quantum Ising model is mapped to that of the 2D Ising model. This new (time-like) dimension along which these classical spins are spaced is called the Trotter dimension. In this paper we assume the model only has nearest neighbor coupling on a square lattice. The coupling along the x-direction and y-direction are denoted as J x and J y respectively. We also define K x = 1/(βJ x ) and K y = 1/(βJ y ). We set N = M in this study. For the two-dimensional classical Ising model, the critical points can be obtained due to the self-dual property. The detail of the dual transformation is shown in the appendix A. III. METHODOLOGY A. Monte Carlo Sampling The spin configurations are generated by using the single spin-flip Metropolis algorithm. In a single spin flip, first we flip the spin of a single lattice site, and then calculate the change in energy ∆E. After that, we use the resulting change in energy to calculate the Metropolis criterion exp(−∆E/T ). If a randomly generated uniform number is in the range of [0, 1), and is smaller or equal to the Metropolis criterion, the configuration is accepted as the new configuration. The code for the simulation is written in Python using the NumPy library [38][39][40][41]. We note that using generative neural network, instead of Monte Carlo method, for sampling has also been proposed recently for the 1D quantum spin models [42,43]. B. Variational Autoencoder Variational autoencoder(VAE) is under the category of generative models. New data can be produced by learning the distribution of the input data [21,44]. We use an encoder-decoder architecture where the encoder maps the input to a latent representation in term of some chosen distributions. The latent distribution is mapped back to reconstruct the input using the decoder. The latent representation in a well-trained model can be used to generate new samples resembling the original training data [45]. In this study, we only consider a multi-dimensional Gaussian distribution for the latent space representation. We briefly explain the main concepts of the VAE in the following. Encoder: Encoder is a neural network that takes the higher dimensional input data into a lower dimensional latent space. For example, an image of the spin configuration of size 32×32 = 1024 lattice points, can be converted into a vector of dimension 8. The encoder is a method for dimensionality reduction. The neural network representing the encoder maps each sample to a distribution and is also called a probabilistic encoder of VAE. The encoder is denoted as q φ (z|x), which is a distribution that maps an input sample x (the Ising spin configurations) to provide a latent representation z (the parameters in the multi-dimensional Gaussian distribution). φ is a set of learnable parameters in the neural network that is varied to produce outputs from the encoder. Latent Space: The latent space is the input for the decoder network, and it is outputted from the encoder network. For the VAE, the latent space is represented by multi-dimensional Gaussian distribution. As the Gaussian distribution is completely specified by its mean and standard deviations, the dimension of the latent space is two times as that of the dimension of the multidimensional Gaussian distribution. An encoded sample is denoted by z. The latent space is regularized and then get penalized for deviating from the prior multi-dimensional Gaussian distribution by the Kullback-Leibler (KL) divergence term [46]. Decoder: From the fig. 1 one can see that the decoder in VAE converts the compressed samples in the latent space back to the original input samples [47] . It is represented as p θ (x|z), a distribution to produce reconstructed samples x conditioned on latent representations z. θ is a learnable parameter in the neural network that can be varied to produce different outputs from the decoder. A sampling distribution provides the input to the decoder which is described by the latent representation obtained from the encoder. Loss Functions in VAE: The VAE contained two sets of trainable parameters, θ and φ for the neural networks of the encoder and decoder respectively. They are trained by minimizing the loss function. The loss functions in VAE consist of two terms. The first one measures the "similarity" of the inputs and the reconstructed outputs. The second one measures the difference between the designated priori distribution, chosen to be multidimensional Gaussian distribution, and the actual distribution of the compressed inputs. For the first term of the loss function, the standard reconstruction loss measures the error between the samples generated by the decoder and the original input samples. It is measured by the binary crossentropy between the encoder input and decoder output in our study. This is expressed as L RC = −E z∼q φ (z|x) [log p θ (x|z)] The expectation, E, is over the representations, z with respect to the encoder's distribution. The second is the Kullback-Leibler divergence (KLD) of the latent representations. KLD measures the divergence between the chosen latent representation p(z) and the approximated distribution from the output of the encoder q φ (z|x). The KLD is defined as L KLD = D KL [q φ (z|x)||p(z)] = − z q φ (z|x) log q φ (z|x) p(z) . (13) It is minimized to optimize the latent representation from the encoder q φ (z|x) to resemble the latent representations from p(z). The total loss is the sum of the reconstruction loss and the KLD [48], L(φ, θ; x, z) = −E z∼q φ (z|x) [log p θ (x|z)] +D KL [q φ (z|x)||p(z)] = L RC + L KLD .(14) The two losses in the VAE together are optimized simultaneously to describe the latent state for an observation with distributions close to the prior approximated distribution but also deviates to describe the salient features of the input. The linear combination of the reconstruction loss and the KLD is often denoted as a variational lower bound or evidence lower bound loss function as both the reconstruction loss and the KLD are non-negative. Minimizing the loss, min θ,φ L(θ, φ; x, z), maximizes the lower bound of the probability of generating new samples [45]. C. β-total correlation VAE A further refinement of the VAE can be achieved by decomposing the KLD into three parts. The three parts describing the index-code mutual information, total correlation, and dimension-wise Kulback-Leibler divergence, which are characterized by parameters often denoted as α, β, and γ respectively. β is the most important one for obtaining good results [48,49]. A β-total correlation VAE (β-TCVAE) is defined as the VAE with the α = γ = 1 and tuning β as a parameter. The detail is given in the appendix B. It is well suited for representation learning of patterns [49]. We fix the parameters of the decomposition with α = γ = 1 and β = 8 in this study, these values were found to be good for finding the phase diagram of the 2D isotropic Ising model [25,50]. Our goal is to map the raw Ising spin configurations to a reduced set of descriptors that discriminate between the samples using the criterion inferred by the β-total correlation VAE [25,50]. The encoder and decoder are implemented as deep convolution neural networks (CNN) to preserve the spatially dependent 2-dimensional structure of the Ising spin configurations [51]. Scaled exponential linear unit (SELU) activation function is used in each convolution layer. The output of the final convolution layer is flattened, and fed into two 2-dimensional dense layers. Then, it is used as the input layer for the decoder CNN and reshaped to match the structure of the output from the final convolution layer in the encoder CNN. Hence, the decoder network is simply the opposite of the encoder network, incorporated with convolution transpose layers in favor of standard convolution layers [50][51][52]. The final output layer from the decoder network is reproduced from the original input configurations obtained from the encoder network which uses a sigmoid activation function. The loss term consists of the reconstruction loss and the β-total correlation KLD (β-TCKLD) term with α = γ = 1 and β = 8 [46,50]. We employ minibatch stratified sampling on the given data while training. To optimize the loss, Nesterov-accelerated Adaptive Moment Estimation(Nadam) was used and it efficiently minimizes the loss during the training of the β-TCVAE model [50,53]. The default parameters provided by the Keras library and a learning rate of 0.00001 were chosen. The training is performed over 100 epochs with a batch size of 33 × 33 = 1089 for both lattice size N = 64 and N = 128 with a number of sample 1024 per phase point. The reduced descriptors of the 2D Ising spin configuration are given by the latent variables [50]. The β-TCVAE model used in this work was implemented using the Keras ML library with TensorFlow as a supporting backhand [54,55]. D. Principal Component Analysis on the Latent Space The principal components analysis (PCA) is applied on the latent means and standard deviations of the Ising spin configurations obtained after fitting the β -TCVAE, independently to produce linear transformations of the Gaussian parameters that discriminate between the samples using the scikit-learn package [16,56]. The PCA performs an orthogonal transformation into a new basis set of linearly uncorrelated features, principal components. Each principal component encompasses the largest possible variance across the sample space under an orthogonality constraint [16]. The latent representation characterizes the structure of the Ising configurations, but the principal components of the latent representation shows greater discrimination between the different structural characteristics of the configurations compared to the raw variable representations [50]. The rationale of using the PCA is to provide a more compact representation which characterizes the different phases of the Ising model. As we will show in the results section, the first and the second dominated components already show good signals to distinguish different phases of the anisotropic Ising model. IV. RESULTS As explained in the previous section, for the VAE models, the samples were drawn from a multi-dimensional Gaussian distribution parametrized by vectors of means and standard deviations denoted as µ i and σ i respectively, where i is the index of the dimension of the distribution. All of the plots in this paper were generated with the MatPlotLib package using a perceptually uniform colormap [57]. In each plot, the coloration of each square pixel on the diagram represents the ensemble average value of the principal components of the latent space mean or variance of the multi-dimensional Gaussian function. We study two different system sizes N = 64, and 128. We will focus on the first two principal components of the latent variance and the second principal component of the latent mean. We denote them as τ 0 , τ 1 , and ν 1 respectively. The first principal component of the latent mean, ν 0 does not capture a clear distinction between the ferromagnetic and paramagnetic phases of the system. ν 1 are plotted in fig. 2. It is apparent that ν 1 is a resemblance of the magnetization, m, of snapshots of the Ising spin configuration. We note that it is not to be expected that any of the latent variables to have the same value of any physical quantities, such as magnetization. Nonetheless, from the plot of ν 1 , it clearly discriminates the ferromagnetic from paramagnetic phases, separating them by the phase transition line. The phase transition line in white corresponds to the analytical solution, eq. A14 in the appendix calculated in the thermodynamic limit. Since the magnetization can be seen as the order parameter for the 2-dimensional Ising model, a reasonable representation of the order parameter is seen to be possibly extracted from the VAE. We note that the simulations and the VAE are done on finite size systems, a true broken symmetry does not occur, that is the reason of the seemingly random values of ν 1 in the ferromagnetic phase. Usually, the amplitude of the magnetization is considered for finite size simulation, as we will show other latent variables from the VAE has similar structure as the amplitude of the magnetization. As the magnetization is a linear feature of the Ising spin configuration, a simpler linear model would be sufficient for extracting the magnetization. The first principal component of the latent variance τ 0 are plotted in fig 3. The white line is the analytical solution for the phase transition points. One can find that the value of τ 0 remains very small in the upper right region of the figures. This can be considered as the reflection of the small changes of amplitude of the energy or amplitude of the magnetization in the paramagnetic phase. Once the systems approach the critical line from the upper right region of the figure, the value of τ 0 increase sharply. This behavior is again consistent with the amplitude of energy or magnetization. In particular, we consider the case for the isotropic limit (K x = K y ), that is the classical (Γ = 0) limit. The critical point is given as [58]. From the fig. 3, we see a sharp change in the τ 0 around the value of K c . This result is consistent with the accomplishments of prior published works for the isotropic Ising model on a square lattice [23][24][25]50]. K c (= K x = K y ) = 2 ln[1+ √ 2] ∼ 2.2721 Given that the paramagnetic samples are essentially noise due to entropic contributions. Therefore, these happens to be easy to discriminate from the rest of the samples using a β-TCAVE model. In view of the fact, the samples with ν 1 values corresponding to nearly zero magnetization and rather high values for τ 0 resemble Gaussian noise with no notable order preference. An interesting question is to look for quantity which may resemble to response function, such as susceptibility and heat capacity. We plot the second principal component of the latent variance τ 1 in fig. 4. The phase transition line is again included to show the transition in the thermodynamic limit for comparison. The values of τ 1 are much larger along the analytical critical line than the other combinations of K x and K y . Thus τ 1 bears a strong resemblance to the magnetic susceptibility. From the plots of τ 0 , τ 1 , and ν 1 , we find that the VAE is capable to generate quantities which resembles to the magnetization, amplitude of magnetization/energy, and magnetic susceptibility/heat capacity, respectively. From these quantities we can draw the phase transition line between the paramagnetic phase and the ferromagnetic phase. The quality of fitting the Ising spin configurations to the present VAE model can be quantified by the values of the loss functions. The three losses represented V. DISCUSSION AND CONCLUSIONS We use the β-TCVAE to extract structural information from raw Ising spin configurations. It exposes interesting derived descriptors of the configurations used to identify the second order phase transition line among other regions. The analysis here shows the interpretation of the extracted sample space as represented by the latent variables. This is done by studying the behavior of the latent variable mappings of the Ising spin configurations with respect to the anisotropic coupling (J x , J y ) associated with the temperatures. We find that ν 1 , the second principal component of the latent mean, is considered to reflect the magnetization of the 2-dimensional anisotropic Ising model, hence it is interpreted as an indicator for the magnetization exhibited by the configurations. In contrast, τ 0 and τ 1 , which are the first two principal components of the latent variance, can be seen as an indicator of the amplitude of magnetization or energy and the magnetic susceptibility or heat capacity. Thus both τ 0 and τ 1 can provide a reasonable estimate of the second order phase transition line. Since the d + 1-dimensional anisotropic Ising model is equivalent to the d-dimensional quantum spin system through the Suzuki-Trotter transformation. This method can be trivially extended to other 1D quantum systems [59]. Moreover, methods for strongly correlated systems, such as the dynamical mean field theory (DMFT) and their cluster generalizations-the dynamical cluster approximation (DCA) and the cellular dynamical mean field theory (CDMFT)-have very similar data structure in their Hirsch-Fye impurity solver [60][61][62]. It is notoriously difficult to obtain the putative quantum critical point from the paramagnetic solution of the Hubbard model as there is no simple quantity to track the transition [63,64]. Therefore, the method developed here would be an important tool for analyzing the data from the DMFT, DCA, and CDMFT. There are many opportunities for further developing this method, not only investigating more complex systems, but also by introducing improvements which are beyond the scope of this work. Finite size scaling turns to be an important approach to address the limitations in solving finite-sized systems for investigating regions near critical phenomena [65,66]. A correspondence established between the VAE encodings of different system sizes is a challenging argument, as different VAE structures need specific training for each system size, which in turn demands for different hyperparameters and training iterations, counts to obtain equivalent analysis and hence similar answers [65]. Numerical difficulties tend to arise while performing finite-size scaling analysis, because the variation of predicted properties with respect to the system size is hard to isolate from the systemic variation due to different neural networks trained with different hyperparameters is being used to extract the specific macroscopic properties. However, this would play a significant role towards improving VAE characterization of critical phenomena. Another interesting direction is to use the generative adversarial network (GAN) [67] instead of VAE as the generative model, promising results have been obtained for the isotropic 2D Ising model [68]. In this appendix, we summarize the derivation of the critical points for the anisotropic 2-dimensiaonl Ising model via the self-duality property of the model [29,69,70]. We closely follow the lecture notes by Muramatsu in the following [70]. Considering the partition function of the Ising model (where K = 1 βJ ) Z = {Sj } e K <j,l> Sj S l = {Sj } <j,l> e KSj S l = {Sj } <j,l> 1 r=0 C r (K)(S j S l ) r (A1) where C 0 (K) = cosh K and C 1 (K) = sinh K. Applying the simple transformation, for each bond < j, l >, a new Z 2 variable, r is introduced. We label the new variable as r µ with µ ≡ (i, < i, j >), labelling it with the site i from which the bond < i, j > emanates. The partition function thus follows Z = {Sj } {rµ} <j,l> C rµ (K) i S <i,r µ )] (A3) We define a dual lattice, where the vertices of the dual lattice are set in the center of the plaquettes defined by the original lattice. We have vanishing contributions due to the presence of the Kronecker delta in many configurations. Defining the new Z 2 variables σ i = ±1 on the sites of the dual lattice. We can associate for each link of the original lattice, there is a pair of σ i 's (e.g. on the sites i and j on the dual lattice). Therefore, the variable r µ is expressed as: r µ = 1 2 (1 − σ i σ j ) (A4) where sites i and j on the dual lattice are those where the link crosses r µ . The sum of r µ is over the four nearest neighbors of a site i, we have <i,j> r µ = 2 − 1 2 (σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 4 + σ 4 σ 1 ) (A5) There are 2 4 possible configurations for the four variables σ 1 , ..., σ 4 . They are grouped in four cases and all the cases lead to an even number for the sum of r µ over the nearest neighbor bonds. The choice of variables needs to satisfy the δ−function. The partition function becomes Z = 1 2 2 N {σi} <j,l> C [(1−σiσj )/2] (K),(A6) where N is the number of sites on the lattice and the product is now over the bonds associated in the dual lattice. The expression of the partition function shows that the weight for each configuration of the σ's is given by the coefficients C(K). Hence, introducing C(K) to a form that resembles a Boltzmann weight. C r (K)= cosh(K)[1 + r(tanh(K) − 1)] (A7) = cosh(K) exp(ln[1 + r(tanh(K) − 1)]) = cosh(K) exp(r ln tanh(K)) = cosh(K) exp[ 1 2 (1 − σ i σ j ) ln tanh(K)] = [cosh(K) sinh(K)] 1/2 exp(− 1 2 ln tanh Kσ i σ j ) The partition function becomes Z = 1/2(sinh 2K) −N {σi} exp(K <j,l> σ j σ l ).(A8) There are 2N bonds and defining the new coupling constant asK ≡ − ln tanh(K/2) (A9) (where K = 1 βJ ) coupling for the Ising model on the dual lattice. The Ising model is self-dual since the duality transformations brings it into itself. We consider the free energy per site f (Z) = − 1 N ln Z. (A10) According to the relation between the partition functions of the original and the dual model, we can write f (K) = ln sinh(2K) + f (K). (A11) This is a strong constraint on the free energy. Since, sinh(2K) is an analytic function, the equation has a singularity in f (K), which corresponds to a singularity in f (K).K(K) is a monotonous function of K, hence it holdsK c = K c . We have K c = 1 2 ln(1 + √ 2).(A12) The self-duality has allowed us to calculate the exact value of the critical temperature in the 2-dimensional isotropic Ising model(K x = K y ), where K x = 1 βJx and K y = 1 βJy . Generalizing the results obtained in the isotropic to the anisotropic one, i.e. when couplings K x = K y in the respective directions. The anisotropic case is as follows K y ≡ − 1 2 ln tanh(K x ),K x ≡ − 1 2 ln tanh(K y ). (A13) Given K x and K y there is only one critical point, summarizing both the equations above with the following condition for a critical line separating the ordered from the disordered phase in the anisotropic Ising model sinh(2K xc ) sinh(2K yc ) = 1. (A14) Appendix B: β-TCVAE Loss The expression of the total loss for the VAE is given by L(φ, θ; x, z) = L RC + L KLD ,(B1) where the reconstruction error (RC) and the Kullback-Leibler divergence (KLD) are defined as L RC = −E z∼q φ (z|x) [log p θ (x|z)](B2) and L KLD = D KL [q φ (z|x)||p(z)](B3) respectively. If the prior distribution of the latent representation is Gaussian, the VAE provides disentangled factors in the latent representation, which means the significant dimension of the latent space is largely independent of each other. In β-total correlation VAE (β-TCVAE), we try to improve the disentanglement of factors in the representation by decomposing the KLD term and apply tuning parameters to them independently [49,50]. Each training sample is identified with a unique integer index n ∈ 1, 2, .., N and assigned a uniform random variable in this decomposition. The aggregated posterior, q φ (z) = n q φ (z|n)p(n) captures the aggregate structure of the latent variables under the distribution of the input, where q φ (z|n) = q φ (z|x n ) and q φ (z, n) = q φ (z|n)p(n) = 1 N q φ (z|n). The decomposition is given as I(z; x) + D KL [q φ (z)|| j q φ (z j )] + j D KL [q φ (z j )||p(z j )] (B4) The first term is the index-code mutual information (MI), I(z; x) = D KL [q φ (z, n)||q φ (z)p(n)], between the input and the latent variable, which is based on the empirical input distribution q φ (z, n). The second term is a mea-sure of the dependence between the latent variables and is called the total correlation(TC). It is important to produce representations which penalize the total correlation and hence force the model towards discovering statistically disentangled factors in the input distribution. The third term prevents the individual latent variables in the representation from deviating far from their priors. It is called the dimension-wise KLD [50]. After adding the tuning parameters to the decomposition, the β-TC modified KLD term becomes :- L β−T C = αI(z; x) + βD KL [q φ (z)|| j q φ (z j )] +γ j D KL [q φ (z j )||p(z j )] (B5) Modulating only the β parameter shows the greatest effect on disentanglement in the latent representation given by empirical evidence [49]. = 1 σ 1i,k =−1 |σ i,k , ..., σ N,k σ i,k , ..., σ N,k | where k = 1, 2, ..., M . Hence, it is the product of M exponentials in Z. FIG. 1 . 1Diagram depicting the structure of the VAE. The left hand side is the encoding part and the right hand side is the decoding part. x is the input Ising spin configuration, q φ (z|x) is the encoder neural network, µ and σ are the latent means and standard deviations of the latent space distribution, z. p θ (x |z) is the decoder neural network and x is the reconstructed Ising spin configuration. FIG. 2 . 2The second principal component of the latent mean, ν1, with respect to the different Kx and Ky (1/(βJx) and 1/(βJy)) for the 2D square lattice. The left panel and right panel are for the system sizes N = M = 64 and N = M = 128, respectively. FIG. 3. The first principal component of the latent variance, τ0, with respect to the different Kx and Ky (1/(βJx) and 1/(βJy)) for the 2D square lattice. The left panel and right panel are for the system sizes N = M = 64 and N = M = 128, respectively. FIG. 4 . 4The second principal component of the latent variance, τ1, with respect to the different Kx and Ky (1/(βJx) and 1/(βJy)) for the 2D square lattice. The left panel and right panel are for the system sizes N = M = 64 and N = M = 128, respectively. FIG. 5. β-TCVAE loss and the reconstruction loss settle to about 0.5 and the latent loss settles close to 0 for 100 epoch cycles. The left panel and right panel are for the square lattice of size N = M = 64 and N = M = 128, respectively. in fig 5 are the VAE loss, reconstruction loss, and the latent loss. The reconstruction loss converges to about 0.5. The latent loss is obtained from the β-TCKLD term. This loss converges quickly to a value near to 0. The total β-TCVAE loss for the 2-dimensional anisotropic Ising model for both lattice size N = 64 and N = 128 settles quickly around 0.5. is based upon work supported by NSF DMR-1728457. This work used the high performance computational resources provided by the Louisiana Optical Network Initiative (http://www.loni.org), and HPC@LSU computing. NW research work at Louisiana State University was supported by NSF DMR-1728457. JM is partially supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0017861. 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[ "INSTITUTE FOR MATHEMATICS AND PHYSICS THEORETICAL PHYSICS DIVISION Towards learning optimized kernels for complex Langevin", "INSTITUTE FOR MATHEMATICS AND PHYSICS THEORETICAL PHYSICS DIVISION Towards learning optimized kernels for complex Langevin" ]
[ "Daniel Alvestad ", "Rasmus Larsen ", "Alexander Rothkopf " ]
[]
[]
We present a novel strategy aimed at restoring correct convergence in complex Langevin simulations. The central idea is to incorporate system-specific prior knowledge into the simulations, in order to circumvent the NP-hard sign problem. In order to do so, we modify complex Langevin using kernels and propose the use of modern auto-differentiation methods to learn optimal kernel values. The optimization process is guided by functionals encoding relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. For the strongly coupled quantum anharmonic oscillator we achieve correct convergence up to three-times the real-time extent of the previous benchmark study. An appendix sheds light on the fact that for correct convergence not only the absence of boundary terms, but in addition the correct Fokker-Plank spectrum is crucial.
10.1007/jhep04(2023)057
[ "https://export.arxiv.org/pdf/2211.15625v2.pdf" ]
254,044,114
2211.15625
0091f006417743279024fee719e9e5017b03f398
INSTITUTE FOR MATHEMATICS AND PHYSICS THEORETICAL PHYSICS DIVISION Towards learning optimized kernels for complex Langevin Daniel Alvestad Rasmus Larsen Alexander Rothkopf INSTITUTE FOR MATHEMATICS AND PHYSICS THEORETICAL PHYSICS DIVISION Towards learning optimized kernels for complex Langevin Alvestad et al. RESEARCH ARTICLE -PREPRINT Full list of author information is available at the end of the articlestochastic quantizationcomplex Langevinsign problemreal-timemachine learning We present a novel strategy aimed at restoring correct convergence in complex Langevin simulations. The central idea is to incorporate system-specific prior knowledge into the simulations, in order to circumvent the NP-hard sign problem. In order to do so, we modify complex Langevin using kernels and propose the use of modern auto-differentiation methods to learn optimal kernel values. The optimization process is guided by functionals encoding relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. For the strongly coupled quantum anharmonic oscillator we achieve correct convergence up to three-times the real-time extent of the previous benchmark study. An appendix sheds light on the fact that for correct convergence not only the absence of boundary terms, but in addition the correct Fokker-Plank spectrum is crucial. Motivation Strongly correlated quantum systems underlie some of the most pressing open questions in modern theoretical physics. Whether it is the transport of highly energetic partons through a liquid of deconfined quarks and gluons [1], created in heavy-ion collisions [2] or the transport of non-relativistic fermions [3], captured in the iconic Hubbard model [4] at low energies. When formulated in Minkowski time, quantum field theories so far have defied a treatment by conventional Monte-Carlo simulation techniques, due to the presence of the notorious sign problem [5,6]. And while progress has been made in extracting real-time dynamics from Euclidean time simulations using e.g. Bayesian inference [7], the sign problem prevails by rendering the extraction ill-posed and equally exponentially hard. The sign problem has been proven to be NP-hard [8], which entails that no generic solution method is likely to exist. In turn, if we wish to make inroads towards overcoming the sign problem, system-specific solutions are called for. Over the past decade, several approaches to tackle the sign problem have been put forward [5,9]. They can be divided into system-specific and system-agnostic arXiv:2211.15625v2 [hep-lat] 18 Apr 2023 approaches. The reformulation strategies discussed e.g. in Refs. [10][11][12] are an example of the former class, where the partition function of the original system is re-expressed in terms of new degrees of freedom, for which no sign problem exists. While highly successful in the systems for which a reformulation has been discovered, no systematic prescription exists to transfer the approach to other systems. The other approaches, among them reweighting, extrapolation from sign-problem free parameter ranges [13][14][15][16], density of states [17][18][19], tensor networks [20,21], Lefschetz thimbles [22][23][24] and complex Langevin (CL) [25,26] all propose a generic recipe to estimate observables in systems with a sign problem. As the NP-hard sign problem however requires system-specific strategies, all of these methods are destined to fail in some form or the other. Be it that their costs scale excessively when deployed to realistic systems (e.g. reweighting, Lefschetz thimbles, tensor networks) or that they simply fail to converge to the correct solution (complex Langevin). Both the Lefschetz Thimbles and complex Langevin belong to the class of complexification strategies [9]. They attempt to circumvent the sign problem by moving the integration of the Feynman integral into the complex plane. After complexifying the degrees of freedom, the former proposes to integrate over a specific subspace on which the imaginary part of the Feynman weight remains constant (thimble), while the latter proposes to carry out a diffusion process of the coupled real-and imaginary part of the complexified degrees of freedom. In this paper our focus lies on the complex Langevin approach, as it has been shown to reproduce correctly the physics of several strongly correlated model systems, albeit in limited parameter ranges [27]. Most importantly in its naive implementation it scales only with the volume of the system, similar to conventional Monte-Carlo simulations. In the past, complex Langevin had suffered from two major drawbacks: the occurrence of unstable trajectories, called runaways and the convergence to incorrect solutions. In a previous publication [28] we have shown how to avoid runaways by deploying inherently stable implicit solvers (c.f. the use of adaptive step size [29]). In this study we propose a novel strategy to restore correct convergence in the complex Langevin approach. One crucial step towards establishing complex Langevin as reliable tool to attack the sign problem is to identify when it converges to incorrect solutions. The authors of ref. [30] and later [31] discovered that in order for CL to reproduce the correct expectation values of the underlying theory, the histograms of the sampled degrees of freedom must fall off rapidly in the imaginary direction. Otherwise boundary terms spoil the proof of correct convergence. The absence of boundary terms has been established as necessary criterion and efforts are underway [32] to compensate for their presence to restore correct convergence. With QCD at the center of attention, the gauge cooling strategy [33,34], based on exploiting gauge freedom, has been proposed. It has recently been amended by the dynamic stabilization approach [35,36], which modifies the CL stochastic dynamics with an additional drift term. Both are based on the idea that by pulling the complexified degrees of freedom closer to the real axis, boundary terms can be avoided. Their combination has led to impressive improvements in the correct convergence of complex Langevin in the context of QCD thermodynamics with finite Baryo-chemical potential [37] and is currently explored in the simulation of real-time gauge theory [38]. We focus here on scalar systems formulated in real-time on the Schwinger-Keldysh contour (for a Lefschetz thimble perspective see [39,40]). For scalars, gauge freedom does not offer a rescue from the convergence problems. The fact that dynamical stabilization introduces a non-holomorphic modification of the drift term means that the original proof of convergence is not applicable, which is why we refrain from deploying it here. Furthermore the boundary term correction requires that the eigenvalues of the Fokker-Planck equation associated with the original system lie in the lower half of the complex plane, which is not necessarily the case in the scalar systems that we investigate. The convergence problem in real-time complex Langevin is intimately connected with the extent of the real-time contour [41]. In a previous publication [28] we showed that for a common benchmark system, the strongly correlated quantum anharmonic oscillator, real-time simulations directly on the SK contour are feasible for times up to mt max = 0.5. Convergence quickly breaks down when extending the contour beyond this point. Within the complex Langevin community, coordinate transformations and redefinitions of the degrees of freedom have been used in the past to weaken the sign problem in a system specific manner (see e.g. discussion in [42]). All of these reformulations can be captured mathematically by introducing a so called kernel for complex Langevin. It amounts to a simultaneous modification of the drift and noise contribution to the CL stochastic dynamics. In the past it has been used to improve the autocorrelation time in real-valued Langevin simulations [26] and has been explored in simple model systems to restore the convergence of complex Langevin (see e.g. [43]). The construction of the kernels, as discussed in the literature applies to a specific system only and so far no systematic strategy exists to make kernels work in more realistic theories. Our study takes inspiration from both conceptual and technical developments in the machine learning community. In machine learning, an optimization functional, based on prior knowledge and data is used to train an algorithm to perform a specific task. The algorithm depends on a set of parameters, e.g. the weights of a neural network, which need to be tuned to minimize the prescribed optimization functional. Highly efficient automatic differentiation programming techniques [44] have been developed to compute the dependence of the outcome of complex algorithms on their underlying parameters. Here we utilize them to put forward a systematic strategy to incorporate prior knowledge about the system into the CL evolution by learning optimal kernels. In section 2 we review the concept of kernelled Langevin, first in the context of Euclidean time simulations and subsequently for use in complex Langevin. In section 3 we show how the concept of a kernel emerges in a simple model system and how it relates to the Lefschetz thimbles of the model. Subsequently we discuss that a constant kernel can be used to restore convergence of real-time complex Langevin for the quantum harmonic oscillator. The kernel found in this fashion will help us to improve the convergence of the interacting theory too. Section 4 introduces the central concept of our study: a systematic strategy to learn optimal kernels for complex Langevin, based on system-specific prior information. Numerical results from deploying a constant kernel to the quantum anharmonic oscillator are presented in section 4.3 (Source code for the kernel optimization and simulation is written in Julia and available at [45]), leading to a significant extension of correct convergence. In the appendices, we discuss some of the limitations of constant kernels and show in the context of simple models that correct convergence requires not only the vanishing of boundary terms but in addition requires the spectrum of the associated Fokker-Plank equation to remain negative. Neutral and non-neutral modifications of Langevin dynamics Stochastic quantization, the framework underlying Langevin simulations, sets out to construct a stochastic process for fields in an artificial additional time direction τ L with a noise structure, which correctly reproduces the quantum statistical fluctuations in the original theory. In the context of conventional Monte-Carlo simulations in Euclidean time, where expectation values of observables are given by the path integral O = 1 Z Dφ O[φ]e −S E [φ] , S E [φ] = d d xL E [φ],(1)0 dτ L O[φ(τ L )]. The simplest stochastic process, which realizes this goal and which is therefore commonly deployed is dφ dτ L = − δS E [φ] δφ(x) + η(x, τ L ) with η(x, τ L ) = 0, η(x, τ L )η(x , τ L ) = 2δ(x − x )δ(τ L − τ L ).(2) Its drift term is given by the derivative of the action S E and the noise terms η are Gaussian. The associated Fokker-Planck equation reads F FP = d d x ∂ ∂φ(x) ∂ ∂φ(x) + δS E [φ] δφ(x) , ∂Φ(φ, τ L ) ∂τ L = F FP Φ(φ, τ L ).(3) For an in-depth review of the approach see e.g. ref. [26]. In the following we will discuss the fact that there exists the freedom to introduce a so called kernel into eq. (3), which as a purely real quantity allows us to modify the above Fokker-Planck equation without spoiling the convergence to the correct stationary solution Φ [φ] = lim τ L →∞ Φ[φ, τ L ] ∝ exp − S E [φ] . One may use this freedom to improve autocorrelation times of the simulation and for other problemspecific optimizations as has been explored in the literature. Subsequently we will turn our attention to the case of complex Langevin, where the simplest stochastic process proposed by stochastic quantization is not guaranteed to converge to the correct solution. In that case we will explore how a reparametrization of the associated Fokker Planck equations through in general complex kernels can be used to not only change the convergence speed but actually to change the stationary distribution itself, allowing us to recover correct convergence where the naive process fails. Kernelled Real Langevin As alluded to above, there exists a freedom to reparametrize the Fokker-Planck eq. (3) by introducing a real-valued kernel function K ij (x, x , φ; τ L ) F FP = i,j d d x d d x ∂ ∂φ i (x) K ij (x, x , φ; τ L ) ∂ ∂φ j (x ) + δS E [φ] δφ j (x ) .(4) Written in its most generic form, it may couple the different degrees of freedom of the system (according to the ij indices), it may couple different space-time points (according to its x and x dependence) and may depend explicitly both on the Langevin time τ L , as well as the field degrees of freedom φ. The corresponding Langevin equation reads dφ i (x, τ L ) dτ L = j − d d x K ij (x, x ; φ) δS E [φ] δφ j (x , τ L ) + d d x δK ij (x, x ; φ) δφ j (x , τ L ) + d d x H ij (x, x ; φ)η(x , τ L ) with K(x, x ; φ) = k d d x H ik (x, x ; φ)H jk (x , x ; φ),(5) where in the last equation we assume that K is factorizable. In practice we will either choose kernels, which can be factorized using the square root of their eigenvalues or will start directly by constructing the function H that can be combined into an admissible K. Let us gain a bit of intuition about the role of the kernel when considering it in its simplest form, a constant scalar kernel, which multiplies each d.o.f. with a real number γ. Inspecting eq. (5) we find that, as it appears in front of the drift term and as square root in front of the noise term, γ simply leads to a redefinition of the Langevin time coordinate τ L = γτ L . While the stationary solution is left unchanged, the convergence time has been modified. Even for more general kernels, the fact that K appears in the generalized Fokker-Planck eq. (4) on the outside of the parenthesis ∂ ∂φi(x) + δS E [φ] δφi(x) tells us that the stationary distribution remains unchanged. It goes without saying that choosing K ij (x, x ; φ) = δ ij δ(x − x) we regain the standard Langevin eq. (2). Kernelled Complex Langevin Let us now consider the application of stochastic quantization to complex-valued path integrals, in particular to those describing real-time physics in Minkowski time. Here the observables are given by Feynman's path integral O = 1 Z Dφ O[φ]e iS M [φ] , S M [φ] = d d xL M [φ],(6) which houses the Minkowski time action of the theory S M . Stochastic quantization in this case proposes to modify the real-valued stochastic process of eq. (2) via the substitution −S E → iS M such that dφ dτ L = i δS M [φ] δφ(x) + η(x, τ L ) with η(x, τ L ) = 0, η(x, τ L )η(x , τ L ) = 2δ(x − x )δ(τ L − τ L ).(7) It is obvious that even if one starts out with purely real degrees of freedom at τ L = 0, the presence of the complex drift term necessitates the complexification φ = φ R + iφ I , each of which will obey a coupled stochastic evolution. In the complexified scenario, the question of correct convergence is not as simple to answer as in the purely real case. The most stringent criterion refers to whether complex Langevin reproduces the correct expectation values lim τL→∞ 1 τ L τL 0 dτ L O[φ R + iφ I ] ? = 1 Z Dφ O[φ]e iS M [φ](8) of the theory, defined on the right. And indeed it has been found that the dynamics of eq. (7) may violate the equal sign of eq. (8). I.e. complex Langevin converges, but it does not converge to the correct solution. In this study we set out to recover correct convergence by introducing kernels into the complex Langevin dynamics. To this end we consider a not-necessarily real kernel function K(x, x ; φ) which enters the complexified dynamics as dφ dτ L = d d x iK(x, x ; φ) δS M [φ] δφ(x , τ L ) + ∂K(x, x ; φ) ∂φ(x , τ L ) + H(x, x ; φ)η(x, τ L ) with η(x, τ L ) = 0, η(x, τ L )η(x , τ L ) = 2δ(x − x )δ(τ L − τ L ) and K(x, x ; φ) = d d x H(x, x ; φ)H(x , x ; φ).(9) Expressed as two separate but coupled stochastic processes for the real-and imaginary part of the complexified field we obtain dφ R dτ L = d d x Re K[φ]i δS M [φ] δφ + δK[φ] δφ + Re [H[φ]] η φ=φ R +iφ I , dφ I dτ L = d d x Im K[φ]i δS M [φ] δφ + ∂K[φ] ∂φ + Im [H[φ]] η φ=φ R +iφ I .(10) Note that at this point we are dealing with two different concepts of Fokker-Planck equations. One describes how the probability distribution Φ[φ R , φ I ] of the real-and imaginary part φ R , φ I of the complexified field evolve under eq. (9) ∂Φ ∂τ L = ∂ ∂φ R H R + ∂ ∂φ I H I 2 − ∂ ∂φ R Re iK ∂S M ∂φ + ∂K ∂φ − ∂ ∂φ I Im iK ∂S M ∂φ + ∂K ∂φ Φ = L K Φ.(11) We define the operator for the real Fokker Planck equation, which has been separated into real and imaginary part, as L K , not to be confused with the original now complex Fokker Planck equation F F P , which was only defined on the real part of φ. For the term quadratic in derivatives, we have split the kernel K into the product of H functions, as shown in eq. (9), such that each derivative acts on either the real or the imaginary part of H respectively. Since it is the noise term of the Langevin eq. (9) that translates into a term quadratic in derivatives in the Fokker-Planck language, it is there that H appears in eq. (11). It is important to recognize that the correct late Langevin-time distribution of this Fokker-Planck equation is purely real and therefore is not related in a trivial manner to the Feynman weight exp[iS M ] of the original path integral, as has been established in simple models in the literature as discussed e.g. in refs. [46][47][48][49][50]. The other Fokker-Plank equation is not a genuine Fokker-Planck equation, in the statistical sense, as it does not describe the evolution of a real-valued probability density P [φ, τ L ] but instead that of a complex-valued distribution ρ(φ, τ L ) ∂ ∂τ L ρ(φ, t) = F F P ρ(φ, τ L ),(12)F F P = i,j d d x d d x ∂ ∂φ i (x) K ij (x, x , φ; τ L ) ∂ ∂φ j (x ) − i δS M [φ] δφ j (x ) . It is this equation whose late time limit we expect to reproduce the Feynman weight lim τ L →∞ ρ(φ, τ L ) = exp[iS M ] and we will refer to in the following as the complex Fokker-Planck equation. Significant progress in the understanding of the convergence properties of complex Langevin had been made starting with ref. [30] in the form of so-called correctness criteria. The criteria most often discussed in the literature are boundary terms (for a detailed exposition see Refs. [30,31]). They tell us if the expectation value calculated from the real distribution Φ(φ R , φ I ; τ L ) (eq. (11)), which we can sample using the CL, is the same as the expectation value obtained from the complex distribution ρ(φ; τ L ). The latter one can only be obtained from solving the complex Fokker-Planck equation, eq. (12). The two expectation values, O Φ (τ L ) = O ρ (τ L ) only agree if Φ(φ R , φ I ; τ L ) falls off exponentially fast. If it does not fall of sufficiently fast, it will produce boundary terms and the equal sign in eq. (8) is not valid. This criterion is however not sufficient as it does not guarantee the equilibrium distribution of the complex Fokker-Planck equation to be exp[iS M ]. These two criteria combined are however sufficient to claim convergence of the CL to the true solution. For a proof that the correctness criterion still holds after introducing a kernel into the CL, we revisit the proof in appendix A. How can a kernel help to restore the correct convergence? Not only do we need to make sure that no boundary terms arise in sampling eq. (10) but also that the complex Fokker-Planck equation has a unique and correct complex stationary distribution. I.e. we need in general a non-neutral modification of the complex Langevin dynamics. If we were to introduce a real-valued kernel, similarly to the case of conventional real-valued Langevin, we will be able to change the speed of convergence but not the stationary solution. On the other hand, since the drift term is complex, there is no reason not to consider also complex valued kernels, which will act differently on the stochastic process for φ R and φ I , representing a genuine non-neutral modification in the corresponding Fokker-Planck eq. (11). Similarly in the complex Fokker-Planck eq. (12) the presence of a complex K ij can change the stationary distribution through a reshuffling of the associated eigenvalues, as is discussed in more detail in appendix A. For a comprehensive discussion of different modifications to complex Langevin, including kernels, see also ref. [42]. In the following sections we will start off with constructing an explicit example of a field-independent kernel that improves convergence in the free theory and find that it can restore correct convergence in the interacting theory to some degree. We will then continue to present our novel strategy to learn optimal kernels for the restoration of correct convergence and showcase their efficiency in a benchmark model system. Subsequently we discuss the limitations of field-independent kernels and shed light on how kernels connect to the correctness criteria. A field independent kernel for real-time complex Langevin In this section, we will manually construct one specific field-independent kernel and demonstrate its use to improve convergence in real-time simulations of the quantum anharmonic oscillator. The form of the kernel is motivated by insight gained in a simple one d.o.f. model and reveals an interesting connection between kernelled Langevin and the thimble approach. Since in the following only low dimensional model systems are considered, we will refer to the dynamical degrees of freedom from now on as x. 3.1 A kernel for the simplest real-time model Following ref. [43] let us investigate the simplest model of real-time physics, the integrals x n = 1 Z dx x n exp[− 1 2 ix 2 ], Z = dx exp[− 1 2 ix 2 ].(13) Attempting to solve this expression using the complex Langevin approach for x(τ L ), leads to a stochastic process dx dτ L = −ix + η,(14) with Gaussian noise η. Equation (14) fails at reproducing correct values of x n . We can understand this failure by recognizing that without regularization the original integral in eq. (13) is not well defined and this lack of regularization is inherited by the Langevin eq. (14). One way to proceed is to explicitly modify the action by introducing a regulator term, such as x 2 . The integral becomes well-defined and its value is obtained when we let → 0 at the end of the computation. In a numerical setting this would require to explicitly include the regulator term, carry out the corresponding simulation for different values of and extrapolate → 0. There are two drawbacks to this strategy: first it requires several evaluations of the simulation, which can be expensive for large systems. The second reason is that the relaxation time for the simulation grows, the smaller becomes, and hence in practice we cannot make arbitrarily small. Let's consider an alternative strategy of solving the integral of eq. (13), which relies on contour deformations, the so-called Lefschetz thimble method. We will carry out a change of variables in the integral which moves the integration path into the complex plane and which in turn will weaken the oscillatory nature of the integrand. This method is based on a continuous change of variables according to the following gradient descent equation dx dτ = dS E [x] dx ,(15) which complexifies the degree of freedom,x = a + ib. Equation (15) evolves the formerly real-valued x towards the so called Lefschetz thimble which is the optimal contour deformation where the imaginary part of the action stays constant. Following the steps outlined in [51], we solve the flow of eq. (15) analytically which givesx(x, τ ) = x(cosh(τ ) − i sinh(τ )). For large values of τ it leads tõ x(x, τ ) τ 1 ≈ x(1 − i) 1 2e −2τ = x 2e −2τ e −i π 4 .(16) The above equation tells us that the optimal thimble in this system lies on the downward 45 • diagonal in the complex plane z(x) = xe −i π 4 . On this contour the integrand of the original integral eq. (14) reduces to a real Gaussian e −x 2 for which no regularization is required. If we flow for just a very small τ = , we obtain on the other hand cosh(τ ) − i sinh(τ ) ≈ 1 − i and dx exp[− 1 2 ix 2 ] = dx ∂x ∂x exp[− 1 2 ix 2 ](17)= (1 − i ) dx exp[− 1 2 ix 2 (1 − i ) 2 ] ≈ (1 − i ) dx exp[− 1 2 ix 2 − x 2 ]. We see that the term here takes on the role of a regulator in the action but due to its presence also in the Jacobian, the value of the integral is not changed. This is different from introducing the regulator only in the action itself. Hence the obvious benefit of the deformation method is that we can introduce a regulator to tame oscillations without the need to extrapolate that regulator in the end. The closer we approach the optimal thimble, the easier the integral will be to solve numerically. How can such a coordinate transformation be implemented in complex Langevin? Intuitively the action in the integral is what influences the drift in complex Langevin and the measure is related to the noise structure. The above tells us that the change we introduced will therefore affect the drift quadratically, while it occurs in the noise linearly. Thinking back to eq. (9), we see that this is just how a field-independent kernel modifies the complex Langevin equations. For the optimal thimble with z(x) = xe −i π 4 the modification in the drift therefore becomes K = e −i π 2 = 1 i and for the noise H = √ K = √ −i. This leads to the 14) and (right) simulation after introducing the kernel in eq. (18). The color of the scatter points refers to the number of measurements recorded at the corresponding position, a lighter color indicates a larger number. Note that the optimal kernel has moved the sampling onto the single thimble present in this simple system. following stochastic process dx dτ L = −x + √ −iη,(18) which had been identified as optimal already in ref. [43]. This stochastic process converges to the correct solution of the integral eq. (13). Interestingly the imaginary unit has disappeared from the drift term since the kernel K exactly canceled it there and instead moved it over into the noise term. As the last step, let us show explicitly that the choice of kernel above indeed amounts to a coordinate transform. Following [42] we have dx dτ L = − HH T ∂S E (x) ∂x + Hη (19) ⇒ H −1 dx dτ L = −H T ∂S E (x) ∂x + η (20) ⇒ du dτ L = −H T (H T ) −1 ∂T (u) ∂u + η = − ∂T (u) ∂u + η.(21) Here x = Hu and T (u) = S E (Hu) = S E (x). We find that introducing a kernel K = HH T in the evolution equation for x has the same effect as carrying out a coordinate transformation to u = H −1 x. As an example of the complex Langevin dynamics in the absence (left panel) and presence (right panel) of the kernel discussed above, we show the corresponding scatter plots in fig. 1. The kernel has indeed rotated the noise into the direction of the thimble, along which the system now samples. Note that while the naive CL dynamics have been implemented using the semi-implicit Euler-Maruyama scheme to avoid runaways, we are able to carry out the kernelled dynamics with a fully explicit solver without adaptive step size. The reason is that on the deformed contour the integral has already been regularized. This result shows that in the simple model discussed here we can find a kernel that both restores correct convergence of the complex Langevin dynamics and at the same time removes the need for a regulator. Both are related to the fact that the kernel effectively instituted a coordinate transform that amounts to a deformation of the integration contour into the complex plane. In the next section, we will consider a similar kernel for the harmonic oscillator. We investigate the relation between the Lefschetz thimbles and kernel controlled complex Langevin in appendix B.2, where a similar analysis is performed in a system where a λ 4 x 4 term has been added to the action. As such a one-degree-of-freedom model has a non-linear drift term, a constant kernel in that case will not suffice to remove the imaginary unit from the drift term, and hence the complex Langevin will not sample directly on the thimble as was the case for the example above. A kernel for the harmonic oscillator When constructing a kernel for the harmonic oscillator, we encounter similar difficulties related to stability and convergence of complex Langevin process as in the previous section. In order to see how an optimal kernel can be chosen we revisit the discussion originally found in refs. [26,43]. The continuum action of this one-dimensional system is given by S M = dt 1 2 ∂x(t) ∂t 2 − 1 2 m 2 x 2 (t) = dt 1 2 x(t) − ∂ 2 t − 1 2 m 2 x(t) ,(22) In quantum mechanics the fields φ are the position x and the coordinates, previously called x, are the time t. The corresponding complex Langevin equation reads dx(t, τ L ) dτ L = −i ∂ 2 t + m 2 x(t, τ L ) + η(t, τ L ).(23) In the absence of a regularization, this stochastic process is unstable and does not show convergence to the correct result. In analogy with the results for the simple model system in the previous section we will argue analytically that correct convergence can be achieved in this system via a kernel with the property − ∂ 2 t + m 2 K(t − t ) = iδ(t − t ) . This kernel will render the drift term trivial, proportional to x itself and move all complex structure into the noise. Following [26,43] we solve eq. (23) analytically and obtain for the two-point correlator in Fourier space x(ω, τ L )x(ω , τ L ) = δ(ω + ω ) i ω 2 − m 2 e i(ω 2 −m 2 )|τ L −τ L | − e i(ω 2 −m 2 )(τ L +τ L ) .(24) Obviously this expression does not have a well defined value in the late Langevintime limit. Introducing an explicit regulator of the form i x(t) 2 yields S M = dx 1 2 ∂ 0 φ(x)∂ 0 φ(x) − (m 2 − i )φ 2 (x)(25) and improves the situation, as now the stochastic process correctly converges to lim τ L →∞ x(ω, τ L )x(ω , τ L ) = δ(ω + ω ) i ω 2 − m 2 + i .(26) A careful analysis of the associated Fokker-Planck equation in ref. [52] however reveals that the relaxation time towards the correct solution scales with 1/ . I.e. carrying out a CL simulation based on a small regulator will lead to slow convergence. In addition one also needs to take the limit → 0 as well as ∆τ L → 0, which may not commute [53]. Since the action of the harmonic oscillator in Fourier space decouples into a collection of non-interacting modes we may deploy a similar strategy for each mode as we considered in the simple model of the preceding section. I.e. we introduce a kernel, which moves the integration onto the single thimble for each mode. ∂ ∂τ L x(ω, τ L ) = iK(ω) δS M [x] δx(ω) + K (ω)ξ(ω, τ L ),(27)ξ(ω, τ L ) = 0, ξ(ω, τ L )ξ(ω , τ L ) = 2δ(ω + ω )δ(τ L − τ L ).(28) This train of thought leads us to choose the following field-independent kernel, which had been explored in ref. [43] beforẽ K(ω) = iA(ω) ω 2 − m 2 + i , K(t) = dω (2π)K (ω)e −iωt ,(29) where A(ω) is a real, positive and even function of ω. Thus for a constant A(ω), K(ω) is nothing but the propagator of the free theory in momentum space. The corresponding correlation function is found to read lim τL→∞ φ(ω, τ L )φ(ω , τ L ) = δ(ω + ω )K (ω) A(ω) = δ(ω + ω ) i ω 2 − m 2 + i(30) which is the correct result. The most important difference to simply introducing a regulator in the action however lies in the fact that now the relaxation time for each mode is proportional to 1/A(ω) and not proportional to 1/ and no extrapolation in needs to be carried out. For completeness let us note the corresponding coordinate space complex Langevin process ∂ ∂τ L x(t, τ L ) = i dt K(t − t ) δS M [x] δx(t ) + χ(t, τ L ),(31)χ(t, τ L ) = dωe iωt K (ω)ξ(ω, τ L ).(32) A kernel for real-time Langevin on the thermal SK contour The analytic study of the one d.o.f. model and the harmonic oscillator have provided us with insight into how a kernel can be used to both satisfy the need for regularization of the path integral and achieve convergence to the correct solution of the associated complex Langevin equation in practice. t x + (t) x -(t) t max β x(ɣ) = {x + , ɣ<t max x -, t max <ɣ<2 tmax x E 2t max < ɣ } x E (-iτ) Re[〈x²(ɣ)〉] Re[〈x(0)x(ɣ)〉] Im[〈x(0)x(ɣ)〉] Im[〈x²(ɣ)〉] real-time domain Euclidean domain Figure 2 (left) Sketch of the Schwinger-Keldysh contour deployed in our study with forward x + (t) and backward x − (t) branches on the real-time axis, connected to an imaginary time branch x E (−iτ ). The contour parameter γ is used to address all branches in a unified manner. (right) Sketch of the visualization of our observables along the contour parameter γ for the example of mtmax = 1. The analytic solution of the real-and imaginary part of the equal time correlator x 2 (γ) and the unequal time correlator x(0)x(γ) will be plotted in the real-time γ < 2mtmax and subsequently Euclidean domain γ > 2mtmax. We will now construct the corresponding kernel for the harmonic oscillator at finite temperature, discretized on the Schwinger-Keldysh contour. Numerical simulations will confirm the effectiveness of the kernel in the non-interacting theory. The Schwinger-Keldysh contour for a quantum system at finite temperature encompasses three branches. The forward branch along the conventional time axis reaches up to a real-time t max and the degrees of freedom associated with it are labeled x + (t). The backward branch with x − (t) returns to the initial time t 0 in reverse and the Euclidean branch which houses x E (−iτ ) and extends along the negative imaginary time axis. The physical length of the imaginary time branch dictates the inverse temperature of the system. A sketch of our contour setup is shown in the left panel of fig. 2. In the action of the system, the integration over time is rewritten into an integration over a common contour parameter γ. The d.o.f. on the different branches are then distinguished by the values of the contour parameter x(γ) and we will drop the superscript in the remainder of the text. As sketched in the right panel of fig. 2, we will refer to the equal-and unequaltime two-point correlation functions along the SK contour in the following, plotted against the contour parameter. The reader can identify the values along the forward and backward branch as being mirrored, connecting to the values on the Euclidean branch that show the expected periodicity of a thermal theory. When discretizing the action for use in a numerical simulation the direction of each branch of the SK contour is encoded in a contour spacing a i ∈ C. Computing the drift term for an arbitrary contour yields i ∂S M [x] ∂x j = i 1 2 (|a j | + |a j−1 |) x j − x j−1 a j−1 − x j+1 − x j a j − 1 2 [a j−1 + a j ] ∂V (x j ) ∂x j .(33) This expression simplifies if we use a constant magnitude step-size |a i | = |a|, such that the prefactor in the above equation can be reduced to i |a| . In that case we can go over to a convenient matrix-vector notation i∇ x S M [x] = 1 |a| iM x,(34) where M jk =        1 aj−1 + 1 aj − 1 2 [a j−1 + a j ] m 2 , j = k − 1 aj , j = k − 1 − 1 aj−1 , j = k + 1.(35) Based on the findings in the previous sections the form of the optimal discrete free theory kernel in coordinate space will be the inverse propagator K = H H T = iM −1 ,(36) where H is the factorized kernel used in the noise term. The form of this kernel relies on the matrix M to be invertible, and M −1 to be factorizable, both of which holds. We obtain H = √ iM −1 by using the square root of the eigenvalues. Written in differential form with Wiener processes dW , the corresponding Langevin equation reads dx = 1 |a| i M iM x dτ L + 2 i M dW = − 1 |a| xdτ L + 2 i M dW ,(37) which leaves us with a complex non-diagonal noise coefficient 2i M and a drift term pointing in the direction of −x. Let us demonstrate the effect of this kernel by carrying out a simulation for the following parameters. We discretize the canonical SK contour with N t = 50 points on the forward and backward branch each and N τ = 5 points on the imaginary branch. Note that we do not introduce any tilt here. Choosing a mass parameter m = 1, the imaginary branch extends up to mτ max = 1. As real-time extent, we choose mt max = 10. The value chosen here is arbitrary as the kernelled dynamics of the free theory are stable and converge for any real-time extent. The results of the simulation without a kernel are given in the top panel of fig. 3 and rely on the implicit Euler-Maruyama scheme to avoid the occurrence of runaway solutions. The results with our choice of kernel are shown in the bottom panel and were obtained using a simple forward-stepping Euler scheme at ∆τ L = 10 −3 without adaptive step size. In each case we generate 100 different trajectories, saving configurations at every m∆ τ L = 0.1 in Langevin time up to a total of mτ L = 100. Each panel in fig. 3 showcases four quantities plotted against the contour parameter γ. Their values for 0 < mγ < 10 are obtained on the forward branch, those for 10 < mγ < 20 on the backward branch and the small piece 20 < mγ < 21 denotes the Euclidean time results. The real-and imaginary part of the equal time expectation value x 2 (γ) are plotted as green and pink data points respectively. The real-and imaginary-part of the unequal time correlator x(0)x(γ) on the other hand are plotted as orange The results without a kernel show both deviations from the correct result and exhibit relatively large uncertainties. The reason lies in the slow relaxation rate to the correct result due to the presence of an explicit regulator. Here the regulator is provided by our use of an implicit numerical scheme (Euler-Maruyama with implicitness parameter θ = 1.0 and an adaptive step-size with a maximum step size of 10 −3 ), but could equally well be introduced by adding a small term i x 2 to the system action. Using a stronger regulator, e.g., tilting the contour, would yield a shorter relaxation time, but any such explicit regulator distorts the results away from the actual → 0 physical solution. It is interesting to note that it is the equal time observable x 2 that is performing the worst. We will see later on that this is the hardest observable to accurately reproduce. For the bottom plot we use the free theory propagator kernel, eq. (36). This simulation now aligns excellently with the true solution for all the observables. After applying the kernel, the problem is regularized and thus less stiff, and we can revert to using a fixed step-size explicit Euler-Maruyama scheme. The step-size here is dτ L = 10 −3 . The fact that we do not need to impose an explicit regulator term is important as in this case we only need to take the limit ∆τ L → 0 to obtain a physical result, and do not need to extrapolate the regulator term to zero ( → 0). This might be important considering the recent work in ref. [54], which shows that one encounters subtleties in taking the limit of the regulator → 0. A kernel for the quantum anharmonic oscillator Not only did the kernel in the free theory change the convergence behavior of the complex Langevin simulation, it also removed the need of a regulator in the action. The obvious next step is to explore the interacting theory where the problem of convergence to the wrong solution is more severe. The potential term in the action is now given by V (x) = 1 2 mx(t) 2 + λ 4! x(t) 4 ,(38) where we use m = 1 and λ = 24. This choice of parameters has been deployed in the past as benchmark for strongly-interacting real-time complex Langevin in refs. [28,41]. As ref. [41] formulated the real-time dynamics on a tilted contour they found correct convergence up to t max = 0.8, while ref. [28] worked with an untilted contour and observed onset of incorrect convergence already above t max = 0.5. In the following we will remain with an untilted contour. We find that using the free kernel of eq. (36) the convergence to the correct solution can be extended slightly to around t max = 0.75. If in addition we modify the free theory kernel by rescaling the contributions from the kinetic term with a common prefactor g and modify the mass term away from the free theory value m M jk (g, m g ) =        g aj−1 + g aj − 1 2 [a j−1 + a j ] m 2 g , j = k − g aj , j = k − 1 − g aj−1 , j = k + 1.(39) convergence can be pushed up to t max = 1.0 by using the heuristically determined parameter values g = 0.8 and m g = 1.8. The CL equation we simulate is given by dx = 1 |a| i M (g, m g ) i M (1, m)x + λ 3 x 2 x dτ L + 2i M (g, m g ) dW .(40) We carry out simulations, assigning N t = 10 points to the forward and backward branches each and N τ = 10 points to the imaginary branch of the contour. Here we use the implicit Euler-Maruyama scheme with implicitness parameter θ = 0.5. Even though we do not need a regulator in the presence of the kernel, the system retains some of its stiffness in contrast to the free theory. The use of an explicit scheme with e.g. adaptive step size is possible, however we find it more efficient to rely on an implicit scheme, as it allows the use of much larger Langevin step sizes. The results of two simulations with a maximum real-time extent t max = 1.0 are shown in fig. 4. One is carried out without a kernel and using an implicit scheme (top) and the other in the presence of a kernel based on the parameters g = 0.8 and m g = 1.8 (bottom). The graphs show the real and imaginary part of the equal time x(t)x(t) (green and pink data points) and unequal time correlator x(0)x(t) (orange and blue datapoints) plotted against the contour parameter γ. For 0 < mγ < 1 and 1 < mγ < 2 it refers to the forward and backward branch of the contour and for 2 < mγ < 2.9 denotes the imaginary time branch. The top panel indicates that naive complex Langevin fails to converge to the correct solution at this real-time extent of mt max = 1. It is interesting to point out the failure of CL at two specific points along the contour, the first one is the starting point at γ = 0, which is connected by periodic boundary condition to the Euclidean path. Then at the turning point of the contour at maximum realtime extent, corresponding to mγ = 1, the real part of the x 2 observable lies significantly away from the true solution. This points seems to be most affected by the convergence problem of the CLE. In the lower panel, the simulation in the presence of the modified free theory kernel is presented. The outcome of the kernelled complex Langevin evolution is very close to the correct solution and shows only small statistical uncertainties. Note however that especially the observable x 2 still shows some deviation from the true result beyond the statistical error bars indicating that exact correct convergence has not yet been achieved [1] The above results are promising, as they indicate that in principle the convergence problem of real-time complex Langevin can be attacked by use of a kernel. At the same time explicitly constructed kernels, such as the modified free theory kernel are limited in the range of real-time extent in which they are effective. The question at hand is how to systematically construct kernels that will restore convergence at even larger real-time extent. Learning optimal kernels In this section, we introduce our novel strategy to systematically construct kernels to improve the convergence of real-time complex Langevin. Our goal is to overcome the limitations of explicitly parametrized kernels, such as the one of eq. (39). While optimal parameter values g and m g were found for this kernel, they only achieved correct convergence for a limited mt max ≤ 1. Most importantly it is not clear how to systematically modify that kernel for realizing convergence at larger real-time extent. Instead we set out to use a generic parametrization of the kernel. We propose to use an expansion in a set of complete basis functions of the dynamical d.o.f. In this study, as a proof of principle, we will restrict ourselves to a field-independent kernel, which can be understood as the first term in an expansion in powers of the field. This field-independent kernel for the quantum anharmonic oscillator on the Schwinger-Keldysh contour will take the form of a τ L independent matrix K with (2N t + N τ ) 2 entries, multiplying the 2N t d.o.f. on the forward and backward contour and the N τ ones on the imaginary time branch. It is the values of these matrix entries that we set out to tune in order to achieve optimal convergence. And even though simple model systems indicate that a field-dependent kernel is needed to achieve correct convergence in case of strong complex drift terms, we find that an optimal field-independent kernel can already extend the range of convergence of the anharmonic oscillator out to mt max = 1.5, three times larger than the previous record set for CL in ref. [41]. In order to obtain kernel values that restore correct convergence, we formulate an optimization problem based on a cost functional, which incorporates prior knowledge about the system of interest. Taking advantage of modern programming techniques that allow us to compute the dependence of a full complex-Langevin simulation on the entries of the kernel we propose to iteratively learn the optimal kernel. The fact that we incorporate prior information into the simulation opens a novel path to beat the notorious sign problem, i.e. for the first time complex Langevin can be amended by system-specific information in order to restore correct convergence. The optimization functional In order to guarantee that a complex Langevin simulation converges to the true solution we must fulfill the correctness criteria of [30]. First we must ensure the absence of boundary terms and second that the late-time distribution of the complex Fokker-Planck equation is indeed exp[iS M ]. Constructing a loss function for both criteria however is only feasible for very low dimensional models, as it entails calculating the eigenvalues and eigenvectors of the complex Fokker-Planck operator, which is prohibitively expensive already for the anharmonic oscillator discussed here. Instead we will retain only the first ingredient of the correctness criteria, the absence of boundary terms and use other prior information in order to guide the kernelled complex FP equation to the correct stationary solution. The boundary terms can be calculated via the expectation value L c O Y where L c is the Langevin operator and O refers to any observable (for a detailed discussion see e.g. [55]). In appendix A we demonstrate that the correctness criterion still holds with a kernel and how to calculate these boundary terms. Besides the boundary terms, we often possess additional relevant prior information about the system at hand. We can e.g. compute correlation functions in Euclidean time using conventional Monte-Carlo methods. In addition, we know that in thermal equilibrium the correlation functions on the forward and backward branch are related due to the KMS relation. In order to exploit this prior information it is vital for the CL equations to be formulated on the canonical SK contour, whose real-time branches lie parallel to each other and connect to the Euclidean branch at the origin. In a tilted contour setup, access to the Euclidean branch is limited and the comparison of the values on the forward and backward branch is much more involved. In addition, symmetries provide powerful constraints to the simulation, as e.g. time-translation invariance in a thermal system renders local observables such as x n (γ) constant along the full contour. We quantify the distance of the simulated result from the behavior dictated by prior knowledge via a loss function L prior . The comparison is carried out on the level of expectation values of observables, where apriori known values from conventional Euclidean simulations are referred to as O MC and those from the complex Langevin simulation in the presence of a kernel by O K . In principle one can distinguish between four categories of prior knowledge: • Euclidean correlators (L eucl ), which are accessible via conventional Monte-Carlo simulations: L eucl = O dτ O(τ ) K − O(τ ) MC 2 /σ 2 O(τ ) K τ ∈ imaginary time • Model symmetries (L sym ), which exploit that the expectation values of observables O must remain invariant under a symmetry transformation T ξ governed by a continuous (or discrete) parameter ξ: L sym = O dξ| T ξ O K − O K | 2 /σ 2 O(τ ) K • Contour symmetries (L rt ), which arise predominantly in systems in thermal equilibrium: L C = O dγ O C + (γ) K −F [ O C − (γ) K ] 2 /σ 2 O(τ ) K with F analytically known • Boundary terms (L BT ), which can be explicitly computed from the outcome of the kernelled Langevin simulation: L BT = O L c (K)O K 2 /σ 2 O(τ ) K In practice one wishes to combine as many of these different contributions as possible. To this end they must be added as dimensionless quantities. This is why each of the terms above is normalized by the variance of the complex Langevin simulation. In order for the combined functional to provide a meaningful distinction of the success of convergence (also in the case of e.g. the free theory as shown in fig. 3) we propose to introduce an overall normalization for the combined prior functional L prior = N tot L eucl + L sym + L C + L BT .(41) There is an element of arbitrariness in what overall normalization to choose, and we find that the best distinction between wrong and correct convergence is achieved if one uses the relative error of the most difficult observable to reproduce. In case of the systems studied here this amounts to the relative error of the equal time correlator obtained in the complex Langevin simulation with respect to the correct known value from Euclidean simulations N tot = max γ {σ x 2 K (γ)/ x 2 MC (γ)}. We carry the subscript K in the expectation values above, in order to emphasize that the loss functional depends implicitly on the choice of kernel used in the underlying complex Langevin simulation. The number of observables O contained in the cost functional is not specified here and depends on the problem at hand. In practice, we find that often including the apriori known Euclidean one-and twopoint functions already allow us to reliably distinguish between correct and incorrect convergence. In the next section, we will discuss both fully general numerical strategies to locate the minimum of the optimization functional, as well as an approximate lowcost approach, which we have deployed in the present study. Let us characterize the kernel via a set of variables κ. We emphasize that this does not limit the general nature of the approach, as κ may refer to the prefactors of a general expansion of the kernel in a complete set of basis functions. To efficiently locate its critical point we deploy standard numerical optimization algorithms, which utilize the information of the gradient of the functional with respect to the parameters of the kernel. The computational challenge lies in determining the gradient robustly. In the continuum, the gradient of the loss reads ∇ κ L prior [K] =2 G[x(K), K] | G[x(K), K] | ∇ κ G[x(K), K](42) =2 G[x(K), K] | G[x(K), K] | { ∇ x G[x(K), K] · ∇ κ x + ∇ κ G[x, K] } (43) In order to evaluate eq. (43) we need to compute the change in the field x(K), which depends on the kernel. This requires taking the gradient of the CL simulation itself. While a demanding task, dedicated methods to evaluate such gradients have been developed, which underpin the recent progress in the machine learning community. They are known as differential programming techniques (for an in-depth review see e.g. ref. [44]). As a first option, we considered using direct auto-differentiation [2] on the full loss function, as we are dealing with the standard setting of estimating the gradient of a highly dimensional functional whose output is a single number. For small systems with a number of degrees of freedom O(10), forward-auto-differentiation is feasible as it requires multiple runs of the full CL simulation. As the number of independent d.o.f. grows, backward-auto-differentiation offers us to reduce the number of necessary simulation runs, trading computational cost for increased memory demands to store intermediate results of the chain rule it computes internally. We find that already for the quantum anharmonic oscillator this direct computation of the gradient is too costly and thus not practical. A more advanced approach, which promises to avoid the cost and memory limitations of direct auto-differentiation are so-called sensitivity analysis methods, such as e.g. adjoint methods for stochastic differential equations. A detailed discussion of these methods is beyond the scope of this paper and the interested reader is referred to refs. [58][59][60] for further details. We find that for the specific case of real-time complex Langevin, these methods in their standard implementation, as provided e.g. in [60] are challenged in estimating the gradient robustly. We believe that the difficulty here lies in the stiffness of the underlying stochastic differential equation. One possible way out is to deploy sensitivity analysis methods specifically developed for chaotic systems, such as Least Square Shadowing algorithms, discussed e.g. in Refs. [61,62]. While these methods at [2] Auto-differentiation is a method to compute derivatives to machine precision on digital computers based on an efficient use of the chain rule, exploiting elementary arithmetic operations in the form of dual variables (see e.g. [56]). We have used the Julia library Zygote.jl [57] and ForwardDiff.jl for computing gradients. this point are still too slow to be deployed in CL simulations, the rapid development in this field over the past years is promising. Our survey of differential programming techniques indicates that while possible in principle, the optimization of the loss functional L(K) is currently plagued by issues of computational efficiency. We believe that implementing by hand the adjoint method for the real-time complex Langevin systems considered here will offer a significant improvement in speed and robustness compared to the generic implementations on the market. This line of work goes beyond the scope of this manuscript and will be considered in an upcoming study. To make headway in spite of these methods limitations we in the following propose an approach to compute an approximate low-cost gradient, which in practice allows us to significantly reduce the values of the optimization functional. A low cost update from an heuristic gradient Our goal is to compute a gradient, which allows us to approximately minimize the cost functional L prior [K] without the need to take derivatives of the CL simulation. The approach we propose here relies on using a different optimization functional, whose form is motivated by the need to avoid boundary terms. While updating the values of the kernel according to a heuristic gradient obtained from this alternative functional, we will monitor the values of the true optimization functional, selecting the kernel which achieves the lowest value of L prior [K]. We saw that the optimal kernel for the free theory reduces the drift term to a term in the direction of −x. This drift term points towards the origin. In this spirit we construct a functional that penalizes drift away from the origin. The starting point is the following expression, where we define D = −iK∂S M /∂x as the drift term modified by the kernel D(x, K) · (−x) = ||D(x, K)||||x|| cos θ. Here cosθ denotes the angle between the drift and the optimal direction. As we wish to align the drift and −x, our optimization problem becomes finding a kernel K such that min K {D(x, K) · (−x) − ||D(x, K)|| ||x||} .(45) We can write down different loss functionals which encode this minimum L D = D(x) · (−x) − ||D(x)|| ||x|| ξ = 1 T D(x(τ L ))·(−x(τ L ))−||D(x)|| ||x|| ξ(46) The choice of ξ determines how steep the gradients on the functional are and we find that in practice a value between 1 < ξ < 2 leads to most efficient minimization, when L D is used to construct the heuristic gradient we describe below. Note that turning the drift towards the origin differs from the strategy employed by dynamic stabilization. The scalar counterpart to minimizing the unitarity norm is driving the values of the complexified x towards the real axis. In addition, in dynamical stabilization a non-holomorphic term is added to the action. Here the CL equation is modified only by a kernel, which still leads to a holomorphic complex Langevin equation that leaves the correctness criteria intact. The exact gradient of the functional L D of eq. (46) also contains the costly derivatives over the whole CL simulation. However we find that in practice for values 1 ≤ ξ ≤ 2 in L D these contributions can be neglected. We believe the reason to lie in the fact that L D consists of the difference between two terms that contain the same powers of x. I.e. we find that carrying out the optimization using only the explicit dependence of L D on the kernel K, which is computed using standard auto-differentiation. The approximate gradient allows us to locate kernel values, which significantly reduce the values of the true optimization functional L prior [K]. The kernels identified in this way in turn achieves correct convergence on contours with larger real-time extent than previously possible. [3] . The full optimization scheme can be summarized as follows: 1 Initialize the kernel parameters yielding the initial kernel K 1 2 Carry out the CL simulation with K 1 and save the configurations {x j } 1 , where the subscript indicates that this is the first iteration 3 Compute the values of the loss functions L D and L prior [K 1 ] 4 Compute the gradients of the loss function L D ({x j } 1 , K 1 ) with respect to the kernel parameters using auto-differentiation 5 Update the kernel parameters using one step of the ADAM optimization scheme 6 Rerun the CL simulation with the new kernel K i+1 and save a new set of configurations {x j } i+1 7 Loop over step 3 -6 for N steps, or until L D have reached a minimum and then select the kernel parameters with the smallest L prior [K i ] We will demonstrate the efficiency of the proposed optimization based on the heuristic gradient in the next section, where we learn optimal kernels for the quantum harmonic and anharmonic oscillator on the thermal Schwinger-Keldysh realtime contour. Learning optimal kernels for the thermal harmonic oscillator To put the strategy laid out in the previous section to a test we set out here to learn a field-independent kernel for the quantum harmonic oscillator on the canonical Schwinger-Keldysh contour at finite temperature. In section 3.3 we had identified one kernel by hand, which actually minimizes the low-cost functional eq. (46). We will compare it to the learned kernel at the end of this section. We simulate on the canonical Schwinger-Keldysh contour with real-time extent mt max = 10 and an imaginary time branch of length mβ = 1. The contour will be discretized with steps of equal magnitude |a i | = |a| such that N t = 25 points are assigned to the forward and backward branch each and N τ = 5 to the imaginary time axis. [3] Note that disregarding the costly terms in the gradient of the true cost functional L prior [K] introduced in section 4.1 did not lead to a viable minimization of its values. The field-independent kernel therefore is a complex 55 × 55 matrix, which we parametrize via two real matrices A and B such that K = e A+iB . This choice is arbitrary and is based on the observation that the minimization procedure is more robust for the exponentiated matrices than when using A + iB directly. The kernel is initialized to unity before the start of the optimization by setting all elements of A and B to zero. The optimization itself, as discussed in the previous section, is carried out using the approximate gradient following from L D with a choice of ξ = 2. One needs to choose the actual cost functional L prior based on prior knowledge through which to monitor the optimization success. We decide to include the known values of the Euclidean two-point correlator x(0)x(−iτ ) and exploit the knowledge about the symmetries of the system, which require that x = x 3 = 0, as well as x 2 (γ) = x 2 (0) . This leads us to the following functional L prior = N tot (47) i∈SK | x(γ i ) K | 2 /σ 2 xK + | x 3 (γ i ) K | 2 /σ 2 x 3 K + | x 2 (0) MC − x 2 (γ i ) K | 2 /σ 2 x 2 K + i∈Eucl. | x(0)x(τ i ) MC − x(0)x(τ i ) K | 2 /σ 2 xxK where the first sum runs over all points of the discretized Schwinger-Keldysh contour, while the second sum only contains the correlator on the Euclidean branch. As discussed before, the overall normalization is based on the uncertainty of the equal-time x 2 correlator. Since we start from a trivial kernel, we must make sure that our simulation algorithm provides a regularization and remains stable even for stiff dynamics. Therefore we solve the complex Langevin stochastic differential equation using the Implicit Euler-Maruyama scheme with implicitness parameter θ = 1.0 and adaptive stepsize. For every update of the CL configurations we simulate 30 different trajectories up to a Langevin time of mτ L = 30, with a thermalization regime of mτ L = 5 in Langevin time before we start collecting the configurations at every m∆ τ L = 0.05 in Langevin time. To calculate the expectation values, we compute sub-averages from the saved configurations in each trajectory separately. The final mean and variance are then estimated from the results of the different trajectories. The iterative optimization of the kernel values, based on the low-cost functional and its approximate gradient, is performed using the ADAM (Adaptive Moment) optimizer with a learning rate of 0.001. This is an improved gradient descent optimizer, which combines gradient descent momentum and an adaptive learning rate. Since we know that the complex Langevin simulation will be the slowest part of the optimization scheme we will only run the full CL simulation for every five optimization steps. For this simple model it would not be a computation time problem to update the expectation values in L D after every kernel update, but for realistic models in higher dimensions this might be too expensive. As the distributions of the observables should be similar for a small change in the kernel we indeed find that not updating the CL configurations at every update steps still allows us to obtain a good estimate of the heuristic gradient. Starting with the unit kernel, the functionals L D = 6.58 × 10 11 and L prior = 107 show appreciable deviation from zero. After 32 steps of the ADAM optimizer we manage to find values of K which reduce the value of L prior = 26.5 indicating that the apriori known information has been well recovered. The results for the simulation with the optimal learned kernel are plotted in fig. 5, based on 100 trajectories (top) and 400 trajectories (bottom) each of which progresses up to mτ L = 100 in Langevin time . The x-axis refers to the contour parameter γ, such that at mγ = 10 we are at the turning point of the real-time branch of the Schwinger-Keldysh contour and at mγ = 20 the contour has returned to the origin, before extending along the imaginary axis to mt = −i. We plot the real-and imaginary part of the unequal time correlator x(0)x(γ) as orange and blue data points, while the real-and imaginary part of the equal time expectation value x 2 (γ) are given in green and pink respectively. The analytically known values from solving the Schrödinger equation are underlaid as black solid lines. How has the learned kernel improved the outcome? When comparing to a simulation without kernel in the top panel of fig. 3 we see that using the same amount In the optimal learned kernel based on the low-cost update we have subtracted the unit matrix from the real-part to avoid it dominating the other structures. We find that the learned kernel exhibits some of the structure of the manually constructed kernel but in general has a more simple form, which nevertheless manages to achieve correct convergence of the complex Langevin dynamics. of numerical resources (i.e. 100 trajectories at mτ L = 100) the learned kernel has reduced the resulting errorbars significantly. On the other hand in the top panel of fig. 5 residual oscillations in x 2 seem to persist. One may ask whether these indicate incorrect convergence, which is why we provide in the lower panel the result after including 400 trajectories at the same Langevin time extent. One can see that not only the errorbars further reduce but also that the oscillatory artifacts have diminished. The improvement amounts to another factor of two in terms of L prior from the value L prior = 26.5 in the top panel to L prior = 13.4 in the lower panel. We emphasize that we did not use the analytically known solution of the system for the optimization procedure. Let us inspect the learned kernel and compare it to the free theory propagator kernel of eq. (36). In fig. 6 we visualize the structures of the kernel by plotting a heat-map of the matrix entries of the complex matrix kernel. The right sketch shows how the matrix is structured, where the top panel refers to the real part and the lower panel to the imaginary part. The entries of the matrices are laid out corresponding to the contour parameter γ. The smaller regions inside the two panels indicate how the kernel mixes points along the time contour. The ++ corresponds to the mixing of the forward branch of the contour, while +− mixes the forward and backward branch time points. There exists also a small strip involving the Euclidean points, mixing with the real-time points (E+ and E−), as well as a small corner (EE) mixing within the Euclidean points. The different regions shown in the sketch can easily be recognized in the two kernel structure plots. Note that we have subtracted the unit matrix from the real-part of the optimized kernel to more clearly expose off-diagonal structures, if present. The manually constructed free theory propagator kernel (middle), as expected from being the inverse free propagator, exhibits an oscillatory pattern. It leads to a significant coupling between the forward and backward time points, due to an anti-diagonal structure in the real and imaginary parts, forming an oscillatory cross pattern. This anti-diagonal behavior is much less pronounced in the optimized kernel (left). In its real-part it mainly exhibits a diagonal which is not as wide as in the manually constructed kernel. There is however a small negative structure present, an off-diagonal band, similar to the black part in the middle panel. For the imaginary part, the patterns close to the diagonal are similar between the manually constructed kernel and the optimal learned kernel. Both possess a diagonal close to zero and a broad sub/super-diagonal that switches sign at the turning points between the ++ and −− part of the time contour. We also see that the anti-diagonal structure is similar for a very short part in the +− and −+ quadrants in the imaginary panel. The rest of the +− and −+ quadrant seems to contain noise. While some similarities exist between the explicit kernel and the optimal learned kernel, it appears that correct convergence requires some non-trivial structure in the imaginary part of K. The learned kernel achieves correct convergence with much less structure than the manually constructed one. Learning optimal kernels for the strongly coupled anharmonic oscillator After successfully testing the learning strategy for a field-independent kernel in the free theory in the previous section, we are now ready to attack the central task of this study: learning an optimal kernel for a strongly coupled quantum system in order to extend the correct convergence of the corresponding real-time complex Langevin simulation. We deploy the same parameter set as before with m = 1 and λ = 24. In section 3.4 we showed that for a real-time extent of mt max = 1 an explicit kernel based on insight from the free theory can be constructed, which allows us to restore correct convergence within statistical uncertainties (see fig. 4). Here we set out to learn an optimal kernel based only on the combination of our low-cost functional and prior knowledge of the Euclidean two-point functions and time-translation invariance of the thermal system. Since we restrict ourselves to a field-independent kernel we expect that our approach will be able to improve on the manually constructed kernel but will itself be limited in the maximum real-time extent up to which correct convergence can be achieved. As testing ground we selected three different real-time extents, mt max = 1, mt max = 1.5 and mt max = 2, all of which show convergence to the wrong solution when performing naive complex Langevin evolution. We discretize the real-time contour with a common magnitude of the lattice spacing |a i | = |a|. I.e. depending on the maximum real-time extent the number of grid points changes. E.g. in case of mt max = 2 we use N t = 20 on the forward and backward part of the real-time contour each, and N τ = 10 for the imaginary part of the contour. Due to the stiffness of the complex Langevin equations in the interacting case, all CL simulations are performed with the Euler-Maruyama scheme with θ = 0.6 and adaptive step-size. We simulate 40 different trajectories up to mτ L = 40 in Langevin time, computing observables at every m∆τ L = 0.02 step. The setup for learning the optimal kernel is very similar to that in the previous section. The kernel parametrization is given by K = e A+iB , where A and B are real matrices. We search for the critical point of the true loss function L prior (see eq. (47)) via the heuristic gradient obtained from the loss function L D of eq. (46). We find that minimization proceeds efficiently, when choosing the parameter ξ = 1 in L D . The optimal kernel is chosen according to the lowest values observed in L prior . We find that for a trivial unit kernel where complex Langevin fails, the cost functional L prior based on prior information indicates values of L prior mtmax=1 = 942, L prior mtmax=1.5 = 597320 and L prior mtmax=2 = 12923. The left column of fig. 7 shows from top to bottom the rows correspond to results of the naive CL simulation for mt max = 1, mt max = 1.5 and mt max = 2 respectively. As in previous comparison plots the real-and imaginary part of the unequal time correlation function x(0)x(γ) is given by orange and blue data points, while the real-and imaginary part of the equal time expectation value x 2 (γ) is represented by the green and pink symbols respectively. The analytically known values from solving the Schrödinger equation are underlaid as black solid lines. The results of real-time CL in the presence of the optimal learned kernel for the anharmonic oscillator are shown in the right column of fig. 7. For mt max = 1 we achieve to lower the value of L prior mtmax=1 = 14.3. At this low value all the correlation functions plotted, agree with the true solution within uncertainties. Note that we manage to restore correct convergence for the unequal time correlation function on the real-time axis, even though no prior information about these points was provided in L prior nor L D . In contrast to the use of the modified free theory kernel, we see here that x 2 does not show a systematic shift on the real-time branches anymore. We continue to the second row, where, via an optimal learned kernel, we achieve extending the correctness of CL into a region inaccessible to the modified free theory kernel at mt max = 1.5. The value of the functional encoding our prior knowledge has reduced to L prior mtmax=1.5 = 48.1. We find that the unequal time correlation function values are reproduced excellently, while the real-and imaginary part of x 2 show residual deviations from the correct solution around those points along the SK contour, where the path exhibits sharp turns, i.e. at the end point γ = t max and the point where the real-time and Euclidean branch meet γ = 2t max . The results shown in the third row clearly spell out the limitation of the fieldindependent kernel we deploy in this study. At mt max = 2 we do not manage to reduce the value of the cost functional below L prior mtmax=2 = 759. Correspondingly in the bottom row of fig. 7 it is clear that CL even in the presence of the field-independent kernel fails to converge to the correct solution. Interestingly the imaginary part of the unequal-time two-point correlator still agrees very well with the true solution on the forward branch while its real part already shows significant deviations from the correct solution. This deviation of the unequal time correlation function affects also the values of the equal-time correlation function which is far from constant and thus leads to a penalty in L prior , correctly indicating failure of correct convergence. There are two possible reasons behind the failure of convergence at mt max = 2. One is that the low-cost gradient obtained from L D is unable to bring the kernel close to those values required for restoring correct convergence. The other is that the field-independent kernel is not expressive enough to encode the change in CL dynamics needed to restore correct convergence. In simple models it is e.g. known from ref. [63] that field-independent kernels may fail to restore correct convergence for large imaginary drift. We believe that, as a next step, the investigation of field dependent kernels is most promising. The unequal time correlation function is most relevant phenomenologically, as it encodes the particle content and occupation numbers in the system. We thus compare in fig. 8 the values of x(0)x(t) along the forward real-time extent of the contour for mt max = 1.0, 1.5 and 2.0 to the correct solution given as black solid line. Here we can see in more detail that for a real-time extent of 1 and 1.5 CL with the optimal learned kernel converges to the true solution within uncertainties. At 2 the real part of the correlator begins to deviate from the correct solution. Note that the most difficult points to achieve convergence at are t = 0 and at t = t max . Similarly we find that these points are also the ones, where the equal time correlator deviates the most from the correct solution, an important fact as this allows this deviation to contribute to the penalty in L prior . In fig. 9 we plot a heat map of the values of the kernels with mt max = 1.5 (center) and mt max = 2 (right) compared to the free theory propagator kernel from eq. (36) (left) for mt max = 1.5. (for a sketch of the structure of the heat map see the left panel of fig. 6). We have subtracted the unit matrix from the real-part of the two optimized kernels. They both exhibit a diagonal band in the real part, which is thinner than the one in the free theory kernel. It is interesting to see that both show non-trivial structures passing through the t max point and when connecting to the Euclidean branch. In the imaginary part the structures have more similarity with the free theory propagator kernel, where a sign change occurs as one moves away from the diagonal. The difference in the optimal kernels between mt max = 1.5 and mt max = 2 is small overall. Summary and Conclusion In this paper we proposed a novel strategy to recover correct convergence of realtime complex Langevin simulations by incorporating prior information into the simulation via a learned kernel. The effectiveness of the strategy was demonstrated for the strongly coupled anharmonic oscillator on the Schwinger Keldysh contour by extending correct convergence in this benchmark system up to mt max = 1.5, three times the previously accessible range of mt max = 0.5. After discussing the concept of neutral and non-neutral modifications of Langevin dynamics by use of real and complex kernels, we demonstrated that an explicitly constructed complex kernel can be used to improve the convergence behavior of realtime complex Langevin on the Schwinger-Keldysh contour. Taking insight from a single d.o.f. model and the harmonic oscillator, approximately correct convergence in the strongly coupled anharmonic oscillator was achieved up to mt max = 1. As no systematic extension to the explicit construction of that kernel exists, we instead proposed to learn optimal kernels using prior information. The ingredients to learning an optimal kernel are prior information and an efficient prescription for computing gradients. Prior information comes in the form of apriori known Euclidean correlation functions, known symmetries of the theory and the Schwinger-Keldysh contour, as well as information on the boundary terms. Here we included only the first two types of information, which sufficed to achieve improvements in convergence. We surveyed different modern differential programming techniques that in principle allow a direct optimization of the kernel based on the full prior information, but found that in their standard implementations they are of limited use in practice due to runtime or memory limitations. Instead we constructed an approximate gradient based on an alternative optimization functional, inspired by the need to avoid the presence of boundary terms. This optimization functional possesses a gradient, which can be approximated with much lower cost than that of the original optimization functional. The low-cost gradient in practice is computed using standard auto-differentiation. By minimizing with this gradient and monitoring success via the full prior information cost functional we proposed, we were able to locate optimal kernels. Our strategy was successfully applied first to the harmonic oscillator on the thermal SK contour. We managed to restore correct convergence with an optimal learned field-independent kernel that shows a simpler structure compared to the manually constructed kernel. This result bodes well for future studies, where we will investigate in detail the structure of the optimal learned kernel to draw conclusions about the optimal analytic structure for extending the approach to a field-dependent kernel. The central result of our study is the restoration of correct convergence in the strongly correlated anharmonic oscillator on the thermal SK contour up to a realtime extent of mt max = 1.5, which is beyond the reach of any manually constructed kernel proposed so far. We find some remnant deviations of the equal-time correlation function x 2 from the true solution at the turning points of the SK contour. The phenomenologically relevant unequal-time correlation function x(0)x(t) on the real-time branch on the other hand reproduces the correct solution within statistical uncertainty. While our strategy based on a field-independent kernel is successful in a range three times the previous state-of-the-art, we find that the restricted choice of kernel limits its success at larger real-time extent. We conclude that our study provides a proof-of-principle for the restoration of correct convergence in complex Langevin based on the inclusion of prior information via kernels. Future work will focus on extending the approach to field-dependent kernels, carefully reassess the discretization prescription of the SK at the turning points and improve the efficiency of the differential programming techniques necessary to carry out a minimization directly on the full prior knowledge cost functional. = ∂ ∂x + 1 2 i ∂S M ∂x K[x] ∂ ∂x − 1 2 i ∂S M ∂x ,(48) which by definition has the same eigenvalues as L. The transformation is carried out here to follow closely the conventional way of proving the correct convergence for a real action S. I.e., when S is real, G becomes a self-adjoint and hence negative semidefinite operator. For complex actions, iS M , this transformation is not necessary for the following arguments. It is however useful in practice as a pre-conditioner for calculating the eigenvalues of the Fokker Planck operator. The complex distribution ρ(x, τ L ) is also transformed based on the same transformation, such that ρ(x, τ L ) = e −i 1 2 S M ρ(x, τ L ), whereρ(x, τ L ) = G(x)ρ(x, τ L )(49) is the Fokker-Planck equation for the transformed operator. Since we are interested in the stationary distribution, we construct the eigenvalue equation G(x)ψ n (x) = λ n ψ n (x).(50) Due to the form of the operator we know that it must have at least one zero eigenvalue, λ = 0, associated with the eigenvector e i 1 2 S M . The formal solution of the Fokker-Planck equation after the similarity transform of eq. (49) is given bỹ ρ(x; τ L ) = e τ L G(x)ρ (x, 0)(51) and by expandingρ(x; 0) in the eigenbasis ψ n , and using ψ 0 e λ0t = e 1 2 iS M we get ρ(x; τ L ) =e 1 2 iS M (x)ρ (x; τ L )(52)=e 1 2 iS M (x) ∞ n=0 a n ψ n (x)e λnτ L = ce iS M (x) + ∞ n=1 a n ψ n (x)e λnτ L (53) such that when τ L → ∞ only the first term is left, namely the equilibrium distribution exp[iS M ]. This is however only true if Re λ n ≤ 0, in which case the spectrum of G provides information of the equilibrium distribution of the Fokker-Planck equation. The second condition, which needs to be satisfied is that the sampling of CL gives the same distribution as the complex Fokker-Planck equation. To establish that it does, we follow the correctness criterion of ref. [30]. Let us show that the criterion also holds in the presence of a kernel by revisiting some central steps of the original proof. We start with the Fokker-Planck equation for complex Langevin eq. (11), which operates on a real distribution Φ(x R , x I ; t) for the complexified degrees of freedom x R and x I . Let us take a look at the Fokker-Planck equation, which evolves the distribution of an observable O ∂ τ L O(x R , x I ) = (H R ∂ x R + H I ∂ x I ) 2 + Re iK[x R + ix I ]∇S M + ∂K[x R + ix I ] ∂x R ∂ x R +Im iK[x R + ix I ]∇S M + ∂K[x R + ix I ] ∂x R ∂ x I O(x R , x I ) = L T K O(x R , x I ),(54) where we can identify the operator L K to be the bilinear adjoint of the Fokker-Planck operator L T K [30]. If we assume that O is holomorphic, we know that ∂ x I O = i∂ x R O → i∂ z O, where for the last equality we have used the following relation between derivatives ∂ x R O(x R + ix I ) → ∂ z f (z) with z = x R + ix I . Replacing derivatives yields the following Langevin equation for the holomorphic observable O expressed in the complex variable z ∂ t O = K[z]∂ 2 z + iK[z]∇S M ∂ z + ∂K[z] ∂z ∂ z f = [∂ z + i∇S M ] K[z]∂ z O =L T K O,(55) where in the last equality we have used that K[z]∂ 2 z + (∂ z K[z])∂ z = ∂ z K[z] ∂ z based on integration by parts. We have now shown that (L T K − L T K )O = 0 for a Fokker-Planck equation with a field dependent kernel. In turn, we conclude that the correctness criterion also holds for a kernelled complex Langevin equation. For the above derivation to hold there may not arise any boundary terms given by [55] B n = dx R dx I Φ(x R , x I ) L T K n O(x R + ix I )(56) whereL T K is the Langevin operator given bỹ L T K = (∂ z + i∇S M )K[z]∂ z .(57) The formal criterion is then that the observable L T K O should be zero. This expression for B includes contributions from the full range of values of the d.o.f. between −∞ to ∞. Including all of these will introduce significant amounts of noise in the expectation value. This can be avoided by introducing a cut-off Ω for the values for x R and x I in the calculation of the observable. The boundary terms of eq. (56) are thus calculated using B Ω n = L T K n O(x R + ix I ) Ω =    L T K n O(x R + ix I ), if x R ≤ Ω x R and x I ≤ Ω x I 0, otherwise(58) where Ω x R and Ω x I denote the individual cutoffs for the real-and imaginary part respectively. In the case of scalar fields (which in contrast to gauge fields do not feature a compact dimension), we need to cut off in both x R and x I direction. We will in this paper stick to considering the cut-off to be a square. For all the values outside the square we set the contributions to the expectation value to zero. Since the observable of interest in the simple models is z 2 (i.e. it is the most difficult to capture accurately), we find the boundary terms observable from eq. (58) to beL T K z 2 =(∇ z + i∇S M )K(z)∇ z z 2 = (∇ z + i∇S M )K(z)2z =2((∇ z K(z))z + K(z) + i∇S M K(z)z) =2K(z)(1 + i∇S M z) + 2(∇ z K(z))z,(59) which for a field-independent kernel reduces to L T K z 2 Ω = 2K + iz∇S M Ω . We have discussed both ingredients necessary to establish correct convergence of our simulation in the presence of a kernel, i.e. the behavior of the Fokker-Planck spectrum and boundary terms. The boundary terms can be calculated in practice without problems, while the eigenvalues of the Fokker-Planck operator of eq. (50) so far remain out of reach for realistic systems, due to computational cost. Appendix B: Constant kernels and correct convergence in simple models In this appendix we investigate concrete examples of our optimization procedure and the corresponding learned kernels in one-degree of freedom models, for which in the literature (see e.g. [43,63]) kernels have been constructed by hand. The motivation behind this appendix is to understand how the kernels affect the behavior of the complex Langevin simulation, in particular how they are connected to the idea of minimizing the drift loss in eq. (46). To this end we connect complex Langevin to the Lefschetz thimbles and the correctness criterion [30]. We investigate the one-degree of freedom model with the action S = 1 2 σx 2 + λ 4 x 4 ,(60) which leads to the following partition function Z = dxe −S ,(61) i.e. we use the same convention as in the literature [43,63,64]. Note that this is a different convention from the main text as S can now have a imaginary part. This model is interesting as it exhibits similar properties as the interacting realtime model: the convergence problem appears, breaking both the boundary term condition and the equilibrium distribution of the Fokker-Planck equation for various parameters. We will therefor take a closer look at two specific sets of parameters. The first one is σ = 4i and λ = 2 where we can find an optimal kernel, and as second parameter we choose σ = −1 + 4i with the same λ = 2, where for correct convergence we have to go beyond a constant, field-independent kernel. In section 3.1 we looked at a variant of this model corresponding to σ = i and λ = 0 in eq. (60). The optimal field independent kernel K = −i transforms the complex Langevin equation such that it samples exactly on the Lefschetz thimble. In contrast, the models considered here have more than one critical point, and hence the relation to the Lefschetz thimbles is not as simple. The critical points for eq. (60), can be found via ∂S(x) ∂x = 0.(62) which are located at x = 0, ± σ/λ [30]. We see that the smaller the real-part of the σ parameter becomes, the further out into the complex plane the two critical points away from the origin are located. B.1 Non-uniqueness of the optimization In this study we used the optimization functional L D = D(x) · (−x) − ||D(x)|| ||x|| ξ(63) with D = KδS/δx, to compute an approximate gradient for the minimization of the true cost functional L prior . L D was constructed with the idea in mind that in order to remove boundary terms we wish to penalize drift away from the origin. In this appendix we discuss the fact that there exist multiple critical points to L D , which may or may not correspond to a kernel that restores correct convergence. In practice we distinguish between these solutions by testing the success of the corresponding kernel in restoring correct convergence via the value of L prior . Let us start with the parameter set σ = 4i and λ = 2 in eq. (60). For this choice ref. [43] showed that a constant kernel can be constructed that restores correct convergence. In a one-degree of freedom model, where the constant kernel is nothing but a complex number, we can optimize by brute force. Using the parametrization K = e iθ , H = √ K = e i θ 2(64) we only have to consider a single compact parameter: θ ∈ [0, 2π). A scan of the θ values reveals two minima of the L D loss function. One at θ 1 = π 3 and one at θ 2 = 2π 3 , where the first one corresponds to the kernel found manually in ref [43]. When deriving the optimal kernel, the authors also obtained two solutions, which correspond to these two kernels. They selected the correct one by requiring the kernel to belong to the first Riemann sheet when taking a square root. In our case, we too need to select the correct one and in this simple model can use the correctness criteria directly to do so. To proceed in this direction, let us take a look at the complex Langevin distribution according to the two kernels found in the optimization process and compare them to the Lefschetz thimble structure of the model. The thimble here consist of three different parts as shown by red lines in fig. 10, together with the critical points (green points). Note that the thimbles always cross through the critical points. The distribution of the complex Langevin evolution is shown as a point cloud. The three different distributions shown in each panel correspond to the case of (top left) K 0 = 1, (top right) K 1 = exp[−iπ/3] and (bottom) K 2 = exp[−i2π/3]. One can clearly see that for the trivial kernel complex Langevin tries to sample parallel to the real axis. As we saw in section 3 the angle parametrizing the kernel translates into a preferred sampling direction. In the top right and bottom of fig. 10, we have plotted the complex Langevin distribution obtained after introducing one of the two kernels that minimize L D . Again we find that the angle of the noise term decides where CL samples. We see that the highest density of the CL distribution lies along the direction in which the thimble passes through the critical point at the origin. Further out from the origin, the distribution follows closely the angle of the noise term, which is H 1 = √ e −iπ/3 = e −iπ/6 for the first kernel (top left) and H 2 = √ e −i2π/3 = e −iπ/3 for the second (bottom). I.e. we can distinguish that sampling with the first kernel leads to samples slightly closer to the thimbles going out along the real-axis, compared to the other kernel which favors sampling more closely along the parts of the thimble that eventually run off to infinity. We will give a formal explanation for this behavior in the next paragraphs. As shown in appendix A the correctness criterion consist of two parts, the first one states that no boundary terms may appear and the second requires that the eigenvalues of the complex Fokker-Planck equation need to have a negative real part. For this simple model we can compute both of these criteria, which is illustrated in fig. 11. The left plot contains the boundary terms for the real-part of the observable x 2 . Each of the curves corresponds to one of the three kernels K i . They are computed using the boundary term expectation value of eq. (58). We see that both of the kernels lead to very small values of the boundary terms for this observable, while the complex Langevin process without kernel exhibits a clear boundary term. However at this point we cannot yet say which of the two kernels produces the correct solution, if any. In order to see which of them is correct, we need to look at the right plot in fig. 11 where the five eigenvalues of the Fokker-Planck operator are plotted, which have the largest real-part (blue lines). They are plotted against different kernel parameters θ and the red lines indicate the position of the two kernels that optimize L D . The eigenvalue calculation is carried out using a restarted Arnoldi method solver, which internally uses a Krylov-Schur method. We see that there is a region of θ where the eigenvalues are all satisfying Re(λ) ≤ 0, which includes the kernel θ 1 = − π 3 . It is exactly this kernel, which, when incorporated into the complex Langevin evolution gives the right solution for the model. For smaller θs, the eigenvalues will eventually cross the zero. This is the region where one finds the second kernel θ 2 = − 2π 3 . We can therefore attribute the failure to restore correct convergence with the second kernel to a violation of the correctness criterion pertaining to the spectrum of the complex Fokker-Planck equation. The interesting point here is that the boundary terms do not seem to distinguish between the two kernels as both lead to quickly diminishing distributions. B.2 Limitation of constant kernels and boundary terms Let us now go to the set of parameters σ = −1 + 4i and λ = 2, for which there does not exist a constant kernel, which restores correct convergence. It is however possible to construct a field-dependent kernel that solves the problem [63]. We can understand this behavior, as the constant kernel that is optimal in the sense of removing boundary terms, does not achieve correct convergence of the complex Fokker-Planck equation to the correct e −S . This can be seen again by plotting the CL distribution for some of the local minima of the L D loss function. For this parameter space there are more than two, but we have picked out two of the solutions which have the interesting property that they both have no boundary terms, and still do not converge to the true solution. The kernels that are picked have the parameters θ 3 = − 3π 4 and θ 4 = π 2 The CL distribution together with the thimble is plotted in fig. 12 for the three different kernels K 0 = 1 (top left), K 3 = exp(iθ 3 ) (top right) and K 4 = exp(iθ 4 ) (bottom). We see that the thimbles show three distinct structures, connecting at infinity. To obtain the thimbles we evolve the gradient flow equation starting from a small offset from the critical points (which all are saddle points) and then combine the six part of the thimbles. The CL distribution without a kernel (top left plot in fig. 12) again favors sampling parallel to the real-axis, while the two other kernels sample completely different parts of the thimbles. The distribution for K 3 is located along the thimble crossing the origin. The other kernel (K 4 ), follows the other two thimbles crossing the critical points away from the origin. We can explain this behavior with the angle of the noise coefficient. For K 3 we have an angle of − 3π 8 against the real axis and for K 4 we have an angle of π 4 against the real axis. In fig. 13 (left) the boundary terms for this set of model parameters is calculated for the observable Re x 2 and plotted for increasing square box cutoff. We see that without a kernel, there are boundary terms present, as the blue datapoints do not go to zero for large cutoff. This can also been seen directly from the distribution in fig. 12 which exhibits a large spread and hence the falloff of the distribution is not fast enough. For the two kernels, K 3 and K 4 , that correspond to a local minimum in L D , the system does not show any boundary terms. This is an important point as even though we have avoided boundary terms, the CL dynamics under the kernels K 3 and K 4 still does not converge to the correct solution. In turn it appears that it is in general not enough to remove the boundary terms to achieve correct convergence. In fact one also needs to be sure that the complex Fokker-Planck equation converges to the desired equilibrium distribution. In fig. 13 (right) we show the five eigenvalues of the Fokker-Planck equation with the largest real-part plotted against the parameter θ which determines the kernel K = e iθ . For the parameters chosen here, we find that both kernels lie outside of the admissible region [4] , where λ ≤ 0. Interestingly at θ = 0 the eigenvalues [4] An interesting observation was made in [63], that combining kernels, which sample different parts of the thimble into a field-dependent kernel seems to work well. The motivation was to find a kernel that would reduce the drift term to −x when either the x 2 or the x 4 term in the action dominates. A similar argument for constructing a field-dependent kernel can now be made via the minima of the L D loss function, which favor sampling different parts of the thimbles. actually all lie in the lower half complex plane but there the boundary criterion is not fulfilled. But as one increases the imaginary part of σ, e.g. at σ = −1 + 5i, one finds that the eigenvalues for the identity kernel K = 1 already take on positive real-parts. Including the calculation of eigenvalues in the cost functional would be possible for simple models such as the one of eq. (60), but for larger, more realistic systems the dimension of the Fokker-Planck operator scales as N d , where N is the number of points along in each dimensions d. Even for the anharmonic oscillator on the SK contour, the calculation of the Fokker-Planck eigenvalues is too costly in practice. We therefore need a different way of distinguishing which kernel leads to correct convergence. As discussed in detail in the main text of this manuscript we thus propose to collect as much prior information about the system as possible in the cost functional L prior , based on which the success of the optimal kernel according to L D is judged. Figure 1 1Distribution of a complex Langevin simulation (scatter points) and the Lefschetz thimble (red line) for the model eq. (13). (left) simulation according to the naive CL eq. ( Figure 3 3The result of a complex Langevin simulation of the real-time harmonic oscillator without a kernel (top) and with a kernel (bottom) for the observables x 2 , and the correlator x(0)x(t) . Simulations are carried out with 100 trajectories and up to mτ L = 100 in Langevin time, saving at every m∆τ L = 0.1 using the implicit scheme Euler-Maruyama with θ = 1.0 with adaptive step-size with tolerance 10 −3 (top) and the explicit scheme Euler-Maruyama with θ = 0 with fixed step-size ∆τ L = 10 −3 (bottom). Values of the correlators from the solution of the Schrödinger equation are given as solid black lines. and blue data points. The analytically known values from solving the Schrödinger equation are underlaid as black solid lines. Figure 4 4Anharmonic oscillator at m = 1 and λ = 24 up to tmax = 1 in real-time without (top) and in the presence (bottom) of the heuristic free theory kernel with g = 0.8 and mg = 1.8 from eq. (40). Both simulations are carried out generating 100 trajectories simulated up to mτ L = 100 in Langevin time, saving configurations at every m∆τ L = 0.01. We deploy the Euler-Maruyama solver with θ = 1.0 without kernel (top), and θ = 0.5 with kernel (bottom). Values of the correlators from the solution of the Schrödinger equation are given as solid black lines. The task at hand is to find the critical point of a cost functional that is comprised of a subset of the contributions listed in the previous section, i.e. of L eucl ,L sym ,L C or L BT . Generically each contribution can be written as the expectation value of a known function G, depending on the dynamical degrees of freedom x and the kernel K, i.e. L prior [K] = | G[x, K] K | 2 . In order to make the dependence of the expectation value on the kernel explicit we consider the d.o.f. within the simulation to explicitly depend on K as x(K). This allows us to remove the subscript K from the expectation value so that L prior [K] = | G[x(K), K] | 2 . Figure 5 5Harmonic oscillator in the presence of our learned optimal kernel, based on the heuristic gradient from the loss function in eq. (46). We simulate on the SK contour with mtmax = 10 in real-time and choose m = 1. Simulating up to mτ L = 100 in Langevin time, we combine samples from 100 (top) and 400 (bottom) different trajectories. Note that improved statistics diminishes the residual oscillatory artifacts in x 2 . Values of the correlators from the solution of the Schrödinger equation are given as solid black lines. Figure 6 6(left) An explanatory sketch of our visualization of the complex kernel used in the simulations. The top major panel denotes the values of the real part and the lower major panel those of the imaginary part of the kernel. Inside each panel the values of the kernel are ordered along the contour parameter, indicating which parts of the kernel couple which range on the contour. (center) The free theory kernel in eq. (36), constructed explicitly in the previous section for mtmax = 10. The repeating pattern indicates an oscillatory behavior in coordinate space arising from the fact that this kernel is just the propagator of the free theory. (right) Figure 7 7Complex Langevin simulation of the strongly coupled anharmonic oscillator on the thermal Schwinger-Keldysh contour in the absence (left) of a kernel and in the presence of the optimal learned field-independent kernel (right). The top row corresponds to results from a contour with real-time extent mtmax = 1, while the center row shows results for mtmax = 1.5 and the bottom row for mtmax = 2. Values of the correlators from the solution of the Schrödinger equation are given as solid black lines. Figure 8 8A detailed comparison of the unequal time correlation functions x(0)x(t) from fig. 7 evaluated in the presence of the optimal learned field-independent kernel on contours with mtmax = 1, 1.5 and 2 respectively.The different colored circles correspond to the real-part while the squares to the imaginary part of the correlator. Values of the correlators from the solution of the Schrödinger equation are given as solid black lines. Figure 9 ( 9left) Free theory kernel for the SK contour with mtmax = 1.5. (center) The optimal learned kernel in the interacting theory for mtmax = 1.5, which achieves correct convergence of CL. The diagonal entries with values close to unity are subtracted from the kernel. (right) The kernel obtained as a result of the optimization procedure in the case of mtmax = 2.0, which does not achieve correct convergence. At the turning point at tmax and when connecting to the Euclidean domain the kernel for the interacting theory shows nontrivial structure not present in the free theory. Figure 10 10Distribution of the complex Langevin simulation and the Lefschetz thimble (red line) for the model of eq. (60) with σ = 4i and λ = 2, using different kernels;K 0 = 1 (top left), K 1 = exp[−iπ/3] (top right) andK 2 = exp[−i2π/3] (bottom).The green points denote the critical points given by the solution to eq. (62). The color in the distribution heat map corresponds to the number of samples at the corresponding position (a lighter color refers to a higher value). Figure 11 ( 11left) Boundary term according to the x 2 observable for the model of eq. (60) with σ = 4i and λ = 2, evaluated for the three different kernels K i discussed in the main text. (right) the five eigenvalues of the Fokker-Planck operator with the largest real part (blue lines) plotted against the kernel parameter θ. The position of the two kernels that optimize L D are indicated by red lines. Figure 12 12Distribution of the complex Langevin simulation and the Lefschetz thimbles (red line) for the model of eq. (60) with σ = −1 + 4i and λ = 2, using different kernels; K 0 = 1 (top left),K 3 = e −i 3π4 (top right) and K 4 = e i π 2 (bottom). The green points are the critical points given by the solution to eq. (62). Figure 13 ( 13left) Boundary term according to the x 2 observable for the model of eq. (60) with σ = −1 + 4i and λ = 2, evaluated for the three different kernels K i discussed in the main text. (right) the five eigenvalues of the Fokker-Planck operator with the largest real part (blue lines) plotted against the kernel parameter θ. The position of the two kernels that optimize L D are indicated by red lines. with Euclidean action S E , the goal thus is to guarantee at late Langevin times a distribution of fields Φ[φ] ∝ exp − S E [φ] .The chain of configurations φ(τ L ) underlying the distribution Φ[φ], can then be used to evaluate the expectation values of observables O from the mean of samples O = lim τL→∞1 τL τL [1] This behavior may be understood in terms of boundary terms. The kernel manages to significantly reduce the magnitude of boundary terms for x 2 where it differs from the true solution the boundary terms, while small, are not exactly zero. Appendix A: Correctness criterion in the presence of a kernelIn this section we discuss the correctness criterion in the presence of a kernel in the CL evolution. As mentioned in section 2.2 there are two parts to the correctness criterion that need to be fulfilled in order for complex Langevin to converge to the correct solution. We must avoid boundary terms for the real-valued distribution Φ(x R , x I , τ L ) and the complex Fokker-Planck eq. (12) must have the correct equilibrium distribution. If both conditions are fulfilled the equal sign of eq. (8) holds.To check if the equilibrium distribution of ρ(x, τ L ) is exp[iS M ] we need to either solve the Fokker-Planck equation explicitly, or inspect the eigenvalue spectrum of the Fokker-Planck equation[46]. To make inference about correct convergence based on the eigenspectrum, the eigenvectors of the Fokker-Planck operator must form a complete set, as otherwise there exist non-orthogonal zero modes competing with the e iS M stationary distribution. For a non-self-adjoint operator this is not always the case.To show the connection between the eigenvalues of the Fokker-Planck equation and the equilibrium distribution we use a similarity transform to define the operator G from the Fokker-Planck operator L including the kernelCompeting interestsThe authors declare that they have no competing interests.Author's contributions • D. Alvestad: concept development, algorithmic development, code implementation, data analysis, writing • R. Larsen: supervising code development • A. Rothkopf: project conceptualization, funding acquisition, supervision, writing Heavy Ion Collisions: The Big Picture, and the Big Questions. W Busza, K Rajagopal, W Van Der Schee, 10.1146/annurev-nucl-101917-020852physics:nucl-th]. Accessed 2022-08-30hep-ph, physics:hep-th, physics:nucl-ex. 68Busza, W., Rajagopal, K., van der Schee, W.: Heavy Ion Collisions: The Big Picture, and the Big Questions. 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[ "Local lower bounds on characteristics of quantum and classical systems", "Local lower bounds on characteristics of quantum and classical systems" ]
[ "M E Shirokov \nSteklov Mathematical Institute\nMoscowRussia\n" ]
[ "Steklov Mathematical Institute\nMoscowRussia" ]
[]
We consider methods for obtaining local lower bounds on characteristics of quantum (correspondingly, classical) systems, i.e. lower bounds valid in the · 1 -norm ǫ-neighborhood of a given state (correspondingly, probability distribution). The main attention is paid to infinite-dimensional systems.
null
[ "https://export.arxiv.org/pdf/2210.11462v2.pdf" ]
253,018,542
2210.11462
c3a2eb49070eec061372a89b1b13ae7066764e9f
Local lower bounds on characteristics of quantum and classical systems 23 Apr 2023 M E Shirokov Steklov Mathematical Institute MoscowRussia Local lower bounds on characteristics of quantum and classical systems 23 Apr 2023arXiv:2210.11462v2 [quant-ph] We consider methods for obtaining local lower bounds on characteristics of quantum (correspondingly, classical) systems, i.e. lower bounds valid in the · 1 -norm ǫ-neighborhood of a given state (correspondingly, probability distribution). The main attention is paid to infinite-dimensional systems. Introduction There are many characteristics used in quantum (resp. classical) information theory that are nonnegative lower semicontinuous functions either on the set of all quantum states (resp. probability distributions) or on some its proper subset. The list of basic examples includes: • the von Neumann entropy (resp. the Shannon entropy); • the quantum relative entropy (resp. the Kullback-Leibler divergence); • the quantum conditional entropy of q-c states (resp. equivocation); • the quantum (resp. classical) conditional mutual information; • several entanglement measures (the entanglement of formation, the relative entropy of entanglement, the squashed entanglement). If a function f defined on a subset S 0 ⊆ S(H) is lower semicontinuous at a state ρ ∈ S 0 then it is easy to see that the quantity L f (ρ|ε) = inf f (σ) | σ ∈ S 0 , 1 2 ρ − σ 1 ≤ ε tends to f (ρ) ≤ +∞ as ε → 0. Naturally, the question arises about the estimation of this quantity. We will say that L f (ρ|ε) is a faithful local lower bound on a function f at a state ρ if L f (ρ|ε) ≤ L f (ρ|ε) ∀ε ∈ (0, 1] and lim ε→0 L f (ρ|ε) = f (ρ) ≤ +∞. In this article we consider general ways of getting faithful local lower bounds on basic characteristics of quantum systems and discrete random variables. Preliminaries Quantum systems Let H be a separable Hilbert space, B(H) the algebra of all bounded operators on H with the operator norm · and T(H) the Banach space of all trace-class operators on H with the trace norm · 1 . Let S(H) be the set of quantum states (positive operators in T(H) with unit trace) [8,13,20]. Write I H for the unit operator on a Hilbert space H and Id H for the identity transformation of the Banach space T(H). We will use the Mirsky inequality +∞ i=1 |λ ρ i − λ σ i | ≤ ρ − σ 1(1) valid for any positive operators ρ and σ in T(H), where {λ ρ i } +∞ i=1 and {λ σ i } +∞ i=1 are the sequences of eigenvalues of ρ and σ arranged in the non-increasing order (taking the multiplicity into account) [12,5]. The von Neumann entropy of a quantum state ρ ∈ S(H) is defined by the formula S(ρ) = Tr η(ρ), where η(x) = −x ln x if x > 0 and η(0) = 0. It is a concave lower semicontinuous function on the set S(H) taking values in [0, +∞] [8,11,18]. The von Neumann entropy satisfies the inequality S(pρ + (1 − p)σ) ≤ pS(ρ) + (1 − p)S(σ) + h 2 (p) (2) valid for any states ρ and σ in S(H) and p ∈ [0, 1], where h 2 (p) = η(p) + η(1 − p) is the binary entropy [14,13,20]. Note that inequality (2) holds with h 2 (p) replaced by the nondecreasing concave continuous functionh 2 (p) on [0, 1] defined as h 2 (p) = h 2 (p) if p ∈ [0, 1 2 ] ln 2 if p ∈ ( 1 2 , 1].(3) We will use the homogeneous extension of the von Neumann entropy to the positive cone T + (H) defined as S(ρ) . = (Trρ)S(ρ/Trρ) = Trη(ρ) − η(Trρ) (4) for any nonzero operator ρ in T + (H) and equal to 0 at the zero operator [11]. By using concavity of the von Neumann entropy and inequality (2) it is easy to show that S(ρ) + S(σ) ≤ S(ρ + σ) ≤ S(ρ) + S(σ) + H({Trρ, Trσ}) (5) for any ρ and σ in T + (H), where H({Trρ, Trσ}) = η(Trρ) + η(Trσ) − η(Tr(ρ + σ)) is the homogeneous extension of the binary entropy to the positive cone in R 2 . Let H be a positive (semi-definite) operator on a Hilbert space H (we will always assume that positive operators are self-adjoint). Write D(H) for the domain of H. For any positive operator ρ ∈ T(H) we will define the quantity TrHρ by the rule TrHρ = sup n TrP n Hρ if suppρ ⊆ cl(D(H)) +∞ otherwise (6) where P n is the spectral projector of H corresponding to the interval [0, n] and cl(D(H)) is the closure of D(H). 1 If H is the Hamiltonian (energy observable) of a quantum system described by the space H then TrHρ is the mean energy of a state ρ. For any positive operator H the set C H,E = {ρ ∈ S(H) | TrHρ ≤ E} is convex and closed (since the function ρ → TrHρ is affine and lower semicontinuous). It is nonempty if E > E 0 , where E 0 is the infimum of the spectrum of H. The von Neumann entropy is continuous on the set C H,E for any E > E 0 if and only if the operator H satisfies the Gibbs condition Tr e −βH < +∞ for all β > 0 (7) and the supremum of the entropy on this set is attained at the Gibbs state γ H (E) . = e −β(E)H /Tre −β(E)H , where the parameter β(E) is determined by the equation TrHe −βH = ETre −βH [18]. Condition (7) can be valid only if H is an unbounded operator having discrete spectrum of finite multiplicity. It means, in Dirac's notation, that H = +∞ k=0 E k |τ k τ k |,(8) where T . = {τ k } +∞ k=0 is the orthonormal system of eigenvectors of H corresponding to the nondecreasing unbounded sequence {E k } +∞ k=0 of its eigenvalues and it is assumed that the domain D(H) of H lies within the closure H T of the linear span of T . In this case TrHρ = i λ i √ Hϕ i 2 for any operator ρ in T + (H) with the spectral decomposition ρ = i λ i |ϕ i ϕ i | provided that all the vectors ϕ i lie in D( √ H) = {ϕ ∈ H T | +∞ k=0 E k | τ k |ϕ | 2 < +∞}. If at least one eigenvector of ρ corresponding to a nonzero eigenvalue does not belong to the set D( √ H) then TrHρ = +∞. We will use the function F H (E) . = sup ρ∈C H,E S(ρ) = S(γ H (E)) = β(E)E + ln Tre −β(E)H .(9) This is a strictly increasing concave function on [E 0 , +∞) [15,21]. It is easy to see that F H (E 0 ) = ln m(E 0 ), where m(E 0 ) is the multiplicity of E 0 . By Proposition 1 in [15] the Gibbs condition (7) is equivalent to the following asymptotic property F H (E) = o(E) as E → +∞.(10) For example, ifN . = a † a is the number operator of a quantum oscillator then FN (E) = g(E), where g(x) = (x + 1) ln(x + 1) − x ln x, x > 0, g(0) = 0.(11) We will often assume that E 0 . = inf ϕ =1 ϕ|H|ϕ = 0.(12) In this case the function F H satisfies the conditions of the following lemma (cf. [21,Corollary 12]). Lemma 1. If f is a nonnegative concave function on R + then xf (z/x) ≤ yf (z/y) for all y > x > 0 and z ≥ 0. Discrete random variables Any vector X 1 ,...,X n of random variables, each of which takes a countable set of values, is described by n-variate probability distributionp = {p i 1 ...in } (i 1 ,..,in)∈N n 0 , where N 0 is the set of non-negative integers. 2 We will denote by P n the set of all such probability distributions equipped with the total variation distance TV, which is defined for anȳ p = {p i 1 ...in } andq = {q i 1 ...in } in P n as follows TV(p,q) . = 1 2 i 1 ,..,in |p i 1 ...in − q i 1 ...in |.(13) For anyp ∈ P n we will writep k and [p k ] i for the marginal distribution of X k and its i-th entry, i.e.p k = {[p k ] i } i , [p k ] i = i 1 ,..,i k−1 ,i k+1 ,..,in p i 1 ...i k−1 i i k+1 ...in . The number of nonzero entries of the distributionp k will be denoted by |p k |. The Shannon entropy of a probability distributionp = {p i 1 ...in } ∈ P n is defined as H(p) = i 1 ,..,in η(p i 1 ...in ), where η(x) = −x ln x if x > 0 and η(0) = 0. It is a concave lower semicontinuous function on the set P n taking values in [0, +∞] [4,8]. The Shannon entropy satisfies the inequality H(λp + (1 − λ)q) ≤ λH(p) + (1 − λ)H(q) + h 2 (λ)(14) valid for any distributionsp andq in P n and λ ∈ (0, 1), where h 2 (λ) = η(λ) + η(1 − λ) is the binary entropy [14,13,20]. Let S = {E i } +∞ i=0 be a nondecreasing sequence of nonnegative numbers such that +∞ i=0 e −βE i < +∞ ∀β > 0.(15) Consider the function F S (E) = sup H({p i }) {p i } ∈ P 1 , +∞ i=0 E i p i ≤ E = β(E)E + ln +∞ i=0 e −β(E)E i , (16) where β(E) is defined by the equation +∞ i=0 E i e −βE i = E +∞ i=0 e −βE i [18],[15, Propo- sition 1]. It is easy to see that the function F S (E) coincides with the function F H (E) defined in (9) provided that H is a positive operator with the spectrum S. If S = {0, 1, 2, ...} then F S (E) = g(E) -the function defined in (11). So, Proposition 1 in [15] shows that condition (15) is equivalent to the asymptotic property F S (E) = o(E) as E → +∞.(17) Basic lemma In this section we describe one general result concerning properties of a real-valued function f on a convex subset S 0 of S(H) satisfying the inequalities f (pρ + (1 − p)σ) ≥ pf (ρ) + (1 − p)f (σ) − a f (p)(18) and f (pρ + (1 − p)σ) ≤ pf (ρ) + (1 − p)f (σ) + b f (p)(19) for all states ρ and σ in S 0 and any p ∈ [0, 1], where a f and b f are continuous functions on [0, 1] such that a f (0) = b f (0) = 0. We will call functions satisfying both inequalities (18) and (19) locally almost affine (briefly, LAA functions). We will assume that 1]. For technical simplicity we will also assume that the functions a f and b f are concave and non-decreasing on [0, 1 2 ]. a f (p) = a f (1 − p) and b f (p) = b f (1 − p) for all p ∈ [0, For any state ρ with the spectral representation ρ = i λ ρ i |ϕ i ϕ i | and arbitrary ε > 0 introduce the trace class positive operators ρ ∧ εI H . = i min{λ ρ i , ε}|ϕ i ϕ i |, [ρ − εI H ] + . = i max{λ ρ i − ε, 0}|ϕ i ϕ i |. (21) It is clear that ρ = ρ ∧ εI H + [ρ − εI H ] + and that ρ ∧ εI H → 0 in the trace norm as ε → 0. For any function f on a convex subset S 0 of S(H) we will use its homogeneous extensionf to the cone S 0 generated by S 0 . It is defined as f (ρ) . = (Trρ)f (ρ/Trρ),f (0) = 0.(22) The following lemma allows us to obtain local lower bounds for nonnegative LAA functions valid for commuting states. Lemma 2. Let f be a function on a convex subset S 0 of S(H) taking values in [0, +∞] and satisfying inequalities (18) and (19) on S 0 with possible values +∞ in both sides. Let ρ be a state in S 0 with the properties (a) h(ρ) ∈ S 0 for any function h : R + → R + such that h(x) ≤ x ∀x ∈ R + , (23) (b) [ρ − σ] ± ∈ S 0 for any state σ in S 0 such that [ρ, σ] = 0 and σ = ρ,(24) where [ρ − σ] ± are the positive and negative parts of the Hermitian operator ρ − σ, [ρ, σ] . = ρσ − σρ. 1 2 ρ − σ 1 = ǫ ≤ ε ≤ 1. Let {ϕ k } +∞ k=0 be an orthonormal basis such that ρ = +∞ k=0 λ ρ k |ϕ k ϕ k | and σ = +∞ k=0 λ σ k |ϕ k ϕ k |, where {λ ρ k } is the sequence of eigenvalues of ρ arranged in the non-increasing order and {λ σ k } is the corresponding sequence of eigenvalues of σ (not nondecreasing, in general). A) Following the Alicki-Fannes-Winter technique (cf. [1,21]) introduce the states τ + = ǫ −1 +∞ k=0 [λ ρ k − λ σ k ] + |ϕ k ϕ k | and τ − = ǫ −1 +∞ k=0 [λ ρ k − λ σ k ] − |ϕ k ϕ k |, where [λ ρ k − λ σ k ] + = max{λ ρ k − λ σ k , 0} and [λ ρ k − λ σ k ] − = max{λ σ k − λ ρ k , 0}. The states τ + and τ − belong to the set S 0 due to the condition (24). Then we have 1 1 + ǫ ρ + ǫ 1 + ǫ τ − = ω * = 1 1 + ǫ σ + ǫ 1 + ǫ τ + , where ω * is a state in S 0 (due to the convexity of S 0 ). The obvious operator inequalities ǫτ + ≤ ρ and ǫτ − ≤ σ along with the assumed finiteness of f (ρ) and f (σ) imply, due to inequalities (18) and (19), the finiteness of f (τ + ), f (τ − ) and f (ω * ). By applying inequalities (18) and (19) to the above decompositions of ω * we obtain (1 − p)(f (ρ) − f (σ)) ≤ p(f (τ + ) − f (τ − )) + a f (p) + b f (p) where p = ǫ/(1 + ǫ). Hence f (ρ) − f (σ) ≤ ǫ(f (τ + ) − f (τ − )) + D f (ǫ) ≤ ǫf (τ + ) + D f (ε),(28) where the last inequality follows from the nonnegativity of f and the fact that the function D f is nondecreasing on [0, 1] due to the assumption (20). Since ǫτ + ≤ ρ and τ + ≤ I H , we have ǫτ + ≤ ρ ∧ ǫI H ≤ ρ ∧ εI H .(29) So, by using inequality (18) and taking the nonnegativity of f into account we obtain ǫf (τ + ) ≤r ε f (r −1 ε ρ ∧ εI H ) +r ε a f (ǫ/r ε ) ≤r ε f (r −1 ε ρ ∧ εI H ) +ã f (ε),(30) wherer ε = 1 − r ε = Tr(ρ ∧ εI H ) and the last inequality follows from Lemma 1 and the assumption (20). This inequality and (28) imply (25). B) Assume that f (ρ) < +∞. Since ρ = ρ ∧ εI H + [ρ − εI H ] + , it follows from inequality (18) that r ε f (r −1 ε ρ ∧ εI H ) ≤ f (ρ) − r ε f (r −1 ε [ρ − εI H ] + ) + a f (1 − r ε ). This inequality and (25) imply (27). Let f (ρ) = +∞. Assume that (27) does not hold for some ε 0 ∈ (0, 1] and a state σ 0 diagonizable in the basis {ϕ k } such that 1 2 ρ − σ 0 1 = ǫ 0 ≤ ε 0 . Consider the sequence of states ρ n = [ρ − γ n I H ] + /Tr[ρ − γ n I H ] + , γ n = 1/n, well defined for all n large enough. It is clear that ρ n → ρ and, hence, ǫ n . = 1 2 ρ n −σ 0 1 tends to ǫ 0 as n → +∞. Condition (26) implies that f (ρ n ) < +∞ for any n and hence (27) holds with ρ = ρ n and ε = ε n . = min{ǫ n + ε 0 −ǫ 0 , 1} by the above part of the proof. Since ρ n and ε n tend, respectively, to ρ and ε 0 as n → +∞, it is easy to see that the sequence of operators [ρ n − ε n I H ] + = c −1 n [ρ − (c n ε n + γ n )I H ] + , c n = Tr[ρ − γ n I H ] + , tends to the operator [ρ − ε 0 I H ] + in the trace norm. Thus, since the functions a f and b f are continuous on [0, 1], to obtain a contradiction to the above assumption it suffices to show that lim n→+∞ f (r −1 εn [ρ n − ε n I H ] + ) = f (r −1 ε 0 [ρ − ε 0 I H ] + ),(31) where r εn = Tr[ρ n − ε n I H ] + and r ε 0 = Tr[ρ − ε 0 I H ] + . To prove this relation note first that condition (23) implies that the set S 0 contains the set S ρ of states proportional to the spectral projectors of ρ corresponding to its positive eigenvalues. Thus, condition (26) and inequality (18) allow us to show that the function f takes finite values on the set S ρ . Let n 0 be a natural number such that c n ε n ≥ ε 0 /2 for all n ≥ n 0 , where c n = Tr[ρ − γ n I H ] + . Then it is easy to see that all the states proportional to the operators [ρ n − ε n I H ] + = c −1 n [ρ − (c n ε n + γ n )I H ] + , n ≥ n 0 , and [ρ − ε 0 I H ] + belong to the convex hull of the finite subset of S ρ corresponding to the eigenvalues of ρ which are not less than ε 0 /2. By Corollary 3 in [16, Section 3.1.2] the LAA-function f is continuous on this convex hull. This implies (31). ✷ Remark 2. Since the proof of Lemma 2 uses the very coarse estimate (29), the lower bounds given by this lemma can not be too accurate. The main advantage of these lower bounds is their universality. Note also that the lower bound given by Lemma 2B is easily computable as the operator [ρ − εI H ] + has finite rank for any ρ and ε > 0. ✷ Applications of Lemma 2 to different characteristics of quantum systems are considered in Section 4. The version of Lemma 2 adapted to characteristics of discrete random variables is presented in Section 5.1 (Lemma 3). Characteristics of quantum systems 4.1 The von Neumann entropy Since the von Neumann entropy is a continuous function on the compact set of states of a finite-dimensional quantum system, it attains its maximal and minimal values on the ε-neighbourhood V ε (ρ) . = σ ∈ S(H) | 1 2 ρ − σ 1 ≤ ε of any state ρ. The states in V ε (ρ) at which these maximal and minimal values are attained are described by Hanson and Datta in [6]. In the infinite-dimensional case the von Neumann entropy is not bounded from above on the ε-neighbourhood V ε (ρ) of any state ρ for all ε > 0, since it takes the value +∞ on the dense subset of S(H). But for any state ρ in S(H) we can consider the quantity L S (ρ|ε) = inf S(σ) | σ ∈ S(H), 1 2 ρ − σ 1 ≤ ε , where the infimum is not attained in general, since the ε-neighbourhood of any state ρ is not compact. The lower semicontinuity of the von Neumann entropy implies that lim ε→0 + L S (ρ|ε) = S(ρ) ≤ +∞. The results of Section 4.1 in [17] imply several computable lower bounds on L S (ρ|ε) expressed via the spectrum of ρ. Proposition 2 in [17] (proved by using Audenaert's continuity bound [2]) shows that L S (ρ|ε) ≥ S(ρ) − ε ln(rankρ − 1) − h 2 (ε)(32) for any finite rank state ρ and any ε ≤ 1 − 1/rankρ. By Lemma 2.2 in [6] the equality holds in (32) if and only if the state ρ has the spectrum (1 − ε, ε/d, ..., ε/d), where d = rankρ − 1. Proposition 1 in [17] shows that L S (ρ|ε) ≥ S(ρ) − εF H (E H, ε (ρ)/ε) − g(ε) ≥ S(ρ) − εF H (E/ε) − g(ε)(33) for any state ρ in S(H) such that TrHρ = E < +∞, where H is any positive operator on H satisfying conditions (7) and (12), F H and g are the functions defined in (9) and (11) correspondingly and 3 E H, ε (ρ) . = TrH(ρ ∧ εI H ) = E − TrH[ρ − εI H ] + = o(1) as ε → 0. The lower bound (33) is faithful, since condition (7) implies the convergence of the r.h.s. of (33) to S(ρ) as ε → +∞ (due to the equivalence of (7) and (10)). The last claim of Proposition 1 in [17] implies that the lower bound (33) is quite accurate: for some states ρ the gap between the l.h.s. and the r.h.s. of (33) is less than g(ε). The lower bound (33) is applicable to any state ρ in S(H) with finite S(ρ), since for any such state there is a positive operator H on H satisfying conditions (7) and (12) such that TrHρ < +∞ [15,Proposition 4]. The operator H can be treated as a "free parameter", with which one can optimize lower bound (33). The second claim of Proposition 1 in [17] and the arguments from its proof show that with this optimization, we can restrict attention to positive operators H having the form (8), in which {τ k } is the basis of eigenvectors of the state ρ corresponding to the nondecreasing sequence {λ ρ k } +∞ k=0 of its eigenvalues. By taking such an operator H we can rewrite (33) as follows L S (ρ|ε) ≥ S(ρ) − εF S (E S, ε (ρ))/ε) − g(ε)(34) where F S (E) is the function defined in (16) via a non-decreasing sequences S = {E k } +∞ k=0 of nonnegative numbers with E 0 = 0 satisfying condition (15) such that +∞ k=0 E k λ ρ k = E < +∞ and E S, ε (ρ) . = +∞ k=0 E k min{λ ρ k , ε} = E − k∈Iε E k (λ ρ k − ε) = o(1) as ε → 0,(35) where I ε is the (finite) set of all k such that λ ρ k > ε. Example 1. Let ρ N be the Gibbs state of one-mode quantum oscillator with the mean number of quanta N that has the spectral representation ρ N = (1 − q) +∞ k=0 q k |τ k τ k |, q = N N + 1 , where {τ k } is the Fock basis [8]. Let S = (0, 1, 2, ..., n, ..). Then E S, ε (ρ N ) = (1 − q) +∞ k=0 kq k = q (1 − q) = N if ε(N + 1) ≥ 1 and E S, ε (ρ N ) = ε nε−1 k=0 k + +∞ k=nε kq k = εn ε (n ε − 1) 2 + q nε n ε 1 − q + q nε+1 (1 − q) 2 if ε(N + 1) < 1, where n ε = [log q (ε(N +1))]+1 is the minimal natural number n such that (1−q)q n < ε. Thus, in the case ε(N + 1) < 1 we have E S, ε (ρ N ) ≤ εR N (ε) = o(1) as ε → 0, where R N (ε) . = A 2 N (ε) 2 + (N + 1) 2 (A N (ε) + N), A N (ε) = log N N+1 (ε(N + 1)) + 1. Since F S (E) = S(ρ N ) = g(E), it follows from (34) that L S (ρ N |ε) ≥ g(E) − εg(R N (ε)) − g(ε) ≥ g(E) − εg(N/ε) − g(ε), ∀ε ∈ (0, (1 + N) −1 ) and L S (ρ N |ε) ≥ g(E) − εg(N/ε) − g(ε), ∀ε ∈ [(1 + N) −1 , 1]. Universal lower bounds on L S (ρ|ε) can be obtained from Lemma 2 in Section 3. Proposition 1. Let ρ be an arbitrary state in S(H). Then L S (ρ|ε) ≥ S([ρ − εI H ] + ) − g(ε)(36) for any ε ∈ (0, 1], where [ρ − εI H ] + is the positive part of the operator ρ − εI H and S(·) is the extended von Neumann entropy defined in (4). If ρ is a state with finite entropy then L S (ρ|ε) ≥ S(ρ) − S(ρ ∧ εI H ) − g(ε)(37) for any ε ∈ (0, 1], where ρ ∧ εI H is the operator defined in (21). The lower bounds (36) and (37) are faithful: their r.h.s. tend to S(ρ) ≤ +∞ as ε → 0. Remark 3. Since ρ = ρ ∧ εI H + [ρ − εI H ] + , it follows from (5) that S(ρ) − S(ρ ∧ εI H ) ≥ S([ρ − εI H ] + ). So, lower bound (37) is sharper than (36). The advantage of lower bound (36) consists in its validity for all states ρ (including states with infinite entropy). Note also that the r.h.s. of (36) is easily calculated, since S([ρ − εI H ] + ) = i∈Iε η(λ ρ i − ε) − η i∈Iε (λ ρ i − ε) , η(x) = −x ln x, where {λ ρ i } is the sequences of eigenvalues of ρ and I ε is the finite set of all i such that λ ρ i > ε. For practical application of the lower bound (36) it may be convenient to use the inequality S([ρ − εI H ] + ) ≥ i∈Iε η(λ ρ i ) − η i∈Iε λ ρ i − ε|I ε | ln |I ε | − h 2 (ε|I ε |)(38) valid for any operator ρ in T + (H) such that Trρ ≤ 1 and ε ∈ (0, Trρ), where |I ε | denotes the cardinality of the set I ε of all i such that λ ρ i > ε. Since [ρ − εI H ] + = i∈Iε λ ρ i |ϕ i ϕ i | − i∈Iε ε|ϕ i ϕ i |, inequality (38) follows from the right inequality in (5) and the inequality (x + y)h 2 x x + y ≤ h 2 (x) valid for any nonnegative numbers x and y such that x + y ≤ 1, which can be proved by noting that {x, 1 − x} = (1 − x − y){0, 1} + (x + y){x(x + y) −1 , y(x + y) −1 } and by using concavity of the Shannon entropy. Proof of Proposition 1. Let ρ and σ be any states in S(H) such that 1 2 ρ − σ 1 ≤ ε. Let {λ ρ i } +∞ i=0 and {λ σ i } +∞ i=0 be the sequences of eigenvalues of these states arranged in the non-increasing order and {ϕ i } +∞ i=0 the basis in H such that ρ = +∞ i=0 λ ρ i |ϕ i ϕ i |. Consider the stateσ = +∞ i=0 λ σ i |ϕ i ϕ i |. The Mirsky ineqiality (1) implies that 1 2 ρ −σ 1 ≤ 1 2 ρ − σ 1 ≤ ε. So, since the states ρ andσ commute and the von Neumann entropy is a concave nonnegative function on S(H) satisfying inequality (2) and taking finite values at finite rank states, Lemma 2B in Section 3 with S 0 = S(H) implies that S(σ) = S(σ) ≥ S([ρ − εI H ] + ) − g(ε), while Lemma 2A implies that S(σ) = S(σ) ≥ S(ρ) − S(ρ ∧ εI H ) − g(ε) provided that S(ρ) < +∞. ✷ Inequality (36) means that S(σ) ≥ S([ρ − εI H ] + ) − g(ε) for any states ρ and σ in S(H) such that 1 2 ρ − σ 1 ≤ ε. Inequality (37) can be written as the one-side continuity bound S(ρ) − S(σ) ≤ S(ρ ∧ εI H ) + g(ε). valid for any states ρ and σ in S(H) such that 1 2 ρ − σ 1 ≤ ε and S(ρ) < +∞. Analysis of concrete examples shows that the accuracy of the local lower bounds given by Proposition 1 is significantly lower than the accuracy of the lower bounds (32) and (34) applicable in the cases rankρ < +∞ and S(ρ) < +∞ correspondingly (in accordance with Remark 2 at the end of Section 3). The advantages of lower bounds (36) and (37) are their universality and simplicity. This is especially true for lower bound (36) valid for all states in S(H) (including states ρ with S(ρ) = +∞) and easily computable. Example 2. Consider the state ρ * = +∞ k=5 |k k| ck ln 2 k , where c = +∞ k=5 1 k ln 2 k < 1. It is easy to see that S(ρ * ) = +∞. For given sufficiently small ε ∈ (0, c] let n ε be the maximal n ∈ N ∩ [5, +∞) such that n ln 2 n ≤ 1/ε. Since n ε ln 2 n ε ≤ 1/ε, by applying inequality (38) we obtain S([cρ * − εI H ] + ) ≥ nε k=5 η 1 k ln 2 k − η nε k=5 1 k ln 2 k − εn ε ln n ε − h 2 (εn ε ) ≥ nε k=5 η 1 k ln 2 k − 1 e − 1 ln n ε − h 2 1 ln 2 n ε .(39) We have nε k=5 ln(k ln 2 k) k ln 2 k ≥ nε+1 5 dx x ln x = ln ln(n ε + 1) ln 5 ≥ ln − ln ε 3 ln 5 , where the obvious inequality n ε + 1 ≥ ε −1/3 was used. Hence, by using (39) we obtain S([ρ * − εI H ] + ) = 1 c S([cρ * − cεI H ] + ) ≥ 1 c ln − ln(cε) 3 ln 5 − 1 e − 1 ln n cε − h 2 1 ln 2 n cε for all sufficiently small ε > 0. Thus, since n ε → +∞ as ε → 0, lower bound (37) implies that L S (ρ * |ε) ≥ 1 c ln − ln(cε) 3 ln 5 − 1 ce + o(1) = O(ln(− ln ε)) as ε → 0. This is a lower estimate on the unknown rate of convergence of L S (ρ * |ε) to S(ρ * ) = +∞ as ε → 0. Energy-type functionals and the quantum relative entropy Let H be a positive operator on a separable Hilbert space H with representation (8 L E H (ρ|ε) = inf E H (σ) | σ ∈ S(H), 1 2 ρ − σ 1 ≤ ε(40) tends to E H (ρ) ≤ +∞ as ε → 0 + . Lemma 2 in Section 3 gives the following Proposition 2. Let H be a positive operator on H with representation (8) and ρ a state in S(H) such that suppρ lies within the closure H T of the linear span of T . = {τ k } +∞ k=0 . 4 Then L E H (ρ|ε) ≥ k∈Iε E k ( τ k |ρ|τ k − ε) ∀ε ∈ (0, 1],(41) where I ε is the set of all k such that τ k |ρ|τ k > ε. The lower bound (41) is faithful: its r.h.s. tends to E H (ρ) ≤ +∞ as ε → 0. Remark 4. The series in the r.h.s. of (41) contains a finite number of nonzero summands and, hence, it is finite and easily calculated for any ε > 0. The condition suppρ ⊆ H T in Proposition 2 is essential. If suppρ H T then it is easy to see that L E H (ρ|ε) = E H (ρ) = +∞ for all sufficiently small ε > 0. Proof. Let σ be any state in S(H) such that 1 2 ρ − σ 1 ≤ ε. Assume that E H (σ) < +∞. This implies that suppσ ⊆ H T . Since the same inclusion holds for the state ρ by the condition, the operatorŝ ρ = +∞ k=0 τ k |ρ|τ k |τ k τ k | andσ = +∞ k=0 τ k |σ|τ k |τ k τ k | are states in S(H) such that E H (ρ) = E H (ρ) and E H (σ) = E H (σ). Sinceρ = Π(ρ) andσ = Π(σ), where Π is the channel ̺ → +∞ k=0 τ k |̺|τ k |τ k τ k | from T(H T ) to itself, we have 1 2 ρ −σ 1 ≤ 1 2 ρ − σ 1 ≤ ε.E H (σ) = E H (σ) ≥ E H ([ρ − εI H ] + ) = k∈Iε E k ( τ k |ρ|τ k − ε). To complete the proof of (41) it suffices to note that the above inequality holds trivially if E H (σ) = +∞. The last claim of the proposition is easily verified. ✷ Example 3. Despite the fact that the lower bound (41) is not too accurate in general, there exist states ρ and ε > 0 for which an equality holds in (41). Assume that the operator H satisfies condition (12) and ρ = |τ k τ k |, k > 0. Then it is easy to see that the infimum in (40) is attained at the state σ = ε|τ 0 τ 0 | + (1 − ε)ρ for any ε ∈ (0, 1] and hence both sides of (41) are equal to E k (1 − ε). The quantum relative entropy for two states ρ and σ in S(H) is defined as D(ρ σ) = i ϕ i | ρ ln ρ − ρ ln σ |ϕ i ,(42) where {ϕ i } is the orthonormal basis of eigenvectors of the state ρ and it is assumed that D(ρ σ) = +∞ if suppρ is not contained in suppσ [8,11,18]. Since the function D ω (ρ) . = D(ρ ω) is lower semicontinuous on S(H) for any given ω ∈ S(H), the quantity L Dω (ρ|ε) = inf D(σ ω) | σ ∈ S(H), 1 2 ρ − σ 1 ≤ ε tends to D(ρ ω) ≤ +∞ as ε → 0 for any ρ ∈ S(H). Our technique allows to obtain lower bounds on L Dω (ρ|ε) which are faithful only if either rankω < +∞ or [ρ, ω] = 0. 5 It also gives faithful lower bounds on the quantities L d Dω (ρ|ε) = inf D(σ ω) | σ ∈ S(H), rankσ ≤ d, 1 2 ρ − σ 1 ≤ ε ,(43) where it is assumed that inf ∅ = +∞, and L com Dω (ρ|ε) = inf D(σ ω) | σ ∈ S(H), [ρ, σ] = 0, 1 2 ρ − σ 1 ≤ ε .(44) Note that we do not impose any constraints on the state ρ in (43) and (44). The quantities L d Dω (ρ|ε) and L com Dω (ρ|ε) are different upper bounds on L Dω (ρ|ε). If ω is a finite rank state then it is easy to see that L d Dω (ρ|ε) = L Dω (ρ|ε) for all d ≥ rankω. The lower semicontinuity of D ω implies that L com Dω (ρ|ε) tends to D(ρ ω) ≤ +∞ as ε → 0. The same is true for L d Dω (ρ|ε) provided that rankρ ≤ d. Proposition 3. Let ω be an arbitrary state with the spectral decomposition ω = +∞ k=0 λ ω k |ψ k ψ k | and ρ a state such that suppρ ⊆ suppω. A) If ρ = +∞ k=0 λ ρ k |ϕ k ϕ k | is a spectral decomposition 6 of ρ then L Dω (ρ|ε) ≥ k∈Iε (λ ρ k − ε) (ln(λ ρ k − ε) − ln ϕ k |ω|ϕ k ) − g(ε) −h 2 (ε) − η(1 − r ε ) (45) for any ε ∈ (0, 1], where I ε is the set of all k such that λ ρ k > ε,h 2 and g are the functions defined in (3) and (11), η(x) = −x ln x and r ε . = Tr[ρ − εI H ] + = k∈Iε (λ ρ k − ε). B) If S(ρ) < +∞ and d ≥ 2 is a natural number then L d Dω (ρ|ε) ≥ k∈Jε (− ln λ ω k )( ψ k |ρ|ψ k − ε) − S(ρ) − ε ln(d − 1) − h 2 (ε)(46) for any ε ∈ (0, 1 − 1/d], where J ε is the set of all k such that ψ k |ρ|ψ k > ε. C) If either D(ρ ω) < +∞ or − ϕ| ln ω|ϕ < +∞ for any eigenvector ϕ of the state ρ corresponding to its nonzero eigenvalue then L com Dω (ρ|ε) ≥ r ε D(r −1 ε [ρ−εI H ] + ω)−g(ε)−h 2 (ε)−h 2 (1−r ε ), r ε . = Tr[ρ−εI H ] + ,(47) for any ε ∈ (0, 1], whereh 2 and g are the functions defined in (3) and (11). The lower bounds (46) and (47) are faithful: their r.h.s. tend to D(ρ ω) ≤ +∞ as ε → 0. The lower bound (45) is faithful if [ρ, ω] = 0 and the state ω is diagonizable in the basis {ϕ k }. In the general case its right hand side tends to D(ρ ω) ≤ D(ρ ω) ≤ +∞ as ε → 0, where ω = +∞ k=0 ϕ k |ω|ϕ k |ϕ k ϕ k |. Remark 5. The series in the r.h.s. of (45) and (46) contain a finite number of nonzero summands and, hence, they are finite and easily calculated for any ε > 0. In the proof Proposition 3 it is shown that the condition in part C implies that the r.h.s. of (47) is finite and determined by a finite sum for any ε > 0. The condition suppρ ⊆ suppω in Proposition 3 is essential. If suppρ suppω then it is easy to see that L Dω (ρ|ε) = L d Dω (ρ|ε) = L com Dω (ρ|ε) = +∞ for all sufficiently small ε > 0. Proof. A) Let σ be a state such that 1 2 ρ − σ 1 ≤ ε. Consider the channel Π(̺) = +∞ k=0 ϕ k |̺|ϕ k |ϕ k ϕ k |, ̺ ∈ S(H). By monotonicity of the relative entropy we have D(σ ω) ≥ D(σ ω)(48)r ε D(r −1 ε [ρ − εI H ] + ω) = k∈Iε (λ ρ k − ε) (ln(λ ρ k − ε) − ln ϕ k |ω|ϕ k ) + η(r ε ) < +∞ for any ε > 0. So, as the states ρ andσ commute and 1 2 ρ −σ 1 = 1 2 Π(ρ) − Π(σ) 1 ≤ 1 2 ρ − σ 1 ≤ ε by the monotonicity of the trace norm under action of a channel, it follows from Lemma 2B in Section 3 with S 0 = ̺ = +∞ k=0 p k |ϕ k ϕ k | | {p k } ∈ P 1 that D(σ ω) ≥ r ε D(r −1 ε [ρ − εI H ] + ω) − g(ε) −h 2 (ε) − h 2 (1 − r ε ) This inequality and (48) implies (45). B) Assume that σ is a state in S(H) such that rankσ ≤ d and 1 2 ρ − σ 1 ≤ ε ≤ 1 − 1/d. Then S(σ) < +∞ and hence D(σ ω) = Trσ(− ln ω) − S(σ) = Trσ(− ln ω) − S(ρ) + [S(ρ) − S(σ)]. By Proposition 2 in [17] (proved by using Audenaert's continuity bound [2]) we have S(ρ) − S(σ) ≥ −ε ln(d − 1) − h 2 (ε)(49) (regardless of the rank of ρ). Thus, Proposition 2 implies that D(σ ω) ≥ k∈Jε (− ln λ ω k )( ψ k |ρ|ψ k − ε) − S(ρ) − ε ln(d − 1) − h 2 (ε). C) The function D ω (̺) = D(̺ ω) is nonnegative convex and satisfies inequality (18) with a f = h 2 [14, Proposition 5.24]. The condition in part C implies that D(r −1 ε [ρ − εI H ] + ω) < +∞ for all ε ∈ (0, 1]. Indeed, if D(ρ ω) < +∞ then this follows from the validity of inequality (18) for the function D ω , since [ρ − εI H ] + ≤ ρ. If − ϕ| ln ω|ϕ < +∞ then this follows from the convexity of the function D ω . Thus, Lemma 2B with S 0 = S(H) shows that D ω (σ) ≥ r ε D ω (r −1 ε [ρ − εI H ] + ) − g(ε) −h 2 (ε) − h 2 (1 − r ε ) for any state σ in S(H) such that [ρ, σ] = 0 and 1 2 ρ − σ 1 ≤ ε ≤ 1. This inequality implies (47). To prove the last claims of the proposition it suffices to note that lim ε→0 + k∈Iε (λ ρ k − ε) (ln(λ ρ k − ε) − ln ϕ k |ω|ϕ k ) = D(ρ ω) ≤ +∞, lim ε→0 + k∈Jε ( ψ k |ρ|ψ k − ε)(− ln λ ω k ) = −Trρ ln ω ≤ +∞ and that lim ε→0 + r ε D(r −1 ε [ρ − εI H ] + ω) = D(ρ ω) ≤ +∞. The first two limit relations are obvious, the last one can be easily shown by using the lower semicontinuity of the nonnegative function D ω (ρ) = D(ρ ω) and the validity of inequality (18) for this function with a f = h 2 mentioned before. ✷ By using the arguments from the proof of Proposition 3B and Proposition 1 in [17] one can obtain the following Proposition 4. Let ω be an arbitrary state in S(H) with the spectral decomposition ω = +∞ k=0 λ ω k |ψ k ψ k | and ρ a state such that suppρ ⊆ suppω. If H is a positive operator on H satisfying conditions (7) and (12) and E > 0 then D(σ ω) ≥ k∈Jε (− ln λ ω k )( ψ k |ρ|ψ k − ε) − S(ρ) − εF H (E/ε) − g(ε)(50) for any state σ in S(H) such that 1 2 ρ − σ 1 ≤ ε ≤ 1 and TrHσ ≤ E, where J ε is the set of all k such that ψ k |ρ|ψ k > ε, F H and g are the functions defined in (9) and (11). The right hand side of (50) tends to D(ρ ω) ≤ +∞ as ε → 0. Proof. To prove (50) it suffices to note that the condition TrHσ ≤ E implies the finiteness of S(σ) and to replace inequality (49) in the proof of Proposition 3B by the inequality S(ρ) − S(σ) ≥ −εF H (E/ε) − g(ε), which follows from Proposition 1 in [17]. The last claim of the proposition follows from the equivalence of (7) and (10). ✷ Note: In Proposition 4 no constrains on the state ρ are imposed. So, the set of states σ such that 1 2 ρ − σ 1 ≤ ε ≤ 1 and TrHσ ≤ E may be empty for some E and ε. Quantum conditional entropy of quantum-classical states The quantum conditional entropy (QCE) of a state ρ of a finite-dimensional bipartite system AB is defined as S(A|B) ρ = S(ρ) − S(ρ B ).(51) Definition (51) remains valid for a state ρ of an infinite-dimensional bipartite system AB with finite marginal entropies S(ρ A ) and S(ρ B ) (since the finiteness of S(ρ A ) and S(ρ B ) are equivalent to the finiteness of S(ρ) and S(ρ B )). For a state ρ with finite S(ρ A ) and arbitrary S(ρ B ) one can define the QCE by the formula S(A|B) ρ = S(ρ A ) − D(ρ ρ A ⊗ ρ B )(52) proposed and analysed by Kuznetsova in [10], where D(· ·) is the quantum relative entropy defined in (42) (the finiteness of S(ρ A ) implies the finiteness of D(ρ ρ A ⊗ ρ B )). The QCE extented by formula (52) to the convex set {ρ ∈ S(H AB ) | S(ρ A ) < +∞} possesses all basic properties of the QCE valid in finite dimensions [10]. In this subsection we consider local lower bounds on the (extended) quantum conditional entropy defined in (52) restricting attention to quantum-classical (q-c) states, i.e. states ρ in S(H AB ) having the form ρ = k p k ρ k ⊗ |k k|,(53) where {p k , ρ k } is an ensembles of states in S(H A ) and {|k } a fixed orthonormal basis in H B . 7 By using definition (52) of QCE one can show (see the proof of Corollary 3 in [19]) that S(A|B) ρ = k p k S(ρ k ).(54) This expression allows us to define the QCE on the set S qc of all q-c states with representation (53) (including the q-c states ρ with S(ρ A ) = +∞ for which definition (52) does not work). It means that the value of QCE at any q-c state ρ determined by an ensemble {p k , ρ k } coincides with the average entropy of this ensemble. The properties of the von Neumann entropy imply that the QCE defined by expression (54) on the convex set S cq satisfies inequalities (18) and (19) with a f = 0 and b f = h 2 with possible values +∞ in both sides. Since for any q-c states ρ and σ determined, respectively, by ensembles {p k , ρ k } and {q k , σ k } we have ρ − σ 1 = k p k ρ k − q k σ k 1 , expression (54) shows, due to the lower semicontinuity of the von Neumann entropy, that the function ρ → S(A|B) ρ is lower semicontinuous on the set S cq . So, the quantity L qc S(A|B) (ρ|ε) = inf S(A|B) σ | σ ∈ S qc , 1 2 ρ − σ 1 ≤ ε tends to S(A|B) ρ ≤ +∞ as ε → 0 for any q-c state ρ in S cq . The results of Section 4.2 in [17] imply easily computable lower bounds on L qc S(A|B) (ρ|ε). Proposition 3A in [17] shows that L qc S(A|B) (ρ|ε) ≥ S(A|B) ρ − ε ln(rankρ A ) − g(ε)(55) for any q-c state ρ in S qc such that rankρ A is finite and arbitrary ε ∈ (0, 1], where g is the function defined in (11). Proposition 3B in [17] shows that L qc S(A|B) (ρ|ε) ≥ S(A|B) ρ − εF H (E qc H, ε (ρ))/ε) − g(ε) ≥ S(A|B) ρ − εF H (E/ε) − g(ε) (56) for any q-c state ρ in S qc such that TrHρ A = E < +∞ and arbitrary ε ∈ (0, 1], where H is any positive operator on H A satisfying conditions (7) and (12), F H is the function defined in (9) and 8 E qc H, ε (ρ) . = k TrH((p k ρ k ) ∧ εI A ) = E − k TrH[p k ρ k − εI A ] + = o(1) as ε → 0. The lower bound (56) is applicable to any q-c state ρ with finite S(ρ A ), since for any such state there is a positive operator H on H A satisfying conditions (7) and (12) such that TrHρ A < +∞ [15,Proposition 4]. As in the von Neumann entropy case (see Section 4.1), the operator H can be treated as a "free parameter", with which one can optimize lower bound (56). Since one-side continuity bounds given by parts A and B of Proposition 3 in [17] are asymptotically tight for large rankρ A and TrHρ A correspondingly, the lower bounds (55) and (56) are quite accurate. They are faithful: the right hand sides of (55) and (56) tends to S(A|B) ρ as ε → 0 (for the last one this follows from the equivalence of (7) and (10)). Coarse but universal and easily computable lower bounds on L qc S(A|B) (ρ|ε) can be obtained by using Lemma 2 in Section 3. In the following proposition we assume that S(A|B) is the homogeneous extension of the QCE to the cone generated by the set S qc of q-c states which is defined according to the rule (22). We use the notation in (21). Proposition 5. Let ρ be an arbitrary q-c state in S qc . Then L qc S(A|B) (ρ|ε) ≥ S(A|B) ρ⊖ε − g(ε) ∀ε ∈ (0, 1],(57) where ρ ⊖ ε . = k [p k ρ k − εI A ] + ⊗ |k k|. If S(A|B) ρ < +∞ then L qc S(A|B) (ρ|ε) ≥ S(A|B) ρ − S(A|B) ρ∧ε − g(ε) ∀ε ∈ (0, 1],(58) where ρ ∧ ε . = k ((p k ρ k ) ∧ εI A ) ⊗ |k k|. The lower bounds (57) and (58) are faithful: their r.h.s. tend to S(A|B) ρ ≤ +∞ as ε → 0. Using expression (54) one can rewrite Proposition 5 as follows. Corollary 1. Let {p k , ρ k } be an arbitrary ensemble of states in S(H). Then k q k S(σ k ) ≥ k S([p k ρ k − εI H ] + ) − g(ε) (59) for any ensemble {q k , σ k } of states in S(H) such that 1 2 k p k ρ k − q k σ k 1 ≤ ε, where S(·) in the r.h.s. is the extended von Neumann entropy defined in (4). If k p k S(ρ k ) < +∞ then k q k S(σ k ) ≥ k p k S(ρ k ) − k S((p k ρ k ) ∧ εI H ) − g(ε) (60) for any ensemble {q k , σ k } of states in S(H) such that 1 2 k p k ρ k − q k σ k 1 ≤ ε. Remark 6. Since p k ρ k = (p k ρ k ) ∧ εI H + [p k ρ k − εI H ] + , it follows from (5) that k p k S(ρ k ) − k S((p k ρ k ) ∧ εI H ) ≥ k S([p k ρ k − εI H ] + ). So, lower bound (60) is sharper than (59). The advantage of lower bound (59) consists in its validity for all ensembles {p k , ρ k } (including ensembles with k p k S(ρ k ) = +∞). Note also that the r.h.s. of (59) is easily calculated, since the series in the r.h.s. of (59) contains a finite number of nonzero summands and all the operators [p k ρ k − εI H ] + have finite rank. Proof of Proposition 5. Let ρ = k p k ρ k ⊗ |k k| and σ = k q k σ k ⊗ |k k| be q-c states in S qc such that ρ − σ 1 = k p k ρ k − q k σ k 1 ≤ 2ε. For each k let {ϕ k i } i be an orthonormal basis in H A such that ρ k = i λ ρ k i |ϕ k i ϕ k i |, where {λ ρ k i } i is a non-increasing sequence of eigenvalues of ρ k . Consider the q-c statẽ σ = k q kσk ⊗ |k k|,σ k = i λ σ k i |ϕ k i ϕ k i |, where {λ σ k i } i is a non-increasing sequence of eigenvalues of σ k for each k. By Mirsky inequality (1) we have ρ −σ 1 = k p k ρ k − q kσk 1 ≤ k p k ρ k − q k σ k 1 ≤ 2ε. The representation (54) implies that S(A|B) σ = k q k S(σ k ) = k q k S(σ k ) = S(A|B)σ. It is easy to see that any q-c state ρ in S qc satisfies conditions (23) and (24) with S 0 = S qc and that ρ ∧ εI AB = k ((p k ρ k ) ∧ εI A ) ⊗ |k k|, [ρ − εI AB ] + = k [p k ρ k − εI A ] + ⊗ |k k|. So, since the states ρ andσ commute and the (extended) quantum conditional entropy S(A|B) defined by expression (54) on the convex set S cq of all q-c states is a concave nonnegative function on S cq satisfying inequality (19) with b f = h 2 and taking finite values at finite rank states in S cq , Lemma 2B in Section 3 with S 0 = S cq implies that S(A|B) σ = S(A|B)σ ≥ S(A|B) ρ⊖ε − g(ε), while Lemma 2A with S 0 = S cq implies that S(A|B) σ = S(A|B)σ ≥ S(A|B) ρ − S(A|B) ρ∧ε − g(ε) provided that S(A|B) ρ < +∞. ✷ 5 Characteristics of discrete random variables On general methods of getting local lower bounds In this section we consider the class of characteristics of discrete random variables X 1 ,...,X n each of which can be considered as a function f on a convex subset P 0 of the set P n of all n-variate probability distributionsp = {p i 1 ...in } (i 1 ,..,in)∈N n 0 (where N 0 = N ∪ {0}) taking values in [0, +∞] and satisfying the inequalities f (λp + (1 − λ)q) ≥ λf (p) + (1 − λ)f (q) − a f (λ)(61) and f (λp + (1 − λ)q) ≤ λf (p) + (1 − λ)f (q) + b f (λ) (62) for any distributionsp andq in P 0 and any λ ∈ [0, 1] with possible value +∞ in both sides, where a f (λ) and b f (λ) are continuous functions on [0, 1] such that a f (0) = b f (0) = 0. Inequalities (61) and (62) are classical versions of inequalities (18) and (19). So, we may call functions satisfying (61) and (62) locally almost affine (briefly, LAA functions) on the set P 0 . As in Section 3 we will assume for technical simplicity that a f (λ) = a f (1 − λ) and b f (λ) = b f (1 − λ) for all λ ∈ [0, 1] and that the functions a f and b f are concave and non-decreasing on [0, 1 2 ]. The list of important entropic characteristics of discrete random variables satisfying inequalities (61) and (62) includes the Shannon (conditional) entropy, the mutual (conditional) information, the Kullback-Leibler divergence (as a function of the first argument), the interaction information, etc. (see the subsections below and Section 4.4 in [17]). We will assume that the set P n of all n-variate probability distributions is equipped with the total variance distance TV defined in (13). Note that many of the characteristics mentioned before are lower semicontinuous functions either on the whole set P n or on proper subsets of P n w.r.t. this distance. Below we will use the notation described in Section 2.2. If a function f defined on a subset P 0 ⊆ P n is lower semicontinuous at a distribution p ∈ P 0 then the quantity L f (p|ε) = inf {f (q) |q ∈ P 0 , TV(p,q) ≤ ε} tends to f (p) ≤ +∞ as ε → 0. We will say that L f (p|ε) is a faithful local lower bound on a function f at a distributionp if Another way to get local lower bounds for characteristics of discrete random variables satisfying inequalities (61) and (62) is given by the classical analogue of Lemma 2 in Section 3 presented below. L f (p|ε) ≤ L f (p|ε) ∀ε ∈ (0, 1] and lim ε→0 L f (p|ε) = f (p) ≤ +∞. For any n-variate probability distributionp . = {p i 1 ...in } (i 1 ,. .,in)∈N n 0 and arbitrary ε in (0, 1] introduce two n-variate arrays of nonnegative numbers p ∧ ε . = {min{p i 1 ...in , ε}} (i 1 ,..,in)∈N n 0 , [p − ε] + . = {max{p i 1 ...in − ε, 0}} (i 1 ,..,in)∈N n 0 .(63) It is clear thatp =p ∧ ε + [p − ε] + and that [p − ε] + tends top as ε → 0. Write P 0 for the cone generated by a subset P 0 ⊆ P n , i.e. P 0 is the set of all n-variate arrays of nonnegative numbers of the form {cp i 1 ...in }, {p i 1 ...in } ∈ P 0 , c ∈ R + . For any function f on a convex subset P 0 ⊆ P n denote byf the homogeneous extension of f to the cone P 0 defined as f (p) . = p 1 f (p/ p 1 ), p 1 . = i 1 ,..,in p i 1 ...in ,(64) for any nonzero arrayp = {p i 1 ...in } in P 0 and equal to 0 at the zero array. Lemma 3. Let f be a function on a convex subset P 0 ⊆ P n taking values in [0, +∞] and satisfying inequalities (61) and (62) with possible values +∞ in both sides. Letp . = {p i 1 ...in } be a probability distribution in P 0 with the properties (a) h(p) ∈ P 0 for any function h : R + → R + such that h(x) ≤ x ∀x ∈ R + ,(65)(b) [p−q] ± = {[p i 1 ...in −q i 1 ...in ] ± } ∈ P 0 for anyq . = {q i 1 ...in } in P 0 s.t.q =p, where h(p) = {h(p i 1 ...in )}, [x] − = max{−x, 0} and [x] + = max{x, 0}. A) If f (p) < +∞ then f (q) ≥ f (p) −f (p ∧ ε) − D f (ε) −ã f (ε) (66) for any probability distributionq in P 0 such that T V (p,q) ≤ ε ≤ 1, wherep ∧ ε is the array defined in (63),f is the homogeneous extension of f defined in (64), D f (ε) = (1 + ε)(a f + b f ) ε 1 + ε andã f (ε) = a f (ε) if ε ∈ [0, 1 2 ] a f ( 1 2 ) if ε ∈ ( 1 2 , 1]. B) If either f (p) < +∞ orf ([p − ε] + ) < +∞ ∀ε ∈ (0, 1],(67) where [p − ε] + is the array defined in (63), then f (q) ≥f ([p − ε] + ) − D f (ε) −ã f (ε) − a f (1 − r ε )(68) for any probability distributionq in P 0 such that T V (p,q) ≤ ε ≤ 1, where r ε = [p − ε] + 1 = i 1 ,..,in [p i 1 ...in − ε] + . Remark 7. All the terms in the r.h.s. of (66) and (68) are well defined and finite. Indeed, the arraysp ∧ ε and [p − ε] + belong to the cone P 0 due to condition (65). If f (p) < +∞ then inequality (61) implies thatf (p ∧ ε) < +∞ andf ( [p − ε] + ) < +∞, sincep ∧ ε + [p − ε] + =p. The lower bounds given by Lemma 3 can not be too accurate (by the reason described in Remark 2 in Section 3). The main advantage of these lower bounds is their universality. Note also that the lower bound given by Lemma 3B is easily computable as the array [p − ε] + has a finite number of nonzero entries for anyp and ε > 0. Proof. The set P n of n-variate probability distributions {p i 1 ...in } (i 1 ,..,in)∈N n 0 is isomorphic to the set S τ of states in S(H A 1 ...An ) diagonizable in the basis {τ 1 i 1 ⊗ · · · ⊗ τ n in } (i 1 ,..,in)∈N n 0 , where {τ 1 i } i∈N 0 ,...,{τ n i } i∈N 0 are orthonormal base in separable Hilbert spaces H A 1 ,.., H An correspondingly. This isomorphism is given by the affine isometry Θ :p = {p i 1 ...in } (i 1 ,..,in)∈N n 0 → i 1 ,..,in p i 1 ...in |τ 1 i 1 τ 1 i 1 | ⊗ · · · ⊗ |τ n in τ n in | from P n onto S τ . Since the set S τ consists of commuting states, the isomorphism Θ allows us to derive all the claims of Lemma 3 from the corresponding claims of Lemma 2 in Section 3. ✷ The Shannon entropy and the equivocation L H (p|ε) ≥ H([p − ε] + ) − g(ε)(69) for anyp in P 1 , where g is the function defined in (11) and H([p − ε] + ) . = c ε H(c −1 ε [p − ε] + ) = i∈Iε η(p i − ε) − η(c ε ), c ε = [p − ε] + 1 = i∈Iε (p i − ε), I ε is the finite set of all i such that p i − ε > 0. The lower bound (69) is faithful: its r.h.s. tends to H(p) ≤ +∞ as ε → 0. for anyq in P 2 such that TV(p,q) ≤ ε, where F S is the function defined in (16) with S = {E i } +∞ i=0 . This shows that L H(X 1 |X 2 ) (p|ε) ≥ H(X 1 |X 2 )p − εF S (E/ε) − g(ε) ∀ε ∈ (0, 1].(73) The equivalence of (15) and (17) imply that the r.h.s. of (73) tends to H(X 1 |X 2 )p as ε → 0. One-side continuity bounds (70) and (72) p 1 = {[p 1 ] i } i . So, lower bound (73) is applicable if H(p 1 ) is finite. Not too accurate but universal and simple lower bounds on L H(X 1 |X 2 ) (p|ε) can be obtained by applying Lemma 3 to the function f (p) = H(X 1 |X 2 )p. Proposition 6. Let H(X 1 |X 2 ) be the homogeneous extension of H(X 1 |X 2 ) to the cone of 2-variate arrays of nonnegative numbers defined according to the rule (64). A) Ifp is an arbitrary probability distribution in P 2 then L H(X 1 |X 2 ) (p|ε) ≥ H(X 1 |X 2 ) [p−ε] + − g(ε) ∀ε ∈ (0, 1],(74) where [p − ε] + is the array defined in (63). B) Ifp is a probability distribution in P 2 such that H(X 1 |X 2 )p < +∞ then L H(X 1 |X 2 ) (p|ε) ≥ H(X 1 |X 2 )p − H(X 1 |X 2 )p ∧ε − g(ε) ∀ε ∈ (0, 1],(75) wherep ∧ ε is the array defined in (63). The lower bounds (74) and (75) are faithful: their r.h.s.tend to H(X 1 |X 2 )p ≤ +∞ as ε → +∞. Proof. The main claims of the proposition follow directly from Lemma 3 with P 0 = P 2 and the properties of the functionp → H(X 1 |X 2 )p mentioned before. We have only to note that condition (67) holds for this function, since it takes finite values at any probability distributions in P 2 with a finite number of nonzero entries. To prove the last claim it suffices to note that H(X 1 |X 2 ) [p−ε] + tends to H(X 1 |X 2 )p as ε → 0 due to the lower semicontinuity of the functionq → H(X 1 |X 2 )q and the inequality H(X 1 |X 2 )p ≥ H(X 1 |X 2 )p ∧ε + H(X 1 |X 2 ) [p−ε] +(76) which follows from the concavity of this function, sincep =p ∧ ε + [p − ε] + . ✷ Remark 8. Inequality (76) implies that lower bound (75) is sharper than (74). The advantage of lower bound (74) consists in its validity for all probability distributionsp (including probability distributions with infinite H(X 1 |X 2 )p). Note also that the r.h.s. of (74) is easily calculated, since the array [p − ε] + contains a finite number of nonzero entries. Affine functionals and the Kullback-Leibler divergence Let S = {E i } +∞ i=0 be a sequence of nonnegative numbers. Then E S (p) = +∞ i=0 E i p i ,p = {p i } +∞ i=0 , is an affine lower semicontinuous function on the set P 1 taking values in [0, +∞). The lower semicontinuity of E S implies that for anyp ∈ P 1 the quantity L E S (p|ε) = inf {E S (q) |q ∈ P 1 , TV(p,q) ≤ ε}(77) tends to E S (p) ≤ +∞ as ε → 0 + . Direct application of Lemma 3 gives the following Proposition 7. Let S = {E i } +∞ i=0 be a sequence of nonnegative numbers andp = {p i } +∞ i=0 a probability distribution in P 1 . Then L E S (p|ε) ≥ i∈Iε E i (p i − ε) ∀ε ∈ (0, 1],(78) where I ε is the set of all i such that p i > ε. The lower bound (78) is faithful: its r.h.s. tends to E S (p) ≤ +∞ as ε → 0. Note: Since the set I ε is finite, the r.h.s. of (78) is easily calculated for any ε > 0. Example 5. Assume that S = {E i } +∞ i=0 is a nondecreasing sequence with E 0 = 0 andp = {p i } +∞ i=0 is a degenerate distribution with p j = 1 for some j > 0. Then it is easy to see that the infimum in (77) is attained at the distributionq = ε{1, 0, 0, ...}+(1−ε)p for any ε ∈ (0, 1] and hence both sides of (78) are equal to E j (1 − ε). The Kullback-Leibler divergence for probability distributionsp = {p i } andq = {q i } is defined as D KL (p q) = i p i ln(p i /q i ), where it is assumed that 0 ln 0 = 0 and D KL (p q) = +∞ if suppp is not contained in suppq [9,4]. 10 For any givenq the function Dq(p) . = D KL (p q) is lower semicontinuous on P 1 , takes values in [0, +∞] and satisfies inequalities (61) and (62) with a f = h 2 and b f = 0 [9,4,14]. The lower semicontinuity of this function implies that for anyp and q the quantity L Dq (p|ε) = inf {D KL (r q) |r ∈ P 1 , TV(p,r) ≤ ε} tends to D KL (p q) ≤ +∞ as ε → +∞. For any natural d consider the quantity L d Dq (p|ε) = inf {D KL (r q) |r ∈ P 1 , |r| ≤ d, TV(p,r) ≤ ε} , where it is assumed that inf ∅ = +∞. This quantity is an upper bound on L Dq (p|ε). It also tends to D KL (p q) ≤ +∞ as ε → +∞ provided that |p| ≤ d. If |q| < +∞ then it is easy to see that L Dq (p|ε) = L d Dq (p|ε) for all d ≥ |q|. B using Lemma 3 and Proposition 7 one can prove the following Proposition 8. Letp = {p i } andq = {q i } be probability distributions in P 1 such that suppp ⊆ suppq. Then L Dq (p|ε) ≥ i∈Iε (p i − ε) ln p i − ε c ε q i − g(ε) −h 2 (ε) − h 2 (1 − c ε ) ∀ε ∈ (0, 1],(79) where I ε is the set of all i such that p i > ε, c ε = i∈Iε (p i − ε) andh 2 is the modification of the binary entropy defined in (3). If H(p) < +∞ and d ≥ 2 is a natural number then L d Dq (p|ε) ≥ i∈Iε (− ln q i )(p i − ε) − H(p) − ε ln(d − 1) − h 2 (ε) ∀ε ∈ (0, 1−1/d]. (80) The lower bounds (79) and (80) are faithful: their r.h.s. tend to D KL (p q) ≤ +∞ as ε → +∞. Remark 9. Since the set I ε is finite for any ε > 0, the right hand sides of (79) and (80) are finite and easily calculated. The condition suppp ⊆ suppq in Proposition 8 is essential. If suppp suppq then it is easy to see that L Dq (p|ε) = L d Dq (p|ε) = +∞ for all sufficiently small ε > 0. Proof. The function Dq(p) = D KL (p q) is nonnegative convex and satisfies inequality (61) with a f = h 2 . It is easy to see that condition (67) with f = Dq is valid for any probability distributionp such that suppp ⊆ suppq. Thus, Lemma 3B with P 0 = P 1 shows that Dq(r) ≥ c ε Dq(c −1 ε [p − ε] + ) − g(ε) −h 2 (ε) − h 2 (1 − c ε ), c ε . = [p − ε] + 1 , for any probability distributionr in P 1 such that TV(p,r) ≤ ε ≤ 1, where [p − ε] + is the array defined in (63). This implies the lower bound (79). Assume thatr is a probability distribution in P 1 such that |r| ≤ d and TV(p,r) ≤ ε ≤ 1 − 1/d. Then H(r) < +∞ and hence D(r q) = +∞ i=0 (− ln q i )r i − H(r) = +∞ i=0 (− ln q i )r i − H(p) + [H(p) − H(r)]. The classical version of Proposition 2 in [17] shows that H(p) − H(r) ≥ −ε ln(d − 1) − h 2 (ε) (regardless of the value of |p|). Thus, Proposition 7 implies that D(r q) ≥ i∈Iε (− ln q i )(p i − ε) − H(p) − ε ln(d − 1) − h 2 (ε). To prove the last claim of the proposition it suffices to note that lim ε→0 c ε D(c −1 ε [p − ε] + q) = D(p q) ≤ +∞ and that lim ε→0 i∈Iε (− ln q i )(p i − ε) = +∞ i=0 (− ln q i )p i ≤ +∞. The second limit relation is obvious, the first one can be easily shown by using the lower semicontinuity of the nonnegative function Dq(p) = D KL (p q) and the validity of inequality (61) for this function with a f = h 2 mentioned before. ✷ The mutual information The mutual information of random variables X 1 and X 2 with joint probability distributionp = {p i 1 i 2 } is defined as I(X 1 : X 2 )p = D KL (p p 1 ×p 2 ) = H(p 1 ) + H(p 2 ) − H(p), where the second formula is valid provided that H(p) is finite. By the properties of the Kullback-Leibler divergence the functionp → I(X 1 : X 2 )p is lower semicontinuous on P 2 and takes values in [0, +∞]. It satisfies inequalities (61) and (62) with a f = b f = h 2 with possible value +∞ in both sides. 11 The lower semicontinuity of this function implies that for anyp ∈ P 2 the quantity L I(X 1 :X 2 ) (p|ε) = inf {I(X 1 : X 2 )q |q ∈ P 2 , TV(p,q) ≤ ε} tends to I(X 1 : X 2 )p ≤ +∞ as ε → +∞. Below we will obtain several faithful lower bounds on L I(X 1 :X 2 ) (p|ε). The well known inequality I(X 1 : X 2 )p ≤ H(p 1 ) and the validity of inequalities (61) and (62) for the functionp → I(X 1 : X 2 )p with a f = b f = h 2 mentioned before imply that this function belongs to the class T 1 2 (1, 2) in the notation used in Section 4.4 in [17]. Thus, ifp is such that either |p 1 | or |p 2 | is finite then Proposition 5B in [17] and the symmetry arguments imply that I(X 1 : X 2 )p − I(X 1 : X 2 )q ≤ ε ln min{|p 1 |, |p 2 |} + 2g(ε) for anyq in P 2 such that TV(p,q) ≤ ε (|p k | denotes the number of nonzero entries of the marginal distributionp k , g(ε) is the function defined in (11)). This shows that L I(X 1 :X 2 ) (p|ε) ≥ I(X 1 : X 2 )p − ε ln min{|p 1 |, |p 2 |} − 2g(ε) ∀ε ∈ (0, 1]. Ifp is such that +∞ i=0 E i [p k ] i = E < +∞, where k is either 1 or 2 and {E i } +∞ i=0 is a nondecreasing sequence of nonnegative numbers with E 0 = 0 satisfying condition (15), then Proposition 6B in [17] and the symmetry arguments imply that I(X 1 : X 2 )p − I(X 1 : X 2 )q ≤ εF S (E/ε) + 2g(ε) for anyq in P 2 such that TV(p,q) ≤ ε, where F S is the function defined in (16) with S = {E i } +∞ i=0 . This shows that L I(X 1 :X 2 ) (p|ε) ≥ I(X 1 : X 2 )p − εF S (E/ε) − 2g(ε) ∀ε ∈ (0, 1]. The equivalence of (15) and (17) imply that the r.h.s. of (84) tends to I(X 1 : X 2 )p as ε → 0. Thus, both lower bounds (82) and (84) Coarse but universal and easily computable lower bounds on L I(X 1 :X 2 ) (p|ε) can be obtained by applying Lemma 3 to the function f (p) = I(X 1 : X 2 )p. Proposition 9. Let I(X 1 : X 2 ) be the homogeneous extension of I(X 1 : X 2 ) to the cone of 2-variate arrays of nonnegative numbers defined according to the rule (64). A) Ifp is an arbitrary probability distribution in P 2 then L I(X 1 :X 2 ) (p|ε) ≥ I(X 1 : X 2 ) [p−ε] + − 2g(ε) −h 2 (ε) − h 2 (1 − r ε ) ∀ε ∈ (0, 1], where [p − ε] + is the array defined in (63), r ε = [p − ε] + 1 = i 1 ,i 2 [p i 1 i 2 − ε] + andh 2 is the function defined in (3). B) Ifp is a probability distribution in P 2 such that I(X 1 : X 2 )p < +∞ then L I(X 1 :X 2 ) (p|ε) ≥ I(X 1 : X 2 )p − I(X 1 : X 2 )p ∧ε − 2g(ε) −h 2 (ε) ∀ε ∈ (0, 1], wherep ∧ ε is the array defined in (63). The lower bounds (85) and (86) are faithful: their r.h.s. tend to I(X 1 : X 2 )p ≤ +∞ as ε → +∞. Proof. The main claims of the proposition follow directly from Lemma 3 with P 0 = P 2 and the properties of the functionp → I(X 1 : X 2 )p mentioned at the begin of this subsection. We have only to note that condition (67) holds for this function, since it takes finite values at any probability distributions in P 2 with a finite number of nonzero entries. To prove the last claim it suffices to note that I(X 1 : X 2 ) [p−ε] + tends to I(X 1 : X 2 )p as ε → 0 due to the lower semicontinuity of the functionp → I(X 1 : X 2 )p and the inequality I(X 1 : X 2 )p ≥ I(X 1 : X 2 )p ∧ε + I(X 1 : X 2 ) [p−ε] + − h 2 (r ε ),(87) which follows from the validity of inequality (61) for this function with a f = h 2 mentioned before, sincep =p ∧ ε + [p − ε] + . ✷ Remark 10. Inequality (87) implies that lower bound (86) is sharper than (85). The advantage of lower bound (85) consists in its validity for all probability distributionsp (including probability distributionsp with infinite I(X 1 : X 2 )p). Note also that the r.h.s. of (85) is easily calculated, since the array [p − ε] + contains a finite number of nonzero entries. I am grateful to A.S.Holevo and to the participants of his seminar "Quantum probability, statistic, information" (the Steklov Mathematical Institute) for useful discussion. So, since the statesρ andσ commute and E H is an affine nonnegative function on S(H T ) taking finite values at finite rank states in S(H T ) diagonizable in the basis {τ k }, Lemma 2B in Section 3 with S 0 = S(H T ) implies that The function Dω(̺) = D(̺ ω) is nonnegative convex and satisfies inequality(18) with a f = h 2[14, Proposition 5.24]. Since the states ρ andω are diagonizable in the basis {ϕ k } and suppρ ⊆ suppω by the condition suppρ ⊆ suppω, we have Faithful local lower bound for characteristics of discrete random variables satisfying inequalities (61) and (62) and such that|f (p)| ≤ C f m k=1 H(p k ) ≤ +∞, C f > 0, m ≤ n,for any distributionp in the domain of f (wherep k denotes the marginal distribution ofp corresponding to the k-th component and H(·) is the Shannon entropy) can be obtained by using the one-side continuity bounds presented in Propositions 5 and 6 in [17]. The concrete examples are considered in Sections 5.2 and 5.4 below. The Shannon entropy H(p) = − i p i ln p i of 1-variate probability distributionp = {p i } is a concave lower semicontinuous function on the set P 1 taking values in [0, +∞] and satisfying inequality(14). The lower semicontinuity of the Shannon entropy implies that the quantityL H (p|ε) = inf {H(q) |q ∈ P 1 , TV(p,q) ≤ ε}tends to H(p) ≤ +∞ as ε → +∞ for any probability distributionp in P 1 .Sincefor anyp = {p i } andq = {q i } we have H(p) = S(ρ), H(q) = S(σ) and TV(p,q) = 1 2 ρ − σ 1 , where ρ = i p i |τ i τ i | and σ = i q i |τ i τ i | are states on a separable Hilbert space H determined by a given orthonormal basic {τ i } in H, lower bounds on L H (p|ε) can be obtained by "rewriting" the lower bounds on L S (ρ|ε) presented in Section 4.1. In particular, it follows from Proposition 1 that Example 4 . 4Consider the probability distributionp * = (ci ln 2 i) −1 +∞ i=5 , where c = +∞ i=5 1 i ln 2 i < 1, with infinite Shannon entropy. Lower bound (69) and the estimates made in Example 2 in Section 4.1 show that L H (p * |ε) ≥ 1 c ln − ln(cε) 3 ln 5 − 1 ce + o(1) = O(ln(− ln ε)) as ε → 0. ). Then the function E H (ρ) = TrHρ defined according to the rule (6) is an affine lower semicontinuous function on S(H) taking values in [0, +∞]. If H is the Hamiltonian (energy observable) of some quantum system described by the space H then E H (ρ) is the mean energy of a state ρ. The lower semicontinuity of E H implies that for any state ρ in S(H) the quantity are faithful (provided thatp satisfies the corresponding conditions). By Proposition 4 in [15] the existence of a nondecreasing sequence {E i } +∞ i=0 of nonnegative numbers satisfying condition (15) such that +∞ i=0 E i [p k ] i < +∞ is equivalent to the finiteness of the Shannon entropy of the distribution {[p k ] i } i . So, the lower bound (84) is applicable if either H(p 1 ) or H(p 2 ) is finite. The support suppρ of a state ρ is the closed subspace spanned by the eigenvectors of ρ corresponding to its positive eigenvalues. We will assume that the values of a discrete random variable is indexed by the set N 0 = N ∪ {0}. This simplifies the transition from the case of quantum systems to classical ones. ρ ∧ εI H and [ρ − εI H ] + are the operators defined in(21). The support suppρ of a state ρ is the closed subspace spanned by the eigenvectors of ρ corresponding to its positive eigenvalues. In the case rankω < +∞ faithful lower bounds on L Dω (ρ|ε) can be also derived from the continuity bounds for the function ρ → D(ρ ω) in the finite-dimensional settings obtained in [7, Section 5.8] and in[3, Section 5.2.4]. We assume that {ϕ k } is basis in H, so some λ ρ k in this decomposition may be equal to zero. An ensemble {p k , ρ k } is a collection {ρ k } of quantum states with a probability distribution {p k }. (p k ρ k ) ∧ εI A and [p k ρ k − εI A ] + are the operators defined in(21). Inequality (61) with a f = 0 means concavity of the conditional entropy, inequality (62) with b f = h 2 can be proved easily for probability distributionsp andq such that H(p 2 ) and H(q 2 ) are finite by using concavity of the Shannon entropy and inequality(14). The validity of this inequality in the general case can be proved by approximation. The support suppp of a probability distributionp = {p i } is the set of all i such that p i > 0. This follows, in particular, from the inequalities (86) and (87) in[16]. This is a lower bound on the unknown rate of convergence of L H (p * |ε) to H(p * ) = +∞ as ε → 0.The Shannon conditional entropy H(X 1 |X 2 ) (also called equivocation) of random variables X 1 and X 2 with joint probability distributionp = {p i 1 i 2 } is defined aswhere the second formula is valid if H(p 2 ) is finite (while the first one is well defined for anyp in P 2 , since all the terms in the series are nonnegative)[4]. The function p → H(X 1 |X 2 )p is lower semicontinuous on the set P 2 and takes values in [0, +∞]. It satisfies inequalities(61)and(62)with a f = 0 and b f = h 2 .9The lower semicontinuity of this function implies that the quantitytends to H(X 1 |X 2 )p ≤ +∞ as ε → +∞ for anyp ∈ P 2 . Below we will obtain several lower bounds on L H(X 1 |X 2 ) (p|ε). The well known inequalityand the validity of inequalities (61) and (62) for the functionp → H(X 1 |X 2 )p with a f = 0 and b f = h 2 mentioned before imply that this function belongs to the class T 1 2 (1, 1) in the notation used in Section 4.4 in[17]. Thus, ifp is such that the marginal distributionp 1 has a finite number of nonzero entries (denoted by |p 1 |) then Proposition 5B in[17]implies thatfor anyq in P 2 such that TV(p,q) ≤ ε, where g is the function defined in(11). This shows thatis a nondecreasing sequence of nonnegative numbers with E 0 = 0 satisfying condition(15), then Proposition 6B in[17]implies that H(X 1 |X 2 )p − H(X 1 |X 2 )q ≤ εF S (E/ε) + g(ε)(72) Continuity of quantum conditional information. R Alicki, M Fannes, J. Phys. A Math. Gen. 375R.Alicki, M.Fannes, "Continuity of quantum conditional information", J. Phys. A Math. Gen. 37(5), L55-L57 (2004). A sharp continuity estimate for the von Neumann entropy. K M R Audenaert, J. Math. Phys. A: Math. Theor. 4028K.M.R.Audenaert, "A sharp continuity estimate for the von Neumann entropy", J. Math. Phys. A: Math. Theor. 40(28), 8127-8136 (2007). A Bluhm, A Capel, P Gondolf, A Perez-Hernandez, arXiv:2208.00922Continuity of quantum entropic quantities via almost convexity. A.Bluhm, A.Capel, P.Gondolf, A.Perez-Hernandez, "Continuity of quantum en- tropic quantities via almost convexity". arXiv:2208.00922 (2022). Elements of information theory. T M Cover, J A Thomas, New YorkT.M.Cover, J.A.Thomas, "Elements of information theory", New York (2006). Existence and continuity of differential entropy for a class of distributions. H Ghourchian, A Gohari, A Amini, IEEE Commun. Lett. 217H.Ghourchian, A.Gohari, A.Amini, "Existence and continuity of differential en- tropy for a class of distributions", IEEE Commun. Lett. 21(7), 1469-1472 (2017); . 10.1109/LCOMM.2017.2689770https://doi.org/10.1109/LCOMM.2017.2689770. Maximum and minimum entropy states yielding local continuity bounds. E P Hanson, N Datta, arXiv:1706.02212Journal of Mathematical Physics. 59442204E.P.Hanson, N.Datta, "Maximum and minimum entropy states yielding local continuity bounds", Journal of Mathematical Physics 59(4), 042204 (2018); arXiv:1706.02212. E P Hanson, arXiv:2010.02408Entropic Continuity Bounds and Eventually Entanglement-Breaking Channels. E.P.Hanson, "Entropic Continuity Bounds and Eventually Entanglement- Breaking Channels", arXiv:2010.02408. Quantum systems, channels, information. A mathematical introduction. A S Holevo, Berlin, DeGruyterA.S.Holevo, "Quantum systems, channels, information. A mathematical introduc- tion", Berlin, DeGruyter (2012). On information and sufficiency. S Kullback, R A Leibler, Annals of Mathematical Statistics. 221S.Kullback, R.A.Leibler, "On information and sufficiency", Annals of Mathemat- ical Statistics. 22 (1), 79-86 (1951) Quantum conditional entropy for infinite-dimensional systems. A A Kuznetsova, Theory of Probability and its Applications. 55A.A.Kuznetsova, "Quantum conditional entropy for infinite-dimensional systems", Theory of Probability and its Applications. 55(4), 709-717 (2011). Expectation and Entropy Inequalities for Finite Quantum Systems. G Lindblad, Comm. Math. Phys. 392G.Lindblad, "Expectation and Entropy Inequalities for Finite Quantum Systems", Comm. Math. Phys., 39(2), 111-119 (1974). Symmetric gauge functions and unitarily invariant norms. L Mirsky, Quart. J. Math. Oxford. 211L.Mirsky, "Symmetric gauge functions and unitarily invariant norms", Quart. J. Math. Oxford, 2(11), 50-59 (1960). M A Nielsen, I L Chuang, Quantum Computation and Quantum Information. Cambridge University PressM.A.Nielsen, I.L.Chuang, "Quantum Computation and Quantum Information", (Cambridge University Press, 2000). Quantum Entropy and Its Use. M Ohya, D Petz, Theoretical and Mathematical Physics. Berlin HeidelbergSpringerM.Ohya, D.Petz, "Quantum Entropy and Its Use", Theoretical and Mathematical Physics (Springer Berlin Heidelberg, 2004). Entropy characteristics of subsets of states I. M E Shirokov, arXiv:quant-ph/0510073Izv. Math. 706M.E.Shirokov, "Entropy characteristics of subsets of states I", Izv. Math., 70(6), 1265-1292 (2006); arXiv: quant-ph/0510073. Quantifying continuity of characteristics of composite quantum systems. M E Shirokov, Physica Scripta. 98442002M.E.Shirokov, "Quantifying continuity of characteristics of composite quantum systems", Physica Scripta, 98(4), 042002 (2023). Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique. M E Shirokov, arXiv:2207.08791M.E.Shirokov, "Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique", arXiv:2207.08791. General properties of entropy. A , Rev. Mod. Phys. 50A.Wehrl, "General properties of entropy", Rev. Mod. Phys. 50, 221-250 (1978). Optimal uniform continuity bound for conditional entropy of classical-quantum states. M M Wilde, arXiv:1909.01755Quantum Information Processing. 19M.M.Wilde, "Optimal uniform continuity bound for conditional entropy of classical-quantum states", Quantum Information Processing. 19, Article no. 61 (2020); arXiv:1909.01755. Quantum Information Theory. M M Wilde, Cambridge Univ. PressCambridge, UKM.M.Wilde, "Quantum Information Theory", Cambridge, UK: Cambridge Univ. Press, (2013). Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints. A Winter, arXiv:1507.07775Comm. Math. Phys. 3471A.Winter, "Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints", Comm. Math. Phys., 347(1), 291-313 (2016); arXiv:1507.07775.
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10.5281/zenodo.7750670
[ "https://export.arxiv.org/pdf/2304.00019v1.pdf" ]
257,913,539
2304.00019
794ba58adbce4d1630bbf3e3001347c2643f9a65
November 29 and 30, 2022 R Ferreira Da Silva R M Badia V Bala D Bard T Bremer I Buckley S Caino-Lores K Chard C Goble S Jha D S Katz D Laney M Parashar F Suter N Tyler T Uram I Altintas S Andersson W Arndt J Aznar J Bader B Balis C Blanton K R Braghetto A Brodutch P Brunk H Casanova A Cervera Lierta J Chigu T Coleman N Collier I Colonnelli F Coppens M Crusoe W Cunningham B De Paula Kinoshita P Di Tommaso C Doutriaux M Downton W Elwasif B Enders C Erdmann T Fahringer L Figueiredo R Filgueira M Foltin A Fouilloux L Gadelha A Gallo A Garcia D Garijo R Gerlach R Grant S Grayson P Grubel J Gustafsson V Hayot O Hernandez M Hilbrich A Justine I Laflotte F Lehmann A Luckow J Luettgau K Maheshwari M Matsuda D Medic P Mendygral M Michalewicz J Nonaka M Pawlik L Pottier L Pouchard M Pütz S K Radha L Ramakrishnan S Ristov P Romano D Rosendo M Ruefenacht K Rycerz N Saurabh V Savchenko M Schulz C Simpson R Sirvent T Skluzacek S Soiland-Reyes R Souza S R Sukumar Z Sun A Sussman D Thain M Titov B Tovar A Tripathy M Turilli B Tużnik H Van Dam A Vivas L Ward P Widener S R Wilkinson J Zawalska M Zulfiqar &quot; Workflows November 29 and 30, 202210.5281/zenodo.7750670Community Summit 2022: A Roadmap Revolution", Technical Report, ORNL/TM-2023/2885, March 2023, DOI: 10.5281/zenodo.7750670.Workflows Community Summit 2022 A Roadmap Revolution S p e c i f i c a t i o n s , S t a n d a r d s , a n d A P I s Workflows Community Summit 2022 Disclaimer The Workflows Community Summit was supported by the Exascale Computing Project (17-SC-20-SC), a collabora-tive effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration. License This report is made available under a Creative Commons Attribution 4.0 International Public license (https: //creativecommons.org/licenses/by/4.0). and Chigu , J u s t i n and Coleman , T a i n a and C o l l i e r , Nick and C o l o n n e l l i , I a c o p o and Coppens , F r e d e r i k and Crusoe , M i c h a e l and Cunningham , W i l l and de P a u l a K i n o s h i t a , Bruno and Di Tommaso , P a o l o and D o u t r i a u x , C h a r l e s and Downton , Matthew and E l w a s i f , Wael and E n d e r s , B j o e r n and Erdmann , C h r i s and F a h r i n g e r , Thomas and F i g u e i r e d o , L u d m i l l a and F i l g u e i r a , Rosa and F o l t i n , M a r t i n and F o u i l l o u x , Anne and Gadelha , L u i z and G a l l o , Andy and G a r c i a , A r t u r and G a r i j o , D a n i e l and G e r l a c h , Roman and G r a n t , Ryan and Grayson , Samuel and G r u b e l , P a t r i c i a and G u s t a f s @ t e c h r e p o r t { wcs2022 , a u t h o r = { F e r r e i r a da S i l v a , R a f a e l and Badia , Rosa M. and Bala , V e n k a t and Bard , D e b b i e and Bremer , Timo and Buckley , I a n and Caino − L o r e s , S i l v i n a and Chard , Kyle and Goble , C a r o l e and Jha , S h a n t e n u and Katz , D a n i e l S . and Laney , D a n i e l and P a r a s h a r , Manish and S u t e r , F r e d e r i c and T y l e r , Nick and Uram , Thomas and A l t i n t a s , I l k a y and A n d e r s s o n , S t e f a n and Arndt , W i l l i a m and Aznar , J u a n and Bader , J o n a t h a n and B a l i s , B a r t o s z and B l a n t o n , C h r i s and B r a g h e t t o , K e l l y Rosa and B r o d u t c h , Aharon and Brunk , P a u l and Casanova , H e n r i and C e r v e r a L i e r t a , Alba s o n , J o h a n and Hayot , V a l e r i e and Hernandez , O s c a r and H i l b r i c h , Marcus and J u s t i n e , Annmary and L a f l o t t e , I a n , and Lehmann , F a b i a n and Luckow , Andre and L u e t t g a u , J a k o b and Maheshwari , K e t a n and Matsuda , Motohiko and Medic , D o r i a n a and Mendygral , P e t e and M i c h a l e w i c z , Marek and Nonaka , J o r j i and P a w l i k , M a c i e j and P o t t i e r , L o i c and P o u c h a r d , L i n e and P u t z , M a t h i a s and Radha , S a n t o s h Kumar and R a m a k r i s h n a n , Lavanya and R i s t o v , S a s h k o and Romano , P a u l and Rosendo , D a n i e l and R u e f e n a c h t , M a r t i n and Rycerz , K a t a r z y n a and S a u r a b h , N i s h a n t and Savchenko , Volodymyr and S c h u l z , M a r t i n and Simpson , C h r i s t i n e and S i r v e n t , R a u l and S k l u z a c e k , T y l e r and S o i l a n d −Reyes , S t i a n and Souza , Renan and Sukumar , S r e e n i v a s Rangan and Sun , Z i h e n g and Sussman , Alan and Thain , D o u g l a s and T i t o v , M i k h a i l and Tovar , B e n j a m i n and T r i p a t h y , A a l a p and T u r i l l i , M a t t e o and Tuznik , B a r t o s z and van Dam , H u b e r t u s and Vivas , A u r e l i o and Ward , Logan and Widener , P a t r i c k and W i l k i n s o n , Sean R . and Zawalska , J u s t y n a and Z u l f i q a r , Mahnoor } , t i t l e = {{ Workflows Community Summit 2 0 2 2 : A Roadmap R e v o l u t i o n }} , month = {March } , y e a r = {2023} , p u b l i s h e r = { Zenodo } , number = {ORNL/TM− 2 0 2 3 / 2 8 8 5 } , d o i = { 1 0 . 5 2 8 1 / z e n o d o . 7 7 5 0 6 7 0 } , i n s t i t u t i o n = {Oak R i d g e N a t i o n a l L a b o r a t o r y } } Scientific workflows have become integral tools in broad scientific computing use cases [1]. Science discovery is increasingly dependent on workflows to orchestrate large and complex scientific experiments that range from the execution of a cloud-based data preprocessing pipeline to multi-facility instrument-toedge-to-HPC computational workflows [2,3]. Given the changing landscape of scientific computing (often referred to as a computing continuum [4]) and the evolving needs of emerging scientific applications, it is paramount that the development of novel scientific workflows and system functionalities seek to increase the efficiency, resilience, and pervasiveness of existing systems and applications. Specifically, the proliferation of machine learning/artificial intelligence (ML/AI) workflows, need for processing large-scale datasets produced by instruments at the edge, intensification of near real-time data processing, support for long-term experiment campaigns, and emergence of quantum computing as an adjunct to HPC, have significantly changed the functional and operational requirements of workflow systems. Workflow systems now need to, for example, support data streams from the edge-to-cloud-to-HPC [5], enable the management of many small-sized files [6], allow data reduction while ensuring high accuracy [7], orchestrate distributed services (workflows, instruments, data movement, provenance, publication, etc.) across computing and user facilities [8], among others. Further, to accelerate science, it is also necessary that these systems implement specifications/standards and APIs for seamless (horizontal and vertical) integration between systems and applications [9,10], as well as enable the publication of workflows and their associated products according to the FAIR principles [11]. Summit Organization This document reports on discussions and findings from the 2022 international edition of the "Workflows Community Summit" that took place on November 29 and 30, 2022 [12]. The two-day summit included 106 participants (Figure 1), from a group of international researchers and developers (Australia, Austria, Belgium, Brazil, Canada, Colombia, Egypt, France, Germany, Italy, Japan, Netherlands, Norway, Poland, Spain, Switzerland, United Kingdom, United States) from distinct workflow management systems and users, and representatives from funding agencies and industry. The summit was organized by lead members of the Workflows Community Initiative (WCI) [13], a volunteer effort that brings the workflows community together (users, developers, researchers, and facilities) to provide community resources and capabilities to enable scientists and workflow systems developers to discover software products, related efforts, events, technical reports, etc. and engage in community-wide efforts to tackle workflows grand challenges. These members efforts have been supported by distinct research projects that contributed to the organization of the event, including ExaWorks [10], WorkflowHub [14], eFlows4HPC [15], and Covalent [16]. Summit Structure and Activities Based on the outcomes of the 2021 summits [17][18][19], the community developed a roadmap for workflows research and development [6] (summarized in Section 2). From this roadmap, we have identified six crosscutting research/technical topics for the 2022 edition of the summit, each of which was the object of a focused discussion led by a volunteer community member (Table 1). (Note that the first three topics were discussed in the first day of the event, and the latter three topics in the second day.) For each of the technical topics shown in Table 1, the co-leaders gave a plenary 10-minute lightning talk followed by focused discussions in breakout sessions. The goal of these sessions was to (i) review the previous challenges and proposed solutions; and (ii) identify crucial gaps and potential short-and long-term solutions for enabling emerging and new workflow applications given the rapid evolution of the computing continuum paradigm. The lead then reported on the outcome of the discussion in plenary sessions, after which final remarks were given by the organizers and the summit was adjourned. Additionally, the summit included an invited talk from Prof. Manish Parashar (University of Utah), a renowned researcher in the field of Scientific Computing, entitled "A Translational Perspective on End-to-End Workflows." The cross-cutting theme of Parashar's talk was discussed throughout all breakout sessions. All presentations and videos can be found in the summit website (https://workflows.community/ summits/2022), and videos can be watched from WCI's YouTube channel (https://www.youtube.com/ playlist?list=PLAtmuqHExRvO-l-mACmlK_4mfmG6yLnpN). An Overview of the 2021 Community Roadmap for Scientific Workflows Research and Development The overarching goal of the 2021 Workflows Community Summits was to (i) develop a view of the state of the art, (ii) identify key research challenges, (iii) articulate a vision for potential activities, and (iv) explore technical approaches for realizing (part of) this vision. The summits gathered lead researchers and developers from around the world, and spanning distinct workflow systems and user communities. We synthesized the discussions and outcomes of the previous summits and findings into a community roadmap [6] that presents a consolidated view of the state of the art, challenges, and potential activities. Table 2 presents, in the form of top-level themes, a summary of those challenges and targeted community activities. Table 3 Definition of common workflow patterns and benchmarks • Define small sets of workflow patterns and benchmark deliverables, and implement them using a selected set of workflow systems • Investigate automatic generation of patterns and configurable benchmarks (e.g., to enable weak and strong scaling experiments) • Establish or leverage a centralized repository to host and curate patterns and benchmarks Identifying paths toward interoperability of workflow systems • Define interoperability for different roles, develop a horizontal interoperability (i.e., making interoperable components), and establish a requirements document per abstraction layer • Develop real-world workflow benchmarks, use cases for interoperability, and common APIs that represent workflow library components • Establish a workflow systems developer community Improving workflow systems' interface with legacy and emerging HPC software and hardware stacks • Document a machine-readable description of key properties of widely used sites, and remote authentication needs from the workflow perspective • Identify new workflow patterns (e.g., motivated by AI workflows), attain portability across heterogeneous hardware, and develop a registry of execution environment information • Organize a community event involving workflow system developers, end users, authentication technology providers, and facility operators proposed community roadmap with technical approaches. In the 2022 edition of the Workflows Community Summit, we revisited this roadmap, discussed the relevance of the challenges and proposed milestones given the current landscape of workflow applications needs and emerging infrastructures. In the following sections, we summarize the outcomes of these discussions. Specifications, Standards, and APIs Defining reference specifications, standards, and APIs is paramount to enable interoperability across the workflow software stack; for capturing end-to-end workflow provenance to facilitate user understanding of the scientific output; for enabling composability of underlying workflow system components to enable seamless exchange of functional modules; among others. The workflows community has tackled this problem in many ways ranging from solutions that provide reference implementations of standards and/or specifications [10,20] to define common patterns/motifs [21,22]. In this breakout session, participants focused on identifying the challenges impeding adoption of current standards, specifications, best practices, and APIs developed by the community. Below, we summarize the key outcomes of these discussions, highlighting exemplar efforts, and outline proposed actionable solutions to be tackled by the community. Workflow Stack Despite the vast efforts for providing reference standard implementations and identifying common patterns, defining the levels of the workflow stack which are amenable to sharing technologies (and potentially APIs) is still an open question and potentially one of the critical aspects hindering wide adoption of proposed standards and specifications. For instance, there are several ways to define a workflow stack: (i) consider different types of workflows based on their resource requirements (e.g., in situ, intra-site, inter-site); (ii) con-sider a vertical hierarchical stack starting from the workflow down to computational jobs and individual tasks; (iii) consider the kind of stakeholder and their needs, etc. Given the above, an overarching question is how do these different workflow stacks map to different types of workflow systems? Examples of such aspects include resource authentication/authorization and allocation; the granularity of jobs definition/submission: fine-grained (functions) vs. coarse-grained (long running jobs); data management (from/to external data stores, within the workflow system, and within the tasks/serverless functions); workflow specification, language, and framework; provenance and error reporting; visualization; and application deployment (native installation vs. containerization). The Global Alliance for Genomics & Health (GA4GH) [23] has targeted the development of API standards (and implementations) that make it easier to "send the algorithms to the data" on a cloud-based environment. Their workflow stack includes four definitions of standard APIs for interacting with a data repository, tool registry, task execution, and workflow execution services [24]. The Pulsar-Network project [25] provides a distributed, uniform job execution system across European data centers, in which the execution environment is detached from the Galaxy framework. NERSC's SuperFacility API [26] exposes characteristics of the computational facility's resources that could be leveraged by decision making systems. One limitation of these approaches is the need for describing every single component of the stack. When considering heterogeneous platforms (Cloud, HPC, Edge, or the emerging quantum computing environments) the definition of such standards may become overly specialized to a specific platform or service [21]. The Common Workflow Scheduler [27] API allows workflow systems to exchange scheduling information with resource managers, making them "workflow-aware" and capable of scheduling tasks based on workflow dependencies and optimization goals. This approach eliminates the need for workflow systems to implement resource managerspecific logic and simplifies the separation of scheduling responsibilities between workflow systems and resource managers. Currently, this API is available as a plugin for Nextflow and Kubernetes. Recommendations. There is an imminent need for a description of a common workflow stack that might need to be represented by different sets of characterizations depending on the execution environment, however each described using common patterns identified across distinct environments. Once the fundamental workflow stack is defined and understood, the community could focus on the development of standard APIs as the result of a long-term objective. Standardization Attaining standardization is a foremost requirement to achieve interoperability, portability, and reuse of workflow components and applications. However, full standardization must be accomplished at different levels of the workflow stack. For instance, user-facing standards consider workflow descriptions and workflow input objects, while resource/infrastructure standards target the description of services that can be leveraged by, for example, workflow systems. In addition to standards, shared libraries and designs are also essential for interchanging components and services at horizontal levels of the workflow stack. One of the challenging aspects of defining standards is the lack of a common terminology that is widely adopted by the community. Unfortunately, the workflows literature suffers from an absence of a common vocabulary that defines terms and describes the components of a workflow. Recent efforts to provide a consolidated view [28] or reuse [29] of specification languages for a subset of workflow systems has underlined the discrepancy of term definitions. (Dissimilarities can be even identified at the simple definition of tasks and jobs across workflow systems.) Additionally, it is also necessary to describe the semantics of the components and of the data, as well as the variety of systems on which workflows are executed. Standardization is also required at the metadata level (user-defined metadata prior to data generation, ad-hoc metadata to annotate a dataset [30], and workflow-based metadata) [31], which is key for enabling provenance. The Workflow Run RO-Crate working group [32] is defining profiles for capturing the provenance of an execution of a computational workflow based on different levels of granularity: process (execution of one or more tools that contribute to the same computation), workflow (coordinated execution of the tools driven by a workflow system), and provenance (internal details of each step of the workflow). These profiles are based on the CWLProv profile [33], which are organized based on CWL's workflow mode. Although CWLProv is not a standard such as W3C PROV, it attempts to fill the gaps when translating the latter's definitions to workflow specifications (e.g., traditional standards are hard to adopt into HPC due to their origins in other domains). A remaining open question is how to increase adoption of these standards to other workflow systems, and how to capture specifics of heterogeneous systems without hiding the level of provenance the user cares about? Recommendations. There is a trade-off between sharing technologies vs. standards, within constraints that come from computational facilities. As standards are often constraining and/or hard to implement across facilities, one approach could be to focus on promoting standards of giving data from user to workflow systems/operators, i.e. a standard format for describing workflow input objects. Another recommendation is to develop a "marketplace" for workflow standards that would map them to parts of the workflow stack. The community would then be able to identify relevant standards for their systems and infrastructures, as well as identify gaps in current standards. Finally, there is a pressing need for defining a common vocabulary forcomponents of the workflow stack and even the relationships between them. AI Workflows Artificial intelligence (AI) and machine learning (ML) methods are now mainstream in modern science. As computational power increases, including the recent achievement of exascale computing, AI/ML methods can at present, for example, generate highly accurate models that accelerate the rate of scientific discovery of complex problems running at very large scales. As a result, workflows increasingly integrate ML models to guide analysis, couple simulation and data analysis codes, and exploit specialized computing hardware (e.g., GPUs and neuromorphic chips) [34,35]. These workflows inherently couple various types of tasks such as short ML inference, multi-node simulations, and long-running ML model training [36]. They are also often iterative and dynamic, with learning systems deciding in real time how to modify the workflow, e.g., by adding new simulations or changing the workflow all together. In this breakout session, participants focused on understanding the different characteristics of AI workflows, specifically the role of AI to create workflows or when used within workflows. AI Workflows Characteristics In the previous roadmap [6], discussions focused on identifying challenges inherent to workflows in which (most of) their tasks represent AI methods. The main challenges include the fine-grained data management and versioning features, heterogeneity of resources, integration to widely used ML framework, the iterative nature of ML processes, and the support of dynamic branching (Table 2). Although these challenges are still relevant to the current landscape of AI workflows, the community experience with this ever-increasing class of workflows has identified novel characteristics that hinder the efficient execution of these workflow applications at large scales. For instance, managing many small files (e.g., image training sets) may significantly impair the performance of the shared filesystem. Although solutions such as NVMe (nonvolatile memory express) have significantly improved I/O throughput, the data management aspect in the software component still lacks optimized solutions to tackle the large volumes of data produced during workflow execution. Additionally, the volume of metadata generated for provenance may also become intractable. In addition to the data management issue above, pseudo-random access to datasets leads to another major challenge in which optimizing access to microservices and performing data caching operations becomes a fundamental scheduling and/or resource provisioning problem. This challenge is aggravated by the unreliability of models, i.e., training processes often have a human-in-the-loop decision process that may result in a dramatic change to the shape of the task graph as the workflow execution evolves. Furthermore, coupling AI-coupled HPC workflows introduces additional challenges arising from the coupling of AI/ML models to traditional HPC workflows. Recommendations. There is a need for a better understanding of the requirements of AI workflows. Although there exists a diverse set of AI workflow applications that could be leveraged as benchmarks or proxy/miniapps, the complex specialized deployment (and requirements) of these workflows may prevent their large adoption that would enable comparative studies across computing facilities. Recently, the community has developed workflow benchmarks [37][38][39] that help to understand the requirements of traditional workflow applications; however, these benchmarks do not capture most of the characteristics intrinsic to AI workflows. Thus, there is a pressing need for creating a benchmark suite for representative AI workflows. AI Workflows Categories and Motifs The increasing adoption of AI/ML in modern science has not only enabled the development of novel applications using these methods [40] but also to improvements of systems and problem solving. In the workflows domain, the terminology around 'AI workflows' has become overloaded; thus, it is necessary to clearly and consistently define the different (sub)categories that encompass the distinct and unique use of AI/ML methods in the different aspects of workflow applications and systems. One approach to define these workflows into (sub)categories considering the role of AI would be: (i) Workflows for AI -the workflow is to develop the AI; (ii) AI-enabled-workflows. The latter in turn encapsulates two different sub-classes, viz., the AI-integrated workflows, where the workflow is about problem-solving with AI; and AI-enhanced-workflow systems, where the workflow engine is empowered with intelligence using a form of AI. Within each of the above categories there should be subdivisions that could, for example, consider the dynamic nature of the workflow task graph (conditions, loops) or its adaptive response to events (task prioritization or preemption due to dynamic sampling) that could lead to different design and communication requirements. For the first two categories, the workflows could be classified into applications composed of (i) inner, (ii) outer, or (iii) coupled loops. Examples of inner loops include ML replacement of subroutines, adapting, and parallel training. Outer loops are characterized by reinforcement learning (RL) -there is a central "controller" that defines how the different steps (e.g., rollout and training) interact; and active learning -similar to RL in the sense that there are training, inference, and simulation components with a different class of sampling. Coupled loops can be represented by digital twins, which are a component of the workflow controlled by an AI method. A recent characterization used common motifs of AI-HPC workflows [41] that considers workflows using AI and workflows. These motifs include: (i) AI models steering ensembles of simulations (e.g., advanced sampling, swarm methods); (ii) multistage (and typically multiscale and multiphysics) pipelining (e.g., molecule selection, virtual screening); (iii) inverse design including from observations, or determine causal factors (e.g., molecule or material design given properties such as structure to sequence); (iv) concurrent duality (e.g., concurrent HPC simulation and AI based digital twins); (v) distributed models and dynamic data (e.g., distributed AI based reduction/analysis coupled to HPC simulation, diverse models on edge-to-exascale infrastructure); and (vi) adaptive execution for training (e.g., HPO, NAS, LLM). Recommendations. For each of the above (sub)categories, it is necessary to refine the motifs that would capture the unique requirements of each class; and develop benchmarks based on them that would then help catalyze the development of tailored solutions or example problems. An immediate activity includes spearheading a community effort to formulate a technical/white paper that would provide common terminology and definitions of categories of AI workflows and their unique requirements. As a long-term recommendation, the community needs to determine ways to smooth the integration (along with the deployment) paths across these categories and their associated components. High Performance Data Management and In Situ Workflows In situ workflows aim to overcome I/O limitations in HPC by sharing data between simulation, analysis, visualization and orchestration tasks as data are produced. The term "in situ" has become an umbrella covering approaches well beyond the seminal idea to distribute the load of the traditional post hoc analysis associated with a numerical simulation throughout the execution of that simulation [42]. Nowadays, in situ workflows facilitate data reduction, annotation, and transformation in different stages (e.g., data acquisition, simulation, analysis, visualization). These workflows can rely on multiple components, execution environments, and data transport methods to bypass the file system while delivering data between components. Additionally, modern in situ workflows exhibit different data production and consumption patterns (i.e., volume, frequency, structure) that need to co-exist in a coordinated and efficient manner. This leads to a need for high-performance management of the input, output, intermediate data, and metadata produced and consumed during the workflow execution. In this breakout session, participants focused on identifying challenges related to data management for in situ workflows, including how data abstractions may provide fine data management detached from the computing platform, and challenges related to the broader range of in situ workflows today (e.g., streaming and event-based workflows). In situ workflows have evolved from the traditional simulation and analysis approach. In large-scale science, heavy computations are typically offloaded from the edge where data are produced (e.g., from a large scale scientific instrument such as a particle detector) to HPC infrastructures and cloud resources during burst conditions, leading to complex geographically distributed platforms with diverse performance characteristics. A major challenge in this scenario consists in transferring all this generated data from the edge to the processing facility. An approach to lower the pressure put on network resources is to employ in situ data reduction, an optimization technique in which the size of data is reduced without compromising the quality of the information it carries. This implies that this approach has to control and bound the loss of precision to ensure that end users will still have trust in the data. Approaches leveraging self-descriptive data and metadata could thus be used to define the relevance of pieces of data thanks to a tolerable error bound [43][44][45] Adaptive compression by regions of interest can also be used to reduce the size of data, using techniques similar to those underlying Adaptive Mesh Refinement: regions where information is more important are compressed less than those where information has little scientific significance. The need for high performance data management has significantly increased as modern science applications leverage heterogeneous resources for scaling their computations. The emergence of AI/ML workflows is intertwined with the imminent need for efficiency in situ data management, as these workflows increasingly produce and consume large volumes of data and have new motifs that profoundly differ from traditional workflows (Section 4). Differences in data formats, acquisition and pre-processing methods, training approaches, and provenance needs further exacerbate the complexity of enabling ML/AI for in situ analysis. Additionally, a key element in modern in situ workflows is the need to integrate them into the edge-to-cloud continuum. In this environment, data exchanges are typically performed through data objects in which the storage stack and data location are hidden from the application. Consequently, individual tasks create and work with objects, not files, and are not concerned with where those objects live. This enables storage-centric optimizations transparent to the end user to improve application performance. A potential solution to address the challenges and requirements of geographically distributed in situ workflows is to develop a data management layer abstraction for providing transparent data operations optimization and intelligent decisions for the workflow application. Ideally, workflow tasks would interact with this data management layer responsible for tracking the location of all these objects and performing intelligent data movements to place the necessary objects physically close to the computations using them. However, providing a single abstraction layer can be challenging when dealing with the requirements of a diverse community (e.g., different types of data, performance requisites, latency, consumption rates, availability needs, etc.). Thus, it is necessary to separate the data management paradigm from the workflow orchestration paradigm and move towards a data-centric event-driven data plane for workflows. Note that it is also necessary to detach data from its structure (e.g., file) and associated implementation (e.g., HDF5). Frameworks such as ADIOS2 [46] and Maestro [47], combined with self-descriptive data (metadata) capabilities and the publish/subscribe model, can be used to expose the data interaction mechanisms available to applications when building workflows. Emerging in situ workflows involve streaming and event-based elements. Event-driven workflows have a strong need for tightly-coupled and high performing components that integrate HPC and streaming (especially for urgent computing) [48]. In these workflows, in situ computations are initiated based on events (e.g., changes to a variable or data, or access in a data repository). This class of workflows has been largely adapted to Cloud computing environments based on a service-oriented approach that uses streams of data for composing the workflows. In the HPC environment, streaming workflows are typically limited by policies that restrict connectivity to external data sources or long-running services. One approach to address these limitations is to couple a container-based platform (e.g., NERSC's Spin, OLCF's Slate) to HPC systems, in which services running on these platforms would act as a bridge between the outside world and the HPC ecosystem. Another limitation for enabling urgent computing on HPC is the need for specialized queues that can swiftly allocate resources for these types of jobs. Recommendations. There is a need for determining what are the requirements for upcoming workflow applications, especially those integrating data or moving data across facilities. Dataflow and I/O contention benchmarks are needed to assist in the quantification of performance for workflow and data management solutions in upcoming scenarios. Current efforts [37,49] mostly target traditional workflow motifs, and must be expanded to cover new applications (e.g., AI/ML workflows) and infrastructures (e.g., edge-to-cloud workflows). There is also a need for defining community-driven schemas to describe data and metadata, and to further provide an integrated view of the different views of the data associated to a workflow execution. There is also a need to define a service-oriented composability approach for enabling urgent computing workflows in the HPC ecosystem. To explore this, the community should conduct a comprehensive study to understand the specific needs for this class of applications for HPC, derive lessons learned from the service-oriented Cloud computing solutions, and engage with HPC computing facility operators to design a potential solution that would satisfy both the application's performance requirements as well as guarantee that facilities' security policies are still enforced. HPC and Quantum Workflows Most current workflow tools can operate over classical HPC, providing automated orchestration of tasks and data management. With the emergence of quantum computing (QC) there is a need to bridge these two computing models to improve the efficiency and potential impact of applications that could leverage the capabilities of each model. The need for a hybrid QC-HPC/Cloud approach is motivated for modern applications. In the quantum chemistry domain, for instance,researchers seek to understand and harness the quantum properties of atoms and systems around us. QC has also been increasingly used for ML processing and optimization -conversely, ML has been leveraged for the calibration of quantum processors. In the current state of quantum computing, a.k.a. noisy intermediate-scale quantum (NISQ) era, there is a limited number of noisy physical qubits, thus it is paramount that quantum algorithms be efficient. Recently, variational quantum algorithms (VQAs) [50] have shown modest success in ML and optimization tasks. The approach requires a constant exchange of data between classical and quantum devices, which leads to a model in which quantum devices act as accelerators for classical computing. As a result, managing these sets of distributed, heterogeneous devices motivates the need for workflow solutions that can provide a common abstraction layer to interface with the multitude of specialized APIs and components provided by each platform. Due to the novelty of the topic, this breakout session was structured with a series of short talks that underlined different aspects of quantum computing in workflows research, and identified a list of challenges associated to each of these aspects. Table 4 shows the list of topics. All presentation videos can be found at the WCI YouTube channel (https://www.youtube.com/playlist?list=PLAtmuqHExRvO-l-mACmlK_ 4mfmG6yLnpN). Topic Presenter Tensor State-of-the-art and Challenges Recent efforts target the definition of languages for modeling computations between classical and quantum computing [51]. The open-source Covalent framework [16] provides mechanisms to manage experiments (expressed as workflows) and facilitate access to quantum devices, bridging them to HPC and cloud platforms. RosneT is a library for distributed, out-of-core block tensor algebra [52] built on top of the PyCOMPSs [53] programming model to transform tensor operations into a collection of tasks to be executed on HPC resources. QC has also been leveraged to solve well-known scheduling problems for traditional workflow applications [54]. Despite the above efforts, there are still several challenges that need to be addressed. For instance, (i) the reproducibility aspect in quantum computing requires domain expertise to understand the circuits and qubit topologies; (ii) the scheduling timescales significantly differ between classical and quantum computing -the latter is performed in the order of microseconds; (iii) there is a limited number of available resources, thus waiting time in queues can attain hours and costs to perform substantial computations may become excessive; (iv) programming each quantum device requires expert knowledge for each vendor API; (v) there are no standard representations for intermediate representations (IRs) for quantum programs, which are described as circuits; (vi) although some approaches bridge quantum devices to HPC and cloud systems, there is no tight integration between them (e.g., as between GPUs and CPUs). Recommendations. The community is actively working in multiple objectives to make progress in the different topics of research. The foremost research directions include (i) the optimization of classical simulators based on tensor networks in classical HPC systems; (ii) the need to develop quantum systems with a component-based architecture to facilitate the integration with HPC and cloud environments; (iii) moving the hardware-aware software from the Quantum to the HPC system; (iv) reaching a consensus between QC as accelerators for classical computing (e.g., QPUs as accelerators to combine traditional HPC-QC workflows) or vice versa; (v) predicting optimizability of workflow decisions (e.g., how long will it take to optimize a QC workflow? Should a circuit be optimized?). FAIR Computational Workflows The original FAIR principles [55] laid a foundation for sharing and publishing data assets, emphasizing machine accessibility in that data and all other assets should be: (i) Findable -user of persistent identifies, cataloguing and indexing of data; (ii) Interoperable -machine processable metadata using standards; (iii) Accessible -clear access protocols to access (meta)data; (iv) Reusable -metadata standards, machine accessible usage license and provenance. When considering FAIR computational workflows, both data and software aspects need to be considered. FAIR data principles can be applied to, for example, workflow descriptions and specifications (typically via a domain specific language or API) that can be associated to metadata with Digital Object Identifiers (DOIs) or objects such as test data and parameter files. Workflow software objects, on the other hand, bring additional challenges such as reproducibility, usability, quality, maturity, etc. The recently published FAIR for research software principles [56] can then be applied to both workflows and workflow management systems. In this breakout session, participants focused on the software aspect of workflows and discussed best practices for research software registries and repositories, as well as approaches to build FAIR into workflow systems. FAIR Workflow Repositories and Registries The current state of the art for finding and accessing workflows consists of community repositories (e.g., nf-core [57], snakemake workflow catalog [58]), community platforms (e.g., nextflow tower [59]), data repositories (e.g., Zenodo [60], Dataverse [61]), registries (e.g., WorkflowHub [14], Dockstore [62]), and metadata frameworks (e.g., CWL [20], Workflow RO-Crate [63], Bioschemas profiles [64]). Registries and repositories provide curation and best practices for recording workflows, while metadata frameworks capture workflows description and their associated metadata (e.g., via canonical descriptions). The notion that workflows can encompass both data and software brings several challenges when building FAIR workflow repositories and registries. In addition to capturing the workflow descriptions and (reference to) their input data, it is also necessary to capture metadata, containers, execution information, configurations, etc. The current approach adopted by the community is to separate the workflow definition from its execution -e.g., WorkflowHub records the workflow description and refers to Workflow-RO-Crates objects for execution information; Workflow-Run-RO-Crates collect the actual execution provenance. Although this approach has demonstrated success for traditional DAG-and cloud-based workflows, capturing workflow executions in the edge-to-HPC computing continuum is still an open question -mostly due to their intrinsic configurations, specialized architectures, and unique scientific instruments. The dynamic nature of emerging AI/ML workflows poses an additional challenge to capture executions that match the canonical description of the workflow. An additional feature for registries would be to provide mechanisms to characterize workflows in terms of structures so that a user could explore similar solutions to their problem. Recent approaches on extracting metadata from software and research data repositories [30,65,66] or automated extraction of workflow patterns [67] could be leveraged to measure similarities across workflows. Past work [68] should be revisited. For instance, workflows could be compared by tags or labels, by the workflow structure (i.e., task graph), or by a stored relationship (e.g., sub-workflow, forked variant, etc.). There is also a need for defining persistent identifiers (PIDs) for workflows. DOIs are appropriate for workflows that are "completed and published", i.e., as a snapshot of the workflow -similar to the publication process of research articles. There is a need for defining persistent identifiers (PIDs) for workflows that capture their complex nature. Recommendations. Given the above, an imminent need is to define standards for capturing the metadata for workflows. This is critical for enabling automatic FAIR (unit/end-to-end) testing of workflows to support reuse and composability. There is also a need to expand the first set of recommendations for research software registries and repositories [69] for scientific workflows, especially attempting to address the need for workflow PIDs. A long-term activity is to consider new paradigms for workflow descriptions that could capture the emerging class of dynamic workflows. Building FAIR into Workflow Management Systems In addition to providing repositories and registries to store workflows and their associated metadata/information, it is crucial to empower workflow systems with capabilities to enable support to FAIR data and software throughout the workflow execution. To attain this objective, the community argues that two key features are needed: standards and metadata. Specifically, there is a need for a standard for expressing the inputs of the workflow and how to set them. Unfortunately, this issue is not novel and has not received much attention from the community. In fact, this issue is more a people problem than a technical problem, as is frequently also the case when attempts to implement FAIR practices fall short [70]. Ideally, in terms of FAIRness, documentation is the most important asset for a workflow. It is very difficult to maintain software, but documentation lives on. On the description of the workflow itself, there are system agnostic languages (e.g., CWL and OpenWDL [71]) that have been adopted by several workflow systems to foster workflow portability, though these may be limited in the workflows that they can describe. The goal is to provide a mechanism to describe workflows so that they can be compared. Portability at the system level remains an open question, especially in edge and HPC environments. The availability of metadata is another key feature necessary to enable FAIR within workflow systems. To this end, FAIR registration needs to be automatic and cannot introduce overhead in recording provenance information (i.e., the facts that link the inputs and outputs of the workflow) when running the workflow -recording of provenance information needs to be transparent and scalable [72]. Thus, FAIR data and FAIR workflows are intertwined. A solution would be to leverage Workflow-Run RO-Crate profiles, where the workflow system would export these profiles that would capture the provenance of an execution of a computational workflow. As a result, a coupling between the workflow and its associated data and metadata would be properly documented. Recommendations. An immediate action is to continue the efforts to define methods for fostering portability across workflow systems, instead of defining the "standard" language for expressing workflows. Another key direction is to invest into people and communities rather than specific tools. The community strongly believes that the development of people's and communities' skills will outlast workflow systems, thus the incentive for FAIRness will be built intrinsically into these systems. Last, there is a need to assess the limitation of FAIR for different kinds of workflow types such as streaming and IoT workflows, or HPC workflows that are monolithic and tied to a particular architecture. Workflows for Continuum and Cross-Facility Computing Continuum and cross-facilities workflows are becoming more prevalent in the computational sciences. As these workflow paradigms have recently emerged, their definitions are evolving. The current understanding is that continuum workflows represent analysis pipelines that require continuous access to computing (e.g., urgent computing). Examples of workflows that fall under this category include in situ workflows, components that talk to each other within the same system (Section 5); geographical computing, i.e., edge-to-cloud-to-HPC computing; and high throughput computing, e.g., long-running computing campaigns over long periods of time. Cross-facility workflows represent analysis pipelines that hit more than a single site, which may include an experiment and a computing site, multiple experiment sites, or multiple computing facilities [8] -a site can be defined as a local compute, HPC, cloud, edge, campus cluster, and sensors at the edge. Cross-facility workflows can be seen as a solution for the needs of a continuum workflow (e.g., commercial cloud providers can be considered effectively cross-facility) offering resiliency for the computing needs of real time workflows. Another view is to consider continuum and cross-facility computing by the data plane, i.e., workflow systems should be able to tolerate different representations and underlying storage systems. In this breakout session, participants focused on identifying challenges to enable these categories of workflows and potential actions for the community. Characteristics and Challenges Data management is a major challenge in modern science, which involves operating data streams (e.g., from an instrument to a computing resource), data staging (e.g., processing fragments of data from different locations), or data movement (e.g., wide area transfers). The need for running workflows on multiple computing sites has increased as, for example, nowadays instruments can generate an ever-growing volume of data (in the order of TB/s) [73]. These workflows are characterized by their unique requirements to access and operate over computing resources regardless of the time or location sensitivity. For instance, existing computing facilities provide no mechanisms for scheduling and/or provisioning I/O capacity. Enabling a data-centric approach (i.e., doing computation on data quanta) may make the problem more tractable. To this end, it is necessary that tools to measure system contention/resource utilization and capture I/O and data movement performance are made available for measuring this end-to-end performance at multiple levels: computing system, across the network (local and wide area), and across the multiple compute and user facilities (e.g., node memory at one site to a file system at another site). Current systems perform matchmaking of task requirements (CPU/GPU/RAM) across multiple HPC platforms (e.g., RADICAL-Cybertools [74], HTCondor [75], etc.), however I/O requirements are neglected due to the lack of control over the I/O performance of an HPC machine at a given point in time. A precondition to attain the envisaged level of fine-grained workflow orchestration described above is to define the requirements of a workflow task. Specifically, what metadata does the workflow task need to have to request resources (computing, storage, network) from the appropriate site? To answer this question, it is necessary to convey that tasks are less portable than commonly assumed (including containers). The process for labelling every single resource a task needs is still an open question, and requirements may significantly diverge considering the targeted computing environment (e.g., compiler optimization for a determined architecture, data location and network and I/O bandwidths, policies, etc.). In addition to the above challenges, cross-facility workflows also face challenges associated to autonomous administration domains, i.e., different policies and security models at different sites (authorization, access control), different software stacks, etc. Approaches for providing resource information in a machine-readable format (e.g., NERSC's superfacility project [26]) or enabling federated identification (e.g., DOE's OneID [76]) are a first step for enabling distributed workflow orchestration and intelligent decisions at the workflow management system level. A collaboration between the Computational Science Initiative (CSI), the Center for Functional Nanomaterials (CFN), and the National Synchrotron Light Source (NSLS-II) attempt to facilitate using larger institutional compute resources to support the beamlines. The idea is to go from giving users access to beamlines and sending them home with their data, to giving users access to experiments and sending them home with answers to their science questions. Multi-site workflows are a mainstream component to this effort. A different approach is letting users embrace and exploit heterogeneity from the design phase through hybrid workflows [77]. Traditional workflow models, which describe steps and data dependencies, can be flanked by a topology of execution locations, encoding execution environments, available resources, and communication channels. Users can then explicitly map the different steps of the workflow onto different locations of the topology in an N-to-M relationship. Hybrid workflows are heterogeneous by design and do not require a homogeneous representation of locations' capabilities. As a result, site-specific plugins can be used to handle orchestration aspects such as authentication, scheduling, data transfers, and task offloading. Hybrid workflows have already successfully orchestrated cloud-HPC and cross-HPC distributed workflows using batched [77] and interactive [78] execution paradigms. The main drawback is that the additional complexity introduced in the design phase steepens the learning curve for domain experts with little knowledge of distributed execution environments. Recommendations. Continuum and cross-facility workflows are becoming mainstream in modern science. These new classes of workflows present, in addition to the typical challenges inherent to computing on a single site, new challenges that require coordination and cooperation among computing and experimental facilities. An immediate action is to enable tracking the metadata associated with tasks in workflow across different sites. To this end, it is necessary to define a standard/specification for task descriptions, including a way to represent the overall I/O requirements of an entire workflow. There is also a need for a community effort to define what a computing site needs to provide to enable a workflow orchestrator to make intelligent decisions about where to place tasks in a workflow. Another recommendation is to factor in the need for cross-facility computing from the conceptualization/design phase of computer systems and experimental facilities -typically, personnel involved in the design of experiment facilities are not necessarily computing experts. . . . . . . . . . . . . . . . . . . Figure 1 : 1Screenshot of the 2022 edition of the Workflows Community Summit participants. (The event was held virtually via Zoom on November 29 and 30, 2022.) Table 1 : 1Workflows Community Summit topics and discussion co-leaders. summarizes a Metrics to measure the "FAIRness" of a workflow • Principles, policies, and best practices• Review prior and current efforts for FAIR data and software with respect to workflows, and outline principles for FAIR workflows • Define recommendations for FAIR workflow developers and systems • Automate FAIRness in workflows by recording necessary provenance data AI Workflows• Support for heterogeneous compute resources and finegrained data management features, versioning, and data provenance capabilities • Capabilities for enabling workflow steering and dynamic workflows • Integration of ML frameworks into the current HPC landscape • Develop comprehensive use cases for sample problems with representative workflow structures and data types • Define a process for characterizing the challenges for enabling AI workflows • Develop AI workflows as a way to benchmark HPC systems Workflow systems differ by design, thus interoperability at some layers is likely to be more impactful than others • Workflow standards are typically developed by a subset of the community • Quantifying the value of common representations of workflows is not trivial • Identify differences and commonalities between different systems • Identify and characterize domain-specific efforts, identify workflow patterns, and develop case-studies of business process workflows and serverless workflow systemsTheme Challenges Community Activities FAIR Computational Workflows • FAIR principles for computational workflows that consider the complex lifecycle from specification to execution and data products • Exascale Challenges and Beyond • Resource allocation policies and schedulers are not designed for workflow-aware abstractions, thus users tend to use an ill- fitted job abstraction • Unfavorable design of resource descriptions and mechanisms for workflow users/systems, and lack of fault-tolerance and fault-recovery solutions • Develop documentation in the form of workflow tem- plates/recipes/miniapps for execution on high-end HPC sys- tems • Specify benchmark workflows for exascale execution • Include workflow requirements as part of the machine pro- curement process APIs, Reuse, Interoperability, and Standards • Training and Education • Many workflow systems have high barriers to entry and lack training materials • Homegrown workflow solutions and constraints can prevent users from reproducing their functionality on workflow sys- tems developed by others • Unawareness of the workflow technological and conceptual landscape • Identify basic sample workflow patterns, develop a commu- nity workflow knowledge-base, and look at current research on technology adoption • Include workflow terminology and concepts in university cur- ricula and software carpentry efforts Building a Workflows Community • Diverse definitions of a "workflows community" • Remedy the inability to link developers and users to bridge translational gaps • Pathways for participation in a network of researchers, devel- opers, and users • Establish a common knowledge-base for workflow technol- ogy • Establish a Workflow Guild: an organization focused on in- teraction and relationships, providing self-support between workflow developers and their systems Table 2 : 2Summary of workflows research and development challenges and proposed community activities identified during the 2021 summits [6]. Table 3 : 3Summary of technical roadmap milestones per research and development thrust proposed during the 2021 summits [6]. Table 4 : 4HPC and quantum workflows breakout topics. Sean Wilkinson Oak Ridge National Laboratory Justyna Zawalska AGH University of Science and Technology, ACC Cyfronet Mahnoor Zulfiqar Friedrich Schiller University 12:15-1:35pm EST Breakout sessions • Topic 4: HPC and Quantum workflows -Tensor Networks for Quantum Simulation Artur Garcia Saez (BSC) -Advances in Hybrid Quantum-Classical High-Performance Computing Stefan Andersson (ParTec AG), Mathias Pütz (ParTec AG) -Workflow Scheduling Using Quantum Devices Justyna Zawalska (Academic Computer Centre CYFRONET AGH) -Mechanisms for Enhancing Reliability/Recovery and Performance of Future Workloads Ryan E. Grant (Queen's University) -HPCQC System Workflows Martin Ruefenacht (Leibniz Supercomputing Centre) -Multi-Core Quantum Computing Aharon Brodutch (Entangled Networks) • Topic 5: FAIR workflows • Topic 6: Workflows for continuum and cross-facility computing 1:30-1:45pm EST 15min Break 1:45-2:00pm EST Reports from breakout sessions Workflows are the new applications: Challenges in performance, portability, and productivity. T Ben-Nun, T Gamblin, D S Hollman, H Krishnan, C J Newburn, 2020 IEEE/ACM International Workshop on Performance, Portability and Productivity in HPC (P3HPC). IEEET. Ben-Nun, T. Gamblin, D. S. Hollman, H. Krishnan, and C. J. Newburn, "Workflows are the new applications: Challenges in performance, portability, and productivity," in 2020 IEEE/ACM International Workshop on Performance, Portability and Productivity in HPC (P3HPC). IEEE, 2020, pp. 57-69. Workflows for science: A challenge when facing the convergence of hpc and big data. 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Cham: Springer International Publishing, 2020. doi: 10.1007/978-3-030-63393-6 15. ISBN 978-3-030-63393-6 pp. 226-239. Radical-cybertools: Middleware building blocks for scalable science. V Balasubramanian, S Jha, A Merzky, M Turilli, arXiv:1904.03085arXiv preprintV. Balasubramanian, S. Jha, A. Merzky, and M. Turilli, "Radical-cybertools: Middleware building blocks for scalable science," arXiv preprint arXiv:1904.03085, 2019. Distributed computing in practice: the condor experience. D Thain, T Tannenbaum, M Livny, Concurrency and computation: practice and experience. 17D. Thain, T. Tannenbaum, and M. Livny, "Distributed computing in practice: the condor experience," Concurrency and computation: practice and experience, vol. 17, no. 2-4, pp. 323-356, 2005. OneID. 2023"OneID," https://eams-hub.oneid.energy.gov/hub, 2023. StreamFlow: cross-breeding cloud with HPC. I Colonnelli, B Cantalupo, I Merelli, M Aldinucci, 10.1109/TETC.2020.3019202IEEE Transactions on Emerging Topics in Computing. 94I. Colonnelli, B. Cantalupo, I. Merelli, and M. Aldinucci, "StreamFlow: cross-breeding cloud with HPC," IEEE Transactions on Emerging Topics in Computing, vol. 9, no. 4, pp. 1723-1737, 2021. doi: 10.1109/TETC.2020.3019202 Distributed workflows with Jupyter. I Colonnelli, M Aldinucci, B Cantalupo, L Padovani, S Rabellino, C Spampinato, R Morelli, R Di Carlo, N Magini, C Cavazzoni, 10.1016/j.future.2021.10.007Future Generation Computer Systems. 128I. Colonnelli, M. Aldinucci, B. Cantalupo, L. Padovani, S. Rabellino, C. Spampinato, R. Morelli, R. Di Carlo, N. Magini, and C. Cavazzoni, "Distributed workflows with Jupyter," Future Generation Computer Systems, vol. 128, pp. 282-298, 2022. doi: 10.1016/j.future.2021.10.007
[ "https://github.com/SooLee/Benchmark,", "https://github.com/qc-lab/workflows-dwave/blob/" ]
[ "An isomorphism of unitals, and an isomorphism of classical groups", "An isomorphism of unitals, and an isomorphism of classical groups" ]
[ "Markus J Stroppel " ]
[]
[]
An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).In the present paper, we use an isomorphism between two hermitian unitals to treat isomorphisms of classical groups that are related to the isomorphism between the simple real Lie algebras of type A C,1 3 and D H 3 (in the notation of Tits [18, pp. 28, 40], Helgason [6, X § 2.1, § 6.2] denotes the algebras in question by su(3, 1) and so * (6), respectively).Our incidence geometric approach complements the algebraic approach used in [12, 2.14] by a geometric explanation for the exceptional isomorphism of classical groups. That algebraic approach works in much greater generality, including certain characteristic two cases where the unital over the quaternions collapses into a line, and cannot be used for our purposes.
null
[ "https://export.arxiv.org/pdf/2304.08573v1.pdf" ]
258,187,339
2304.08573
37e15692db7b6d5d159b0748fd260b618d403b49
An isomorphism of unitals, and an isomorphism of classical groups 17 Apr 2023 Markus J Stroppel An isomorphism of unitals, and an isomorphism of classical groups 17 Apr 2023arXiv:2304.08573v1 [math.GR] An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).In the present paper, we use an isomorphism between two hermitian unitals to treat isomorphisms of classical groups that are related to the isomorphism between the simple real Lie algebras of type A C,1 3 and D H 3 (in the notation of Tits [18, pp. 28, 40], Helgason [6, X § 2.1, § 6.2] denotes the algebras in question by su(3, 1) and so * (6), respectively).Our incidence geometric approach complements the algebraic approach used in [12, 2.14] by a geometric explanation for the exceptional isomorphism of classical groups. That algebraic approach works in much greater generality, including certain characteristic two cases where the unital over the quaternions collapses into a line, and cannot be used for our purposes. Hermitian unitals We generalize the notion of finite hermitian unital (see [1, p. 104]) to the case of hermitian forms over infinite (and not necessarily commutative) fields, as follows. 1.1 Definitions. Let K be any (not necessarily commutative) field, and let σ be an antiautomorphism of K, with σ 2 = id = σ. If V is a vector space over K, and h : V × V → K is a non-degenerate σ-hermitian form, we define the set U h := {Kv ∈ Gr 1 (V ) | v ⊥ h v} = {P ∈ Gr 1 (V ) | P ≦ P ⊥ h } of absolute points (with respect to h). If d := dim V is finite, the hermitian form h defines a polarity π h of the projective space PG(V ) ∼ = PG(d − 1, K) (see [2, I, § 5, p. 9 ff], [7, II.6, p. 45 ff]). The set U h then consists of all points of PG(V ) that are incident with their image under that polarity. Consider a line L ∈ Gr 2 (V ). If the set U h ∩ Gr 1 (L) of absolute points on L contains more than one point then it is called a block of U h . The set of all these blocks is denoted by B h . Clearly, any two points of U h are joined by a unique member of B h . If the form h has Witt index 1, we call (U h , B h , ∈) the hermitian unital with respect to h. Lemma. Assume that h : V × V → K is a non-degenerate σ-hermitian form of Witt index 1. If h is trace-valued then the set of blocks through a given point P ∈ U h is U h ∩ Gr 1 (L) L ∈ Gr 2 (V ), P < L ≦ P ⊥ h . Proof. We write h : V × V → K : (x, y) → x|y . Recall (see [2, I, § 10, p. 19]) that h is trace-valued if, and only if, the set { v|v | v ∈ V } is contained in {s + s σ | s ∈ K}. Consider any line L ∈ Gr 2 (V ) through P ∈ U h . Then P = Kv with v ∈ V {0} such that v|v = 0. If L ≦ P ⊥ h then every w ∈ L Kv satisfies w|w = 0 because h has Witt index 1. So P is the unique absolute point in L, and U h ∩ Gr 1 (L) contains no block. If L ≦ P ⊥ h , we pick any x ∈ L P ; then x|v = 0. Replacing x by a suitable scalar multiple, we achieve x|v = −1. For each s ∈ K, we now have K(sv + x) ∈ L and sv + x|sv + x = sv|sv + sv|x + x|sv + x|x = s v|v s σ + s v|x + x|v s σ + x|x = 0 − s − s σ + x|x . If the form h is trace-valued, we find s such that s + s σ = x|x , and K(sv + x) is a second absolute point on L. So U h ∩ Gr 1 (L) is indeed a block in that case. From [2, I, § 10, p. 19] we recall that every σ-hermitian form over a field K with char K = 2 is trace valued. Also, if σ acts non-trivially on the center of K (in particular, if K is commutative) then every σ-hermitian form is trace-valued. Examples. Let C|R be a separable quadratic extension of commutative fields, and let σ be the generator of Gal(C|R). Then the form h : C 3 × C 3 → C : (x 0 , x 1 , x 2 ), (y 0 , y 1 , y 2 ) → x 0 y σ 2 + x 1 y σ 1 + x 2 y σ 0 is not degenerate, trace-valued, and has Witt index 1. If C is finite of order e then the hermitian unital (U h , B h , ∈) is the finite hermitian unital of order e. Definitions. Let U := (U h , B h , ∈) be the hermitian unital with respect to a non-degenerate hermitian form h : V × V → K of Witt index 1, let X ∈ U h be a point of U, and let (P, L, I) be any incidence structure. A map η : U h → P is called an isomorphism from U onto (P, L, I) if η is bijective, for every block B ∈ B h there exists a unique block B ′ ∈ L with B η = X ∈ P (X, B ′ ) ∈ I , and the resulting map β : B h → L : B → B ′ is a bijection. As usual, an automorphism of U is an isomorphism of U onto U itself. An automorphism of U is called a translation of U with center X if it leaves invariant every block through X. We write T [X] for the set of all translations of U with center X. If h : V × V → K is a σ-hermitian form of Witt index 1, then clearly the group PΓU(V, h) of collineations induced by semi-similitudes acts by automorphisms of the hermitian unital (U h , B h , ∈). See 2.4 and 2.7 below for examples of translations. Theorem. Consider an anti-automorphism σ of a (not necessarily commutative) field K, with σ 2 = id = σ. Let h : V × V → K : (v, w) → v|w be a non-degenerate σ-hermitian form of Witt index 1. If the form is trace-valued (in particular, if char K = 2 or if K is commutative) and dim V is finite then the group PU(V, h) acts two-transitively on U h , and thus transitively both on B h and on the set of flags of (U h , B h , ∈). Proof. As h has Witt index 1, there exists a ∈ V {0} with a|a = 0, so Ka lies in U h . As h is not degenerate, there exists x ∈ V with a|x = 0. In L := Ka + Kx there is a second absolute point Kb, see 1.2. Let P, Q be two arbitrary points in U h . Then there are v, w ∈ V {0} with v|v = 0 = w|w such that P = Kv and Q = Kw. As h has Witt index 1, we have v|w = 0. Replacing v by a suitable scalar multiple, we achieve v|w = 1. Now Witt's Theorem (see [2, § 11, p. 21]) asserts that there exists A ∈ U(V, h) with aA = v and bA = w. The induced collineation ⌊A⌋ ∈ PU(V, h) then maps the pair (Ka, Kb) to (P, Q), and maps the block joining Ka and Kb to the block joining P and Q. Proof. The set T [X] is the kernel of the action of the stabilizer Aut(U) X of X in Aut(U) on the set B X of all blocks through X. So T [X] is a normal subgroup of Aut(U) X . Pick v, w ∈ V such that X = Kv and B = U h ∩ Gr 1 (L), where L = Kv + Kw. Then v|v = 0, and without loss of generality, we may assume w|w = 0 and v|w = 1. An easy computation shows that B = {Kv} ∪ {K(pv + w) | p ∈ K, p + p σ = 0}. For each p ∈ K with p + p σ = 0, the linear map M ′ defined by vM ′ = v and wM ′ = pv + w is an isometry of the restriction of h to L × L. As that restriction is not degenerate, the space L ⊥ is a vector space complement to L in V . We extend M ′ to a linear map M that acts trivially on L ⊥ . Then M belongs to U(V, h), and induces a collineation ⌊M ⌋ ∈ T [X] ∩PU(V, h) that maps Kw to K(pv + w). This shows that T [X] ∩ PU(V, h) is transitive on B {X}, as claimed. Two hermitian forms, and their unitals Let R be a commutative field, and let C|R be a quadratic field extension. Then the Galois group Gal(C|R) has order two, and is generated by an involution σ : x → x. We choose an element i ∈ C {0} with i σ = −i. (If char R = 2 then j lies in R; we will exclude that case later on.) We assume that there is an anisotropic σ-hermitian form on C 2 . Without loss of generality (i.e., up to similitude) we may assume that this form has Gram matrix N = ( 1 0 0 s ). We consider the quaternion field H := H s C|R = a x −sx a a, x ∈ C . Using w := 0 1 −s 0 and the embedding c → c 0 0 c of C into H, we obtain H = C + wC with the multiplication rule (a + wb)(c + wd) = ac − sbd + w(ad + bc), for a, b, c, d ∈ C. 2.1 Lemma. The map α : a+wb → a+wb (where a, b ∈ C) is an involutory anti-automorphism of H, the fixed points are those in R + wC. We have (a + wb) + (a + wb) α = a + a + 2wb and (a + wb)(a + wb) α = aa − sbb + 2wab. Proof. In fact, we have X α = i −1 X κ i for each X ∈ H, where κ : a + wb → a − wb is the standard involution of H. So α is the composition of an anti-automorphism (namely, κ) and an (inner) automorphism of H. Straightforward calculations yield the remaining assertions. We note that α is the standard involution if char R = 2. A unital in projective space 2.2 Definitions. On C 4 , we consider the σ-hermitian form g : C 4 × C 4 → C : (x 0 , x 1 , x 2 , x 3 ), (y 0 , y 1 , y 2 , y 3 ) → x 0 y σ 3 + x 3 y σ 0 + x 1 y σ 1 + sx 2 y σ 2 . This form has Witt index 1 because the norm form of H is anisotropic. We assume char R = 2 (so i / ∈ R), and consider Ξ : = ξ(u, p) u ∈ C 2 , p ∈ Ri ⊆ PGL(4, C), where ξ (u 0 , u 1 ), p :=         1 u 0 u 1 p − 1 2 N (u 0 + wu 1 ) 0 1 0 −u σ 0 0 0 1 −su σ 1 0 0 0 1         . (For any matrix A ∈ GL(4, C), we denote by ⌊A⌋ the corresponding element in PGL(4, C), obtained as the coset modulo scalars.) Proposition. a. We have U g = {C(0, 0, 0, 1)} ∪ {C(1, x 1 , x 2 , x 3 ) | x 3 + x σ 3 = −x 1 x σ 1 − sx 2 x σ 2 } . b. The set Ξ is a subgroup of PSU(C 4 , g) . That subgroup fixes the point C(0, 0, 0, 1), and acts sharply transitively on U g {C(0, 0, 0, 1)}. In fact, for u, v ∈ C 2 and p, q ∈ Ri the product in Ξ is obtained as ξ(u, p) ξ(v, q) = ξ u + v, p + q + 1 2 (vM u σ − uM v σ ) , where M = ( 1 0 0 s ). Proof. Consider x = (x 0 , x 1 , x 2 , x 3 ) ∈ C 4 {(0, 0, 0, 0)} with Cx < x ⊥g . If x 0 = 0 then 0 = x 1 x σ 1 + sx 2 x σ 2 = N (x 1 + wx 2 ) , and Cx = (0, 0, 0, 1) because the norm form N is anisotropic. If x 0 = 0 then we may assume x 0 = 1, and x 3 + x σ 3 = −x 1 x σ 1 − sx 2 x σ 2 follows, as claimed. It is easy to verify Ξ SU(C 4 , g), and that each element of Ξ fixes the point C(0, 0, 0, 1). We note M (u 0 , u 1 ) σ = M u σ 0 u σ 1 = u σ 0 su σ 1 . Straightforward calculations now yield N (u 0 + wu 1 ) = (u 0 , u 1 )M (u 0 , u 1 ) σ , and then −(u + v)M (u + v) σ + 2(vM u σ − uM v σ ) = −uM u σ − vM v σ − uM v σ leads to ξ(u, p) ξ(v, q) =     1 u p − 1 2 N (u 0 + wu 1 ) 0 E −M u σ 0 0 1         1 v q − 1 2 N (v 0 + wv 1 ) 0 E −M v σ 0 0 1     =     1 u + v p + q + 1 2 (vM u σ − uM v σ ) − 1 2 N (u 0 + v 0 + w(u 1 + v 1 )) 0 E −M (u + v) σ 0 0 1     , where E = ( 1 0 0 1 ). As z := vM u σ − uM v σ satisfies z + z σ = 0, we obtain ξ(u, p) ξ(v, q) = ξ u + v, p + q + 1 2 (vM u σ − uM v σ ) , as claimed. So Ξ is closed under multiplication. The inverse of ξ(u, p) is ξ(−u, −p) ∈ Ξ. Finally, we note that ξ(u, p) maps C(1, 0, 0, 0) to C 1, u 0 , u 1 , p − 1 2 N (u 0 + wu 1 ) . This shows that Ξ acts sharply transitively on U g . Remarks. The set ξ((0, 0), p) p ∈ Ri forms both the center and the commutator group of the group Ξ. That commutator group is the group T [C(0,0,0,1)] of translations of the unital U g = (U g , B g , ∈) with center C(0, 0, 0, 1). For the point C(1, 0, 0, 0) ∈ U g , we obtain T [C(1,0,0,0)] =              1 0 0 0 0 1 0 0 0 0 1 0 p 0 0 1       p ∈ Ri        . A unital in the quaternion plane 2.5 Definitions. We continue to assume char R = 2. On H 3 , we consider the α-hermitian form h : H 3 × H 3 : (X 0 , X 1 , X 2 ), (Y 0 , Y 1 , Y 2 ) → X 0 Y α 2 + X 1 Y α 1 + X 2 Y α 0 , here α is the involution introduced in 2.1. The form h has Witt index 1. We consider the subset Ψ := ψ(X, p) X ∈ H, p ∈ Ri of the group PGL (3, H), where ψ(X, p) :=      1 X p − 1 2 XX α 0 1 −X α 0 0 1      . (Again, for A ∈ GL(3, H), we denote by ⌊A⌋ the corresponding element in PGL(3, H), obtained as the coset modulo central scalars in this case.) Proposition. a. We have U h = {H(0, 0, 1)} ∪ {H(1, X, Y ) | Y + Y α = −XX α } . b. The set Ψ is a subgroup of PU(H 3 , h) . That subgroup fixes the point H(0, 0, 1), and acts sharply transitively on U h {H(0, 0, 1)}. The multiplication in Ψ is given by ψ(X, p) ψ(Y, q) = ψ X + Y, p + q + 1 2 (Y X α − XY α ) . Proof. The proof is quite analogous to the proof of 2.3. Remarks. The center and the commutator group of the group Ψ both coincide with ψ(0, p) p ∈ Ri . That group is the group T [H(0,0,1)] of translations of the unital U h = (U h , B h , ∈) with center H(0, 0, 1). For the point H(1, 0, 0) ∈ U h , we obtain T [H(1,0,0)] =        1 0 0 0 1 0 p 0 1     p ∈ Ri    . Remark. The groups Ξ and Ψ are examples of generalized Heisenberg groups (cp. [15], [5], [11]). In fact, they are both isomorphic to GH(R 4 , R, β), where β is any non-degenerate alternating form on R 4 . We give a direct isomorphism explicitly, in 2.10 below. An isomorphism of unitals 2.9 Definition. For each u = (u 0 , u 1 ) ∈ C 2 and each p ∈ Ri, we define the point C 1, u 0 , u 1 , p − 1 2 N (u 0 + wu 1 ) η := H 1, u 0 + wu 1 , p − 1 2 (u 0 + u 1 w)(u 0 + u 1 w) α . Moreover, we put C(0, 0, 0, 1) η := H(0, 0, 1). Thus we obtain a bijection η : U g → U h : P → P η , see 2.3 and 2.6. Theorem. We assume char R = 2, and use the notation introduced in 2.1, 2.2, 2.5, and 2.9 above. a. The map ϕ : Ξ → Ψ : ξ (u 0 , u 1 ), p → ψ u 0 + wu 1 , p is an isomorphism of groups. b. For each u = (u 0 , u 1 ) ∈ C 2 , each p ∈ Ri, and each point P ∈ U g we have P η ψ(u 0 +wu 1 ,p) = P ξ(u,p) η ; here η : U g → U h is the map introduced in 2.9. c. The map η : U g → U h induces an isomorphism of incidence structures from (U g , B g , ∈) onto (U h , B h , ∈). Proof. We use the multiplication formulae given in 2.3 and 2.6 to prove assertion a. It suffices to verify (v 0 + wv 1 )(u 0 + wu 1 ) α − (u 0 + wu 1 )(v 0 + wv 1 ) α = (v 0 + wv 1 )(u 0 + wu 1 ) − (u 0 + wu 1 )(v 0 + wv 1 ) = v 0 u 0 − u 0 v 0 + w 2 (v 1 u 1 − u 1 v 1 ) = vM u σ − uM v σ ; here we use wc = cw (for c ∈ C) and w 2 = −s. Assertion b is easily checked. As any two points in a hermitian unital are joined by a unique block, it remains to verify that B η ∈ B h holds for each block B ∈ B g . Using transitivity of Ξ on U g {C(0, 0, 0, 1)}, we see that it suffices to consider blocks through C(0, 0, 0, 1), and blocks through C(1, 0, 0, 0). Any block through C(0, 0, 0, 1) is of the form B = U g ∩ L, where L = C(0, 0, 0, 1) + C(1, u 0 , u 1 , u 2 ). We may assume C(1, u 0 , u 1 , u 2 ) ∈ U g . Then u 2 = p − 1 2 N (u 0 + wu 1 ) holds for some p ∈ Ri. So the block in question is B = C(1, u 0 , u 1 , p − 1 2 N (u 0 + wu 1 )) p ∈ Ri ∪ {C(0, 0, 0, 1)}, and its image B η = H(1, u 0 + wu 1 , p − 1 2 (u 0 + wu 1 )(u 0 + wu 1 ) α ) p ∈ Ri ∪ {H(0, 0, 1)} belongs to B h . Now consider a block B through C(1, 0, 0, 0). There exist u = (u 0 , u 1 ) ∈ C 2 and x ∈ Ri such that C 1, u 0 , u 1 , x − 1 2 N (u 0 + wu 1 ) ∈ B {C(1, 0, 0, 0)}. We abbreviate n := N (u 0 + wu 1 ). Every point in B {C(1, 0, 0, 0)} is of the form P a := C 1, au 0 , au 1 , a(x − n 2 ) , where a = a 0 + a 1 i ∈ C (with a 0 , a 1 ∈ R) satisfies aan + 2a 1 ix − a 0 n = 0 . ( * ) So P a = C 1, au 0 , au 1 , y a − aa n 2 , with y a := a(x − n 2 ) + aa n 2 . Note that y a ∈ Ri. We abbreviate Z : = u 0 + wu 1 , so C(1, u 0 , u 1 , x − n 2 ) η = H(1, Z, x − 1 2 ZZ α ). For each a ∈ C satisfying condition ( * ) from above, we obtain C 1, au 0 , au 1 , a(x − n 2 ) η = C 1, u 0 a, u 1 a, y a − aa n 2 η = H 1, Za, y a − 1 2 Za(Za) α = H 1, Za, y a − 1 2 aaZZ α . Each one of those points is contained in U h = U η g . In order to see that it is actually contained in the block H(1, 0, 0) + H(1, Z, x − 1 2 ZZ α ) ∩ U h , it remains to check that there exists Y ∈ H such that Y (Z, x − 1 2 Y Y α ) = (Za, y a − 1 2 aaZZ α ). The entry on the le yields Y = ZaZ −1 . Using ZZ α = (u 0 + wu 1 )(u 0 + wu 1 ) = u 0 u 0 −su 1 u 1 +2wu 0 u 1 and ZaZ −1 = (u 0 a + wu 1 a)(u 0 − wu 1 ) 1 n = a 0 + a 1 i ZZ α 1 n , we compute ZaZ −1 (x − 1 2 ZZ α ) = ZaZ −1 x − 1 2 ZaZ α = a 0 x + a 1 i ZZ α x 1 n − 1 2 a 0 ZZ α − 1 2 a 1 in = a 0 (x − 1 2 ZZ α ) + a 1 i(ZZ α x 1 n − 1 2 n) = a 0 (x − 1 2 ZZ α ) + a 1 i(xZZ α 1 n − 1 2 n) ; we have used i F i −1 = F α and x ∈ Ri. On the other hand, we find H(1, 0, 0), see 1.6 and 2.7. In particular, we obtain that the block y a − 1 2 aaZZ α = a(x − n 2 ) + aa n 2 − 1 2 aaZZ α = ax − 1 2 an + 1 2 a 0 n − a 1 ix − 1 2 (a 0 − 2a 1 ix 1 n )ZZ α = a 0 x − a 1 i n 2 − 1 2 (a 0 − 2a 1 ix 1 n )ZZ α ,B ′ = H(1, 0, 0) + H(1, Z, x − 1 2 ZZ α ) ∩ U h equals the set {H(1, 0, 0)} ∪ H(1 + (x − 1 2 ZZ α )p, Z, x − 1 2 ZZ α p ∈ Ri . So it suffices to show that for each p ∈ Ri there exists a ∈ C satisfying condition ( * ) and such that H 1 + (x − 1 2 ZZ α )p, Z, x − 1 2 ZZ α = H 1, Za, ZaZ −1 (x − 1 2 ZZ α ) ; the description on the right hand side then yields that the point in question lies in B η . We need to find a ∈ C with 1 + ( x − 1 2 ZZ α )p = (ZaZ −1 ) −1 = Za −1 Z −1 . We write b := a −1 as b = b 0 + b 1 i with b 0 , b 1 ∈ R, and compare 1 + (x − 1 2 ZZ α )p = (1 + xp) − 1 2 ZZ α p with ZbZ −1 = b 0 + b 1 i ZZ α 1 n . Since 1 + xp lies in R and 1 2 ZZ α p ∈ Ri + wC, we obtain 1 + xp = b 0 and − 1 2 ZZ α p = b 1 i ZZ α 1 n , so b 1 i = − 1 2 pn, and b = 1 + xp − 1 2 pn. Condition ( * ) for a means n − 2b 1 ix − b 0 n = 0, and is easily verified. Groups of translations, and an isomorphism of groups 3.1 Definition. Let (P, L, I) be an incidence structure such that through any two points in P there is at most one line in L incident with both of those points. An O'Nan configuration in (P, L, I) consists of 4 lines meeting in 6 points (see Fig. 1 below). In particular, any two of those four lines have a (unique) point in common. These configurations are named in honor of Michael O'Nan, who used the finite case of the following result 3.2 in his study of the automorphisms of finite hermitian unitals, see [13]. In the (axiomatic) context of projective spaces, O'Nan configurations are called Veblen-Young figures. The proof of the following result is taken from [4, 2.2]. Proposition. Let b 0 , F b 1 , F (b 0 + b 1 ), F b 2 , F (b 0 + b 2 ) and F (b 1 − b 2 ) , respectively. If these points belong to U h then b n |b n = 0 and b n |b m = − b m |b n holds for all m < n < 3. The matrix ( b m |b n ) m,n<3 has determinant 0 (here we use that F is commutative). Hence f is degenerate, and the restriction of h to F b 0 + F b 1 + F b 2 has Witt index at least 2. But then the Witt index of h is greater than 1, contradicting our assumption. Remark. Kestenband [10] claims that 3.2 holds even for hermitian unitals over skew fields. This claim is false. For instance, consider the quaternion field H := H 1 C|R = C + jC over the real number field R, constructed from C = R + Ri with j 2 = −1, the standard involution κ : x → x, and the hermitian form given by (u 0 , u 1 , u 2 )|(v 0 , v 1 , v 2 ) = u 0 iv 1 + u 0 jv 2 − u 1 iv 0 − u 1 jiv 2 − u 2 jv 0 + u 2 jiv 1 . That form is not degenerate, and has Witt index 1. Proof. We already know from 1. 6 that T [X] is transitive on B {X}. B X Y = Y τ Z W Z τ W τ B Z B W D D τ Corollary. center X = Kv is induced by a transvection τ λ,v ∈ U(V, h) with ker λ = v ⊥ h . Explicitly, we obtain for the two unitals considered here: the commutator groups Ξ ′ = ξ((0, 0), p) p ∈ Ri and Ψ ′ = ψ(0, p) p ∈ Ri of 2.4 and 2.7 are full translation groups, with centers C(0, 0, 0, 1) and H(0, 0, 1), respectively. Proof. Recall that the groups EU(C 4 , g) and EU(H 3 , h), respectively, are generated by all unitary transvections; those transvections induce the translations of the unital. Conjugation by the isomorphism η : U g → U h maps Aut(U g ) onto Aut(U h ), and maps the group T [X] to T [X η ] , for each point X ∈ U g . So conjugation by η induces an isomorphism from PEU(C 4 , g) onto PEU(H 3 , h). Example. We take the field C of complex numbers for C, with the standard involution σ : c → c generating Gal(C|R), and the field H = H 1 C|R = C + jC of Hamilton's quaternions. The involution α from 2.1 represents the unique class of involutory anti-automorphisms of H apart from the standard involution κ. For the forms g and h introduced in 2.2 and 2.5, respectively, we obtain the groups PEU(C 4 , g) ∼ = PSU 4 (C, 1) and PEU(H 3 , h) ∼ = PSαU 3 (H) (in the notation of [14, 94.33], in Tits [18, pp. 28, 40], these occur as the groups of type A C, 1 3 and D H 3 , Helgason [6, X § 2.1, § 6.2] denotes the corresponding algebras by su(3, 1) and so * (6), respectively). Remarks. For the commutative field C, one knows that EU(C 4 , g) = SU(C 4 , g), so PEU(C 4 , g) = PSU(C 4 , g). Also, it is known that the groups PEU(C 4 , g) and PEU(H 3 , h) are simple: see [2, II § 4] for a general result, cf. [17, 10.20] or [3, 11.26] for the case of a commutative ground field. As we restrict our investigation to cases where the characteristic is different from two, all the forms in question are trace valued forms. Remarks. As the field C is commutative, the involution σ of C is an involution of the second kind (in the sense of Dieudonné [2, § 10, p. 19]). According to [16, 5.6c], every reflection in the group PU(C 4 , g) is thus admissible, and we obtain Aut(U g , B g , ∈) = PΓU(C 4 , g). From our result 2.10 we then also infer Aut(U h , B h , ∈) ∼ = PΓU(C 4 , g). Remark. Let F be a commutative field, and let Q be a quaternion algebra over F . Then Q is a central simple F -algebra (cp. [9, 4.5, Lemma 3, p. 232], and every F -linear automorphism is inner (by the Skolem-Noether Theorem, see [9, p. 222], or see [8, Theorem 2, p. 67] for a direct proof). It then follows that every F -linear anti-automorphism β is the product of the standard involution and some inner automorphism, say x → i −1 xi with i ∈ F \ {0}, so x β = i −1 xi. We obtain that β is an involution precisely if i 2 ∈ F , i.e., if either i ∈ F or i = −i. If i ∈ F then β is the standard involution. If i / ∈ F , we form the quadratic extension C = F + F i. The restriction σ of the standard involution of Q then is the generator of Gal(C|F ), and β is obtained as in 2.1. 1. 6 6Lemma. Let U := (U h , B h , ∈) be the hermitian unital with respect to a non-degenerate σ-hermitian form h : V × V → K of Witt index 1. a. For each point X ∈ U h , the set T [X] is a subgroup of Aut(U), and a normal subgroup in the stabilizer of X in Aut(U). b. For each block B ∈ B h through X, the subgroup T [X] acts transitively on the set B {X}. In fact, the intersection T [X] ∩ PU(V, h) acts transitively on that set. and this equals ZaZ −1 (x − 1 2 ZZ α ), as required.So we have established that B η is contained in some block B ′ of B h , for each B ∈ B g .It remains to show that B η fills all of B ′ . To this end, we use the fact that the group T :Ri of translations with center H(1, 0, 0) acts transitively on D {H(1, 0, 0)}, for each block D ∈ B h through V be a vector space over a commutative field F , and assume that there is a non-trivial involutory automorphism σ of F . Let h : V × V → F : (u, v) → u|v be a non-degenerate σ-hermitian form of Witt index 1. Then the hermitian unital U = (U σ , B σ , ∈) does not contain any O'Nan configurations.Proof. Consider an O'Nan configuration in the projective space PG(V ). Then the six points of the configuration are contained in the projective plane spanned by any two of the lines inside PG(V ).Therefore, there are linearly independent vectors b 0 , b 1 , b 2 in V such that the six points of the configuration are F However, the corresponding hermitian unital contains the O'Nan configuration with the points H(1, 0, 0), H(0, 1, 0), H(0, 0, 1), H(1, 1, 0), H(1, 0, 1), and H(0, 1, −1). 3.4 Proposition. Let U = (U h , B h , ∈) be a hermitian unital, and let X be any point in U h . If U contains no O'Nan configurations then the translation group T [X] acts sharply transitively on B {X}, for each block B through X. Figure 1 : 1Constructing an O'Nan configuration from a translation with a fixed point.If the action is not sharply transitive then there exists τ ∈ T[X] {id} such that τ fixes some point Y ∈ B {X}. Let Z be any point in U h B with Z τ = Z. As τ is a translation of the unital U, the block B Z joining X and Z is invariant under τ , and contains Z τ . In the block D joining Y and Z, choose a third point W . Then W τ lies in the intersection of D τ and the block B W joining X and W . So the six points X, Y , Z, Z τ , W , W τ and the four blocks D, D τ , B Z , B W form an O'Nan configuration in the unital, contradicting our assumption. SeeFig. 1. 3. 7 7Theorem. The groups PEU(C 4 , g) and PEU(H 3 , h) are isomorphic. Let h : V × V → K be a non-degenerate σ-hermitian form of Witt index 1. If the corresponding hermitian unital U has no O'Nan configurations then T [X] = T [X] ∩ PU(V, h) holds for each point of the unital. 3.6 Corollary. Let h : V × V → K be a non-degenerate σ-hermitian form of Witt index 1. If the corresponding hermitian unital U has no O'Nan configurations then every translation of the unital U is induced by a transvection of the projective space PG(V ); in fact, each translation with P Dembowski, 10.1007/978-3-642-62012-6Finite geometries. BerlinSpringer-Verlag44P. Dembowski, Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer-Verlag, Berlin, 1968, doi:10.1007/978-3-642-62012-6. MR 0233275 . Zbl 0865.51004. J A Dieudonné, 10.1007/978-3-662-59144-4La géométrie des groupes classiques. BerlinSpringer-Verlag5J. A. 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Stroppel, Compact projective planes, de Gruyter Expositions in Mathematics 21, Walter de Gruyter & Co., Berlin, 1995, doi:10.1515/9783110876833. MR 1384300 . Zbl 0851.51003. Homogeneous symplectic maps and almost homogeneous Heisenberg groups. M J Stroppel, 10.1515/form.1999.018Forum Math. 116M. J. Stroppel, Homogeneous symplectic maps and almost homogeneous Heisenberg groups, Forum Math. 11 (1999), no. 6, 659-672, doi:10.1515/form.1999.018. MR 1724629 . Zbl 0928.22008. Automorphisms of unitals. M J Stroppel, H Van Maldeghem, Bull. Belg. Math. Soc. Simon Stevin. 125M. J. Stroppel and H. Van Maldeghem, Automorphisms of unitals, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 5, 895-908, http://projecteuclid.org/euclid.bbms/1136902624. MR 2241352 . Zbl 1139.51002. The geometry of the classical groups. D E Taylor, MR 1189139 . Zbl 0767Sigma Series in Pure Mathematics. 9Heldermann VerlagD. E. Taylor, The geometry of the classical groups, Sigma Series in Pure Mathematics 9, Heldermann Verlag, Berlin, 1992. MR 1189139 . Zbl 0767.20001. J Tits, 10.1007/BFb0080324Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen. BerlinSpringer-VerlagJ. Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Springer-Verlag, Berlin, 1967, doi:10.1007/BFb0080324. MR 0218489 . Zbl 0166.29703. . Markus J Lexmath, Fakultät für Mathematik und Physik Universität StuttgartMarkus J. Stroppel LExMath, Fakultät für Mathematik und Physik Universität Stuttgart
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[ "A DERIVED CONSTRUCTION OF EIGENVARIETIES", "A DERIVED CONSTRUCTION OF EIGENVARIETIES" ]
[ "Weibo Fu " ]
[]
[]
We construct a derived variant of Emerton's eigenvarieties using the locally analytic representation theory of p-adic groups. The main innovations include comparison and exploitation of two homotopy equivalent completed complexes associated to the locally symmetric spaces of a quasi-split reductive group G, comparison to overconvergent cohomology, proving exactness of finite slope part functor, together with some representation-theoretic statements. As a global application, we exhibit an eigenvariety coming from data of GLn over a CM field as a subeigenvariety for a quasi-split unitary group.IndP0 π(G, K p ) by Prop 3.13. We may replace the right hand side by an injective resolution of admissible Banach representations of G 0 , and apply our eigenvariety machine in §6 to obtain the induction eigenvarieties E * (Ind, G, K p , m) for a maximal ideal m in various degrees. As we have seen, these eigenvarieties E * (Ind, G, K p , m) come from a parabolic induction of the completed cohomology complexes associated to G. Our main theorem in this section ( §7, Theorem 7.3) establishes a type of functoriality between E d G ( K p , m) and E d (Ind, G, K p , m). Theorem 1.3. Suppose F + = Q, d is the middle dimension of symmetric space of G, and m satisfies the condition of [CS19, Thm 1.1]. The induction eigenvariety E d (Ind, G, K p , m) is embedded as a subvariety of the G-eigenvariety E d G ( K p , m). A highlight is that the eigenvariety E d (Ind, G, K p , m) only depends on G. Although there seems to be serious difficulties to figure out the exact relationship between such an eigenvariety and the GL n -eigenvariety at the moment.NotationIf R is a (possibly non-commutative) ring, then we will write Mod(R) for the abelian category of (left unital) R-modules and D(R) for the derived category of R-modules. If K is a field, we use Vect K to denote the abelian category of K-vector spaces. If A is an additive category, we will write Ch(A), K(A) and Ch b (A), K b (A) for the category of chain complexes, the homotopy categories and their full subcategory consisting of bounded complexes of A. If A is an abelian category, we will write D(A) and D b (A) for the derived category and the bounded
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[ "https://export.arxiv.org/pdf/2110.04797v2.pdf" ]
238,582,882
2110.04797
79b4bf870c7ee71111d4b1b1b1b9114a2233c2ff
A DERIVED CONSTRUCTION OF EIGENVARIETIES 16 Oct 2022 Weibo Fu A DERIVED CONSTRUCTION OF EIGENVARIETIES 16 Oct 2022 We construct a derived variant of Emerton's eigenvarieties using the locally analytic representation theory of p-adic groups. The main innovations include comparison and exploitation of two homotopy equivalent completed complexes associated to the locally symmetric spaces of a quasi-split reductive group G, comparison to overconvergent cohomology, proving exactness of finite slope part functor, together with some representation-theoretic statements. As a global application, we exhibit an eigenvariety coming from data of GLn over a CM field as a subeigenvariety for a quasi-split unitary group.IndP0 π(G, K p ) by Prop 3.13. We may replace the right hand side by an injective resolution of admissible Banach representations of G 0 , and apply our eigenvariety machine in §6 to obtain the induction eigenvarieties E * (Ind, G, K p , m) for a maximal ideal m in various degrees. As we have seen, these eigenvarieties E * (Ind, G, K p , m) come from a parabolic induction of the completed cohomology complexes associated to G. Our main theorem in this section ( §7, Theorem 7.3) establishes a type of functoriality between E d G ( K p , m) and E d (Ind, G, K p , m). Theorem 1.3. Suppose F + = Q, d is the middle dimension of symmetric space of G, and m satisfies the condition of [CS19, Thm 1.1]. The induction eigenvariety E d (Ind, G, K p , m) is embedded as a subvariety of the G-eigenvariety E d G ( K p , m). A highlight is that the eigenvariety E d (Ind, G, K p , m) only depends on G. Although there seems to be serious difficulties to figure out the exact relationship between such an eigenvariety and the GL n -eigenvariety at the moment.NotationIf R is a (possibly non-commutative) ring, then we will write Mod(R) for the abelian category of (left unital) R-modules and D(R) for the derived category of R-modules. If K is a field, we use Vect K to denote the abelian category of K-vector spaces. If A is an additive category, we will write Ch(A), K(A) and Ch b (A), K b (A) for the category of chain complexes, the homotopy categories and their full subcategory consisting of bounded complexes of A. If A is an abelian category, we will write D(A) and D b (A) for the derived category and the bounded Introduction Eigenvarieties have been playing important roles in the developments of the Langlands program, such as the developments of the Fontaine-Mazur conjecture ( [Boc01], [Kis03], [Eme11] for examples) and the symmetric power functoriality [NT21a], [NT21b], [NT22], both for modular forms. They were previously constructed for instances by [AIP15], [AS08], [BHS16], [Buz07], [Eme06b], [Han17], [Urb11] using different approaches. The goal of this article is to introduce a generalization of Emerton's construction of eigenvarieties for a quasi-split reductive group in his seminal paper [Eme06b]. For a general quasi-split reductive group over a number field, especially for groups with nonzero defects, we believe that our eigenvarieties are quite different than the ones constructed in [Eme06b] due to existence of nonvanishing higher cohomology of a complex of locally analytic representations of a p-adic group and the corresponding higher locally analytic Jacquet modules. Date: Oct, 2022. If G is a quasi-split reductive group over a number field F . We may form a complex to compute the completed cohomology of the locally symmetric spaces of G. We have two ways of picking such a complex given the locally symmetric spaces ( §5): one canonical way is to consider the set of all singular cochains of the spaces, and we get an enormous complex of Banach representation A • of the group G(F p ), another way is to only use cochains given by a fixed choice of finite triangulation of the Borel-Serre compactification of X G,KpK p for a base level subgroup K p K p ⊂ G(F p )× G(A p F ), accordingly we get a complex S • of admissible Banach representations of the compact group K p . For defining Jacquet modules associated to a tame level K p , both ways alone seem to be concerning: an arbitrary Banach representation does not have good functional analysis properties, while the action of K p on S • does not extend to G(F p ) and the Hecke action for S • is missing before passing to cohomology. Using ingredients and notions in §3 and §4, we illustrate how to make use of both A • and S • via a homotopy equivalence A • ⇆ S • to construct derived Jacquet modules associated to the tame level K p . Eventually we construct eigenvarieties out of these essentially admissible representations. The idea of using a finite triangulation leads to admissibility of completed cohomology by [Eme06b] and constructions of eigenvarieties by [AS08], [Urb11], [Han17]. Hao Lee [Lee22] studies a similar but different question, and he constructs the derived Jacquet-Emerton modules using a different approach. We establish some technical results concerning locally analytic representations in §3 for uses in further sections. In §4, we prove derived Jacquet modules associated to A • ⇆ S • are canonically defined independent of various choices and they are essentially admissible representations of the p-adic maximal torus of G in Prop 4.9. A key ingredient Theorem 4.5 in §4 is to establish exactness of the finite slope part functor introduced in [Eme06a]: let T 0 be a compact p-adic torus, Y + be T 0 × Z + and Y be T 0 × Z, we have Theorem 1.1. Let 0 → U → V → W → 0 be a short exact sequence of certain Y + representations of compact type such that U has subspace topology and W has quotient topology. Then the finite slope part of the sequence in the sense of [Eme06a] 0 → U fs → V fs → W fs → 0 is exact in the category of locally analytic representations of Y , where U fs is a closed subspace and W fs has quotient topology. If I p is an Iwahori level at p with a maximal unipotent subgroup N p , in §5, we give a comparison of cohomology Theorem 5.2 of X G,IpK p with coefficient rising from a locally analytic representation Ind la Ip := {f ∈ C la (I p , E)|f (gn) = f (g), for ∀ n ∈ N p } of I p to the cohomology of N p invariants of locally analytic vectors of S • (or A • ). Theorem 1.2. There is a canonical isomorphism of cohomology H • (X G,IpK p , Ind la ) ∼ − → H • ((S • ⊗ OE E) la ) Np . And a similar comparison holds for compactly supported cohomology. These comparisons help us deduce vanishing results for applications. For example, we are able to deduce vanishing of compactly supported cohomology of N p invariants of locally analytic vectors of A • with the comparison using the main result of Caraiani-Scholze [CS19]. For our main global application in §7, let G be a quasi-split unitary group with Siegel parabolic P whose Levi component is G = GL n over a totally imaginary CM field. Let m be a maximal ideal of derived Hecke algebra of G with certain good tame level K p , and m be the pullback of m via the Satake morphism of derived Hecke algebra of G with tame level K p compatible with K p . Let G p := G(F p ), P p := P (F p ), G p := G(F p ) G 0 := G(O Fp ), P 0 := P (O Fp ), G 0 := G(O Fp ). In §6, we will construct eigenvarieties E * G ( K p , m) for the unitary group G and a suitable maximal ideal m and a suitable tame level K p . For such data, the most interesting eigenvariety is E d G ( K p , m), where degree d is the middle dimension of symmetric space of G. Let π(G, K p ) ⇆ π(G, K p ) be two complexes computing completed cohomology of G and tame level K p as we introduced in the previous paragraph (π(G, K p ) is an admissible Banach representation of G 0 and π(G, K p ) is a Banach representation of G p ). The homotopy equivalence induces the corresponding one on inductions derived category of A. Note that in §5, we actually consider cochain complexes like A • r,s , S • r,s , we consider them as chain complexes up to the usual normalization. If K is a local field, then we will write O K as its ring of integers with an uniformizer ̟ K , and residue field k K . If V and W are two LB-spaces ([Eme17, Def 1.1.16]), the inductive tensor product topology and projective tensor product topology agree on V ⊗ K W by [Eme17, Prop 1.1.31] if V and W are semi-complete. We will write V⊗ K W for completion for the case. If V is a Banach space, then we will write V • as its unit ball consisting of vectors with norms at most 1. If G is a locally profinite group, and U ⊂ G is an open compact subgroup, then we write H(G, U ) for the algebra of compactly supported, U -biinvariant functions f : G → Z, with multiplication given by convolution with respect to the Haar measure on G which gives U volume 1. If X ⊂ G is a compact U -biinvariant subset, then we write [X] for the characteristic function of X, an element of H(G, U ). If L is a p-adic local field with a finite extension L ⊂ K, V is a Hausdorff convex K-vector space and X is a locally-L analytic manifold, then we write C la (X, V ) for the V -valued locally analytic functions on X ([Eme17, §2.1]). We write C la c (X, V ) for the subspace of functions with compact supports. For a locally L-analytic group G, we write D(G, K) := C la (G, K) ′ b as the K-locally analytic distribution algebra of G ([ST02b, §2], denoted as D la (G, K) in [Eme17, §5]). If H is a locally (Q p -)analytic group and K is a p-adic local field, we use Ban H (K) to denote the additive category of K- We also define ∂X G := X G − X G and X G := G(F )\ X G × G(A ∞ F ) , ∂X G := G(F )\ ∂X G × G(A ∞ F ) . Local results Let O be a topological ring, X be a topological space, and M be a topological Omodule. We write C(X, M ) as the topological O-module of continuous maps from X to M with compact open topology. By results in [Sta, Tag 0B1Y], projective limits exist in the category of topological O-modules. Let G be a topological group with a closed subgroup P . Let V be a topological O-module with continuous P action (P × V → V is continuous). Lemma 3.1. For M ≃ lim ← − i M i as an isomorphism in the category of topological O- modules, there is a canonical isomorphism C(X, M ) ≃ lim ← − i C(X, M i ) in We form the continuous induction ct-Ind G P (V ) := {f : G → V |f continuous, f (gp) = p −1 · f (g) , ∀p ∈ P, g ∈ G}, endowed with subspace topology of C(G, V ). Let L be a finite extension of Q p , and let K be an extension of L. For the rest of the section, let G be a locally L-analytic group with an analytic subgroup P . Suppose G 0 , P 0 are compact subgroups of G, P such that G 0 /P 0 ∼ − → G/P . Proposition 3.2. If V is a K-Banach representation of P , the group action G × ct-Ind G P (V ) → ct-Ind G P (V ) is continuous. Proof. It is reduced to the fact that the group action G × C(G, V ) → C(G, V ) is continuous by [Eme17, Lem 3.1.1, Prop 3.1.5]. Lemma 3.3. If V is a K-Banach representation of P , then ct-Ind G P (V ) is a K- Banach representation of G. Moreover, if both P and G are compact and V is admissible ( [ST02a], [Eme17]), then ct-Ind G P (V ) is admissible as a K-Banach representation of G. Proof. There is a topological isomorphism ct-Ind G0 P0 (V ) ∼ − → ct-Ind G P (V ). When V is Banach, the topology of C(G 0 , V ) coincides with the maximal norm, where C(G 0 , V ) is Banach. By [Eme17, Prop/Def 6.2.3], if V is admissible then there exists a P 0 -equivariant closed embedding V → C(P 0 , K) n for some n. The admissibility is therefore guaranteed by the G 0 -equivariant closed embedding ct-Ind G0 P0 (V ) → ct-Ind G0 P0 (C(P 0 , K) n ) ≃ C(G 0 , K) n . Proposition 3.4. The continuous induction functor commutes with projective limits. Equivalently, ct-Ind G P (lim ← − i V i ) ≃ lim ← − i ct-Ind G P (V i ) for a projective system (V i ) of topological O-modules with continuous P action. Proof. By [Fea99, 4.1.1], the map G → G/P admits an analytic splitting ı : G/P × P → G as locally analytic manifolds. If V is a topological O-module with continuous P action, ct-Ind G P (V ) ≃ C(G/P, V ) as topological O-modules. The claim then follows from Lemma 3.1. Lemma 3.5. Any locally analytic representation of G on a vector space of compact type over K restricts to a locally analytic representation of P . Proof. If V is such a representation of G, then the orbit map gives us V ֒→ C la (G, K)⊗V . Locally analytic functions on G pull back to locally analytic functions on P , there is a induced orbit map V ֒→ C la (P, K)⊗V for P . For a locally analytic representation V of P , we have the locally analytic induction as follows ([Fea99, §4.1.1]): la-Ind G P (V ) := {f : G → V |f locally analytic, f (gp) = p −1 · f (g), ∀p ∈ P, g ∈ G}. Theorem 3.6. Let V be a K-Banach space equipped with a continuous P -action. The locally analytic vectors of ct-Ind G P (V ) are identified with the locally analytic induction of locally analytic vectors of V as a topological isomorphism: (ct-Ind G P (V )) la ≃ la-Ind G P (V la ). Proof. We adapt some notation in [Eme17], and use I to denote ct-Ind G P (V ). By [Eme17, §3.5], there is a continuous P -equivariant injection of locally convex Hausdorff K-spaces V la ֒→ V . We know I is a K-Banach closed subrepresentation of the space of continuous V -valued functions C(G, V ) on G. By [Fea99, §4.1.1], la-Ind G P (V la ) is a closed subspace of C la (G, V la ). There is a natural continuous injection la-Ind G P (V la ) ֒→ I, following from continuous injections by [ C la (G, V la ) ֒→ C(G, V la ) ֒→ C(G, V ). Let H be an analytic open subgroup of G, and let H, H P be rigid analytic group corresponding to H and H ∩P . Any group element g ∈ G defines a map E g : I → V sending f ∈ I to f (g) ∈ V . Consider I H−an → C an (H, I) Eg − − → C an (H, V ), we know that V HP −an is a closed subspace of C an (H P , V ). Both V (with orbit map) and C an (H P , V ) are embedded into C(H P , V ). For f ∈ I H−an : G → V , we have f (gp) = p −1 f (g) for p ∈ P from definition. Evaluating at identity of P ∩ H, the target of f lies in V ∩ C an (H P , V ) = V HP −an by [Eme17, Thm 3.3.16]. f induces an analytic map H → V Hp−an by (1). We can claim the topological isomorphism V la HP −an ≃ V HP −an form [Eme17, Prop 3.4.14]. Then by the same argument we know (la-Ind G P (V la )) H−an ∼ − → I H−an , hence (la-Ind G P (V la )) la ∼ − → I la . We finally win by applying [Eme17, Thm 3.6.12]. Recall that from notation Ban G (K) is the category of K-Banach representation of G, and Ban ad G (K) is the abelian category of admissible continues K-Banach representations of G considered in [Eme17, §6.2]. We use Rep a K (G) to denote the abelian category of admissible locally analytic representations over K in [Eme17, §6.1] (and equivalently in [ST03,§6] in the compact case). We use Rep la K (G) to denote the category of locally analytic G representations on compact type locally convex topological K-vector spaces. Lemma 3.7. If G is compact, Ban ad G (K) has enough injective objects. Proof. If G is compact, Ban ad G (K) is anti-equivalent to the category of finitely generated (left unital) K[[G]]-modules via duality by [ST02a, Thm 3.5], which has enough projective objects. Let Rep top.c K (T + ) be the additive category whose objects are Hausdorff locally convex K-vector spaces of compact type, equipped with a topological action of T + , and whose morphisms are continuous T + -equivariant K-linear maps ([Eme06a, §3.1], by a "topological action" of T + on a topological vector space V we mean an action of T + on V by continuous endomorphisms). Consider the following left exact functor Γ la N0 : Ban ad G (K) → Rep top.c K (T + ) defined as Γ N0 • la, where la : Ban ad G (K) → Rep a K (G) V → V la is the exact functor of passing to locally analytic vectors ([ST03, Thm 7.1]). We define Hecke action T + on V N0 to be t · v := [N 0 : tN 0 t −1 ] −1 n∈[N0:tN0t −1 ] ntv, for ∀t ∈ T + , ∀v ∈ V N0 . We refer to [Eme06a,§3.4] for the Hecke action of T + . Γ N0 : Rep a K (G) → Rep top.c K (T + ) V → V N0 is the left exact functor of taking N 0 -invariants. If V ∈ Ban ad G0 (K), then V la is a D(G 0 , K) module by [ST02b, Prop 3.2]. Lemma 3.8. Let n be the Lie algebra of N 0 over L. If I is an injective object in Ban ad G0 (K), then H i (n, I la ) vanish for i ≥ 1. Proof. Let I * be the corresponding finitely generated K[[G 0 ]]-module, I * is projective. There exists a complement M * such that I * ⊕ M * ≃ K[[G 0 ]] r for some rank r ∈ Z ≥1 . As the dual of K[[G 0 ]] is space of continuous functions C(G 0 , K) on G, there exists a closed complement M such that I ⊕ M ≃ C(G 0 , K) r . By taking locally analytic vectors, I la ⊕ M la ≃ C la (G 0 , K) r . As locally analytic manifolds, G 0 decomposes as N 0 × N 0 \G 0 , C la (G 0 , K) ≃ C la (N 0 , K)⊗ K C la (N 0 \G 0 , K). Since • ⊗ K C la (N 0 \G 0 , K) preserves exactness of locally convex K-vector spaces, so does •⊗ K C la (N 0 \G 0 , K) by Prop 4.2. It suffices to prove H i (n, C la (G 0 , K)) vanish for i ≥ 1, which turns out to be the Poincaré lemma ( (T + ) → Mod(K[T + ]). For a complex C • ∈ K b (Ban ad G0 (K)), we choose an injective resolution C • → I • (the existence follows from Lem 3.7). We have R i (for • Γ la N0 )(I • ) (derived functors for for • Γ la N0 ) coincide with Ext i D(N0,K) (1, I la • ) (derived functors of Hom functor) as abstract K[T + ]-modules. Proof. By [Koh11, Thm 4.8, Thm 4.10], Ext i D(N0,K) (1, V la ) ≃ H i (n, V la ) N0 for any V ∈ Ban ad G0 (K) . Therefore Ext i D(N0,K) (1, I la ) vanish for any injective object I in Ban ad G0 (K) and i ≥ 1 by Lem 3.8. Both delta-functors are effaceable and calculate cohomology of (I la • ) N0 . Definition 3.10. Suppose C • ∈ K b (Ban ad G0 (K)). By a G-extension of C we mean the data (C ′ , i, p) such that C ′ • ∈ K b (Ban G (K)), with a G 0 -equivariant homotopy equivalence C • p ⇆ i C ′ • in K b (Ban G0 (K)). In particular, H * (p • i) = id, H * (i • p) = id. If C 1,• , C 2,• are two homotopic equivalent objects in Ch b (Ban ad G0 (K)) and C ′ 1,• , C ′ 2,• are two homotopic equivalent objects in Ch b (Ban G (K)), then a given G-extension C ′ 2,• of C 2,• transfer to a G-extension C ′ 1,• of C 1,• . Definition 3.11. For a morphism f : C 1,• → C 2,• in K b (Ban ad G0 (K)), by a G- extension f ′ of f we mean G-extensions (C 1,• , i, p) of C 1,• , (C 2,• , j, q) of C 2,• and f ′ : C ′ 1,• → C ′ 2,• in K b (Ban G (K)) , with the following diagrams in K b (Ban G0 (K)). C 1,• i / / f C ′ 1,• f ′ C 2,• j / / C ′ 2,• C 1,• i / / f C ′ 1,• f ′ C 2,• C ′ 2,• q o o C 1,• f C ′ 1,• f ′ p o o C 2,• j / / C ′ 2,• C 1,• f C ′ 1,• f ′ p o o C 2,• C ′ 2,• q o o for G-extension C ′ 1,• of C 1,• and C ′ 2,• of C 2, • , commutative up to homotopy equivalences (these diagrams become commutative in the homotopy category). Commutativity for these four diagrams are equivalent. For example, if j • f is homotopic equivalent to f ′ • i, then f is homotopic equivalent to q • f ′ • i. Any G-extension (C ′ , i, p) of C • ∈ K b (Ban ad G0 (K)) defines a map G → Aut K b (BanK ) C • compatible with the natural action of G 0 (a possible misunderstanding is that a G-extension induces G → Aut K b (Ban ad G 0 (K)) C • , this is not true as a general element of G does not commute with G 0 ). If we fix a choice of representative (C ′ • , i, p), then for any group element g ∈ G, we define a representativeg : C • → C • to be p • g • i lifting g ∈ Aut K b (BanK ) C • . If g ∈ G 0 , this lift is given by G 0 structure regarding C • as a complex of G 0 Banach representations. It is easy to establish some first properties of such an extension structure. Proposition 3.12. We fix representatives of G-extension (C ′ • , i, p) of C • ∈ Ch b (Ban ad G0 (K)). • For any two g 1 , g 2 ∈ G,g 1 •g 2 is homotopic equivalent to g 1 g 2 . • Different choices of p : C ′ • → C • with i fixed or different choices of i : C • → C ′ • with p fixed induce differentg, but they all agree up to homotopy equivalences. • We haveg 1 •g 2 = g 1 g 2 if either g 1 or g 2 belongs to G 0 . In particular, a G-extension (C ′ • , i, p) of C • induces admissible Banach representations of G on cohomology. In other words, H i (C • ) ∈ Ban ad G (K) for any i, extending the natural G 0 action to a G action. Proof. The first two claims are clear by [Sta,Tag 010W]. The last one holds since both i and p are G 0 -equivariant. Proposition 3.13. If G is a quasi-split reductive linear algebraic group over L, with a parabolic subgroup P, such that G is a Levi of P. Let G = G(L) with compact open subgroup G 0 . And let P = P(L), P 0 = G 0 ∩ P , G = G(L), G 0 = G 0 ∩ G. Let C • ∈ K b (Ban ad G0 (K)) with a G-extension (C ′ • , i, p). By inflation we view C • ∈ K b (Ban ad P0 (K)) with a P -extension (C ′ • , i, p). Suppose that the projection of P onto G sends P 0 inside G 0 and G 0 P = G, then there is a natural G-extension (Ind G P C ′ • ,ĩ,p) ∈ K b (Ban G (K)) of Ind G0 P0 C • ∈ K b (Ban ad G0 (K)). Proof. By Lem 3.3, Ind G P C ′ • ∈ K b (Ban G (K)) and Ind G0 P0 C • ∈ K b (Ban ad G0 (K)). We define Ind G0 P0 C •p ⇆ i Ind G P C ′ • , whereĩ sends f to an unique function f ′ in Ind G P C ′ • such that f ′ (g) = i(f (g)) for i : C • → C ′ • and any g ∈ G 0 ,p sends f ′ to f satisfying f (g) = p(f ′ (g)) for p : C ′ • → C • and any g ∈ G 0 . We check thatĩ is well-defined as if g 1 p 1 = g 2 p 2 for g 1 , g 2 ∈ G 0 and p 1 , p 2 ∈ P , then g −1 2 g 1 = p 2 p −1 1 ∈ G 0 ∩ P = P 0 , f ′ (g 1 p 1 ) = p −1 1 i(f (g 1 )) = p −1 2 g −1 2 g 1 i(f (g 1 )) = p −1 2 i(f (g 2 )) = f ′ (g 2 p 2 ) . It can also be checked thatĩ,p are G 0 -equivariant, andĩ•p,p•ĩ are both homotopic to identity. We check forĩ •p as follows: note that for each n ∈ Z we have a G 0equivariant map h n : C ′ n → C ′ n−1 such that i • p − 1 = d n−1 • h n + h n+1 • d n , where d n : C ′ n → C ′ n+1 is the differential map for C ′ • at degree n. Letd n : Ind G0 P0 C ′ n → Ind G0 P0 C ′ n+1 andh n : Ind G0 P0 C ′ n → Ind G0 P0 C ′ n−1 be the natural maps induced from d n and h n . We therefore defined n andh n on Ind G P C ′ • via the natural G 0 -equivariant isomorphism Ind G P C ′ • ∼ − → Ind G0 P0 C ′ • , with the identitỹ i •p − 1 =d n−1 •h n +h n+1 •d n . We do not assert functoriality for induction of G-extension of the above proposition. Lemma 3.14. Let notation be as in Prop 3.13. If I is an injective object in the category of admissible continuous Banach representations of P 0 , then Ind G0 P0 I is an injective object in the category of admissible continuous Banach representations of G 0 . Proof. As in the proof of Lem 3.8, there exists a complement M such that I ⊕ M ≃ C(P 0 , K) r . It suffices to prove ct-Ind G0 P0 C(P 0 , K) ≃ C( G 0 , K) as G 0 -representations. There is a natural G 0 -equivariant map ct-Ind G0 P0 C(P 0 , K) → C( G 0 , K) by sending f → f such that f (g) =f (g)(identity) for any g ∈ G 0 . Any f defines a P 0 - equivariant function on G 0 valued in C(P 0 , K) and vice versa, we get the desired isomorphism. Suppose we have a G-extension (with fixed representatives) (C ′ • , i, p) of C • ∈ Ch b (Ban ad G0 (K)) . For any t ∈ T + , we define (derived) Hecke operators as follows. Lemma 3.15. For any g ∈ G,g stabilizes C la • and acts continuously on it. Proof. Taking locally analytic vectors is functorial, each of i, g, p preserves locally analytic vectors ([Eme17, §3]). Lemma 3.16. The operator U t := 1 [N 0 : tN 0 t −1 ] n∈N0/tN0t −1 nt ∈ End K ((C la • ) N0 ), where nt = p • nt • i, is a well-defined endomorphism of (C la • ) N0 . Proof. For any v ∈ C N0 i and n ′ ∈ N 0 , sinceñ •g =ñg,g •ñ =gn for any n ∈ N 0 , g ∈ G, we have n ′ · n∈N0/tN0t −1ñ t(v) = n∈N0/tN0t −1ñ t • n k (v) = n∈N0/tN0t −1ñ t(v), where the sets {n ′ nt} = {ntn k } are equal (a permutation of elements), and all n k ∈ N 0 . By the previous lemma, the operator stabilizes the closed subcomplex (C la • ) N0 of C la • . Let T be the rigid analytic space over L of locally L-analytic characters on T ([Eme17, Prop 6.4.5]), and let C an ( T , K) denote the nuclear Fréchet algebra of K-valued rigid analytic functions on T . Evaluation induces a natural map T → C an ( T , K). Pick t ∈ T such that ∩ n≥1 t n N 0 t −n = {1}, we may replace t by a sufficient large power of t to make it satisfy the assumption in [Eme06a, Prop 4.1.6 (vi)]. Let Y + (resp. Y ) be the submonoid (subgroup) generated by T 0 and t. Definition 3.17. If V is any object of Rep top.c K (Y + ), then we write V fs := L b,Y + (C an ( Y , K), V ) ∈ Rep la K (Y ). The subscript refers to "finite slope part" of [Eme06a, Def. 3.2.1]. Given a G-extension (C ′ • , i, p) of C • , and t ∈ T , there is a U t action on (C la • ) N0 by Lem 3.16. We prescribe t i (i ≥ 0) action as (U t ) i . By [Eme06a, Lem 3.4.4], U t i = (U t ) i for i ≥ 0 up to homotopy equivalence. By (3) of Prop 3.12, U t operator commutes with T 0 , therefore an authentic action of Y + on (C la • ) N0 is defined: for any y = t 0 t i ∈ Y + , t · v := t 0 (U t ) i · v for any v ∈ (C la • ) N0 . We define Jacquet functor J B on the G-extension (C ′ • , i, p) of C • to be J B (C • ) := (C la • ) N0 fs ∈ Ch(Rep la K (Y )), where fs is with respect to the semigroup generated by U t . By Lem 3.16, for any t ∈ T + , t acts on (C la • ) N0 via the Hecke operator U t . These operators do not commute and are not associative in general, but they commute up to homotopy equivalences. The Hecke operators associated to the monoid T + defines a topological action on cohomology H * (J B (C • )) by [Eme06a,Lem 3.4.4]. The fact that cohomology of J B (C • ) are independent of various choices follows from Prop 4.9 and Cor 4.10 proven in the next section. If Z is a topologically finitely generated abelian locally L-analytic group, let Rep es (Z) be the abelian category of essentially admissible locally analytic repre- sentations of Z over K ([Eme17, §6.4]). Lemma 3.18. If W is a locally analytic T -representation, then the action of T on W extends to an action of K [T ] → C an ( T , K) such that C an ( T , K) × W → W is separately continuous. Proof. Let T n be a sequence of abelian rigid analytic group corresponding to a cofinal sequence of analytic open subgroups T n (L) of T 0 . Let W n be the BHsubspace of W obtained as the image of the natural map W Tn−an → W . Clearly W = ∞ n=1 W n , and each W n is invariant by T -action since T is abelian. W satisfies condition (iii) of [Eme17, Prop 6.4.7], and W admits an action of C an ( T , K). Lemma 3.19. If W is a locally analytic T -representation over K such that it restricts to an essentially admissible representation of Y , then W is an essentially admissible representation of T . Proof. By definition, W ′ b is coadmissible as a C an ( Y , K)⊗ K D la (T 0 , K)-module. By [Eme17, Prop. 6.4.6], D la (T 0 , K) is identified with C an ( T 0 , K) as a subalgebra of C an ( Y , K) for T 0 ֒→ Y . We have that W ′ b is coadmissible as a C an ( Y , K)-module ([Eme06b, Prop 2.3.2], remarks before [Eme17, Prop. 6.4.10]). Let C an ( Y , K) ∼ − → lim ← −n A n be a Fréchet-Stein structure of C an ( Y , K) rising from an admissible cover { Y n } of Y with A n := O(Y n ). Then each A n⊗ C an ( Y ,K) W ′ b is finitely generated as an A n -module. By [Eme17,Prop 6.4.7] and the previous Lem 3.18, the C an ( Y , K) action on W ′ b extends (separately) continuously to C an ( T , K). Therefore A n⊗ C an ( Y ,K) W ′ b is finitely generated as a separately continuous A n⊗ C an ( Y ,K) C an ( T , K)-module. We have A n⊗ C an ( Y ,K) C an ( T , K) → End An (A n⊗ C an ( Y ,K) W ′ b ) , factoring through affinoid algebras B n which are finite over A n , defining coherent sheafs on Y n × Y T supported on affinoid spaces Sp(B n ) ֒→ Y n × Y T (as B n is a quotient of a Fréchet-Stein algebra by a closed ideal, one can prove this by proving that any continuous map of K{{z 1 , z −1 1 , · · · , z m , z −1 m }} → B n factors through an affinoid algebra of a p-adic annulus of certain radii). These sheafs are compatible with respect to n, hence we obtain a coherent sheaf on T whose global section agrees with W ′ b by varying n. The claim then follows from coadmissibility of W ′ b with respect to the Fréchet-Stein algebra C an ( T , K). Proposition 3.20. We use notation after Def 3.17. For each i, J B (C i ) = (C la i ) N0 fs is an essentially admissible locally analytic representation of Y , equivalently, J B (C • ) ∈ Ch(Rep es (Y )). Proof. This follows from Emerton's strategy [Eme06a], we give a summary (and a simplification in our setting) as a proof. We refer to [Eme06a], [Eme17] for related notation. , as the Hausdorff N 0 -coinvariants of the completed tensor product Let V be C la i . Let {H n } n≥0D(H • n , H 0 )⊗ D la (H0,K) V ′ b , with kernel K n fitting into 0 → K n → D(H • n , H 0 )⊗ D la (H0,K) V ′ b → U n → 0.(2) Since V consists of locally analytic vectors in an admissible Banach representa- tion C i of G 0 , (C i ) ′ b is, V ′ b ≃ D la (H 0 , K) ⊗ K[[H0]] (C i ) ′ b is a finitely presented D la (H 0 , K)-module. The inverse system lim ← −n K n has dense transition maps as (N 0 − 1)V ′ b is dense in D(H • n , H 0 )⊗ D la (H0,K) V ′ b (and D la (H 0 , K) is dense in D(H • n , H 0 )),0 → lim ← − n K n → V ′ b → lim ← − n U n → 0. The strong dual to V N0 is naturally identified with the Hausdorff N 0 -coinvariants Prop 4.1.6 (vi)]), the Hecke action of t (explicitly expressed as (4)) given by G- (V ′ b ) N0 of the strong dual to V , we have (corresponding to [Eme06a, (4.2.6)]) a Y + -equivariant isomorphism (V N0 ) ′ b ∼ − → lim ← − n U n .extension (C ′ • , i, p) of C • on U n factors through D(T • n , T 0 ) ⊗ D(T • n+1 ,T0) U n+1 / / t U n t x x q q q q q q q q q q q q q q q q q q q q q q q q D(T • n , T 0 ) ⊗ D(T • n+1 ,T0) U n+1 / / U n .(3) Let H(t) n := (t −1 H n t) ∩ H n , H(t) n,n+1 := (t −1 H n t) ∩ H n+1 with underlying strictly σ-affinoid analytic groups H(t) • n , H(t) • n,n+1 as in [Eme06a, §4.1]. We define U (t) n,n+1 := D(H(t) • n,n+1 , H 0 )⊗ D la (H0,K) V ′ b N0 , U (t) n := D(H(t) • n , H 0 )⊗ D la (H0,K) V ′ b N0 . Note that we have to modify the map (4.2.9) U n → U (t) n of [Eme06a]: namely the map (4.2.16) should be changed to µ ⊗ v ′ → δ(z) x s ′ n,k (µ) ⊗ i * • t −1 n −1 • p * v ′ ,(4) where i * , p * are the dual maps of i, p (and we replace z, x in (4.2.16) of [Eme06a] by t, n in line with notions in Lem 3.16) in our setting. We list the properties for Emerton's arguments to work: the dual of Hecke operator U t of Lem 3.16 on U n is continuous and i * , p * are G 0 -equivariant. The same arguments in [Eme06a, Lem 4.2.11, Lem 4.2.19] define the map (4.2.9) so that the diagram (4.2.10) of [Eme06a] is commutative. For the commutativity of (3), we decompose the diagram by the following one (diagram (4.2.29) of [Eme06a]): D(T • n , T 0 ) ⊗ D(T • n+1 ,T0) U (t) n,n+1 / / t U (t) n t w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ D(T • n , T 0 ) ⊗ D(T • n+1 ,T0) U n+1 / / U n . Commutativity of the upper left triangle of (3) follows from directly verifying the formula (4). We finally can apply [Eme06a, Prop 3.2.24] to conclude coadmissibility of (V N0 fs ) ′ b over C an ( Y , K), or equivalently, V N0 fs is an essentially admissible Y -representation. Exactness of finite slope part and applications on Jacquet functors In this section we establish an exactness result of taking finite slope part on locally convex vector spaces. We apply this result to define derived Jacquet functors and obtain further results on functoriality in §7. It is worth noting that Hao Lee defines derived Jacquet functors using a different approach. We use notation in §3. Proof. This is an abstraction of proof of [ST03, Thm 4.11]. Let J ⊆ R be a ideal. As R is Noetherian, A ⊗ R J is finitely presented and therefore coadmissible by [ST03,Cor 3.4]. By the definition of coadmissibility and [ST03, Cor 3.1], V /{0} V ։ W/{0} W . Proof. It is direct to verify 0 → U/{0} U → V /{0} V is injective, V /{0} V → W/{0} W → 0A ⊗ R J ≃ lim ← − n A n ⊗ A (A ⊗ R J) ≃ lim ← − n A n ⊗ R J. As R → A n is flat, A n ⊗ R J → A n is injective, we conclude that A ⊗ R J → A is injective since the projective limit is left exact. (1) holds by left exactness of finite slope part. For (2), d n factors through im(d n ), it suffices to prove (V n ) fs → im(d n ) fs is an epimorphism in Rep la K (Y ). It suffices to consider the dual side by [ST02b, Prop 1.2], namely we want to prove that if ı : U → V is a strict map of spaces of compact type with closed image W , and if ı ′ : 0 → (W fs ) ′ b → (V fs ) ′ b → (U fs ) ′ b →0 → K{{t, t −1 }}⊗ K[t] W ′ b → K{{t, t −1 }}⊗ K[t] V ′ b → K{{t, t −1 }}⊗ K[t] U ′ b → 0, which is the Hausdorff completion of 0 → K{{t, t −1 }} ⊗ K[t] W ′ b → K{{t, t −1 }} ⊗ K[t] V ′ b → K{{t, t −1 }} ⊗ K[t] U ′ b → 0V ′ b → U ′ b is injective, then ı is surjective. By [ST02b, Prop 1.2], V ′ b → W ′ b is a topological quotient with the topological embedding W ′ b → U ′ b . By definition, V ′ b → U ′ b is strict. Consider the strict quotient V → V /ı(U ) → 0, (V /ı(U )) ′ b is in the kernel of V ′ b → U ′ b . If V ′ b → U ′ b is injective, then (V /ı(U )) ′ b = 0. V /ı(U ) =K{{t, t −1 }}⊗ K[t] im(d n ) ′ b → K{{t, t −1 }}⊗ K[t] (V n ) ′ b is injective. As im(d n ) ′ b → (V n ) ′ b is injective, K{{t, t −1 }} is flat over K[t],(T + ) in §3), H * h (V • ) fs are locally analytic T representations since T is generated by T + and t −1 . We win by applying Lem 3.19. Corollary 4.10. If f : C • → D • ∈ K b (Ban ad G0 (K)) with a G-extension f ′ : C ′ • → D ′ • ∈ K b (Ban G (K) ) in the sense of Def 3.11, with a diagram (abbreviation of all four diagrams in Def 3.11) C • p / / f C ′ • f ′ i o o D • q / / D ′ • , j o o where C • p ⇆ i C ′ • and D • q ⇆ j D ′ • are G--equivariant morphism H * h (C la • ) N0 → H * h (D la • ) N0 in Rep top.c K (T + ). We therefore have H * (C la • ) N0 fs ≃ H * h (C la • ) N0 fs as essentially admissible locally analytic T -representations by Prop 4.9. The Hausdorff cohomology H * h (C la • ) N0 does not depend on choices of p and i. Comparisons of completed cohomology and overconvergent cohomology In this section we relate overconvergent cohomology considered by [AS08], [Han17] and the completed cohomology. Let F be a number field. Let G be a connected linear algebraic group over F . Let S p denote the set of places of F dividing p, and let G p be v∈Sp G(F v ). Let E be a big enough extension of Q p which contains the images of all embeddings F ֒→ Q p . For any good subgroup of G(A ∞ F ) in the sense of [ACC+18, §2.1], we write X K := X G K := G(F )\X G × G(A ∞ F )/K. (5) Fix a tame level K p = v / ∈Sp∪∞ K v (K p is a compact open subgroup of G(A ∞,p F ), K v is a compact open subgroup of G(F v )), we use K p K p to denote the good subgroup with tame level K p and level K p at p. For the rest of the section, all the level subgroups considered are assumed to be good. Let C la (K p , E) be the space of locally analytic E-coefficient functions on K p , hence a (left) admissible locally analytic representation of K p . By weak approximation, G(F )\G(A ∞ F )/K is finite, X K = [x]∈G(F )\G(A ∞ F )/K Γ G x,K \X G , X K = [x]∈G(F )\G(A ∞ F )/K Γ G x,K \X G , where Γ G x,K := G(F ) ∩ xKx −1 . We fix a countable basis of open normal subgroups of K p K p = K 0 p ⊃ K 1 p ⊃ . . . ⊃ K r p ⊃ . . . , corresponding to a sequence of continuous covering maps of topological spaces · · · → X r → · · · → X 1 → X 0 , with K p -equivariant actions on the right and their Borel-Serre compactifications · · · → X r → · · · → X 1 → X 0 , where X r := X K r p K p and each X r ֒→ X r is a homotopy equivalence by [BS73, §11] for a good subgroup K 0 p K p . For K = K p K p , let T • (X 0 ) be the set of singular simplices of Borel-Serre compactification X 0 of X 0 := X K . Fix a choice of finite triangulation of X 0 , we let T • n (X 0 ) be the collection of n-dimensional singular simplices occuring in the finite triangulation, which induces a triangulation of the boundary ∂X 0 . Let T • (X r ) (resp. T • • (X r )) be the set of simplices of X r (resp. the pullback of T • • (X 0 ) for the corresponding covering X r → X 0 ). If ∆ ∈ T • n (X 0 ) for some n then we let T • n (X r ) /∆ denote the set of simplices in T • n (X r ) lying over ∆; the set T • n (X r ) /∆ is then a principal homogeneous K p /K r p -set acting on the right. We letT n/∆ denote the projective limit lim ← − r T • n (X r ) /∆ ; this is a profinite set which is principal homogeneous with respect to its natural K p -action. Let T Int • (X r ) be the subset of T • • (X r ) whose interiors do not intersect with ∂X 0 , and T ∂ • (X r ) be the complement of T Int • (X r ) in T • • (X r ), i.e., T • • (X r ) = T Int • (X r ) ⊔ T ∂ • (X r ). For each ∆ ∈ T ∂ • (X r ), ∆ has image in ∂X 0 . On each finite level, the two complexes computing O E /p s O E -coefficient cohomology via T • (X r ) and T • • (X r ) are denoted as A • r,s and S • r,s whose n-th terms are Hom Z Z∆ ′ , O E /p s O E ∼ −→ ∆∈Tn(X0) ∆ ′ ∈Tn(Xr ) /∆ Hom Z Z∆ ′ , O E /p s O E . S n r,s := ∆ ′ ∈T • n (Xr ) Hom Z Z∆ ′ , O E /p s O E ∼ −→ ∆∈T • n (X0) ∆ ′ ∈T • n (Xr) /∆ Hom Z Z∆ ′ , O E /p s O E . The natural map A n r,s → S n r,s is a quasi-isomorphism and admits a homotopy inverse. We set S • s := lim − → r S • r,s , S • := lim ← − s S • s , A • s := lim − → r A • r,s , A • := lim ← − s A • s . We endow discrete topology for A n r,s , A n s , S n r,s , S n s and projective limit topology for A n , S n . By [Eme06b, Prop 1.2.12], A • computes the completed cohomology. A choice of compatible homotopy equivalences ∆ ′ ∈T • n (Xr ) Z∆ ′ ⇆ ∆ ′ ∈Tn(Xr ) Z∆ ′(6) induces K p -equivariant homotopy equivalence S • ⇆ A • . We also have the following complex computing compactly supported completed cohomology S n c,r,s := ∆ ′ ∈T Int n (Xr ) Hom Z Z∆ ′ , O E /p s O E ∼ −→ ∆∈T Int n (X0) ∆ ′ ∈T Int n (Xr ) /∆ Hom Z Z∆ ′ , O E /p s O E , with a short exact sequence of chain complexes 0 → S • c,r,s → S • r,s → S • ∂,r,s → 0,(7) where S n ∂,r,s := ∆ ′ ∈T ∂ n (Xr) Hom Z Z∆ ′ , O E /p s O E ∼ −→ ∆∈T ∂ n (X0) ∆ ′ ∈T ∂ n (Xr ) /∆ Hom Z Z∆ ′ , O E /p s O E . We set S • c,s := lim − → r S • c,r,s , S • c := lim ← − s S • c,s , S • ∂,s := lim − → r S • ∂,r,s , S • ∂ := lim ← − s S • ∂,C(T n/∆ , O E ) S n c ∼ −→ ∆∈T Int n (X0) C(T n/∆ , O E ), where C(T n/∆ , E) is the space of continuous functions on C(T n/∆ , E) with the right regular K p -actions: K p acts onT n/∆ on the right, inducing right actions on C(T n/∆ , E). Taking limits of the short exact sequences (7), we get a short exact sequence of chain complexes of admissible continuous E-Banach representations of K p : 0 → S • c → S • → S • ∂ → 0.(8) Similarly we describe natural right G p -actions on A • s and A • induced by the G pactions on lim ← − r T n (X r ). Lemma 5.1. For each n ≥ 0, A n is a p-adic separated, complete and torsion free O E -module. Therefore A • ⊗ OE E is a complex of E-Banach space representation of G p . Proof. It suffices to prove A n /p s A n ≃ A n s for any s ≥ 1. Consider the natural surjection A n /p s A n ։ A n s , we want to prove it is injective. For f ∈ A n in the kernel, the image is the zero function on the inverse system lim ← −r T n (X r ) as it induces injective transition maps between A n r,s . And for any s ′ ≥ s, the value of f at ∆ ′ ∈ T n (X r ) divided by p s is well defined in O E /p s ′ O E , giving rise to f ′ ∈ A n such that f = p s f ′ . It is also not hard to check A n is torsion free. If G is a quasi-split connected reductive group over F equipped with a Borel pair (B, T), and K p = I p is the Iwahori level with respect to the Borel pair at p-adic places, the group I p admits an Iwahori decomposition I p ≃ N p × T p × N p (with respect to the Borel pair). C la (I p , E) has a closed subrepresentation Ind la Ip := {f ∈ C la (I p , E)|f (gn) = f (g), for ∀ n ∈ N p }. When K p = I p , we use S • N to denote (S • ⊗ OE E) la Np , the N p invariant locally analytic vectors of S • ⊗ OE E, and we similarly define S • c,N . On the other hand, we can form the complex C • and C • c for computing the overconvergent cohomology and compactly supported overconvergent cohomology with coefficient Ind la Ip (the term overconvergent cohomology was given in [Han17,§3], but we will not use the exactly same definition here in this article). For [x] ∈ G(F )\G(A ∞ F )/I p K p , we define the arithmetic group Γ G x,IpK p := G(F )∩ xI p K p x −1 . Each ∆ ∈ T • n (X 0 ) is contained in a component of X 0 , indexed by some [x] ∈ G(F )\G(A ∞ F )/I p K p , and ∆ determines a rank one free Z[Γ G x,IpK p ]-module F ∆ formed by summing (taking Z-linear combination) over all lifts of ∆ to the universal cover of the component. And we define C • := ∆∈T • n (X0) Hom Z[Γ G x,IpK p ] (F ∆ , Ind la Ip ) C • c := ∆∈T Int n (X0) Hom Z[Γ G x,Ip K p ] (F ∆ , Ind la Ip ), where the action of Γ G x,IpK p on Ind la Ip comes from the (left) action of I p . We remark that the cohomology of C • may be considered as a variation of overconvergent cohomology defined in [Han17,§3]. Theorem 5.2. Using notation as above, we suppose K p = I p as Iwahori level at p-adic places. There is a canonical isomorphism of the two complexes C • ∼ −→ S • N , C • c ∼ −→ S • c,N . Proof. To construct such a map is the same to assign f ∆ ∈ Hom Z[Γ G x,IpK p ] (F ∆ , Ind la Ip ) an image in Ind la Ip for any ∆ ∈ T n (X 0 ). To do so, we choose a specific lift∆ to X G for each ∆. Consider the diagram (for any subgroup K p ⊂ I p ), G(F )\G(F )xK p K p × X G /K p K p X G 4 4 4 4 ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ * * * * ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ G(F )\G(F )xI p K p × X G /I p K p ∆ determines a compatible sequence ∆ r ∈ T n (X r ) lifting ∆, hence an element ∆ ∈T n/∆ , f ∆ (∆) ∈ Ind la Ip . For any ∆ g ∈T n/∆ , there exists g ∈ I p such that ∆ g =∆ · g. Let f ∆ map to a function h ∆ h ∆ :T n/∆ → E h ∆ (∆ g ) := f ∆ (∆)(g), hence the image is in (S • ⊗ OE E) la Np . If∆ ′ is another choice of lift of ∆, there exists γ ∈ Γ G x,IpK p such that∆ ′ = γ ·∆. By unraveling definition,∆ ′ =∆·x −1 γ −1 x, h ∆ (∆) = f ∆ (∆ ′ )(x −1 γx) = x −1 γxf ∆ (∆)(x −1 γx) = f ∆ (∆)(1). Therefore our map is well defined and clearly an isomorphism with respect to a given degree. For a n-dimensional simplice ∆, we use ∂ i ∆ (0 ≤ i ≤ n) to denote its ordered (n− 1)-dimensional faces with i-vertex deleted. For any f ∈ ∆∈Tn−1(X0) Hom Z[Γ G x,IpK p ] (F ∆ , Ind la Ip ), its image via the differential map is h = (h ∆ ) ∆∈Tn(X0) ∈ S n N such that h ∆ (∆) = n i=0 (−1) i f ∂i∆ (∂ i∆ )(1). The identification commutes with differentials since if∆ = (∆ r ) ∈T n/∆ , then ∂ i∆ = (∂ i ∆ r ) ∈T n−1/∆ for any 0 ≤ i ≤ n. Remark 5.3. Using finiteness of direct sum, C • c computes compactly supported cohomology H * (X G,K , Ind la Ip ) of X G,K for K = I p K p with coefficient local system arising from the representation Ind la Ip . We now illustrate how to relate the overconvergent cohomology in our sense to the usual cohomology with finite torsion coefficients. We fix a countable basis of open normal subgroups of I p I p = I 0 p ⊃ I 1 p ⊃ . . . ⊃ I r p ⊃ . . . . And let Ind s Ip := {f ∈ C s,an (I p , E)|f (gn) = f (g), for ∀ n ∈ N p }, where C s,an (I p , E) consists of locally analytic functions on I p which restrict to analytic functions on I s p -cosets. We have Ind la Ip ≃ lim − →s Ind s Ip , identifying Ind s Ip with I s p -analytic vectors of the locally analytic representation Ind la Ip . If we want to prove vanishing of cohomology with coefficient Ind la Ip : we may reduce to prove vanishing of cohomology with coefficient Ind s Ip , which has an integral structure as a representation of I p . Lemma 5.4. Let G be a group, V be a Q p -linear unitary Banach representation of G. Let V • be the unit ball in V . An algebra T acts Q p -linearly on RΓ G (V ) in the derived category of Q p vector spaces. Let m be a prime ideal of T . Assume • V • /p · V • ≃ Z F p (Z copies of F p as a direct sum) as F p vector space. • There exists a normal subgroup N G, such that T acts compatibly on RΓ G (F p ) m Res − − → RΓ N (F p ) m G/N (T acts on RΓ N (F p ) m , T action com- mutes with the G/N action on RΓ N (F p ) m , Res is T -equivariant) and R i Γ N (F p ) m = 0 for ∀i < d. The N action on V • /p · V • is trivial. Then we have R i Γ G (V ) m = 0 for ∀i < d. Furthermore, we assume additionally that there exists a connected manifold M of dimension N with homotopy type K(G, 1). We use V to denote the local system associated to V on M . If i > N − d, we have R i Γ c (M, V ) m = 0. Proof. For ∀i < d, R i Γ N (V • /p·V • ) m = 0 since N acts trivially on V • /p·V • ≃ Z F p and direct sum commutes with cohomology and localization. By our assumptions and the Lyndon-Hochschild-Serre spectral sequence, R i Γ G (F p ) m = 0 hence R i Γ G (V • /p n ·V • ) m = 0 for any n ∈ Z + and ∀i < d. The inverse system {V • /p n ·V • } satisfies the Mittag-Leffler condition, therefore by [Sta, Tag 02N1], R i Γ G (V • ) m = 0, R i Γ G (V ) m = 0 for ∀i < d. By Poincaré duality, R i Γ c (M, F p ) m = 0 for ∀i > N − d hence R i Γ c (M, Z F p ) m = 0, R i Γ c (M, V • /p n · V • ) m = 0 for ∀i > N − d. Applying the same Mittag-Leffler condition for the left exact functor Γ c yields the Proof. The congruence subgroup Γ G x,I n p K p < G acts trivially on the reduction of V • if n ≫ s. result R i Γ c (M, V ) m = 0 for ∀i > N − d. Construction of eigenvarieties Definition 6.1. An eigenvariety data is a tuple D = (T, M, A, ψ), where T is a topologically finitely generated abelian locally Q p -analytic group ([Eme17, Prop 6.4.1]), M is an essentially admissible representation of T , A is a commutative Q p -algebra, and ψ is a Q p -algebra homomorphism ψ : A → End T (M ). In practice T will be a maximal p-adic torus of our interested reductive group, M will be a Jacquet module introduced in §3, A will be a Hecke algebra. For the strong dual M ′ b of M , we define the T -action t · λ(m) := λ(t · m) for ∀λ ∈ M ′ b , ∀t ∈ T on M ′ b (it is direct to check that t 1 t 2 · λ = t 1 (t 2 · λ) as T is abelian). is surjective. Taking Sp of the morphism, we construct a natural map from E (D ′ ) to E (D). In the rest two sections, we adapt all notation and terminologies in §2 of [ACC+18] and the previous section. We define our derived eigenvarieties using results of previous sections as follows. Let S be a finite set of "bad" finite places containing S p such that K v is hyperspecial for v / ∈ S. Let A Kp be the abelian category of admissible O E [K p ]-modules introduced in [Eme06b, §1.2] for the compact locally Q p -analytic group K p . As in §5, we define complexes π(G, K p , r, m), π(G, K p , m),π(G, K p , m) whose n-terms are (these are denoted as S • m and A • m in §5) π n (G, K p , r, m) := ∆ ′ ∈T • n (Xr ) Hom Z Z∆ ′ , O E /p m O E , π n (G, K p , m) := lim − → r π n (G, K p , r, m), π n (G, K p , m) := lim − → r ∆ ′ ∈Tn(Xr ) Hom Z Z∆ ′ , O E /p m O E . π(G, K p , m) is a well defined object in K b (O E /p m [G p ]) and π • (G, K p ) := lim ← − m π(G, K p , m) ∈ K b (A Kp ), π(G, K p ) := π • (G, K p ) ⊗ OE E ∈ K b (Ban ad Kp (E)), π • (G, K p ) := lim ← − mπ (G, K p , m) ∈ K b (O E [G p ]), π(G, K p ) :=π • (G, K p ) ⊗ OE E ∈ K b (Ban Gp (E)). By [Eme17, Prop. 6.2.4], the locally Q p -analytic vectors π(G, K p ) la is a complex of admissible locally Q p -analytic representations of K p and their cohomology groups H * (K p ) la are admissible locally Q p -analytic representations of G p . We have an K p -equivariant homotopy equivalence π(G, K p , m) p ⇆ iπ (G, K p , m), compatible with m, as discussed in section §5 (essentially coming from the homotopy equivalence of Z-complexes (6)). We can take a projective limit (with respect to m) of such K p -equivariant homotopy equivalences to get K p -equivariant continuous mapping i, p π • (G, K p ) p ⇆ iπ • (G, K p ), π(G, K p ) p ⇆ iπ (G, K p ). In other words, i, p are morphisms in Ch b (Ban Kp (E)). Lemma 6.4. The K p -equivariant continous homotopy equivalence π(G, K p ) p ⇆ ĩ π(G, K p ) defines a G p -extension (π(G, K p ), i, p) of π(G, K p ) in the sense of Def 3.10. Proof. We have thatπ(G, K p ) is a Banach representation of G p by Lem 5.1. In the following we illustrate how to decompose π • (G, K p ) and π(G, K p ) in terms of maximal ideals of derived Hecke algebra considered in [GN16]. For a compact open subgroup K r p ⊂ K p , we form the following complexes with natural transitions in r and m C(G, K p , r, m) := ∆∈T • • (X0) ∆ ′ ∈T • • (Xr ) /∆ (O E /p m O E ) · ∆ ′ ∈ K b (O E [[K p ]]), C(G, K p , r, m) := ∆ ′ ∈T•(Xr) (O E /p m O E ) · ∆ ′ ∈ K b (O E [[K p ]]). The Iwasawa algebra O E [[K p ]]-action on these complexes factors through the O E /p m [K p /K r p ]action. As discussed in §5, there is a choice of compatible homotopy equivalences in r and m C(G, K p , r, m) ⇆ C(G, K p , r, m). We will sometimes regard C(G, K p , r, m) and C(G, K p , r, m) as complexes or as the same object in For each pair of r and m, we have a decomposition (we identify C(G, K p , r, m) and C(G, K p , r, m) in the derived category, and C(G, K p , r, m) m is defined to be an object in the derived category) D(O E [[K p ]]) or D(O E /p m [K p /K r p ]).C(G, K p , r, m) = m∈MaxSpec(T S,• G ) C(G, K p , r, m) m ∈ D(O E [[K p ]]), compatible in terms of r and m. By [Sta, Lem 091D], we have C(G, K p ) ≃ R lim ← − m lim ← − r C(G, K p , r, m) ∈ D(O E [[K p ]]). By the compatibility of C(G, K p , r, m), we will exhibit a "uniform finiteness" for defining a complex representing C(G, K p , r, m) m (which does not depend on r, m) as follows. Lemma 6.6. We have C(G, K p , r, m) m ≃ C(G, K p ) m ⊗ L OE[[Kp]] (O E /p m O E [K p /K r p ]) ∈ D(O E [[K p ]]) . Proof. As complexes we have C(G, K p , r, m) ≃ C(G, K p ) ⊗ OE[[Kp]] (O E /p m O E [K p /K r p ]) . We get the desired isomorphism by passing to the derived category and applying the idempotent decomposition (9). By Prop 6.5, π(G, K p ) m is a complex of injective objects for each m. The complex π(G, K p ) la Np is homotopic equivalent to m π(G, K p ) la m Np (as N p in §5). The cohomology H * (π(G, K p ) la m ) Np decomposes as H * (π(G, K p ) la ) Np ≃ m H * (π(G, K p ) la m ) Np . Hecke operators at p commute with the spherical Hecke action, therefore preserving each H * h (π(G, K p ) la m ) Np -component. According to Prop 4.9, we may define H * J Bp (π(G, K p )) m to be the finite slope part of cohomology H * h (π(G, K p ) la m ) Np of the complex π(G, K p ) la m Np . As direct summands, H * J Bp (π(G, K p )) m are essentially admissible as well. Similarly, we have ψ m : T S G → End O( Tp) (H * J Bp (π(G, K p )) m ) ′ b , ψ n,m : T S G → End O( Tp) (H n J Bp (π(G, K p )) m ) ′ b . Let D = (T p , H * J Bp (π(G, K p )) , T S G , ψ), D n = (T p , H n J Bp (π(G, K p )) , T S G , ψ n ), D m = (T p , H * J Bp (π(G, K p )) m , T S G , ψ m ), D n,m = (T p , H n J Bp (π(G, K p )) m , T S G , ψ n,m ) be the eigenvariety data defined by Def 6.1. Definition 6.8. We define the eigenvarieties E G (K p ), E n G (K p ), E G (K p , m), E n G (K p , m) to be the eigenvarieties associated to D, D n , D m , D n,m by Prop 6.2. 7. An application to eigenvariety of quasi-split unitary group Let F be an (imaginary) CM number field with maximal totally real subfield F + . Let E be a big enough extension of Q p which contains the images of all embeddings F ֒→ Q p . Let G /O F + be the quasi-split unitary group with respect to the extension F/F + and the Hermitian form J n = 0 Ψ n −Ψ n 0 , where Ψ n is the matrix with 1's on the anti-diagonal and 0's elsewhere. We choose a standard Siegel parabolic subgroup P containing upper triangular Borel pair (B, T ) and maximal F + -split torus S ⊆ T of G. Similarly let T ⊂ G ⊂ P ⊂ G be the standard Levi subgroup of P . We write P = U ⋊G for the unipotent radical U of P , B = N ⋊ T for the unipotent radical N of B. Here we have G ≃ Res OF /O F + GL n . We write X = X G and X = X G . Similarly, we will use the symbols K and K to denote good subgroups of G(A ∞ F + ) and G(A ∞ F + ), respectively. Let I p and I p be the Iwahori subgroups of G p and G p , compatible with respect to the Borel pair. Let P p : = P (F + p ). Let G 0 = G(O F + p ) (resp. G 0 = G(O F + p ) ) be a hyperspecial maximal compact subgroup of G p (resp. G p ). Let P 0 := P (O F + p ), N 0 := G 0 ∩ N (F + p ). Let K ⊂ G(A ∞ F + ) be a good subgroup which is decomposed with respect to P ([ACC+18, §2.1.2]), and K P := K ∩ P (A ∞ F + ) = K U ⋊ K. For any finite place v of F + , we use K v (resp. K v ) to denote the level K at v (level K at v). Let S be a finite set of bad finite places containing S p such that K v (resp. K v ) is hyperspecial for v ∈ S. We let T S := T S G and T S := T S G . There is an unramified Satake transform S : T S → T S defined in [NT15, §2.2.6] (see [NT15,§5.1] for more explicit descriptions). With notation in §6, we define π(K p ) := π(G, K p ) ∈ D b (Ban ad G0 (E)), Π( K p ) := π( G, K p ) ∈ D b (Ban ad G0 (E)) equipped with T S → End D b (Ban ad G 0 (E)) (π(K p )), T S → End D b (Ban ad G 0 (E)) (Π( K p )).(17) Recall ∂X G as in §2. We define Π ∂ ( K p ) ∈ K b (Ban ad G0 (E)) to be a chosen complex of admissible continuous Banach representations of G 0 arising from a choice of finite triangulation of Borel-Serre boundary of locally symmetric space of G with level G 0 · K p (similar to S • as in §5). We define π ∂ ( K p ) ∈ K b (Ban Gp (E)) to be the complex of continuous Banach representations of P p arising from all possible singular simplices (similar to A • as in §5). They are equipped with Hecke actions: T S → End D b (Ban ad G 0 (E)) (Π ∂ ( K p )), T S → End K b (Ban Gp (E)) ( π ∂ ( K p )). If m ⊂ T S is a maximal ideal, let m = S * (m) ⊂ T S be the pullback of m in T S . As T S / m ֒→ T S /m as a subring of a finite field (therefore a subfield), m is a maximal ideal of T S . By the same reason before Prop 6.5 in §6, we can decompose π(K p ), Π( K p ), Π ∂ ( K p ) and π(K p ) m ∈ D b (Ban ad G0 (E)), Π( K p ) m , Π ∂ ( K p ) m ∈ D b (Ban ad G0 (E)) are all well defined for each relevant maximal ideal of the corresponding derived Hecke algebras. Proposition 7.1. Suppose m is a non-Eisenstein ideal ([ACC+18, Def 2.3.6]). Let K ⊂ G(A ∞ F + ) be a good subgroup which is decomposed with respect to P . We have Ind G0 P0 π(K p ) m is a T S -equivariant direct summand of Π ∂ ( K p ) m in D b (Ban ad G0 (E)). Here and below we will write Inf(C • ) when viewing a complex C • of representations of G 0 as a complex of representations of P 0 via P 0 → G 0 . We also write π(K p ) m as the inflation complex Inf(π(K p ) m ) of representations of P 0 whose underlying complex is π(K p ) m in order to lighten the notation. Proof. The claim follows from a combination of proof of [ACC+18, Thm 5.4.1], uniform finiteness of Lem 6.6, Lem 6.7, together with Prop 3.4. Let π P ( K p ) := π(P, K p ∩ P (A ∞,p F + )) ∈ K b (Ban ad P0 (E)), and π P ( K p ) := π(P, K p ∩ P (A ∞,p F + )) ∈ K b (Ban Pp (E)) as in §6. There is a P p -extension π P ( K p ) p ⇆ i π P ( K p )(18) constructed in §5 by Lem 6.4. There are corresponding Hecke actions ψ ∂ : T S → End D b (Ban ad P 0 (E)) (π P ( K p )), ψ ∂ : T S → End K b (BanP p (E)) ( π P ( K p )) and decomposition of π P ( K p ) into components indexed by maximal ideals similarly to (17). For every finite level K P = K U ⋊ K, we have a torus bundle X P KP ։ X G K . And taking levels at p as open compact subgroups of G 0 , we get a T S -equivariant natural map (similar to [ACC+18,(5.4.2)]) Inf(π(K p )) → π P ( K p ) ∈ D b (Ban ad P0 (E)), where the Hecke action on the left hand side factors through the Satake morphism S. By decomposing with respect to m, we have Inf(π(K p ) m ) → π P ( K p ) m . They are quasi-isomorphisms since completed cohomology for U (whose locally symmetric space is torus) concentrates in degree 0. We may represent Inf(π(K p ) m ) or π P ( K p ) m by an injective resolution of objects in Ban ad P0 (E), so does Ind G0 P0 π(K p ) m as a complex of injective objects in Ban ad G0 (E) by Lem 3.14. There is a G p -extension Ind G0 P0 π P ( K p )p ⇆ i Ind Gp Pp π P ( K p )(19) by Prop 3.13 induced from (18). Taking locally analytic vectors, (19) induces a G 0 -equivariant homotopy equivalence la-Ind G0 P0 (π P ( K p ) la )p ⇆ i la-Ind G P ( π P ( K p ) la ) by Thm 3.6. We can form essentially admissible representations H * J Bp (Ind G0 P0 π P ( K p )) m similar to the discussion before Def 6.8. Definition 7.2. We call the induction eigenvariety E d (Ind, G, K p , m) for G ≃ Res OF /O F + GL n of degree n, type m for the eigenvariety associated to the eigenvariety data (recall m is the pullback of m) D Ind G,d := (T p , H d J Bp (Ind G0 P0 π P ( K p )) m , T S , ψ ∂ ). This induction eigenvariety E d (Ind, G, K p , m) should be closely related to eigenvariety for G. But we do not know the exact relationship. Let E d G ( K p , m) be the eigenvariety for G and m constructed in §6. Consider the short exact sequence (8) of admissible continuous E-Banach representations of I p for G with tame level K p of §5: 0 → S • c ( G, K p ) → S • ( G, K p ) → S • ∂ ( G, K p ) → 0, where S • ( G, K p ) is homotopy equivalent to Π( K p ) and S • ∂ ( G, K p ) is homotopy equivalent to Π ∂ ( K p ). By the long exact sequence of N 0 cohomology associated to this short exact sequence and Thm 5.2, it suffices to prove vanishing of H d+1 c (X G, Ip K p , Ind la Ip ) m . This follows from the main result of [CS19, Thm 1.1] and corollary 5.5 in §5. We win by applying Prop 6.3 to the quotient essentially admissible representation H d J Bp (Ind G0 P0 π P ( K p )) m of H d J Bp (Π( K p )) m . for any H-analytic vector f ∈ I H−an : G → V , as a map it is H-analytic by evaluating at different points. Therefore the map H × (P ∩ H) V is H × H P -analytic. Such a map is represented as an element in C an (H × H P , V ) ≃ C an (H × H P , K)⊗V ≃ C an (H, K)⊗C an (H P , K)⊗V ≃ C an (H, C an (H P , V )), As it is remarked in the (end of) proof of [Eme07, Prop. 2.1.2], continuous parabolic induction preserves admissibility for p-adic reductive groups. Now fix a connected quasi-split reductive linear algebraic group G over a padic field L, as well as a Borel pair (B, N) with Levi decomposition B = TN. Let N be the opposite unipotent of N. Write G = G(L), B = B(L), T = T(L), and N = N(L), N = N(L). Let G 0 be a compact open subgroup of G and N 0 := G 0 ∩N , N 0 := G 0 ∩ N , T 0 := G 0 ∩ T . Let T + be the submonoid of T which stabilizes the closed subgroup N 0 ⊂ G. be a decreasing cofinal sequence of good analytic open subgroups of G 0 satisfying conditions in [Eme06a, Prop 4.1.6], with rigid analytic groups H n underlying H n . Let T n := H n ∩ T , and T n be the rigid analytic group underlying T n . Let H • n , T • n be the strictly σ-affinoid analytic open subgroup of H n , T n ([Eme06a, §4.1]). Let D(H • n , H 0 ) be the strong dual to the nuclear Fréchet space C la (H 0 , K) H • n −an . For each n ≥ 0, we define U n := D(H • n , H 0 )⊗ D la (H0,K) V ′ b N0 The proof of[Eme06a, Cor 4.2.26] (replace C an (Ẑ G,n , K) †⊗ K D(M • n , M 0 ) by D(T • n , T 0 ) and C an (Ẑ G,n , K) †⊗ K D(H • n , H 0 ) by D(H • n , H 0 )) shows the map D(T • n , T 0 ) ⊗ D(T • n+1 ,T0) U n+1 → U n is a D(T • n , T 0 )-compact map in the sense of [Eme06a, Def 2.3.3]. The proof of [Eme06a, Prop 4.2.28] shows that for t (suitably chosen in the sense of [Eme06a, For a locally convex vector space V over a p-adic local field K, let {0} V be the closure of 0 in V , let V /{0} V be the Hausdorff quotient of V , and letV be the Hausdorff completion of V ([Sch01, §7]). Lemma 4. 1 . 1Let 0 → U → V → W → 0 be a short exact sequence of locally convex spaces such that U has the subspace topology and W has quotient topology with respect to V . Then 0 → U/{0} U → V /{0} V → W/{0} W → 0 is exact at the first and last term with subspace and quotient topology. For the middle term, the image of U/{0} U is dense in the kernel of is surjective with quotient topology, and the sequence 0 → {0} U → {0} V → {0} W is exact. Quotient maps of locally convex vector spaces are automatically open. If v ∈ V has image in {0} W , then for any open lattice Λ, U ∩ (v + Λ) = ∅, implying v ∈ U , which verifies our claim. Proposition 4. 2 . 2If 0 → U → V → W → 0 is a short exact sequence of locally convex spaces such that U has subspace topology and W has quotient topology, the sequence 0 →Û →V →Ŵ is exact with strict morphisms.Proof. By Lem 4.1, we may assume U, V, W are Hausdorff. The sequence 0 → U → V → W → 0 is exact at the first and last term with subspace and quotient topology, and the image of U is dense in the kernel Z of V ։ W . For any open lattice Λ ⊂ V , projection of Λ is open as quotient map of locally convex spaces is open, we claim that 0 → U/(Λ ∩ U ) → V /Λ → W/Λ → 0 is exact. For any v, whose residue class is in the kernel of V /Λ ։ W/Λ, v ∈ Z + Λ = U + Λ as U is dense in Z. The corresponding sequence of Hausdorff completion ([Sch01, by [Sta, Tag 02N1]. The kernel between Hausdorff spaces is closed, and the strictness of maps can be directly checked as the set of open lattice {Λ ⊂ V open} defines the topology of everything here. Let K{{z, z −1 }} be the ring of entire functions of G rig m . For a sequence of increasing radius {r n } greater than 1 and lim n r n = ∞, the affinoid algebras associated to closed rigid annuli A n := O({r −1n ≤ |z| ≤ r n }) define a Fréchet-Stein algebra structure of K{{z, z −1 }} = lim ← −n A n in the sense of[ST03]. Proposition 4. 3 . 3Let R be a Noetherian (left and right for the noncommutative case) ring, and A = lim ← −n A n be a Fréchet-Stein algebra, together with a compatible family R → A n of flat maps. Then R → A is flat. Definition 4. 6 . 6If C • is a chain complex of objects in Rep top.c K (Y + ) with differential maps d i : C i → C i+1 , then we define i-th Hausdorff cohomology H i h (C • ) to be H i h (C • ) := ker(d i )/im(d i−1 ). Remark 4.7. A morphism C • → D • induces a natural map H * h (C • ) → H * h (D • )on Hausdorff cohomology, invariant under homotopy equivalence. Lemma 4 . 8 .Λ 48Let 0 → U → V be an injective map of Hausdorff locally convex spaces such that the induced mapÛ →V on completions is strict, then U has subspace topology in V and 0 →Û →V is injective. ′ ⊂V open U/(Λ ′ ∩ U ). For any open latticeΛ in U , let π(Λ) ⊂ lim ← − Λ ′ ⊂V open U/(Λ ′ ∩U ) be the image ofΛ under π. SinceÛ →V is strict, π(Λ)is open with respect to the subspace (of lim ← −Λ ′ ⊂V open V /Λ ′ ) topology. We deduce that π(Λ) ∩ U =Λ is open in U with respect to the subspace topology of lim ← − Λ ′ ⊂V open V /Λ ′ and V . Therefore U itself has subspace topology of V .Suppose we have a G-extension (with fixed representatives) (C ′ • , i, p) of C • ∈ Ch b (Ban ad G0 (K)). As in §3, we have defined J B (C • ) out of these data, with the Hecke action of T + on H * (J B (C • )), similarly there is a natural Hecke action of T + on Hausdorff cohomology H * h (C la • ) N0 . We may define the finite slope part of H * h (C la • ) N0 fs using the Y + action via Def 3.17. By Prop 3.20, J B (C • ) is a chain complex of essentially admissible representations of Y . By[Eme17, Prop 6.4.11], the differentials d fs n : (V n ) fs → (V n+1 ) fs are necessarily strict with closed images, the topology of H * (J B (C • )) are therefore canonically defined by subspace topology and quotient topology from J B (C • ). The T + actions on cohomology of the complex J B (C • ) extends to essentially admissible T -representations by Lem 3.19. Proposition 4 . 9 . 49We have H * (J B (C • )) ≃ H * h (C la • ) N0fs as essentially admissible locally analytic T -representations, i.e., taking Hausdorff cohomology commutes with taking finite slope part.Proof. We use V • to denote (C la • ) N0 in the proof. The Hausdorff cohomology do not depend on choices of i and p, so we only have to establish the claimed isomorphism. To apply Thm 4.5 to prove H * (V • ) fs ≃ H * h (V • ) fs , we have to prove (1): ker(d n ) fs agrees with ker(d fs n ), (2): image of d fs n coincides with im(d n ) fs , both as objects of Rep la K (Y ). extensions such that the diagram is commutative up to homotopy. Then it induces a natural morphism on cohomology H * (J B (C • )) → H * (J B (D • )) as essentially admissible locally analytic T -representations. In particular, H * (J B (C • )) are independent of choices of p : C ′ • → C • with i fixed or choices of i : C • → C ′ • with p fixed of the extension and choices of Y + . Proof. The diagram induces a natural T + Corollary 5 . 5 . 55Suppose G = Γ G x,IpK p , M = Γ G x,IpK p \X G , V = Ind s Ip as a Banach representation of I p , T acts on H * c (M, V ) satisfying the conditions of Lem 5.4. The compactly supported cohomology H * c (M, V ) m of M localized at m vanish above the degree N − d. Proposition 6. 2 . 2Let T be the character variety of T parameterizing continuous characters of T . Given an eigenvariety data D = (T, M, A, ψ), there is a rigid analytic space E = E (D) together with a locally finite morphism E → T , an algebra homomorphism ψ E : A → O(E ). , For each r and m, we may consider the derived Hecke algebra T SG (K p , r, m) as the image of H(G(A ∞,S F ), K S ) ⊗ Z O E → End D(OE /p m OE [Kp/K r p ]) ( C(G, K p , r, m)) ([GN16, Def 2.1.11]). The (big) derived Hecke algebra Lem 2.1.14] (we note that although[GN16] is set up for projective general linear groups, both of the proof of [GN16, Lem 2.1.14] and [KT17, Lem 2.5] hold true in general).As complexes, let C(G, K p ) G, K p , r, m),with C(G, K p ) ⇆ C(G, K p ).It is key to notice that C(G,K p ) is a perfect object of D(O E [[K p ]]),each term of the perfect complex (representing) C(G, K p ) is a finite free O E [[K p ]]-module. We will sometimes regard C(G, K p ) as a perfect complex or as an object of D(O E [[K p ]]). Theorem 7.3. Suppose F + = Q and m satisfies the conditions of [CS19, Thm 1.1]. If d = [F + : Q]n 2 is the middle dimension of X G , there is a closed embbeding fromE d (Ind, G, K p , m) to the G-eigenvariety E d G ( K p , m).Proof. First, we exhibit H d J Bp (Ind G0 P0 π P ( K p )) m as a quotient ofH d J Bp (Π( K p )) m. By exactness of finite slope part of Thm 4.5 and Prop 7.1, it suffices to prove the surjectivity ofH d (Π( K p ) la ) N0 m → H d (Π ∂ ( K p ) la ) N0 m(and therefore the surjectivity of corresponding Hausdorff cohomology). Banach representation of H (with H-equivariant continuous maps as morphisms), and Ban ad H (K) to denote the abelian category of admissible continuous Banach H-representations ([ST02a, §2], [Eme17, §6.2, Cor. 6.2.16]).If G is a connected linear algebraic group over a number field F , we refer to X G as the symmetric space for Res F/Q G in the sense of[BS73][ §2] and[NT15][ §3.1].And let X G denote the partial Borel-Serre compactification of X G ([BS73][ §7.1]). M i ) → C(X, M ). Let p i : M → M i be the natural projection, and let U be an open in M i . One can check that it is continuous by checkingthe category of topological O-modules. Proof. By universal property, there is a canonical map C(X, M ) → lim ← − i C(X, M i ), which is injective. By [Sta, Lem. 5.30.11, Tag 0B1Y], M ≃ lim ← − i M i also holds as abstract O-modules. Given a sequence of compatible continuous functions (f i ) i , f i ∈ C(X, M i ), we hence obtain a function f : X → M by M ≃ lim ← − i M i . Since M has the limit topology, f ∈ C(X, M ) is continuous. This constructs the reverse arrow lim ← − i C(X, it on a topological basis V(K, p −1 i (U )) (space of continuous functions carrying K to p −1 i (U )) for any compact K ⊂ X, we're done. a finitely presented K[[H 0 ]]-module over the Iwasawa algebra K[[H 0 ]]. By [ST03, Thm 7.1] we can apply a topological version of Mittag-Leffler condition used by [ST03, §Thm(B)], which refers to [EGA61, Chapter III, §0.13.2] so that Corollary 4.4. The natural map K[z] → K{{z, z −1 }} is flat.Proof. We choose r n to be p n . Each map O K [z] → O K [p n z, p n z −1 ] p is a composition of localization and completion with respect to p. such that U has subspace topology and W has quotient topology.Then 0 → U fs → V fs → W fs → 0 is exact in Rep la K (Y ),and U fs is a closed subspace and W fs has quotient topology. Proof. We pick t ∈ T satisfying [Eme06a, Prop 4.1.6 (vi)] as before. The weight space associated to Z · t in [Eme17, §6.4] is G rig m . By [Eme06a, Prop 3.2.6], U fs ֒→ V fs is a closed embedding. Locally convex topological K-vector spaces of compact type are reflexive. By Hahn-Banach ([Sch01, Cor 9.4]), it suffices to show thatBase change to K, we have K[z] → O({r −1 n ≤ |z| ≤ r n }) is flat. Now we can apply Prop 4.3. Theorem 4.5. Let 0 → U → V → W → 0 be a short exact sequence in Rep top.c K (Y + ) 0 is exact and has subspace and quotient topology. By [Eme06a, Prop 3.2.27] and [Eme06a, Lem 3.2.3], the sequence is the same as . It is exact by Cor 4.4, and equipped with subspace and quotient topology. We conclude by applying Prop 4.2. By [Eme06a, Lem 3.1.1], Rep top.c K (Y + ) is an additive category with images as closures of the usual images. and the above map is strict again by[Eme17, Prop 6.4.11], we can apply Lem 4.8 to deduce (2). Therefore we have proved the isomorphism H * (V • ) fs ≃ H *h (V • ) fs in Rep la K (Y ). As H * h (V • ) are T + continuous representations on spaces of compact type (objects of Rep top.c K s . The proof of [Eme06b, Thm 2.1.5] tells us that S • ⊗ OE E and S • c ⊗ OE E are complexes of admissible continuous E-Banach representations of K p with the rightaction and S n ∼ −→ ∆∈Tn(X0) Acknowledgments. I would like to thank my advisor Richard Taylor for suggesting me to think about functoriality of eigenvarieties and checking my arguments with numerous discussions. I would also like to thank Matthew Emerton and Hao Lee for helpful discussions, as well as the anonymous referee for detailed corrections and expository suggestions.A closed point x ∈ E (Q p ) gives rise to homomorphisms λ : A → Q p and δ : T → Q × p . And there is a natural bijection of sets (the natural map from left to right is constructed in the proof ) Let A be the sheaf of O( T )-algebra generated by ψ. We construct E to be the relative Sp of A, together with ψ E : A → O(E ). Local finiteness of E → T follows from the fact that the homomorphism group between two finite modules is still finite. Given a closed pointWe also illustrate how to construct morphisms out of "sub" or "quotient" eigenvariety data as follows.[Sta,Tag 0BK7], for each maximal ideal m, Choose a perfect complex C(G, K p ) m , we have the following definition of-modules with only finitely many nonzero terms. If we choose different perfect complex for C(G, K p ) m , the resulting π • (G, K p ) m would only differ by a homotopy equivalence by[Sta,Lem 064B]. We may pass π • (G, K p ) m to D(A Kp ).By dualizing the spherical Hecke action on C(G, K p , r, m), C(G, K p ) and C(G, K p ) m , we have the Hecke actionsin various bounded derived categories of abelian categories of representations of K p . In particular, π • (G, K p ) m ∈ D(A Kp ) is equipped with a Hecke action of T S,• G factoring through T S,• G,m such that the decomposition (13) is T S,• G -equivariant. The previous discussions give rise to the following proposition.is a complex of injective admissible Banach representation of K p . Passing to D b (Ban ad Kp (E)), it is equipped with the induced Hecke action of T S G := T S,• G ⊗ OE E. There is an isomorphism of objects in the derived category:for m running through maximal spectrum of T S,• G .Therefore if C(G, K p ) m is a perfect complex, we can use the complex, the rank of such a module does not depend on r and m.We define We may represent π(G, K p , r, m) m by HomThen as an object of Ch(A Kp ), π • (G, K p ) m can be chosen asregarded as chain complexes.Let B p := v∈Sp B(F v ), T p := v∈Sp T(F v ). Let T p denote the rigid analytic variety over Q p that parametrizes the locally Q p -analytic characters of T p . By Prop 4.9, H * (J Bp (π(G, K p ))) are essentially admissible representations of T p . Recall T S G = T S,• G ⊗ OE E in Prop 6.5, we have the Hecke actions ψ : T S G → End O( Tp) (H * (J Bp (π(G, K p ))) ′ b ), ψ n : T S G → End O( Tp) (H n (J Bp (π(G, K p ))) ′ b ). Patrick Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, Jack Thorne, Potential automorphy over CM fields. to appearPatrick Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne, Potential automorphy over CM fields, Ann. of Math., to appear. p-adic families of Siegel modular cuspforms. Fabrizio Andreatta, Adrian Iovita, Vincent Pilloni, Ann. of Math. 2Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni, p-adic families of Siegel mod- ular cuspforms, Ann. of Math. (2) 181 (2015), no. 2, 623-697. Avner Ash, Glenn Stevens, p-adic deformations of arithmetic cohomology. PreprintAvner Ash and Glenn Stevens, p-adic deformations of arithmetic cohomology, Preprint, 2008. Une interprétation modulaire de la variété trianguline. Christophe Breuil, Eugen Hellmann, Benjamin Schraen, Math. Ann. 367Christophe Breuil, Eugen Hellmann, and Benjamin Schraen, Une interprétation mod- ulaire de la variété trianguline, Math. Ann. 367 (2017), 1587-1645. On the density of modular points in universal deformation spaces. Gebhard Böckle, Amer. J. Math. 123Gebhard Böckle., On the density of modular points in universal deformation spaces, Amer. J. Math. 123 (2001), 985-1007. Corners and arithmetic groups. Armand Borel, Jean-Pierre Serre, Comment. Math. Helv. 48Avec un appendice: Arrondissement des variétés a coins, par A. Douady et L. HéraultArmand Borel and Jean-Pierre Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436-491, Avec un appendice: Arrondissement des variétés a coins, par A. Douady et L. Hérault. 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Hao Lee, Towards the derived Jacquet-Emerton module functor. Hao Lee, Towards the derived Jacquet-Emerton module functor, https://arxiv.org/abs/2205.13107. Symmetric power functoriality for holomorphic modular forms. James Newton, Jack A Thorne, Torsion Galois representations over CM fields and Hecke algebras in the derived category. 4Publ. Math. IHES.James Newton and Jack A. Thorne, Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum of Mathematics, Σ, vol. 4, 2016. [NT21a] , Symmetric power functoriality for holomorphic modular forms, Publ. Math. IHES. 134 (2021), 1-116. Symmetric power functoriality for holomorphic modular forms. Publ. Math. IHES. II, Symmetric power functoriality for holomorphic modular forms, II, Publ. Math. IHES. 134 (2021), 117-152. Adjoint selmer groups of automorphic Galois representations of unitary type. Journal of the EMS. to appear, Adjoint selmer groups of automorphic Galois representations of unitary type, Journal of the EMS. to appear. Nonarchimedean Functional Analysis. Peter Schneider, Springer Science & Business MediaPeter Schneider, Nonarchimedean Functional Analysis, Springer Science & Business Media, 2001. Banach space representations and Iwasawa theory. Peter Schneider, Jeremy Teitelbaum, Israel J. Math. 127Peter Schneider and Jeremy Teitelbaum, Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359-380. Locally analytic distributions and p-adic representation theory, with applications to GL 2. Journal of the AMS. 15, Locally analytic distributions and p-adic representation theory, with applica- tions to GL 2 , Journal of the AMS. 15 (2002), 443-468. Algebras of p-adic distributions and admissible representations. Invent. Math. 153, Algebras of p-adic distributions and admissible representations, Invent. Math. 153 (2003), 145-196. Duality for admissible locally analytic representations. Representation Theory. 9, Duality for admissible locally analytic representations, Representation Theory 9 (2005), 297-326. The Stacks Project Authors, Stacks Project. The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu . Eigenvarieties for reductive groups. Eric Urban, Ann. of Math. 174Email address: [email protected] Urban, Eigenvarieties for reductive groups, Ann. of Math. 174 (2011), 1685-1784. Email address: [email protected]
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[ "Von Neumann ", "Betti Numbers ", "Michael Farber " ]
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In this paper we show that Novikov type inequalities for closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a consequence we obtain a vanishing theorem for L 2 cohomology. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings.
10.1090/s0002-9939-00-05340-5
[ "https://export.arxiv.org/pdf/math/9810114v1.pdf" ]
11,841,902
math/9810114
82ca81bcaaa7f42f1b12dd3f5ad388caf3f8ab8f
18 Oct 1998 Von Neumann Betti Numbers Michael Farber 18 Oct 1998arXiv:math/9810114v1 [math.DG] AND NOVIKOV TYPE INEQUALITIES In this paper we show that Novikov type inequalities for closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a consequence we obtain a vanishing theorem for L 2 cohomology. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings. §0. Introduction S. Novikov and M. Shubin [NS] proved that Morse inequalities for smooth functions remain true with the usual Betti numbers being replaced by the von Neumann Betti numbers. Novikov [N] initiated an analog of the Morse theory for closed 1-forms. He showed that the Morse inequalities for functions can be generalized to closed 1-forms if instead of the Betti numbers one uses the Novikov numbers (they will be briefly reviewed in §3.1 below). In this paper we show that the inequalities of Novikov and Shubin [NS] hold for closed 1-forms as well. This result gives new Novikov type inequalities for closed 1-forms. Viewed differently, this gives a vanishing theorem for L 2 cohomology, generalizing a theorem of W. Lück [L3]. The proof of our Theorem 1 uses an idea of Gromov and Eliashberg [EG], which consist in counting additional critical points which appear when transforming the given closed 1-form into a function; the proof also uses the multiplicativity of the von Neumann Betti numbers under finitely sheeted coverings. In papers [BF2] and [MS] different Novikov type inequalities using the von Neumann Betti numbers were put forward. These inequalities involve cohomology of flat bundles of Hilbertian modules twisted by a generic line bundle determined by the closed 1-form (similarly to the finite dimensional case). Our approach in this paper does not require the twisting, and therefore it is simpler. On the other hand V. Mathai and M. Shubin [MS] allow more general Hilbertian flat bundles. It is clear that one may easily generalize Theorem 1 below for closed 1-forms with Bott type singularities, in a fashion similar to [BF1], [BF2]. I would like to thank Andrew Ranicki for stimulating discussions. §1. The main results First we recall the basic notions. A closed 1-form ω locally is a differential of a function ω| U = df U , where f U is a smooth function on U , determined up to a constant. Hence, all local properties of functions (e.g. the notions of a critical point, Morse singularities, indices) immediately generalize to closed 1-forms. For example, a point p ∈ U is a critical point of ω if p if it is a critical point of f U (which is equivalent to the requirement that ω p = 0); p a nondegenerate (or Morse) critical point of the form ω if it is a Morse critical point of the function f U , etc. 1. Theorem. Let M be a closed smooth manifold and let ω be a closed 1-form on M , having only Morse singularities. Let π ′ ⊂ π = π 1 (M ) be a normal subgroup with the following property: for any loop γ ∈ π ′ holds γ ω = 0. Then i k=0 (−1) k c i−k (ω) ≥ i k=0 (−1) k b (2) i−k (M ; π/π ′ ), i = 0, 1, 2, . . . , (1-1) where c i (ω) denotes the number of critical points of ω having index i,M denotes the covering of M corresponding to π ′ , and b (2) i (M ; π/π ′ ) denotes the von Neumann Betti number ofM . In particular, (1-1) implies c i (ω) ≥ b (2) i (M ; π/π ′ ). (1-2) For the definitions of von Neumann Betti numbers and their main properties we refer to [A], [CG], [L2], [F2]. Theorem 1 obviously implies Corollaries 2 and 3, cf. below. These Corollaries generalize a theorem of Wolfgang Lück [L1] (conjectured by M. Gromov), stating that L 2 -Betti numbers of a manifold, fibering over the circle, vanish. 2. Corollary (Vanishing Theorem). Let M be a closed smooth manifold admitting a closed 1-form with Morse type singularities having no critical points of index j for some 0 < j < n. Then for any normal subgroup π ′ ⊂ π, lying in the commutator subgroups of π = π 1 (M ), the j-dimensional von Neumann Betti number vanishes b (2) j (M ; π/π ′ ) = 0. (1-3) 3. Corollary (Morse Lacunary Principle). Let M be a closed manifold admitting a closed 1-form ω with Morse type singularities so that all the critical points of ω have even indices. Then for any normal subgroup π ′ ⊂ π, lying in the commutator subgroups of π = π 1 (M ), the odd-dimensional von Neumann Betti numbers vanish b (2) 2j−1 (M ; π/π ′ ) = 0 and the even-dimensional von Neumann Betti numbers are given by the formula b (2) 2j (M ; π/π ′ ) = c 2j (ω). (1-4) Note that Theorem 0.3 of [BF1] implies that under the conditions of Corollary 2 the number c 2j (ω) equals the Novikov number b 2j (ξ), where ξ ∈ H 1 (M ; R) is the cohomology class of ω. Hence we obtain the equality b (2) 2j (M ; π/π ′ ) = b 2j (ξ), compare Theorem 5 below. Definitions of the Novikov numbers can be found in the literature [BF1], [F1], [N]. Paper [BF1] contains a stronger version of Novikov type inequalities and a more general notion of Novikov numbers. In this paper we will use only the standard Novikov numbers b i (ξ). It will be convenient for our purposes to be based on the definition given in [BF1], cf. §3.1. Corollary (Symplectic circle actions). Let M be a closed symplectic manifold admitting a symplectic circle action with isolated fixed points. Then for any normal subgroup π ′ ⊂ π, lying in the commutator subgroups of π = π 1 (M ), the odd-dimensional von Neumann Betti numbers vanish b (2) 2j−1 (M ; π/π ′ ) = 0, and the even-dimensional von Neumann Betti numbers coincide with the Novikov numbers b (2) 2j (M ; π/π ′ ) = b 2j (ξ). (1-5) Here ξ ∈ H 1 (M ; R) denotes the cohomology class of the generalized moment map, and b i (ξ) denotes the i-dimensional Novikov number, corresponding to ξ. Proof of Corollary 4. Let us first explain the terms used in Corollary 4. Suppose that Ω denotes the symplectic form of M . The S 1 action is assumed to be symplectic, which means that for any g ∈ S 1 holds g * Ω = Ω. Let X denote the vector field generating the S 1 -action. Then ω = ι(X)Ω (1-6) is a closed 1-form on M , which is called the generalized moment map. We consider its De Rham cohomology class ξ = [ω] ∈ H 1 (M ; R) of ω. Recall that a symplectic circle action is called Hamiltonian if ξ = 0. In this case the Novikov numbers b i (ξ) coincide with the Betti numbers b i (M ). The critical points of the generalized moment map ω are precisely the fixed points of the circle action. It is well known that ω has Morse type singularities, assuming that the fixed points are isolated; moreover, all the critical points have even indices, cf. [Au]. Hence from Corollary 3 we obtain that b (2) 2j−1 (M ; π/π ′ ) = 0 and the evendimensional von Neumann Betti numbers are given by the formula (1-4). Now we will use the main theorem 0.3 of [BF1], which gives c 2j (ω) = b 2j (ξ). (1-7) The following Theorem states that in the special case when π ′ coincides with the ker(ξ) the inqualities of Theorem 1 are precisely the Novikov inequalities. 5. Theorem (Novikov numbers equal L 2 Betti numbers). Let M be a closed smooth manifold and let ξ ∈ H 1 (M ; R) be a nontrivial cohomology class. Let π ′ be the kernel of the homomorphism ξ : π = π 1 (M ) → R determined by ξ. Then the i-dimensional Novikov number b i (ξ) coincides with the i-dimensional von Neumann Betti number of the coveringM → M corresponding to π ′ : b i (ξ) = b (2) i (M ; π/π ′ ). (1-8) The plan of the paper is as follows. Proof of Theorem 1 is presented in §2. Proof of Theorem 5 is described in §3. §2. Proof of Theorem 1 Let ω be a closed 1-form on a closed manifold M . First, we will assume that the form ω has integral periods, i.e. its cohomology class is integral ξ = [ω] ∈ H 1 (M ; Z); the general case will be treated at the end of the proof. There exists a smooth map into the circle f : M → S 1 with ω = f * (dθ), where dθ denotes the standard angular form on the circle. This map is given by f (x) = exp(2πi x x 0 ω), where x 0 ∈ M is a base point. We will also denote by ξ : π = π 1 (M ) → Z the induced homomorphism [γ] → γ ω, where γ is a loop in M . Given a positive integer m, we will denote by π m the preimage π m = ξ −1 (mZ). It is a normal subgroup containing π ′ . Denote byM m the cyclic m-sheeted covering M m → M corresponding to the subgroup π m . It is clear that the pullbackω m of the form ω toM m has the following Morse numbers c j (ω m ) = m · c j (ω). (2-1) Claim. There exists a constant C, independent of m, so that on each manifoldM m there exists an exact 1-formω ′ m c j (ω m ) ≤ c j (ω ′ m ) ≤ c j (ω m ) + C, j = 0, 1, . . . (2-2) We will postpone the proof of the Claim and will continue the proof of the Theorem. We have the inequality i k=0 (−1) k c i−k (ω ′ m ) ≥ i k=0 (−1) k b (2) i−k (M ; π m /π ′ ), i = 0, 1, 2, . . . , (2-3) which is just the Novikov -Shubin inequality [NS] (cf. also [MS]) applied to the exact formω ′ m on the compact manifoldM m . Multiplicativity of the von Neumann Betti numbers under finitely sheeted coverings gives b (2) j (M ; π m /π ′ ) = m · b (2) j (M ; π/π ′ ),(2-4) cf. [L2], Theorem 1.7, statement 7. Here we use our assumption that π ′ ⊂ ker(ξ). Combining (2-1), (2-2), (2-3), (2-4) we will have i k=0 (−1) k c i−k (ω) = 1 m i k=0 (−1) k c i−k (ω m ) ≥ 1 m [ i k=0 (−1) k c i−k (ω ′ m ) − (i + 1) · C] ≥ 1 m [ i k=0 (−1) k b (2) i−k (M ; π m /π ′ ) − (i + 1) · C] = i k=0 (−1) k b (2) i−k (M ; π/π ′ ) − (i + 1) · C m . (2-5) Hence, i k=0 (−1) k c i−k (ω) ≥ i k=0 (−1) k b (2) i−k (M ; π/π ′ ) − (i + 1) · C m (2-6) for arbitrary m > 0. Since C is independent of m, taking the limit m → ∞ in the obtained inequality (2-6) proves (1-1). Now we want to prove the Claim. Suppose that exp(2πiθ 0 ) ∈ S 1 is a regular value of the map f : M → S 1 , where 0 < θ 0 < 1. Let V ⊂ M denote f −1 (θ 0 ). Consider the cylinder V × [−1, 1] and find an arbitrary Morse function φ : V × [−1, 1] → R (2-7) with the following properties: (a) φ assumes values in the open interval (0, 1); (b) φ(v, t) = 1 + δt for all t ∈ [−1, −1/2) and v ∈ V , where 0 < δ < 1 is a fixed number; (c) φ(v, t) = δt for all t ∈ (1/2, 1] and v ∈ V . We will denote by C the total number of critical points of φ. g m (v, t) = exp(2πi[θ 0 /m + φ(v, t)]), where φ : V × [−1, 1] → R is the function constructed above. Setω ′ m = g * m (dθ). Then the obtained 1-formω ′ m is exact since the map g m : M m → S 1 is null homotopic (it does not assume the value exp(2πiθ 0 /m) ∈ S 1 ). The inequality (2-2) is clearly satisfied since the formsω m andω ′ m have the same critical points on the complement of V ×[−1, 1] and on V ×[−1, 1] the formω m has no critical points, while the formω ′ m may have at most C critical points. This proves the Claim and hence completes the proof of Theorem 1 in the case of closed 1-forms ω having integral cohomology classes. Consider now a closed 1-form ω representing an arbitrary cohomology class ξ = [ω] ∈ H 1 (M ; R) and having Morse type singularities. It clearly defines a homomorphism ξ : H 1 (M ; Z) → R and the image im(ξ) is a finitely generated free abelian group. We will denote the rank of this image by r = rank(im(ξ)) = rank(ω). Note that any closed 1-form ω having rank 1 can be represented as ω = λω 0 , where λ > 0 and ω 0 represents an integral class. Since the forms ω and ω 0 have the same critical points, we conclude that Theorem 1 holds for all closed 1-forms of rank 1. Suppose now that ω is a closed 1-form having rank r > 1. Then we may find a sequence of closed 1-forms ω n on M with the following properties: (i) ω n converges to ω in the C 0 -norm; (ii) there exists a neighbourhood U of the set S of critical points of ω such that ω| U = ω n | U for all n; (iii) ω n has rank 1 for every n; (iv) the homomorphism [ω n ] : H 1 (M ; Z) → R vanishes on ker(ξ). Because of (i) and (ii), for large n the form ω n has the same critical points as ω and they have the same indices c j (ω) = c j (ω n ). Because of properties (iii) and (iv) we have the inequality i k=0 (−1) k c i−k (ω n ) ≥ i k=0 (−1) k b (2) i−k (M ; π/π ′ ), i = 0, 1, 2, . . . , which obviously implies (1-1). Let us show how to construct the sequence of forms ω n . Let N ξ ⊂ H 1 (M ; R) be the subspace formed by classes η with η| ker(ξ) = 0; here r = rank(ξ). Let η 1 , . . . , η r be a basis of N ξ such that rank(η j ) = 1 for j = 1, . . . , r. We may realize each class η j by a closed 1-form ν j which vanishes identically in a neighbourhood of the zeros of ω. We may write ξ = r j=1 α j η j , where α j ∈ R. Now, choose a sequence of rational numbers α n j converging to α j as n → ∞. Then the sequence of closed 1-forms ω n = ω + r j=1 (α n j − α j )ν j satisfies all the requirements. This completes the proof. §3. Proof of Theorem 5 The proof of theorem 5 will use basic harmonic analysis. Before starting the proof we recall the definition of the Novikov numbers. 3.1. The Novikov numbers b i (ξ). Let ξ ∈ H 1 (M ; R) be a cohomology class. We will denote by ker(ξ) the kernel of the induced homomorphism ξ : H 1 (M ; Z) → R. Let us consider complex flat line bundles L → M with the following property: the monodromy along any loop γ ∈ ker(ξ) is identity. We will denote by V ξ the set of isomorphism classes of all such flat line bundles. Given L ∈ V ξ , it has the monodromy homomorphism Mon L : H 1 (M ; Z)/ ker(ξ) → C * , which completely determines the isomorphism type of L. Hence, there is a one-to-one correspondence between V ξ and the complex torus (C * ) r . Here r is the rank of class ξ, i.e. r = rank(H 1 (M ; Z)/ ker(ξ)) = rank(ξ). This identification V ξ ≃ (C * ) r allows to view V ξ as an affine algebraic variety. Fix some i (with 0 ≤ i ≤ dim M ) and consider the function V ξ ∋ L → dim C H i (M ; L). (3-1) It is well know in the algebraic geometry (cf. [H], chapter 3, §12) that function (3-1) has the following property: there exists a proper algebraic subvariety V = V i (M ) ⊂ V ξ such that the dimension dim C H i (M ; L) is constant for all L / ∈ V and for L ′ ∈ V dim C H i (M ; L ′ ) > dim C H i (M ; L). (3-2) Definition. The Novikov number b i (ξ) is defined as dim C H i (M ; L) for L ∈ V ξ − V . 3.2. Unitary flat bundles. Consider the subset T ξ ⊂ V ξ consisting of isomorphism classes of flat line bundles L admitting flat Hermitian metrics. Under the identification V ξ ≃ (C * ) r given by the monodromy representation, the subset T ξ corresponds to the real torus (S 1 ) r ⊂ (C * ) r . This shows, in particular, that T ξ is a real analytic subvariety of V ξ . It is easy to see that any complex Laurent polynomial p(z 1 , . . . , z r , z −1 1 , . . . , z −1 r ), which vanishes on the torus (S 1 ) r ⊂ (C * ) r , is identically zero. Hence, the intersection of any non-empty Zariski open subset of (C * ) r with the torus T ξ is non-empty. This implies the following interpretation of the Novikov numbers: 3.3. Corollary. For a fixed M and ξ ∈ H 1 (M ; R), consider the function T ξ ∋ L → dim C H i (M ; L). (3-3) There exists a proper real analytic subvariety V = V i (M ) ⊂ T ξ such that for all L / ∈ V the dimension dim C H i (M ; L) equals to the Novikov number b i (ξ) and for L ′ ∈ V dim C H i (M ; L ′ ) > b i (ξ). (3-4) 3.4. Proof of Theorem 5. Let M be a closed manifold and let ξ ∈ H 1 (M ; R) be a real cohomology class. Consider the coveringM → M corresponding to the subgroup π ′ = ker(ξ) ⊂ π = π 1 (M ). Then π/π ′ is isomorphic to Z r for some r. Recall that the von Neumann Betti number b (2) i (M ; π/π ′ ) is defined as follows. Fix a smooth triangulation of M and consider the induced (equivariant) triangulation of the covering spaceM . Let C * (M ) denote the simplicial chain complex of this triangulation; it consists of free finitely generated Z[Z r ]-modules and Z[Z r ]-module homomorphisms. Consider the Hilbert space L 2 (Z r ) with its canonical Z r -action and form the complex of Hilbert spaces L 2 (Z r ) ⊗ Z r C * (M ). (3-5) The von Neumann Betti number b (2) i (M ; π/π ′ ) is defined as the von Neumann dimension of the reduced homology of complex (3-5), cf. [A], [CG], [F2], [L2]. Let us now remind the standard construction of the harmonic analysis, providing an isomorphism between the space of L 2 -functions on the torus (S 1 ) r with respect to the Lesbegue measure dµ (which will be assumed to be normalized so that the total torus (S 1 ) r has measure 1) and the space of L 2 -functions on the lattice Z r . Points of the lattice Z r we will denote by r-tuples n = (n 1 , n 2 , . . . , n r ), where n 1 , . . . , n r ∈ Z. Points of the torus (S 1 ) r we will denote by z = (z 1 , z 2 , . . . , z r ), where z 1 , . . . , z r ∈ S 1 ⊂ C. The symbol z n will denote the product z n = z n 1 1 · z n 2 2 · . . . · z n r r ∈ S 1 . Any L 2 -function f ∈ L 2 ((S 1 ) r , dµ) can be uniquely represented by its Fourier series f (z) = n∈Z r a n z n . The correspondence f → (a n ) n∈Z r defines an isometry L 2 ((S 1 ) r , dµ) ≃ L 2 (Z r ). It is important that under this isometry the multiplication by z m in L 2 ((S 1 ) r , dµ) (i.e. the map f (z) → z m f (z)) transforms into the shift (a n ) n∈Z r → (a n ) n+m∈Z r in L 2 (Z r ); here m ∈ Z r . The above identification allows to view the Hilbert space L 2 (Z r ) as the space of L 2sections of a vector bundle over the torus T ξ , which we will now describe. Consider the real analytic line bundle L → T ξ of unitary Z r -module structures on C. It is trivial as a vector bundle and the fiber over a point z ∈ T ξ has the following Z raction: a point of the lattice n ∈ Z r acts as multiplication by z n ∈ S 1 . We may form the chain complex L ⊗ Z r C * (M ) (3-6) which is a real analytic family (parametrized by the points of the torus T ξ ) of chain complexes of finite dimensional vector spaces. The L 2 -complex (3-5) is isomorphic to the complex of L 2 -sections of (3-6). Hence, applying Theorem 4.11 from [F3] we obtain that the von Neumann Betti number b (2) i (M ; π/π ′ ) coincides with the generic Betti number dim C H i (M ; L) for L ∈ T ξ − V , where V is a proper real analytic subvariety. This may be also stated as the equality b (2) i (M ; π/π ′ ) = T ξ dim C H i (M ; L)dµ(L). (3-7) By Corollary 3.3 the generic Betti number is precisely the Novikov number b i (ξ). This completes the proof. We obtainω m = f * m (dθ), where f m :M m → S 1 is a smooth map. It is clear that for any m we may smoothly imbedV × [−1, 1] intoM m so that on the image of V × [−1, 1] the map f m is given by f m (v, t) = exp(2πi[θ 0 /m + δt]). We will construct now a new map g m :M m → S 1 as follows:(a) g m coincides with f m on the complement of V × [−1, 1] inM m ; (b) on V × [−1, 1] the map g m is given by Elliptic operator, discrete groups and von Neumann algebras. M Atiyah, Astérisque. 32M. Atiyah, Elliptic operator, discrete groups and von Neumann algebras, Astérisque 32 (1976), 43 -72. M Audin, The Topology of Torus Action on Symplectic Manifolds. BirkäuserM. Audin, The Topology of Torus Action on Symplectic Manifolds, Birkäuser, 1991. Novikov type inequalities for differential forms with non-isolated zeros. M Braverman, M Farber, Math. Proc. of the Cambridge Phil. Society. 122M. Braverman, M. Farber, Novikov type inequalities for differential forms with non-isolated zeros, Math. Proc. of the Cambridge Phil. Society 122 (1997), 357 -375. Novikov-Bott inequalities. M Braverman, M Farber, C.R.Acad. Sci. Paris. 321M.Braverman, M.Farber, Novikov-Bott inequalities., C.R.Acad. Sci. Paris 321 (1995), 895- 902. L 2 -cohomology and group cohomology. J Cheeger, M Gromov, Topology. 25J. Cheeger and M. Gromov, L 2 -cohomology and group cohomology, Topology 25 (1986), 189 -215. . Y Eliashberg, M Gromov, Lagrangian Intersection Theorey. PreprintY. Eliashberg, M. Gromov, Lagrangian Intersection Theorey, Preprint (1996). Exactness of the Novikov inequalities. M Farber, Functional Analysis and its Applications. 19M. Farber, Exactness of the Novikov inequalities, Functional Analysis and its Applications 19:1 (1985), 40 -49. Homological algebra of Novikov -Shubin invariants and Morse inequalities. M Farber, GAFA. 6M. Farber, Homological algebra of Novikov -Shubin invariants and Morse inequalities., GAFA 6 (1996), 628 -665. Von Neumann categories and extended L 2 cohomology. M Farber, Journal of K-theory. to appearM. Farber, Von Neumann categories and extended L 2 cohomology, Journal of K-theory (to appear). Algebraic geometry. R Hartshorne, Springer-VerlagR. Hartshorne, Algebraic geometry, Springer-Verlag, 1977. L 2 -Betti numbers of mapping tori and groups. W Lück, Topology. 33W. Lück, L 2 -Betti numbers of mapping tori and groups, Topology 33 (1994), 203 -214. L 2 -invariants of regular coverings of compact manifolds and CW-complexes. W Lück, PreprintW. Lück, L 2 -invariants of regular coverings of compact manifolds and CW-complexes, Preprint (1997). Twisted L 2 invariants of non-simply connected manifolds. V Mathai, M Shubin, Russian Journal of Math. Physics. 4V. Mathai, M. Shubin, Twisted L 2 invariants of non-simply connected manifolds, Russian Journal of Math. Physics 4 (1996), 499 -527. Morse inequalities and von Neumann II 1 -factors. S Novikov, M Shubin, Soviet Math. Dokl. 34S. Novikov and M. Shubin, Morse inequalities and von Neumann II 1 -factors., Soviet Math. Dokl. 34 (1987), 79 -82. The Hamiltonian formalism and a multivalued analogue of Morse theory. S P Novikov, Russian Math. Surveys. 37S.P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Rus- sian Math. Surveys 37 (1982), 1-56. . Ramat-Aviv. 69978School of Mathematical Sciences, Tel-Aviv UniversityIsrael E-mail address: [email protected] of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel E-mail address: [email protected]
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We present the first detailed photometric analysis of ATO J108.6991+27.8306 (hereinafter as J108). The short-period close binary J108 was observed by the Nanshan 1-m Wide Field Telescope (NOWT) of the Xinjiang Astronomical Observatory. The obtained BV RI-band light curves were used to determine the photometric solution by using the 2003 version of the Wilson-Devinney code. J108 is a typical deep (f > 50%), low mass ratio (q < 0.25) (DLMR) overcontact binary system with a mass ratio of q = 0.1501 and a fill-out factor of f = 50.1%, suggesting that it is in the late evolutionary stage of contact binary systems. We found the target to be a W-type W UMa binary and provided evidence for the presence of starspots on both components. From the temperature-luminosity diagram, the main component is the evolved main sequence star with an evolutionary age of about 7.94 Gyr.
10.1088/1674-4527/acb97f
[ "https://export.arxiv.org/pdf/2303.07832v1.pdf" ]
256,658,430
2303.07832
a97e5477ed5c31819c048456670216b3cc507735
14 Mar 2023 Raa Vol XNo.XX14 Mar 2023Received 20XX Month Day; accepted 20XX Month DayResearch in Astronomy and Astrophysics First detailed analysis of a relatively deep, low mass-ratio contact binary: ATO J108.6991+27.8306binaries: close -binaries: eclipsing -stars: evolution -stars: individual (ATO We present the first detailed photometric analysis of ATO J108.6991+27.8306 (hereinafter as J108). The short-period close binary J108 was observed by the Nanshan 1-m Wide Field Telescope (NOWT) of the Xinjiang Astronomical Observatory. The obtained BV RI-band light curves were used to determine the photometric solution by using the 2003 version of the Wilson-Devinney code. J108 is a typical deep (f > 50%), low mass ratio (q < 0.25) (DLMR) overcontact binary system with a mass ratio of q = 0.1501 and a fill-out factor of f = 50.1%, suggesting that it is in the late evolutionary stage of contact binary systems. We found the target to be a W-type W UMa binary and provided evidence for the presence of starspots on both components. From the temperature-luminosity diagram, the main component is the evolved main sequence star with an evolutionary age of about 7.94 Gyr. INTRODUCTION W Urase Majoris-type binaries are usually contact binary systems in which both components fill the Roche lobe during their main sequence evolutionary phase and share a common envelope (Lucy 1968;Stepien 2006), so that both components have approximate surface temperatures. W UMa binaries, whose component spectral types typically range from F to K, may have magnetic activity, such as starspots (Poro et al. 2021;Panchal & Joshi 2021;Ma et al. 2022). In addition, W UMa binaries with more massive components but at lower temperatures are defined as W-type W UMa binaries, and the opposite as A-type (Binnendijk 1970). The orbital period of the W UMa system is usually variable. The mass transfer between the two components (Hoffman et al. 2006;Meng et al. 2021) can cause the orbital period to increase or decrease. The magnetic braking effect (Lanza et al. 1998;Belloni et al. 2020) causes a loss of orbital angular momentum, which is expressed as a decrease in the orbital period. The magnetic cycle (Borkovits et al. 2005;Khaliullina 2018) and the third body (Shi et al. 2020) are responsible for the cyclic variation of the orbital period. There are deep (f > 50%), low mass ratio (q < 0.25) (DLMR) overcontact binaries (Qian et al. 2006;Yang & Qian 2015) among W Urase Majoris-type binaries, which are at the evolutionary stage of late short-period close binaries. These binaries are important for the study of stellar astrophysics because they may be the progenitors of single rapidly rotating stars (Jiang 2020), such as Blue Straggler (BS) stars (Perets & Fabrycky 2009;Wei et al. 2021) and FK Com-type stars (Bopp & Stencel 1981;Sikora et al. 2020). ATO J108.6991+27.8306 (TIC 155709042, LPSEB3, UCAC4 590-039991, ASASSN-V J071447.79+274950.6) was defined as close binary with an orbital period of 0.393204 days by Heinze et al. (2018). Furthermore, J108 was first identified as an EW type binaries by All-Sky Automated Survey for Supernovae (ASAS-SN, Jayasinghe et al. 2019) and confirmed by Chen et al. (2020). At the same time, this target was included in the LAMOST catalog (Yang et al. 2020) and no more studies have been done on J108 since then. In this paper, we found that J108 is a member of the DLMR and attempted to perform the first public analysis of this interesting and unusual target. OBSERVATIONS AND DATA REDUCTION J108 was observed on Dec. 23, 2020 using the Nanshan One-meter Wide-field Telescope (Bai et al. 2020, hereinafter as NOWT) of the Xinjiang Astronomical Observatory. During our observations, a standard Johnson-Cousins BV RI filter was used. Individual observations included 107 in the B filter, 107 in V, 105 in R, 112 in I and a total of 426 CCD images were acquired. These CCD images correspond to exposure times of 70,40,45,50 s for the four bands of BV RI, respectively, with the observational accuracy better than 0.006 mag. The observed CCD images were reduced in a standard manner using IRAF 1 . In our work, the differential photometry method was adopted. The basic information about variable, comparison, and check stars are compiled in Table 1 With our observations, the eclipsing times were obtained through the K-W method (Kwee & van Woerden 1956) in Nelson's program 2 from the BV RI-bands light curves. The new eclipsing times and their mean values are shown in Table 2. PHOTOMETRIC ANALYSIS To further understand J108, we analyzed our BV RI-band light curves using the 2013 version of the W-D program (Wilson & Devinney 1971;Wilson 1979Wilson , 1990Wilson , 2008Wilson , 2012Wilson et al. 2010). We obtained several J108 spectra from LAMOST, as detailed in Table 3 (Paegert et al. 2022) provided effective temperatures of 6214 ± 395 K, 5888.2 ± 17.1 K, and 6264.9 ± 18.1 K, respectively, the average value of 6208 ± 110 K calculated from these temperatures was taken as the effective temperature of the binary. We assumed for the time being that the surface temperature T 1 of the primary component is consistent with this effective temperature. Based on this temperature, the gravitydarkening coefficients and the bolometric albedo were set at g 1,2 = 0.32 (Lucy 1967) and A 1,2 = 0.5 (Ruciński 1969) in either binary system, respectively. The bolometric and bandpass limb-darkening were adopted the square-root functions laws from the values of van Hamme (1993). The mass ratio is an important parameter for obtaining a credible photometric solution. Due to the lack of sufficient spectral observations, we used the q-search method to determine the mass ratio. We searched for the mass ratio of the two components between 0 and 1, i.e. primary component with more massive mass. The results are shown in Figure 2, where the initial value of the mass ratio q entered into the W-D code as an adjustment parameter is 0.168, and this value corresponds to the smallest residual Σω i (O − C) 2 i . During the search for the best mass ratio, we found that the massive component has a lower temperature, which means that it is a W-type W UMa binary. Meanwhile, these two components always fill their Roche lobes. We decided to use the W-D program in mode 3 (contact state). In fact, the effective temperature of the binary star differs somewhat from the temperature of its components, and we calculate the surface temperature of each component using the following equation (Christopoulou & Papageorgiou 2013). T 1 = 1 + k 2 T 4 eff / 1 + k 2 (T 2 /T 1 ) 4 0.25 ,(1)T 2 = T 1 (T 2 /T 1 ) .(2) The k in the equation is the radius ratio (r 2 /r 1 ), T ef f is the effective temperature of the binary, and T 2 /T 1 is the temperature ratio of the two components in combination with the preliminary fitting results of the W-D program. After several iterations, the main component temperature tends to a definite value. During the solution process, the orbital inclination i, the average temperature of the secondary component T 2 , the monochromatic luminosity of the primary component L 1 in the BV RI bands, and the dimensionless potential Ω 1 = Ω 2 of the two components are adjustable parameters. We found that the light curves are asymmetric (O'Connell effect, O'Connell 1951) and explain this phe- Notes: a The BV -band magnitudes of the variable, comparison and check stars were determined from the AAVSO Photometric All-Sky Survey DR9 (APASS9; Henden et al. 2016) b The JHK-band magnitudes were determined from Two Micron All Sky Survey (2MASS; Cutri et al. 2003) Table 4. The corresponding theoretical light curves that fit the observations better are displayed in Figure 3. From the results in Figure 3, introducing a hot spot in the primary component provides better convergence results, and if another cold spot is added to the secondary component, a better solution with a fill-out factor of f = 50.1% and a mass ratio of q = 0.1501 was obtained, as confirmed by the residual values in Table 4. The error in the filling factor of the binary is larger at low mass ratios, because the surface potential through the inner Lagrangian point (L 1 ) and the outer Lagrangian point (L 2 ) are numerically very close. The geometrical structures of J108 at phases of 0.00, 0.25, 0.50, and 0.75 are plotted in Figure 4. DISCUSSION AND CONCLUSION Although J108 had been monitored by many sky surveys, it was neglected for further study. The photometric solutions of this eclipsing binary J108 is presented for the first time. Using the W-D program, we analyzed one set of complete multiple color light curves of target. Our photometric solutions suggest that J108 is a DLMR binary system with a high degree of overcontact (f = 50.1%) and a de- rived mass ratio (q = 0.1501). The asymmetry of the light curve is explained by a hot spot on the primary and a cold spot on the secondary component. J108 is a W-type W UMa system. According to the photometric results of the three groups of component temperatures, through Table 5 of Pecaut & Mamajek (2013), we could conclude that the spectral type of the primary component is F8, and the spectral type of the secondary component ranges from F5 to F7. The spectral type of these daughter stars differs slightly from the LAMOST results for the spectral type of this binary system in Table 3, which is due to the different spectral type and temperature correspondence criteria. Absolute parameters Accurate absolute parameters are obtained for binary star systems with radial velocity curve. In the absence of RV curve, we could use an estimation method to obtain the ab- solute parameters of the binary system. Since entering the Gaia era, we are able to obtain more accurate distance information. Based on the distance and photometric observations, we are able to obtain the absolute bolometric magnitudes of the components after extinction and reddening corrections. We adopted a similar approach, assuming that the massive component of W UMa is the main sequence star (Yakut & Eggleton 2005). Firstly, the luminosity of the primary star is determined by these equations: (a) m Vi − m Vmax = −2.5 log (L i /L), (b) M Vi = m Vi − 5 log D + 5 − A V , (c) M boli = M Vi + BC Vi , (d) M bol1 Evolutionary States Based on the absolute parameters, the primary component is plotted in a temperature-luminosity diagram, which is shown in Figure 5. Combined with the [Fe/H] information provided by the LAMOST spectra in Table 3, corresponding to a Z of about 0.008±0.002, the tracks of stellar evolution taken from Girardi et al. (2000) with metal abundance Z = 0.008 were chosen. The primary is located between the zero-age main sequence (ZAMS) and the terminal-age main sequence (TAMS) lines, which suggests that the primary component is an evolved MS star. Based on the evo-lutionary tracks of the single stars, we estimated the possible evolutionary age of J108 to be ∼7.94 Gyr. ORJ7 ORJ// =$06 7$06 - Fig. 5 Primary component for J108 in the H-R diagram. the zero-age main sequence (ZAMS), the terminal-age main sequence (TAMS), and the evolutionary tracks of each initial mass (from 0.9 M to 1.8 M ) and isochrones around the primary component are taken from Girardi et al. (2000) with metal abundance Z = 0.008. The red dotted line are isochrone line. The mass-radius (M-R) and mass-luminosity (M-L) relations were used to understand the evolutionary state of our target binary. J108 with from Table 7 of Li et al. (2021) for 94 A-type and 85 W-type binaries are plotted in Figure 6. In the figure, the target, like most binary systems, has the higher mass component as the evolved main-sequence star, while the lower mass component of the target has left the main-sequence phase. This could be a mass transfer leading to a mass ratio reversal (Guinan & Bradstreet 1988), meaning that the current low-mass component is the initial high-mass component of the binary system, and this component underwent a rapid mass transfer process as it left the main-sequence phase. Determining the initial masses of the two components is useful to study the evolution of contact binaries. Since binaries are influenced by their companions during their evolution, the method of luminosity excess (Yildiz & Dogan 2013) was used to calculate the initial mass of the initial massive component (currently the lower mass component) before the mass transfer process occurred.The initial mass of the secondary M 2i = 1.83 ± 0.04 M was obtained using the following equation: The initial mass of 0.985 M obtained from Figure 5 is in this initial mass range, assuming that the primary star is a main sequence single star. The thermal timescale of the more massive star is shorter. When the mass difference between the two components of a binary system is relatively large (about 2.5 times), the evolution of the massive component is faster (Sen et al. 2022). The outer layers of the more massive component expand and break through its Roche radius. The less massive component is subject to unstable dynamical mass transfer and gains mass. The previous time scale required for the thermal adjustment of the smaller mass component is longer than the time scale required for the mass loss of the larger mass component, so the less massive star will expand. When both components expand beyond the Roche critical volume, the binary star evolves into a system with a common envelope. (Taam & Ricker 2010;Ivanova et al. 2013) It can be seen that the initial massive component lost a large amount of mass (about 1.6 M ) when it left the main sequence phase, part of which was transferred to the present primary star (about from 0.1 to 0.66 M ), and another more mass was lost from the binary system (about from 1 to 1.5 M ), which will cause a decrease in the orbital angular momentum of the binary system. The process could be called as the roche lobe overflow (Hurley et al. 2002;Hirai & Mandel 2022), during which the initial less massive component gains a small fraction of the mass lost by the massive component, and more mass is spilled from the outer Lagrangian points in order to form short-period binary systems, a theory corroborated by our observations. M 2i = M 2 + 2.50(δM − 0.07) 0.64 ,(3) The equation Jspin Jorb = 1+q q (k 1 r 1 ) 2 + (k 2 r 2 ) 2 q was given by Yang & Qian (2015) to calculate the ratio of the spin angular momentum J spin to the orbital angular momentum J orb , which reflects the stability of the binary evolution. In this work, k 2 i is set as 0.06 (Li & Zhang 2006) and Jspin Jorb = 0.151 was obtained, J spin /J orb < 1/3 indicate that J108 are currently in a stable evolutionary state. This paper presents the first detailed photometric analysis of ATO J108.6991+27.8306 using multimetric photometric data from NOWT observations. This is a neglected, deep (f = 50.1 %), low mass ratio (q = 0.1501) W-type W UMa binary system, which implies that it is at a late evolutionary stage of contact binary systems. We have derived the initial masses of the two components and inferred an evolutionary age of the system of about 7.94 Gyr. The minima of light times we have collected so far are not enough to support our work on long-term orbital period analysis, so more observations are necessary and the variation of the orbital period needs to be further in- Fig. 6 The mass-radius diagram (left) and mass-luminosity diagram (right). The solid line and the dashed line, which represent the ZAMS and TAMS respectively, are constructed by the binary system evolution code provided by Hurley et al. (2002). Red regular triangles represent our own two targets. The A-type and the W-type contact binaries from Li et al. (2021) are displayed as yellow circles and blue circles, respectively. vestigated. The continuous loss of orbital angular momentum is visualized by the decreasing orbital period of the system, which may lead to an increase of the filling factor and a decrease of the inter-binary distance, which will evolve into a rapidly rotating star. Conversely, if the orbital period of the target is continuously increasing, then ATO J108.6991+27.8306 will be more valuable as an idiosyncratic target for research, and more detailed data are needed to support the construction of the theory. Fig. 1 1Multicolour light curves of J108. Fig. 2 2Relationships between sum of squares of residuals ω i (O − C) 2 i and the mass ratios q for J108.nomenon by introducing a starspot model. The photometric solutions are listed in with one hot spot with one hot spot and one clod spot Fig. 3 3The photometric precisions of J108 in BV RI band. Fig. 4 4Geometrical structure of J108 with one hot spot on the primary and one cool spot on the secondary at phases of 0.00, 0.25, 0.50, and 0.75. (L 1,2 /L )(Torres 2010). In our work, the interstellar extinction coefficient A v S&F =0.1701±0.0022 (mag)(Schlafly & Finkbeiner 2011) from IRSA database 3 , the bolometric corrections BC v =-0.042 (mag) fromPecaut & Mamajek (2013), distances obtained by Gaia Collaboration (2022), and the maximum visual magnitude m Vmax obtained by fitting ASAS-SN data.Christopoulou et al. (2022) provided a mass-luminosity relationship log L = log b + a log M (b = 0.63 ± 0.04 and a = 4.8 ± 0.2). The equations M 1 + M 2 = 0.0134a 3 /P 2 and R i = ar i were used. The calculated results are shown inTable 5. Through the mass luminosity relationship, M 1 = 1.331(1) M and M 2 = 0.200(2) M were obtained. ∆M 2i = 2i∆M 2 + 1.59 (δM − 0.07) 0.36 0.22∆L 2 L 0.76 2 + ∆M 2 , where δM = M L − M 2 , M 2 is the current mass of the secondary component, M L = (L 2 /1.49) 1/4.216 , ∆ denote the errors of the corresponding quantity. Based on the possible initial mass ratios, the initial mass range of the primary star was determined (M 1i from 0.67 to 1.21 M o dot). Table 1 1Observing details of the variable, comparison and check starsTargets Name α 2000 β 2000 B mag a V mag a J mag b H mag b K mag b Variable star ATO J108.6991+27.8306 07 14 47.80 +27 49 50.5 14.764 14.224 13.163 12.916 12.925 The comparison 2MASS 07151009+2743415 07 15 10.09 +27 43 41.5 15.145 13.982 11.922 11.297 11.200 The Check 2MASS 07143242+2751055 07 14 32.42 +27 51 05.5 14.956 14.434 13.330 13.070 13.022 Table 2 2Newly Obtained Minima Times of J108 in the BV RI BandsHJD(B) HJD(V) HJD(R) HJD(I) HJD(average) Min. 2459000+ 2459000+ 2459000+ 2459000+ 2459000+ 207.14545±0.00036 207.14488±0.00043 207.14586±0.00022 207.14452±0.00053 207.14518±0.00039 primary 207.34242±0.00052 207.34242±0.00044 207.34296±0.00035 207.34241±0.00019 207.34255±0.00035 secondary Table 3 3LAMOST spectral information of J108UT Date HJD phase Subclass Teff log g [Fe/H] Radial velocity (yyyy-mm-dd) (d) (K) (km·s −1 ) 2011 Nov 14 2455880.390798 0.250407 F0 2011 Dec 07 2455903.249277 0.385092 F4 6407.81 ± 96.88 4.236 ± 0.155 -0.250 ± 0.102 70.63 ± 8.42 2014 Dec 18 2457010.244898 0.744818 F5 6267.75 ± 24.80 4.066 ± 0.033 -0.235 ± 0.018 54.21 ± 4.05 Table 4 4Photometric Elements of J108.Parameters Without spots With one spot With two spots g 1 = g 2 0.32 0.32 0.32 A 1 = A 2 0.5 0.5 0.5 Ω in 2.07094 2.10336 2.10336 Ω out 1.98144 2.00648 2.00648 q(M 2 /M 1 ) 0.1383 (± 0.0026) 0.1501 (± 0.0016) 0.1501 (± 0.0013) T 1 (K) 6130 K 6130 K 6130 K T 2 (K) 6569 (± 20) K 6359 (± 9) K 6254 (± 10) K i • 71.244 (± 0.603) 72.522 (± 0.345) 72.586 (± 0.345) Ω 1 = Ω 2 2.02343 (0.00822) 2.05270 (0.00519) 2.05473 (0.00435) L 1B /L B 0.7859 (± 0.0067) 0.8049 (± 0.0031) 0.8202 (± 0.0029) L 1V /L V 0.8028 (± 0.0057) 0.8137 (± 0.0028) 0.8249 (± 0.0025) L 1R /L R 0.8121 (± 0.0052) 0.8185 (± 0.0026) 0.8275 (± 0.0024) L 1I /L I 0.8189 (± 0.0049) 0.8221 (± 0.0026) 0.8294 (± 0.0023) r 1 (pole) 0.5269 (± 0.0017) 0.5209 (± 0.0011) 0.5200 (± 0.0009 ) r 1 (side) 0.5864 (± 0.0026) 0.5778 (± 0.0016) 0.5763 (± 0.0014 ) r 1 (back) 0.6103 (± 0.0027) 0.6025 (± 0.0017) 0.6008 (± 0.0015 ) r 2 (pole) 0.2225 (± 0.0088) 0.2300 (± 0.0049) 0.2300 (± 0.0040 ) r 2 (side) 0.2333 (± 0.0108) 0.2415 (± 0.0061) 0.2414 (± 0.0049 ) r 2 (back) 0.2831 (± 0.0288) 0.2937 (± 0.0164) 0.2933 (± 0.0131 ) f 53.1 % (± 9.2 %) 52.3 % (± 5.4 %) 50.2 % (± 4.5 %) Spot 1 Primary Primary θ(radian) 0.9291 (± 0.0534) 0.9291 (± 0.0534) φ(radian) 3.7145 (± 0.0513) 3.7145 (± 0.0513) r(radian) 0.3290 (± 0.0092) 0.3290 (± 0.0092) T f (T d /T 0 ) 1.1959 (± 0.0083) 1.1959 (± 0.0083) Spot 2 Secondary θ(radian) 1.9256 (± 0.0992) φ(radian) 3.3127 (± 0.0518) r(radian) 0.2318 (± 0.0211) T f (T d /T 0 ) 0.9058 (± 0.0185) ΣW (O − C) 2 0.00324 0.00203 0.00194 Table 5 5Absolute Parameters of J108.Parameters Value Error D(pc) 1179.078 25.544 mV max(mag) 14.104 0.025 mV 1 (mag) 14.313 0.028 MV 1 (mag) 3.785 0.078 M bol 1 (mag) 3.743 0.078 L1(L ) 2.482 0.177 L2(L ) 0.527 0.047 M1(R ) 1.331 0.001 M2(R ) 0.200 0.002 a(R ) 4.800 0.200 R1(M ) 1.470 0.004 R2(M ) 0.660 0.019 the Image Reduction and Analysis Facility http://iraf. noao.edu/ 2 https://www.variablestarssouth.org/ software-by-bob-nelson/ https://irsa.ipac.caltech.edu/applications/ DUST/ Acknowledgements This work received the generous support of the National Natural Science Foundation of China under grants U2031204. 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[ "Front ", "Zhen-Fan Phys ; Yan -Zhi Meng ", "Tang " ]
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There are some similarities between bursts of repeating fast radio bursts (FRBs) and giant pulses (GPs) of pulsars. To explore possible relations between them, we study the cumulative energy distributions of these two phenomena using the observations of repeating FRB 121102 and the GPs of Crab pulsar. We find that the power-law slope of GPs (with fluence ≥130 Jy ms) is 2.85 ± 0.10. The energy distribution of FRB 121102 can be well fitted by a smooth broken power-law function. For the bursts of FRB 121102 above the break energy (1.22 ×10 37 erg), the best-fitting slope is 2.90 +0.55 −0.44 , similar to the index of GPs at the same observing frequency (∼1.4 GHz). We further discuss the physical origin of the repeating FRB 121102 in the framework of the super GPs model. And we find that the super GPs model involving a millisecond pulsar is workable and favored for explaining FRB 121102 despite that the magnetar burst model is more popular.
10.1007/s11467-020-1039-4
[ "https://arxiv.org/pdf/2012.07303v1.pdf" ]
229,156,668
2012.07303
9a6e1f898c90213f904ea8e637348e4898837bec
Front Zhen-Fan Phys ; Yan -Zhi Meng Tang Research article A comparison between repeating bursts of FRB 121102 and giant pulses from Crab pulsar and its applicationsPulsarsRadio sourcesgeneral -methods: statistical PACS numbers 9760Gb9870Dk0270Rr There are some similarities between bursts of repeating fast radio bursts (FRBs) and giant pulses (GPs) of pulsars. To explore possible relations between them, we study the cumulative energy distributions of these two phenomena using the observations of repeating FRB 121102 and the GPs of Crab pulsar. We find that the power-law slope of GPs (with fluence ≥130 Jy ms) is 2.85 ± 0.10. The energy distribution of FRB 121102 can be well fitted by a smooth broken power-law function. For the bursts of FRB 121102 above the break energy (1.22 ×10 37 erg), the best-fitting slope is 2.90 +0.55 −0.44 , similar to the index of GPs at the same observing frequency (∼1.4 GHz). We further discuss the physical origin of the repeating FRB 121102 in the framework of the super GPs model. And we find that the super GPs model involving a millisecond pulsar is workable and favored for explaining FRB 121102 despite that the magnetar burst model is more popular. Introduction Fast Radio Bursts (FRBs) are mysterious, bright astronomical millisecond-duration radio pulses [1][2][3], occasionally discovered in pulsar searches. Up to date, more than one hundred FRBs have been reported [4] 1 . They are expected to be of cosmological origin due to high dispersion measures (DM, 10 2 − 10 3 pc cm −3 ) in excess of the DM contribution from the Milky Way. And it is further confirmed by the localization of the host galaxies [5][6][7][8]. Among the moderately observed large sample of FRBs, most of them are one-off events, and only twenty of them sporadically show repeating bursts [9][10][11]. Moreover, FRB 180916.J0158+65 detected by CHIME [12] exhibits a ∼16 day period acitvity with unknown mechanism. Most recently, a bright FRB 200428 has been reported to be spatially coincident with the galactic Soft Gamma-ray Repeater (SGR) 1935+2154 [13][14][15][16], also associated with a hard X-ray burst [14,15,17,18]. The repetition of FRB 200428 is very rare, which may be different from other extragalactic repeating FRBs [13]. The progenitors of FRBs are still under heated debates, and many theoretical models have been proposed (see [19] and references therein 2 ), despite some of them have confronted serious challenges. Repeating FRBs are the best candidates to explore physical nature due to its repetition. FRB 121102 is the first observed repeating FRB and has been detected in the frequency range from 600 MHz [20] to 8 GHz [21]. For repeating FRBs, some models involving catastrophic processes have been ruled out, e.g., binary neutron star (NS) mergers [22,23], and binary white dwarf mergers [24]. Many non-catastrophic models have been put forward, such as the flaring magnetars model [25][26][27][28], accretion process by a neutron star in a binary system [29], the interaction of NS with an asteroid belt [30], the interaction between the NS magnetosphere and cosmic winds [31], and the giant pulses (GPs) from pulsars [32,33]. The central engine of FRB 121102 is questionable, which can be constrained from the persistent radio nebula of FRB 121102 for the magnetar-type model [34][35][36][37][38]. GPs were first discovered in the Crab pulsar (PSR B0531+21), and then were found in several energetic pulsars (see Table 1 of [39] and reference therein). Different from the regular pulses, the typical duration of GPs is very short, with no clear periodicity, ranging from nanoseconds to microseconds. The flux density could be 2 or 4 orders of magnitude higher than the average pulseintegrated flux of regular energetic pulses (e.g., [40,41]). Among the pulsars emitting the GPs, the largest value of the magnetic field at the light cylinder B lc are the Crab pulsar and PSR B1937+21 [39]. The Crab pulsar is a remarkable source emitting numerous GPs and simultaneously has been detected across a broadband range of radio frequencies from 20 MHz [42] to 46 GHz [43]. The distinctive feature of GPs is the energy of GPs obeys a power-law distribution, whereas the regular ones follow a Gaussian or log-normal distribution (e.g., [40,44]), indicating that the emission mechanism of GPs may be different from that of regular pulses. Meanwhile, GPs are also crucial to understand the radiation mechanism of pulsars. Repeating FRBs and GPs from pulsars share similarities in many observational aspects, such as brightness temperature, duration time, sub-bursts, polarization, and frequency drift. Moreover, the emission mechanism of both of them is coherent radiation, supporting that they may have a similar physical origin. Actually, in some theoretical models, it has been suggested that FRBs can be treated as GPs from young rapidly rotating pulsars [32,33]. The statistical properties, such as the energy distributions, of repeating FRBs and GPs might help us to understand FRBs progenitors. In this paper, we focus on the statistical properties of the repeating FRB 121102 and GPs from the Crab pulsar. The paper is organized as follows. In Section 2, we introduce the sample selection. The statistical analysis of the isotropicequvivalent energy E is shown in Section 3. Discussion and conclusions are given in Section 4 and Section 5. We assume a flat ΛCDM universe with Ω m = 0.27, and H 0 = 70 km s −1 Mpc −1 throughout this work. Sample selection The Crab Nebula pulsar is well known for its GPs, with detected GPs numerous enough for detailed studies (e.g., [40,41,45]). Here, GPs refer to GPs generated by the main pulse (MP) and interpulse (IP) phase. In our analysis, we adopt a complete sample of 1153 bright GPs (with fluence ≥130 Jy ms to avoid the incomplete effect near the detection threshold) from the Crab pulsar [45]. These GPs were detected by the 15-m telescope in Pune, India (National Centre for Radio Astrophysics) in a 260-hours observation from February to April 2019. The observation was conducted between 1280 MHz and 1380 MHz, with 65 MHz bandwidth usable. Each GP was detected with a signal-to-noise ratio (S/N) of ≥ 10. Among 20 repeating FRBs have been reported, only two of repeating FRBs have host galaxy identified and redshift measured (FRB 121102 and FRB 180916.J0158+65; [5,12]). FRB 121102 is the first and currently the best representative of repeating bursts. More than one hundred bursts from FRB 121102 have been detected by several different surveys, including Green Bank Telescope (GBT) in C-band (4-8 GHz; [46]) and Arecibo telescope at 1.4 GHz [47]. The repeating FRB 121102 has a very large and variable Faraday rotation measure (RM ∼ 10 5 rad m −2 ) which means the burst inhabits in an extreme magneto-ionic environment [3]. To compare with the Crab GPs sample under the same condition, we use the sample of FRB 121102 detected at almost the same frequency [47]. The observing frequency of Arecibo in the source frame is 1.4 * (1 + z) GHz. Due to the redshift 0.193 for FRB 121102 [48], the frequency in the source rest frame approximatively is ∼ 1.4 GHz. In this sample, there are 41 bursts from FRB 121102 observed by the 305-m Arecibo telescope. The data is from two observations taken on 2016 September 13/09:47:07 and 14/09:50:12, and lasting 5967 s and 5545 s. Each observation detected 18 and 23 bursts with S/N ≥10, respectively. Moreover, we also investigate the energy distribution of the repeating FRB 121102 detected by GBT telescope at C-band [46]. There are 93 bursts in total in 5 hours with a convolutional neural network technique. The cumulative energy E distributions Energy is a crucial parameter to identify the progenitor and to explore the radiation mechanism. We perform an analysis on the energy distribution of the repeating FRB 121102 and GPs to explore the progenitor of FRB 121102, despite there are many works devoted to the statistical properties of the repeating FRBs [46,[49][50][51][52]. The isotropic-equvivalent energy E in the source rest frame within the observing bandwidth for the selected sample can be calculated by E = 4πd 2 L f ν ∆ν/(1 + z) ,(1) where d L is the luminosity distance of the source, with 2.0 kpc for the Crab pulsar, f ν is the fluence density of each burst, and ∆ν is the corresponding bandwidth of every burst. In this paper, the bandwidth for FRB 121102 is derived from the burst frequency edges (f high Research article and f low ) from the table given in [47]. Here we adopt a simple power-law model, N (> E) ∝ E −α to fit the cumulative energy distributions of bright GPs (fluence ≥ 130 Jy ms) from the Crab pulsar. While for the repeating FRB 121102, we fit it by the smooth broken power-law function, i.e., N cum (> E) = A E E b α1ω + E E b α2ω −1/ω ,(2) where A is the ampltitude, α 1 , α 2 are slopes of the two segments, E b is the break energy, ω is the sharpness (or smoothness) of the break of energy distribution, and it usually can be set as a constant (ω = 3). We can now obtain the goodness-of-fit χ d.o.f for the difference between the observed cumulative distribution N cum,obs and the theoretical distribution function N cum for the cumulative distribution (with n par = 4 parameters, E 0 , α 1 , E b , α 2 ) χ d.o.f = 1 (Ntot − npar) N tot i=1 [Ncum (Ei) − N cum,obs (Ei)] 2 σ 2 cum,i ,(3) where N tot is the number of bursts dectected by the same instrument at the same frequency for the same source. Note that the uncertainty of N cum (> E) is taken as σ cum,i = N cum,obs (E i ). The best-fitting parameters and the uncertainties can be obtained by minimizing χ d.o.f via the python package emcee (a Bayesian MCMC method by sampling the affine-invariant for Markov Chain Monte Carlo chains, [53]). When fitting variables via a MCMC approach, we assume the various parameters in the fitting function are uniform distribution in the prior probability distribution. Here are the range of parameters for every sample: 1 < α< 5 for GPs; 50 < A < 200, -2.5 <α 1 < -0.5, 36 < lgE b <38 and 2 < α 2 < 5 for Arecibo sample; 50 < A< 200, -3< α 1 < -0.01, 36 < lgE b <38.5 and 1< α 2 < 3 for GBT sample. GPs are generally considered as the power-law tail of the regular pulses for the pulsar. In the upper panel of Figure 1, the yellow-green points represent the GPs from the Crab pulsar, which is found to be well fitted by the power-law model. The rollover does not appear at the high-energy end, while it does show for the flux density distribution as stated in some previous studies [40,44]. The best-fit slope is 2.85 ± 0.10 (χ d.o.f = 0.45), which is well consistent with that of the fluence distribution (2.8 ± 0.3) reported in [45]. In Figure1, we present the cumulative energy (E) distribution for 41 Arecibo bursts from FRB 121102 (blue spots). Fitted by the broken power-law function given in equation (2) [47], the threshold energy was set at E th = 2 × 10 37 erg considering the sensitivity limit and the observed turnover. Above the E th , the best fit is dR dE ∝ E −2.8 , which is generally consistent with our fitting result within the uncertainty. All the uncertainties quote a 68% confidence interval for the best-fitted parameters. To investigate what causes the different power indices of the energy distribution of FRB 121102 detected by the Arecibo telescope at L-band and that by GBT telescope at C-band and the meaning of the break energy E b in Figure 1, we also fit the energy distribution for 93 bursts from FRB 121102 by GBT telescope with broken powerlaw function. As shown in Figure 2 (Figure 1). At a different observed frequency (C-band, 4-8 GHz), the energy distribution of FRB 121102 detected by the GBT telescope is flatter (Figure 2) than that of the Arecibo telescope with more energetic bursts. There is a lack of bursts below near the break energy ∼ 1.60 × 10 37 erg (the black dashed line) due to the instrument sensitivity and observational limit. So maybe the incomplete sampling below the threshold is the major reason that evokes the different power-law indices of energy distribution for FRB 121102 in Figure1 and Figure 2. The power-law index of energy distribution with repeating FRB 121102 may depend on the energy band, the burst is more energetic at a higher observational frequency (see Figure 1 and Figure 2), which can be further tested by detecting more bursts at different energy bands. Note that the energy distribution of GPs might also depend on the energy band [40,44], and can have the index ∼ 1.8 for some pulsars (such as PSR 0631+1036; [54]). The power-law index of energy distribution and the statistical characteristics of the other observable quantities of FRB121102 are claimed to be similar to those of soft gamma repeater bursts from magnetars [55]. In addition, there are several statistical analyses [50,[55][56][57][58][59][60] dedicated to the self-organizing critical (SOC) study, especially some researches [50,55,60,61] that indicate that the repeating FRB 121102 is a SOC system [62]. Thus, the repeating FRB 121102, is likely to be an SOC system due to some kind of instability driven above the instability threshold value. If so, the break energy in Figure 1 should be the threshold energy, which determines the trigger mechanism of the system. It is an important quantity that affects the slope of the power-law in energy distribution, which is also discussed in [60]. Interestingly, the power-law slope of FRB 121102 omitting bursts that fall bellow the break energy is generally consistent with that of the GPs from the Crab as shown in Figure 1, which implies they may share a similar origin. If FRB 121102 is extragalactic GPs, which energies can power it? We will discuss this in the next section below. Discussion As mentioned in the introduction, neutron stars are the most popular FRB central engine. Especially the super GPs model involving pulsars [32,33,63] and the giant flare model involving magnetars [26][27][28][64][65][66] Research article been widely discussed. Specifically for FRB 121102, it was found that its host is a dwarf galaxy with a high star formation rate [5,48], which implies that the central engine of FRB 121102 may locate in the remnant of the death of a massive star. Moreover, the polarization observations give that the Faraday rotation measure of FRB 121102 is surprisingly high ∼ 10 5 rad m −2 , indicating that FRB 121102 should be surrounded by a strongly magnetized environment [3]. These observational evidences support that the central engine of FRB 121102 may be a magnetized NS. In the following, we explore the physical origin of the repeating FRB 121102 in the framework of the super GPs model. As shown in the upper panel of figure 1, the typical energy of the FRB 121102 and GPs are 10 37 ∼ 10 38 erg and 10 29 ∼ 10 30 erg. On the other hand, the typical time scale of the FRB 121102 and GPs are ms and µs, hence the typical luminosity are 10 40 ∼ 10 41 erg/s and 10 35 ∼ 10 36 erg/s, respectively. Following [67], based on the typical luminosity of FRB 121102 and that of GPs, we have ξ = L FRB L GP ∼ 10 5 .(4) Since the GPs are powered by the spin down of the pulsar L sd , the constraint on the magnetic field strength B FRB of the FRB source is B ≈ 0.3 ξ 1/2 5 B Crab P −3 P Crab 2 ,(5) By scaling the power of the Crab GPs to the level of FRB 121102 i.e. L sd,FRB = LFRB LGP L sd,GP , where B Crab ∼ 4 × 10 12 G and P Crab ∼ 33 ms are the magnetic field strength and the period of the Crab pulsar, respectively [68]. It is seen that the millisecond pulsars may produce GPs with luminosity comparable to FRB 121102. On the other hand, the active time scale is the spin-down time scale of the pulsar, that is t active ∼ 50 ξ −1 5 I 45 R −6 6 B −2 Crab P 4 Crab P −2 −3 yr.(6) which can in principle explain FRB 121102 so far. Considering the condition of t active > 7 yrs, we constraints that the period of the initial spin P should be smaller than 3 ms. In addition, if the radiation efficiency of FRB η FRB is higher than that of GPs η GP , for example an order of magnitude higher, then we would have a longer active time scale t active ∼ 500 (η FRB /10η GP ). Then the constraint to the period of the initial spin is P ≤ 8 ms in this case. For comparison, the constraint from the persistent radio nebula gives P ≤ 7 ms [37]. Moreover, we note that both repeating FRBs and GPs have high brightness temperature (≥ 10 35 K), inferring that their radiation mechanism should be coherent emission [69,70]. GPs from Crab pulsar and FRBs 121102 share similar complex pulses morphology and a similar repetition rate in a single pulsar (∼ 1-100 per hour, see Figure 1). Furthermore, similar to FRBs [71,72], GPs have strong polarization signals, which is either lefthanded or right-handed polarization [73] between linear polarization and circular polarization. Besides, spectral structures in repeating FRBs resemble those seen in Crab GPs [74]. Therefore, the similarities between the GPs and FRB 121102 possibly link to the same origin and mechanism of these two phenomena. Conclusions In this paper, we have made a comparison of the energy distributions between the repeating FRBs and the GPs from the Crab pulsar. We find that FRB 121102 and Crab GPs share similar power-law indices in the energy distribution at the same observed frequency ∼1.4 GHz 3 . Furthermore, we explored the physical origin of the repeating FRB 121102 in the framework of the giant pulses model. We find that the millisecond pulsar is possible to produce super GPs with luminosity comparable to FRB 121102 in a reasonable active time scale. Therefore, we argue that the super GPs model is workable and favored to explain FRB 121102 despite that the magnetar burst model is more popular. , the best fits for the lower and higher energy are −1.49 +0.48 −0.56 and 1.33 +0.18 −0.14 (χ d.o.f = 0.63), respectively, while the break energy is 1.60 +0.94 −0.44 ×10 37 erg. Interestingly, with the same observed frequency (∼ 1.4 GHz), FRB 121102 and Crab GPs share the power-law form of similar indices (∼ 2.8) Fig. 1 1Upper panel: The cumulative energy Eiso distributions of the GPs from the Crab pulsar (yellow-green), FRB 121102 detected by Arecibo (blue circle). The vertical blue dotted line represents the break energy of FRB 121102 fitted, and the red line represents the best fitting line; Lower panel: 1D, 2D posterior marginalized probability distributions of the fitting parameters obtained using MCMC method for FRB 121102 detected by Arecibo telescope. The contours in the 2D plots are 68% and 95% confidence intervals from inside to outside. Fig. 2 2Upper panel: The cumulative energy distribution of FRB 121102 detected by GBT telescope at C-band, the solid red line denotes the best fit line using equation(2)with the best-fit parameters labeled in the lower panel, and the black dashed line is the break energy; Lower panel: 1D, 2D posterior marginalized probability distributions of the fitting parameters obtained using MCMC method. The contours in the 2D plots are 68% and 95% confidence intervals from inside to outside. https://www.frbcat.org https://frbtheorycat.org/index.php/Main_Page © Higher Education Press and Springer-Verlag Berlin Heidelberg 2020 Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020) ??-3 ??-4 Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020) Although the fitting results of GBT data and Arecibo data for repeating FRB 121102 are not consistent with each other, we have discussed some possible reasons for the inconsistency in section 3 such as under different observation frequency, a lack of the bursts below the break energy for Arecibo data, or the intrinsic physical mechanism.Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020)??-5 Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020)??-7 Acknowledgements We thank the anonymous referee for constructive comments and suggestions that improved the paper. 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[]
[ "Clustering in nuclei from ab initio nuclear lattice simulations", "Clustering in nuclei from ab initio nuclear lattice simulations" ]
[ "Ulf-G Meißner [email protected] \nHelmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nInstitute for Advanced Simulation\nInstitut für Kernphysik\nJülich Center for Hadron Physics\nUniversität Bonn\n53115BonnGermany\n\nJARA-HPC and JARA-FAME\nForschungszentrum Jülich\n52425Jülich, PisaGermany, Italy\n" ]
[ "Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics\nInstitute for Advanced Simulation\nInstitut für Kernphysik\nJülich Center for Hadron Physics\nUniversität Bonn\n53115BonnGermany", "JARA-HPC and JARA-FAME\nForschungszentrum Jülich\n52425Jülich, PisaGermany, Italy" ]
[ "The 8th International Workshop on Chiral Dynamics" ]
Nuclear Lattice Effective Field Theory is a new many-body approach that is firmly rooted in the symmetries of QCD. In particular, it allows for truly ab initio calculations of nuclear structure and reactions. In this talk, I focus on the emergence of α-clustering in nuclei based on this approach. I also discuss various recent achievements, the deficiencies of the chiral forces used at present and the prospects to improve upon these and the calculations of nuclear properties and dynamics.
10.22323/1.253.0038
[ "https://arxiv.org/pdf/1509.08290v1.pdf" ]
119,303,866
1509.08290
7fadd19f7e8bf3a5bc5e45490c89ee3ecae2563b
Clustering in nuclei from ab initio nuclear lattice simulations 29 June 2015 -03 July 2015 28 Sep 2015 Ulf-G Meißner [email protected] Helmholtz-Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics Institute for Advanced Simulation Institut für Kernphysik Jülich Center for Hadron Physics Universität Bonn 53115BonnGermany JARA-HPC and JARA-FAME Forschungszentrum Jülich 52425Jülich, PisaGermany, Italy Clustering in nuclei from ab initio nuclear lattice simulations The 8th International Workshop on Chiral Dynamics 201529 June 2015 -03 July 2015 28 Sep 2015 Nuclear Lattice Effective Field Theory is a new many-body approach that is firmly rooted in the symmetries of QCD. In particular, it allows for truly ab initio calculations of nuclear structure and reactions. In this talk, I focus on the emergence of α-clustering in nuclei based on this approach. I also discuss various recent achievements, the deficiencies of the chiral forces used at present and the prospects to improve upon these and the calculations of nuclear properties and dynamics. Short introduction Clustering is one of the most fascinating and intricate features of nuclear physics. A special role is played by the α-particle, which is not only very strongly bound (E/A 7.1 MeV) but also spin-and isospin-saturated. While the theoretical foundations for clustering in nuclei have already been laid out in the seminal work of Wheeler [1], there is still lots of on-going research on this phenomenon. Here, I just mention a few recent works related to the α-cluster structure in nuclei, namely the purely geometrical approach of Ref. [2] to explain the spectrum of 16 O 1 , the work on rod-like alpha structures in carbon isotopes at extreme spin and isospin [3] or the work of Ref. [4] that performed systematic studies of α-clustering based on density functional theory. Still, one important question remains unanswered: Can we understand this phenonenom from ab initio studies? By this I mean a framework in which the two-and three-nucleon forces are fixed in few-nucleon systems and based on these forces, the nuclear A-body problem is solved exactly and clustering emerges naturally. As I will show, that it is indeed possible by combining the so successful nuclear chiral Effective Field Theory (EFT) (for a review, see [5], and the talk by Epelbaum for the most recent developments [6]) with lattice simulations, from here on called Nuclear Lattice EFT (NLEFT) or nuclear lattice simulations, for short. Basics of nuclear lattice simulations In this section, I briefly review the main ingredients of nuclear lattice simulations, omitting all technical details. The basic ingredient in this framework is the discretization of space-time, see Ref. [7] for details. Space-time is represented by a grid. This lattice serves as a tool to compute the nuclear matrix elements under consideration. As a first step, we have to perform a Wick rotation in time so that the time evolution operator behaves as exp(−Ht), with H the nuclear Hamiltonian derived within chiral nuclear EFT, and t is the Euclidean time. As a result, space-time is coarsegrained. In the three spatial directions, the smallest distance on the lattice is given by the lattice spacing a, so that the four-dimensional volume is L × L × L × L t , with L = Na and N an integer, whereas in the time direction one uses a different spacing a t , and L t = N t a t is chosen as large as possible. Typical values are N = 6...8 and N t = 10... 15. As the Euclidean time becomes very large, one filters out the ground state as it has the slowest fall-off with Eucldean time ∼ exp(−E 0 t), with E 0 the ground state energy. Excited states can also be investigated. This, however, requires some more effort, but it is considerably easier than extracting excited states in lattice QCD. The nucleons are placed on the lattice sites. Their interactions are given by one-and two-pion exchanges and multi-nucleon operators, properly represented in lattice variables. So far, nuclear lattice simulations have been carried out at next-to-next-to-leading order (NNLO) in the chiral expansion, incorporating two-and three-nucleon forces. The Coulomb force between protons and isospinbreaking strong interaction effects are also included, thus one has all the required ingredients to describe the structure of nuclei. The lattice is used to perform a numerically exact solution of the A-nucleon system, where the atomic number A = N + Z counts the number of neutrons and protons in the nucleus under investigation. It is important to realize that the finite lattice spacing entails a maximum momentum p max = π/a, so that for a typical lattice spacing of a = 2 fm, one has a maximum momentum of about 314 MeV. This means that one deals with a very soft interaction. The main advantage of this approach is, however, the approximate spin-isospin SU(4) symmetry of the nuclear interactions already realized by Wigner in 1937 [8]. Because of this approximate symmetry, the malicious sign oscillations that plague any fermion simulation at finite density are very much suppressed, quite in contrast to the situation in lattice QCD. A lucid discussion of this issue has been given in Ref. [9]. Consequently, α-type nuclei with N = Z, and spin and isospin zero can be simulated most easily. However, in the mean time our collaboration has developed a method that allows for a remarkable suppression of the remaining sign oscillations in nuclei with N = Z, as discussed below. One more important remark on the formalism is in order. The simulation algorithms sample all possible configurations of nucleons, in particular one can have up to four nucleons on one lattice site, see Fig. 1. Thus, the so important phenomenon of clustering in nuclei arises quite naturally in this novel many-body approach, as discussed in more detail later. Most of the results presented in this talk have been obtained with the following lattice set-up: a = 1.97 fm, N = 7, a t = 1.32 fm. The forces have been obtained at NNLO, with 9 two-nucleon low-energy constants (LECs) fixed from fits to S-and P-wave np phase shifts, 2 isospin-breaking NN LECs determined from the nn and pp scattering lengths and two 3N LECs fixed from the triton binding energy and the axial-vector contribution to triton β -decay [10]. Further, there is some smearing required in the LO S-wave four-nucleon terms with its size parameter determined from the average np S-wave effective range [7]. Results from nuclear lattice simulations α-clustering is known to play an important role in the carbon nucleus 12 C, see e.g. Refs. [11,12,13,14]. Ground-breaking studies using nuclear lattice simulations have been performed for the spectrum and structure of 12 C [15], presenting the first ab initio calculation of the so-called Hoyle state [16], that plays a decisive role in the generation of carbon and oxygen in hot old stars. Further, in this framework it is possible to calculate the dependence of the various fine-tunings in the tripleα process as a function of the fundamental parameters of the Standard Model, in particular of the light quark masses and the electromagnetic fine-structure constant [17,18]. A detailed discussion of these issues and the relation to the anthropic principle have been given in Ref. [19]. In the following, I concentrate on more recent results concerning the spectrum and structure of 16 O [20] and the ground-state energies of the α-type nuclei up to 28 Si [21]. I further discuss a method [22] that allows to tame the remaining sign oscillations, so that systematic calculations of halo nuclei or the exploration of the limits of nuclear stability become possible. An important α cluster-type nucleus is 16 O, that also plays an important role in the formation of life on Earth. Since the early work of Wheeler [1], there have been several theoretical and experimental works that lend further credit to the cluster structure of 16 O, see e.g. Refs. [23,24,11,25]. However, no ab initio calculation existed that gave further support to these ideas. This gap was filled in Ref. [20] where nuclear lattice simulations have been used to investigate the low-lying even-parity spectrum and the structure of the ground and first few excited states. It is found that in the spin-0 ground state the nucleons are arranged in a tetrahedral configuration of α-clusters, cf. Fig. 2. For the first 0 + excited state, the predominant structure is given by a square arrangement of α-clusters, as also shown in Fig. 2. There are also rotational excitations of this square configurations that include the lowest 2 + excited state. Note, however, that these snapshots of the wave functions depend on the lattice spacing a. These cluster configurations can be obtained in two ways. First, one can investigate the time evolution of the various cluster configurations and extract e.g. the corresponding energies as the Euclidean time goes to infinity. Second, one can also start with initial states that have no clustering at all. One can then measure the four-nucleon correlations. For such initial states, this density grows quickly with time and reaches a high level. For the cluster initial states, these correlations start out at a high level and stay large as a function of Euclidean time. This is a clear indication that the observed clustering is not built in by hand but rather follows from the strong four-nucleon correlations in the 16 O nucleus. Or, stated differently, if one starts with an initial wave function without any clustering, on a short time scale clusters will form and make up the most important contributions to the structure of 16 O or any type of nucleus, where α-clustering is relevant. We have also performed investigation of the response of 16 O to electromagnetic probes. In Table 1 the NLEFT results at leading order (LO) for the electric quadrupole moment of the 2 + 1 state, the electric quadrupole (E2) transition probabilities, and the electric monopole (E0) matrix element are given. Since the LO charge radius r LO of the ground state is smaller than the empirical value r exp , a systematic deviation appears, which arises from the overall size of the second moment of the charge distribution. To compensate for this overall scaling mismatch, "rescaled" quantities multiplied by powers of the ratio r exp /r LO , according to the length dimension of each observable, were also calculated. With the scaling factor included, we find that the NLEFT predictions for the E2 and E0 transitions are in good agreement with the experimental values. In particular, NLEFT is able to explain the empirical value of B(E2, 2 + 1 → 0 + 2 ), which is 30 times larger than the Weisskopf single-particle shell model estimate. This provides further confirmation of the interpretation of the 2 + 1 state as a rotational excitation of the 0 + 2 state. Finally, we provide a prediction for the quadrupole moment of the 2 + 1 state. In Ref. [21], we addressed the questions: How large a nucleus can be calculated from first principles using the framework of chiral nuclear EFT, and what are the remaining challenges? For finding answers to these questions, we calculated the ground states of α-type nuclei from 4 He LO rescaled Exp. Table 1: NLEFT results for the charge radius r of the ground state of 16 O, the quadrupole moment Q, and the electromagnetic transition amplitudes for E2 and E0 transitions, as defined in Ref. [26]. We compare with empirical (Exp.) values where these are known. For the quadrupole moment and the transition amplitudes, we also show "rescaled" LO results, which correct for the deviation from the empirical value of the charge radius at LO (see main text). The uncertainties are one-standard-deviation estimates which include the statistical Monte Carlo error as well as the errors due to the N t → ∞ extrapolation. r(0 + 1 ) [fm] 2.3(1) - 2.710(15) [27] Q(2 + 1 ) [e fm 2 ] 10(2) 15(3) - B(E2, 2 + 1 → 0 + 2 ) [e 2 fm 4 ] 22(4) 46(8) 65(7) [28] B(E2, 2 + 1 → 0 + 1 ) [e 2 fm 4 ] 3.0(7) 6.2(1.6) 7.4(2) [29] M(E0, 0 + 2 → 0 + 1 ) [e fm 2 ] 2.1(7) 3.0(1.4) 3.6(2) [30] to 28 Si. We note that NLEFT differs from other ab initio methods in that it is an unconstrained Monte Carlo calculation, which does not require truncated basis expansions, many-body perturbation theory, or any constraint on the nuclear wave function. Thus, our NLEFT calculations are truly unbiased Monte Carlo simulations, and the results obtained and presented here are an important benchmark for ab initio calculations of larger nuclei using chiral nuclear EFT. Any deficiencies are indicative of shortcomings in the specific nuclear interactions, rather than of errors generated by the computational method. Such a definitive analysis would be difficult to achieve using other methods. In Ref. [21], a new method to better perform the extrapolation to large Euclidean time in NLEFT was suggested, denoted as "triangulation". The idea is to start from a set of inexpensive low-energy SU(4) symmetric filters that differ by the strength of the SU(4) coupling, given by H SU(4) ≡ H free + 1 2 C SU(4) ∑ n, n : ρ( n) f ( n − n )ρ( n ) : , (3.1) where f is a Gaussian smearing function with its width set by the average effective range of the two S-wave interaction channels, n, n are points on the lattice and ρ is the total nucleon density. For a precise definition of this SU(4) filter and its role in performing nuclear lattice simulations, see [7]. In Fig. 3, the LO ground state energies for 16 O, 20 Ne, 24 Mg and 28 Si are shown for three or four trial SU(4) symmetric states with different (unphysical) coupling constants C SU(4) . One clearly observes that this methods works and it is furthermore encouraging to note that these new extrapolations are consistent with our earlier results for 12 C in Refs. [15,17], which were computed using delocalized plane-wave as well as α-cluster trial wave functions. As can be seen from Fig. 3, there is some overbinding at LO that increases with increasing atomic number A. This feature persists up to NNLO, as shown in Tab. 2. This clearly is a systematic effect that can be traced back to some deficiencies of our soft NNLO forces. In this context, we note that other ab initio methods using soft potentials encounter similar problems in the description of both light and medium-mass nuclei using the same set of interactions [31,32,33]. For making progress, it is useful to explore the nature of the missing physics. As we ascend the α ladder from 4 He to 28 Si, the lighter nuclei can be described as collections of alpha clusters [15,16]. As the number of clusters increases, they become increasingly densely packed, such that a more uniform liquid of nucleons is approached. This increase in the density of α-clusters appears correlated with the gradual overbinding we observe at NNLO for A ≥ 16. As this effect becomes noticeable for 16 O, we can view it as a problem which first arises in a system of four α-clusters. Following Refs. [34,35], which removed discretization errors associated with four nucleons occupying the same lattice site, we can attempt to remove similar errors associated with four α-clusters in close proximity on neighboring lattice sites. The simplest interaction which permits a removal of the overbinding associated with such configurations is of the form V (4N eff ) = D (4N eff ) ∑ 1≤( n i − n j ) 2 ≤2 ρ( n 1 )ρ( n 2 )ρ( n 3 )ρ( n 4 ),(3.2) with ρ( n) the total nucleon density. The summation includes nearest or next-to-nearest neighbor (spatial) lattice sites. One can tune D (4N eff ) e.g. to give approximately the correct energy for the ground state of 24 Mg. With V (4N eff ) included, a good description of the ground-state energies is obtained over the full range from light to medium-mass nuclei, with a maximum deviation from experiment no larger than ∼ 3%, see Tab. 2. This lends support to the qualitative picture that the overbinding of the NNLO results in Table 2 is associated with the increased packing of αclusters and the eventual crossover to a uniform nucleon liquid. The missing physics would then be comprised of short-range repulsive forces that counteract the dense packing of alpha clusters. From this analysis, the path forward for ab initio calculations of heavier nuclei using chiral nuclear EFT appears clear. The softening of the two-nucleon interaction should not be pushed so far that heavier nuclei become significantly over-bound by the two-nucleon force alone. This is not merely an issue for NLEFT, but would appear to be a universal criterion for all ab initio methods. A concerted effort should be made to improve the current computational algorithms to handle interactions with more short-range repulsion. The NLEFT collaboration is now exploring this approach for simulations of larger nuclei. We are now in the process of improving the lattice algorithms for simulations at smaller lattice spacings, and extending NLEFT to N3LO in the chiral expansion. Projection Monte Carlo (PMC) calculations of lattice chiral EFT suffer from sign oscillations to a varying degree dependent on the number of protons and neutrons, which are minimal for the α-type nuclei discussed so far. In Ref. [22], we have introduced the "symmetry-sign extrapolation" (SSE) method, which allows us to use the approximate Wigner SU(4) symmetry of the nuclear interaction to systematically extend the PMC calculations to nuclear systems where the sign problem is severe. For that, one defines the "interpolating Hamiltonian" H as H ≡ d h H LO + (1 − d h )H SU(4) ,(3.3) which depends on d h as well as the (unphysical) coupling constant C SU(4) , cf. Eq. (3.1). This can also be viewed as giving the interaction parameters a linear dependence on d h . By taking d h < 1, we can always decrease the sign problem to a tolerable level, while simultaneously tuning C SU(4) to a value favorable for an extrapolation d h → 1. Most significantly, we can make use of the constraint that the physical result at d h = 1 should be independent of C SU (4) . The dependence of calculated matrix elements on d h is smooth in the vicinity of d h = 1. We note that an extrapolation technique similar to SSE has been used in Shell Model Monte Carlo calculations for over two decades [36,37]. In that case, the extrapolation is performed by decomposing the Hamiltonian into "good sign" and "bad sign" parts, H G and H B , respectively. The calculations are then performed by multiplying the coefficients of H B by a parameter g and extrapolating from g < 0, where the simulations are free from sign oscillations, to the physical point g = 1. For SSE, the analysis in terms of "good" and "bad" signs is not the entire story. Most of the interactions can be divided into two groups which are "sign free" by themselves, such that a large portion of the sign oscillations is due to interference between the different underlying symmetries of the two groups of interactions. Since this effect is quadratic in the interfering interaction coefficients, the growth of the sign problem is more gradual. We therefore expect to be able to extrapolate from values not so far away from the physical point d h = 1. We first discuss results for the 12 C nucleus, as this provides us with a convenient test case for the SSE method. We extend the PMC calculation to larger values of the Euclidean projection time than would otherwise be possible, and verify how well these new results agree with earlier calculations. Thus, we shall work at finite Euclidean time, extrapolate such data to d h → 1, after which the extrapolation L t → ∞ is performed. For 12 C, we have performed PMC calculations using SSE for Euclidean projection times between N t = 9.0 and 14.5 for one of the trial states used in Ref. [21]. It should be noted that the results of Ref. [21] correspond to d h = 1, and could thus only be extended to N t 10 before the sign problem became prohibitive. In Fig. 4, we extend the dataset of Ref. [21] with the new SSE data. This clearly establishes that the SSE data are consistent with those obtained at d h = 1, when such calculations are not prohibited by the sign problem, for more details, see Ref. [22]. N t -7.0e-5 -7.0e-5 + SSE Figure 4: Comparison of the new PMC data for 12 C from the SSE analysis (red filled squares) and previous calculations [21] for d h = 1 (blue open squares). The contributions from the improved leading order amplitude (LO), the two-nucleon force at next-to-leading order (NLO), the electromagnetic and strong isospin breaking (EMIB) and the three-nucleon force at next-to-next-to-leading order (3NF) are shown separately. The results correspond to a trial state with an SU(4) coupling of −7.0 × 10 −5 MeV −2 , not to be confused with the SU(4) coupling C SU(4) for the SSE analysis. It should be noted that the exponential deterioration of the Monte Carlo error has been circumvented. Note these data should not be interpreted in terms of a plateau. For more details, see Ref. [22]. The sign problem in the A = 6 system with 2 protons and 4 neutrons (or vice versa) is somewhat more severe than for 12 C. Hence, if calculations are performed entirely at d h = 1, the extrapolation to infinite Euclidean time (while still feasible) has to be performed using data with a rather limited range in N t . However, for d h < 1 this situation improves rapidly. For 6 He and 6 Be, one should therefore approach the problem differently than for 12 C. We perform the extrapolation in Euclidean time for each pair of C SU(4) and d h , with the extrapolation d h → 1 as the final step of the analysis. A detailed discussion of the simulations performed in these systems is given in Ref. [22]. Here, we just add some remarks. The SSE method introduced is inspired by the existence of an SU(4) symmetric Hamiltonian which provides a reasonably accurate description of the physics of the full NLEFT Hamiltonian. This has already proven useful in the work discussed above, as it greatly facilitates finding an accurate initial wave function which minimizes the extent of Euclidean time projection necessary with the full Hamiltonian. With the SSE, this concept can be taken one step further, by studying a weighted sum of physical and SU(4) symmetric Hamiltonians. In this way, the sign problem could be arbitrarily ameliorated, at the price of introducing an extrapolation in a control parameter d h → 1. In practice, this means that the SSE method is only useful as long as the extrapolation errors can be kept under control. Naturally, performing simulations at a range of values of d h has the potential to multiply the required CPU time by a large factor. However, we have found that we are able to avoid an exponential increase in computation time as a function of Euclidean projection time, as long as we are able to perform simulations for d h > 0.75, as the accuracy of extrapolation then remains comparable with the statistical errors of typical simulations at d h = 1. We have also explored the freedom in the choice of the SU(4) symmetric Hamiltonian, which clearly plays no role at d h = 1, but which in general gives different results for d h = 1. We have therefore made use of a "triangulation" method to improve the accuracy of the extrapolation d h → 1. An important consideration is whether a continuous shift from an SU(4) symmetric Hamiltonian to the full chiral EFT Hamiltonian can be effected without inducing non-trivial changes in the spectrum. For instance, the appearance of a level crossing at a critical value of d h would clearly limit the applicability of SSE. We note that such level crossings as a function of d h would be quite rare for low-lying nuclear bound states, and furthermore we have the freedom to choose C SU(4) to avoid such level crossings. But if a level crossing were to occur, there would be some subtleties in obtaining accurate and converged results. For such cases, it would be preferable to take the d h → 1 limit first, followed by extrapolation in Euclidean time. An even better solution would entail solving a coupled-channel problem using multiple initial states. This makes it possible to disentangle one or more nearly degenerate states with the same quantum numbers. So far, we have concentrated on systems where such degeneracies are not expected, and where PMC is still possible (though difficult) without the SSE method. One also should explore different "extrapolation Hamiltonians", where the sign oscillations are minimized while retaining as much as possible of the full chiral EFT structure. Further developments Here, I summarize some further important developments which are partly covered by the talks in this session, and provide additional references: • Lattice spacing dependence: Most of the simulations reported here have been performed on a coarse lattice with a lattice spacing a = 1.97 fm. This raises the question whether there are sizeable lattice artifacts related to this spacing. An important first step in this direction is the study on the lattice spacing dependence in Ref. [38]. There, the two-body system for lattice spacings from a = 0.5 fm to a = 2.0 fm at lowest order in the pionless as well as in the pionful theory was investigated. In the pionless case, a simple Gaussian smearing allows to demonstrate lattice spacing independence over a wide range of lattice spacings. It was also shown that regularization methods known from the continuum formulation [39] are necessary as well as feasible for the pionful approach and will lead to a-independent results. In α-cluster models, the a-dependence of various observables was also studied, see Refs. [40,41]. For further work on this issue, see the contribution from Alarcon to these proceedings [42]. • Breaking and restoration of rotational symmetry: On a periodic lattice, the rotational symmetry SO (3) is broken down to the cubic group SO(3, Z), that has five irreducible representations. In Ref. [40] the breaking of rotational symmetry on the lattice for bound state energies and practical methods for suppressing this breaking were explored. The general problems associated with lattice discretization errors and finite-volume errors were discussed using an α-cluster model for 8 Be and 12 C. The focus was put on the lowest states with nonzero angular momentum which split into multiplets corresponding to different irreducible representations of the cubic group. The dependence of such splittings on the lattice spacing and the box size was examined. The lattice spacing errors are closely related to the commensurability of the lattice with the intrinsic length scales of the system. Rotational symmetry breaking effects can be significantly reduced by using improved lattice actions, and the physical energy levels are accurately reproduced by the weighted average of a given spin multiplets. This was extended to the study of irreducible tensor operators in Ref. [41]. The lowest states with non-zero angular momentum were considered and the matrix elements of multipole moment operators were examined. The physical reduced matrix element is well reproduced by averaging over all possible orientations of the quantum state, and this is expressed as a sum of matrix elements weighted by the corresponding Clebsch-Gordan coefficients. For our α-cluster model, we find that the effects of rotational symmetry breaking can be largely eliminated for lattice spacings of a ≤ 1.7 fm, and we expect similar improvement for actual lattice Monte Carlo calculations. For related work on the restoration of rotational symmetry in the continuum limit of lattice field theories see [43]. • Discretized forces to higher orders and extraction of NN phase shifts: Clearly, the forces at NNLO used so far are not sufficiently accurate, as witnessed e.g. by the overbinding problem discussed before. This is to some extent also related to the extraction of the nucleonnucleon phase shifts based on the so-called spherical wall method, that was developed for NLEFT in Ref. [44] (see also Ref. [45] for a similar approach). This method is better suited for nuclear physics as it deals much better with partial-wave mixing than Lüscher's wellknown approach. However, to further improve the precision of the determination of phase shifts and mixing angles on the lattice, a novel approach was presented in Ref. [46]. It starts with angular momentum projection on the lattice wave functions in order to construct lattice radial wave functions. One then imposes spherical wall boundaries together with an adjustable auxiliary potential to determine phase shifts at arbitrary energy, V aux (r) ≡ V 0 exp −(r − R W ) 2 /a 2 , R 0 ≤ r ≤ R W ,(4.1) where R W is the wall boundary, V 0 the strength of the auxiliary potential, a is the lattice spacing and R 0 is chosen to be larger than the range of the interaction but smaller than R W . For coupled partial waves, a complex-valued auxiliary potential is used that breaks time-reversal invariance in order to extract phase shifts and mixing angles from the complex-valued wave functions. The method was benchmarked using a system of two spin-1/2 particles interacting through a finite-range potential with a strong tensor component, also used in Ref. [44]. Scattering parameters can be extracted precisely for all angular momenta and energies, see Fig. 5. Using this new method and an improved smearing, the NLEFT collaboration is presently working out the forces to N3LO. • Ab initio calculation of scattering and inelastic processes: Scattering processes and inelastic reactions like radiative neutron capture on the proton are of major importance to understand the generation of the element in the Big Bang and in stars. Such processes can be tackled using the recently proposed adiabatic projection method (APM) that allows to construct an effective cluster Hamiltonian from the underlying chiral EFT Hamiltonian, see Refs. [47,48] and the talk by Rokash [49]. This method is similar to the NCSM/RGM approach to nuclear scattering developed in Ref. [50]. We are presently working on various scattering reactions. 3 P 0 ) δ( 3 P 1 ) δ( 3 S 1 ) δ( 3 D 1 ) ε 1 ( S -D ) δ( 3 D 2 ) p h a s e s h i f t s δ / m i x i n g a n g l e s ε ( d e g r e e s ) δ( 3 P 2 ) δ( 3 F 2 ) ε 2 ( P -F ) δ( 3 F 3 ) δ( 3 D 3 ) δ( 3 G 3 ) ε 3 ( D -G ) δ( 3 G 4 ) p C M ( M e V ) δ( 3 F 4 ) δ( 3 H 4 ) ε 4 ( F -H ) Outlook NLEFT has enjoyed a number of remarkable successes like the Hoyle state calculation, the investigations of the fine-tunings in the triple-alpha process or the ab initio calculation of 16 O based on the chiral EFT forces at NNLO. After these promising results, it is now time to consolidate this framework in the sense that the remaining inaccuracies of the underlying forces should be eliminated and more detailed investigations of nuclear structure and dynamics have to be performed. Just to mention the rich experimental information from electron scattering, the quest to understand the limits of the nuclear stability or the generation of elements in the Big Bang and in stars through nuclear reactions. Time is ripe to make real progress in all these areas. This requires more manpower than provided by the NLEFT collaboration and I hope that the physics presented here will stimulate other groups to use and further develop this powerful tool. Figure 1 : 1Topologically different configurations to put nucleons on a lattice. From the left upper to the lower right corner: all four nucleons on different sites, two/three/four nucleons on one site, respectively. Figure 2 : 2Tetrahedral (left) and square (right) configurations in16 O. Figure 3 : 3NLEFT results for the LO transient energy E A (t) for A = 16, 20, 24, 28, with C SU(4) given (in MeV −2 ) for each trial state, cf. Eq. (3.1). For definitions, see Ref.[21]. Figure 5 : 5Phase shifts and mixing angles for J ≤ 4 in the S = 1 channel. Full, open and half-open squares denote the results obtained with auxiliary potential strength V 0 = 0, V 0 = −25 MeV and V 0 = −20 MeV, respectively. For V 0 = −20 MeV only selected results are shown. Solid lines denote continuum results. For details, see Ref.[46]. Table 2 : 2NLEFT results for the ground-state energies of the alpha-cluster nuclei. The combined statistical and extrapolation errors are given in parentheses. The rows labelled E(LO) and E(NNLO) give the results at LO and NNLO, in order. The row E(NNLO+4N eff ) includes the effective 4N contribution from Eq. (3.2). Finally, the row Exp. gives the empirical energies. All energies are in MeV. This work was indeed inspired by the nuclear lattice simulation results reported below. Molecular Viewpoints in Nuclear Structure. J A Wheeler, Phys. Rev. 521083J. A. Wheeler, "Molecular Viewpoints in Nuclear Structure," Phys. Rev. 52 (1937) 1083. 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[]
[ "Toroidal Soliton Solutions in O(3) N Nonlinear Sigma Model", "Toroidal Soliton Solutions in O(3) N Nonlinear Sigma Model" ]
[ "A Wereszczyński .27.d*[email protected] \nInstitute of Physics\nJagiellonian University\nReymonta 4KrakówPoland\n" ]
[ "Institute of Physics\nJagiellonian University\nReymonta 4KrakówPoland" ]
[]
A set of N three component unit scalar fields in (3 + 1) Minkowski space-time is investigated. The highly nonlinear coupling between them is chosen to omit the scaling instabilities. The multi-soliton static configurations with arbitrary Hopf numbers are found. Moreover, the generalized version of the Vakulenko-Kapitansky inequality is obtained. The possibility of attractive as well as repulsive interaction between hopfions is shown. A noninteracting limit is also discussed.keyword: Hopf solitons
10.1142/s0217732304015737
[ "https://arxiv.org/pdf/math-ph/0405054v2.pdf" ]
119,143,708
math-ph/0405054
1debc2ebc0e3075de01b3d0d84f239b347724177
Toroidal Soliton Solutions in O(3) N Nonlinear Sigma Model 13 Sep 2004 November 13, 2018 A Wereszczyński .27.d*[email protected] Institute of Physics Jagiellonian University Reymonta 4KrakówPoland Toroidal Soliton Solutions in O(3) N Nonlinear Sigma Model 13 Sep 2004 November 13, 20181 A set of N three component unit scalar fields in (3 + 1) Minkowski space-time is investigated. The highly nonlinear coupling between them is chosen to omit the scaling instabilities. The multi-soliton static configurations with arbitrary Hopf numbers are found. Moreover, the generalized version of the Vakulenko-Kapitansky inequality is obtained. The possibility of attractive as well as repulsive interaction between hopfions is shown. A noninteracting limit is also discussed.keyword: Hopf solitons Introduction Topological defects are one of the main ideas of the temporary theoretical physics. They play crucial role in many models -from cosmological strings and textures [1], [2] to magnetic monopoles and vortices in Quantum Chromodynamics [3], [4]. On the other hand, topological defects are observed in various condense matter experiments -see for instance liquid crystals and 3 He [5] or 4 He quantum liquids [6]. Among quite well understood topological defects like domain walls, strings/vortices and monopoles toroidal-like configurations, that is solitons with nonzero value of the Hopf index, seem to be rather mysterious objects. Since Faddeev, Niemi [7], [8] and Cho [9] have proposed their famous effective model for the low energy gluodynamics, where glueballs (particles which consist of only the gauge fields) are knot-like objects made of self-linking fluxtubes, toroidal solitons, their physical as well as mathematical aspects have been considered by many theoreticians 1 . Unfortunately, due to the rather complicated toroidal symmetry and nonlinearity of equations of motion no analytical results, in recently investigated QCD relevant models [12], [13], [14], have been found. Almost all results have been obtained by means of numerical methods [15], [16], [17], [18]. In order to deal with exact toroidal solutions one is forced to consider various toy models. All such theories, providing analytical description of hopfions, are based on the quite old idea presented by Deser et al. [19]. They have investigated a classical highly nonlinear field theory with a form of the Lagrangian chosen to circumvent Derrick's theorem. In fact a soliton solution with unit Hopf index [20] and then configurations with arbitrary value of the Hopf invariant have been found [21]. Moreover, nontrivial topological solutions have been obtained in more complicated model [22], for example with explicitly broken O(3) symmetry [23]. Using the exact soliton solutions one has checked that the Vakulenko-Kapitanski [24] inequality is fulfilled in all these models. It should be mentioned that soliton solutions with non-trivial Hopf index can be achieved also in another famous model i.e. Skyrme model in (3 + 1) dimensions [25], [26]. Moreover, there is also possibility to analyze hopfions in less dimensional space-time. It is based on the observation that the Faddeev-Niemi knots can be understood as twisted magnetic vortex rings built of so-called new baby skyrmions found in (2 + 1) anisotropic Skyrme model [27]. In the present paper we would like to investigate toroidal soliton solutions in a model which consists of many unit vector fields i.e. allows for many topological charges. The appearance of more than one unit vector field has been observed in recently derived non-Abelian generalization of the color dielectric model [14]. Indeed, there are two unit fields in this model. The main aim, besides finding exact form of the topological solutions and their energies and Hopf charges, is to analyze Vakulenko-Kapitanski formula in case of more than one Hopf index. 2 The model L = N i=1 [ n i · (∂ µ n i × ∂ ν n i )] 2 α i (1) which is a generalized version of the Aratyn-Fereira-Zimerman model with N unit fields. It can be rewritten as L = N i=1 (H (i)2 µν ) α i , where H (i) µν = n i · (∂ µ n i × ∂ ν n i ) are antisymmetric field tensors. In order to omit the Derick arguments for nonexistence of static soliton solutions one has to assume the condition N i=1 α i = 3 4 .(2) In fact, now the total energy becomes invariant under the scale transformations. Now, after taking advantage of the stereographic projection, we can express each of the unit fields in terms of two complex functions u and u * : n i = 1 1 + |u i | 2 (u i + u * i , −i(u i − u * i ), |u i | 2 − 1).(3) Then the Lagrangian density reads L = 8 3/4 N i=1 1 (1 + |u i | 2 ) 4α i (K (i) µ ∂ µ u * i ) α i ,(4) where we have introduced a set of the objects: K (i) µ = (∂ ν u * i ∂ ν u i )∂ µ u i − (∂ ν u i ∂ ν u i )∂ µ u * i ,(5) where i = 1...N. It is straightforward to check that they fulfill the following conditions K (i) µ ∂ µ u i = 0 and Im(K (i) µ ∂ µ u * i ) = 0.(6) The equations of motion take the form ∂ µ     N i=1,i =j (K (i) µ ∂ µ u * i ) α i (1 + |u i | 2 ) 4α i   2α j (K (j) µ ∂ µ u * j ) α j −1 (1 + |u j | 2 ) 4α j K (j)µ   − − N i=1,i =j (K (i) µ ∂ µ u * i ) α i (1 + |u i | 2 ) 4α i (K (j) µ ∂ µ u * j ) α j 1 (1 + |u j | 2 ) 4α j ′ u * j = 0.(7) After some calculation one can rewrite them as follow ∂ µ     N i=1,i =j (K (i) µ ∂ µ u * i ) α i (1 + |u i | 2 ) 4α i   (K (j) µ ∂ µ u * j ) α j −1 (1 + |u j | 2 ) 4α j −2 K (j)µ   = 0,(8) or in the more compact form ∂ µ K (j)µ = 0,(9) where K (j) µ =   N i=1,i =j (K (i) µ ∂ µ u * i ) α i (1 + |u i | 2 ) 4α i   (K (j) µ ∂ µ u * j ) α j −1 (1 + |u j | 2 ) 4α j −2 K (j)µ .(10) It is easy to show that these quantities also fulfill the previously found conditions (6). Because of that we can generalize the procedure defined in [21] and construct the infinity families of the conserved currents J (i) µ = K (i)µ ∂G i ∂u i − K * (i)µ ∂G i ∂u * i ,(11) where G i , i = 1...N are any functions of u j and u * j , j = 1..N. Due to that presented O(3) N invariant model is integrable in the sense that N infinite families of the conserved currents can be found [28], [29]. Let us proceed further and prove that, as in case of soliton models in (1 + 1) and (2 + 1) dimension, the existence of infinite number of the conserved currents leads to topological soliton solutions. We begin with introducing of the toroidal variables x = a q sinh η cos φ, y = a q sinh η sin φ, z = a q sin ξ,(12) where q = cosh η − cos ξ and the constant a > 0 sets the scale of the coordinates. However, one should keep in mind that the model is scale invariant. Due to that, there is no scale on the solution level. All solitons, irrespective of size, have identical energy. In addition we assume a natural generalization of the Aratyn-Feriera-Zimerman Ansatz [21] u i (η, ξ, φ) ≡ f i (η)e i(m i ξ+n i φ) , i = 1...N,(13) where m i , n i , i = 1...N are integral parameters. Inserting the Ansatz into the field equations (9) we derive at the following static equations ∂ η ln     N i=1,i =j (f i f ′ i ) 2α i (1 + f 2 i ) 4α i   (f j f ′ j ) 2α j −1 (1 + f 2 j ) 4α j −2   = = −∂ η ln sinh η N i=1 m 2 i + n 2 i sinh 2 η α i .(14) These equations can be integrate and we obtain that   N i=1,i =j (f i f ′ i ) 2α i (1 + f 2 i ) 4α i   (f j f ′ j ) 2α j −1 (1 + f 2 j ) 4α j −2 = k j sinh η N i=1 m 2 i + n 2 i sinh 2 η −α i ,(15) where the integration constants k j , j = 1...N have been introduced. One can show that (15) can be reduced to the set of N decoupled first order differential equations f j f ′ j (1 + f 2 j ) 2 = 1 k j   N i=1 k 4α i i m 2 i + n 2 i sinh 2 η −2α i   1 sinh 2 η ,(16) with the following general solutions 1 1 + f 2 j = 1 k j   N i=1 k 4α i i m 2 i + n 2 i sinh 2 η −2α i   dη sinh 2 η + l j .(17) Here l j , j = 1...N are integration constants. Now, we will analyze the total energy corresponding to our model (1): E ≡ d 3 xT 00 = 8 3/4 d 3 x N i=1 (K (i) j ∂ j u * i ) α i (1 + |u i | 2 ) 4α i ,(18) where the stereographic projection (3) has been taken into account. Inserting (5) and the Ansatz (13) into (18) we derive that E m,n = (2π) 2 8 · 2 3/4 ∞ 0 dη sinh η N i=1 (f i f ′ i ) 2α i (1 + f 2 i ) 4α i m 2 i + n 2 i sinh 2 η α i(19) Quite surprisingly, using the equations (16) we are able to remove the unknown functions f j from this formula. Then the total energy integral takes the form E m,n = (2π) 2 8 · 2 3/4 N i=1 k 4α i i ∞ 0 dη sinh 2 η N i=1 m 2 i + n 2 i sinh 2 η −2α i ,(20) which can be evaluated, at least by means of some numerical methods, for all values of the parameters m i , n i , α i . One can notice that behavior of the integral (and difficulty of the calculation) strongly depends on a particular value of the following ration n 2 i m 2 i = q 2 i .(21) In the next section the simplest case, when all these numbers are equal will be investigated. 3 Case q 2 = const. Let us focus on the case when n 2 i m 2 i = q 2 = const.,(22) for all i = 1...N. Then the equations of motion (16) can be simplified to the following expressions f j f ′ j (1 + f 2 j ) 2 = 1 k j N i=1 k 4α i i m 2 i + n 2 i sinh 2 η −2α i 1 sinh 2 η .(23) Moreover, these equations can be integrated out and finally we obtain that 1 1 + f 2 j = 1 k j 1 1 − q 2 N i=1 k 4α i i m −4α i i cosh η q 2 + sinh 2 η 1 2 + l j ,(24) where j = 1...N. Here we have taken q 2 < 1. In order to find the value of the integration constants we need to specify the asymptotical conditions for the functions f j . We choice them in the form which admits the nontrivial topological structure [21] n → (0, 0, 1) i.e. f → ∞ as η → 0 and n → (0, 0, −1) i.e. f → 0 as η → ∞. Then we find that N i=1 k 4α i i k j = 1 2 N i=1 m −4α i 1 − q 2 |q| − 1 |q|.(27) This is equivalent to the fact that all integration constants have to take the same value k j = k = const. In addition, one can obtain that l j = l = − 1 |q| − 1 .(29) Finally, the solution is given by the following formula 1 1 + f 2 j = 1 1 − |q| 1 − |q| cosh η q 2 + sinh 2 η .(30) One can recognize here the set of N Aratyn-Fereira-Zimerman toroidal solutions. Let us now show that the solution possesses toroidal symmetry. The components of the unit vector fields read n 1 (i) = 2f i 1 + f 2 i cos(m i ξ + n i φ),(31)n 2 (i) = 2f i 1 + f 2 i sin(m i ξ + n i φ),(32)n 3 (i) = −1 + f 2 i 1 + f 2 i ,(33) where f i is given by (30) i.e. depends only on the radial-like coordinate η. It is clearly visible that surfaces of constant n 3 (i) are torii. After inserting obtained here solution into the total energy integral one finds that E m,n = (2π) 2 4 · 2 1/4 N i=1 m 2α i i 1 + |q| |q|.(34) In the limit m α i = m, i = 1...N the formula founded in [21] can be reproduced. In order to prove that our solutions possess non-trivial topology one has to calculate value of the pertinent topological invariant. In our case static field configurations, which can be viewed as maps from S 3 into S 2 , are classified by so-called Hopf index Q H = π 3 (S 2 ). One should notice that because of the fact that we deal with N fields thus N different topological Hopf charges Q (i) H have to be introduced. We define two additional functions g 2 1 = cosh η − n 2 m 2 + sinh 2 η(35) and g 2 2 = 1 + m 2 n 2 sinh 2 η − cosh η.(36) Then, for every field we introduce four objects Φ (i) ( 1 2 ) =   g 1 g 2 1 + g 2 2   × cos m i ξ sin m i ξ(37) and Φ (i) ( 3 4 ) =   g 2 g 2 1 + g 2 2   × cos n i φ − sin n i φ .(38) They are related with the i-th unit vector field via the standard relations n i = Z †i σZ i , where σ are Pauli matrices and Z i = Z i 1 Z i 2 , Z i † = (Z i * 1 , Z i * 2 )(39) and Z i 1 = Φ (i) 1 + iΦ (i) 2 , Z i 2 = Φ (i) 3 + iΦ (i) 4 .(40) Using these objects one can construct N Abelian vector fields which form the previously introduced antisymmetric field tensors H (i) kl = ∂ k A (i) l − ∂ l A (i) k . In fact one finds that A (i) k = i 2 (Z i † ∂ k Z i − ∂ k Z i † Z i ).(41) Then, using the standard expression for the Hopf index Q (i) H = 1 4π 2 d 3 x A (i) · B (i) ,(42) we arrive at the following result Q (i) H = n i m i 2 ((Φ (i) 1 ) 2 + (Φ (i) 2 ) 2 ) 2 − ((Φ (i) 3 ) 2 + (Φ (i) 4 ) 2 ) 2 ∞ 0 = −n i m i . (43) In other words our soliton solutions are classified by set of N Hopf charges. Every kind of the unit vector field carries its own topological charge which is unique determined by the parameters m i , n i . Now, one could ask about generalization of Vakulenko-Kapitansky inequality for N vector fields. Knowing that 1 + |q| √ q ≥ √ 2|q| 3/4 we obtain the generalized relation E ≥ (2π) 2 4 · 2 3 4 N i=1 Q α i (i) ,(44) where N i=1 α i = 3 4 . If all topological numbers are equal Q (i) = Q, i = 1..N we reproduce the standard Vakulenko-Kapitansky formula. Conclusions In the present work a generalized Aratyn-Ferreira-Zimerman model has been analyzed. This model consists of N unit vector fields and possesses N independent O(3) symmetries. We have proved that this model is integrable in the sense that infinite number of the conserved currents appears. Precisely speaking, one can find a set of N infinite families of the currents. We have also shown that, as in the case of the soliton theories in (2 + 1) dimensions, this property leads to the existence of non-trivial topological field configurations. Our soliton solutions are described by shape functions depending on η coordinate and angular terms characterized by set of two integer parameters m i and n i . The total energy has been also found. In general, it is given by the integral (20). The exact form of the shape functions has been obtained in the limit of the constant ratio of the Ansatz parameters n 2 i /m 2 i = q 2 = const. Then the shape functions take form of the well-know Aratyn-Ferreira-Zimerman solution. However, obtained solutions are much more general due to the angular part which for each of the unit vector field can be different. Because of that the topological behavior of the solutions is changed -now we have N independent topological Hopf indexes. In other words we have obtained multi-soliton configurations classified by N topological numbers Q In our opinion, the most interesting result is the generalized Vakulenko-Kapitanski inequality. As we see this formula can be divided into two parts. The first part contains only numbers and seems to be universal i.e. Lagrangian independent. In fact, it is identical as in the simple one vector field case. The second part, which includes the Hopf indexes, appears to be strongly sensitive to the form of the model. To show it more precisely one can consider a simple sum of N standard Aratyn-Ferriera-Zimerman Lagrangians L = N i=1 (H (i)2 µν ) 3/4 i.e. a model with the noninteracting unit vector fields. One can immediately show that the pertinent Vakuleko-Kapitanski formula reads E ≥ (2π) 2 4 · 2 3 4 N i=1 |Q (i) | 3 4 . We see that the second part of the inequality reflects a form of the Lagrangian. However, a common feature is shared by both inequalities -there is an universal scaling property. Namely, if we transform all topological charges Q (i) → λQ (i) then the total energy will scale as E → λ 3 4 E. Let us now analyze the physical meaning of the parameters α i . In spite of the fact that the form of the solution, at least in the q 2 i = const. regime, does not depend on their values, they play crucial role in the interaction of the toroidal solitons. Indeed, interaction between hopfions in the proposed model is governed by the values of α i and it can be attractive or repulsive. Quite interesting hopfions can even do not interact at all. To be more precisely we will consider two hopfions with the following topological charges Q a = (Q (1) a , .., Q (N ) a ), a = 1, 2. It follows from the formula (44) that if α i < 1 then i-th hopfions built of the i-th vector field should attract each other whereas for α i > 1 this interaction will be repulsive. In the very special case where α i = 1, they do not interact at all. It resembles the Bogomolny limit in the standard soliton systems. As there is only one constrain on the parameters α i (2), all three situations can be realized. Thus, in general, one can deal with a model where every possible type of interaction occurs. As we have mentioned it above, the analytical solutions have been obtained only in the simplest situation i. e. for q 2 i = const.. It constrains also the validity of the generalized Vakulenko-Kapitanski inequality. Thus one should investigate more complicated cases with q 2 i = q 2 j as well. We would like to address this issue in our next paper. This work is partially supported by Foundation for Polish Science and ESF "COSLAB" programme. us consider a set of N unit three component scalar fields n i , i = 1...N, n 2 i = 1, living in the (3 + 1) Minkowski space-time. The Lorentz and O(3) N invariant Lagrangian density is chosen in the following form H , where each of the vector field carries different topological charge. These configurations consist of N toroidal solitons of the different types. It should be stressed that all properties of the solutions (topological charge, total energy), at least in the q 2 i = const. case, are determined only by value of the parameters m i , n i . For example, behavior of the shape functions f i depends only on the ratio between them whereas number of the fields does not affect the shape functions. Regardless of the particular number of the fields N, all functions f i possess identical form. On the other hand, number of the vector fields defines the topological content of the model. 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[ "Stefan Fredenhagen [email protected]&[email protected] \nInstitut des Hautes Etudes Scientifiques\nVolker Schomerus Service de Physique Théorique\nCEA Saclay\nF-91440, F-91191Bures-sur-Yvette, Gif-sur-YvetteFrance, France\n" ]
[ "Institut des Hautes Etudes Scientifiques\nVolker Schomerus Service de Physique Théorique\nCEA Saclay\nF-91440, F-91191Bures-sur-Yvette, Gif-sur-YvetteFrance, France" ]
[]
The c = 1 Liouville theory has received some attention recently as the Euclidean version of an exact rolling tachyon background. In an earlier paper it was shown that the bulk theory can be identified with the interacting c = 1 limit of unitary minimal models. Here we extend the analysis of the c = 1-limit to the boundary problem. Most importantly, we show that the FZZT branes of Liouville theory give rise to a new 1-parameter family of boundary theories at c = 1. These models share many features with the boundary Sine-Gordon theory, in particular they possess an open string spectrum with band-gaps of finite width. We propose explicit formulas for the boundary 2-point function and for the bulk-boundary operator product expansion in the c = 1 boundary Liouville model. As a by-product of our analysis we also provide a nice geometric interpretation for ZZ branes and their relation with FZZT branes in the c = 1 theory.SPhT-T04/121hep-th/0409256 IHES/P/04/42
10.1088/1126-6708/2005/05/025
[ "https://export.arxiv.org/pdf/hep-th/0409256v2.pdf" ]
18,120,604
hep-th/0409256
6bab628b48f4f9c9aee81ee07990c0d1ed1e32b0
September, 2004 Stefan Fredenhagen [email protected]&[email protected] Institut des Hautes Etudes Scientifiques Volker Schomerus Service de Physique Théorique CEA Saclay F-91440, F-91191Bures-sur-Yvette, Gif-sur-YvetteFrance, France September, 2004arXiv:hep-th/0409256v2 27 Apr 2005 Boundary Liouville theory at c = 1 The c = 1 Liouville theory has received some attention recently as the Euclidean version of an exact rolling tachyon background. In an earlier paper it was shown that the bulk theory can be identified with the interacting c = 1 limit of unitary minimal models. Here we extend the analysis of the c = 1-limit to the boundary problem. Most importantly, we show that the FZZT branes of Liouville theory give rise to a new 1-parameter family of boundary theories at c = 1. These models share many features with the boundary Sine-Gordon theory, in particular they possess an open string spectrum with band-gaps of finite width. We propose explicit formulas for the boundary 2-point function and for the bulk-boundary operator product expansion in the c = 1 boundary Liouville model. As a by-product of our analysis we also provide a nice geometric interpretation for ZZ branes and their relation with FZZT branes in the c = 1 theory.SPhT-T04/121hep-th/0409256 IHES/P/04/42 Introduction The study of tachyons and their condensation processes has been a longstanding challenge in string theory. During recent years, the interest in branes has added significantly to the relevance of this issue since open string tachyons occur in various configurations of stable branes, the most important being the brane-antibrane system. Fortunately, unstable brane configurations turned out to be much more tractable than bulk backgrounds with instabilities such as e.g. the 26-dimensional bosonic string. In fact, various different approaches have been employed over the last five years, including string field theory (see e.g. [1] and [2] for a review), effective field theory models (see [3,4] for some early work), and world-sheet conformal field theory (see e.g. [5,6] and references therein). In [7], Sen initiated the study of exact time-dependent solutions. One goal of his work was to find a world-sheet description of so-called S-branes [8], i.e. of branes that are localized in time. In this context he proposed to add a cosh-shaped boundary interaction to the usual world-sheet model of D-branes in a flat space-time, S S [X] = 1 4π Σ d 2 z η µν ∂X µ∂ X ν + ∂Σ du λ cosh X 0 (u) . (1.1) Here, X 0 denotes the time-coordinate and we have chosen units in which α ′ = 1. With such an interaction term the open string tachyon becomes large at early and late times and hence, according to an earlier proposal of Sen [9], is believed to dissolve the brane for |x 0 | → ∞. Unfortunately, the world-sheet model (1.1) appears to be ill-defined without an additional prescription to select the proper vacuum. To resolve this issue, Sen argued that the appropriate choice for super-string computations would be determined by Wick rotation from the corresponding Euclidean theory. This suggestion relates the study of open string tachyon condensation to the so-called boundary Sine-Gordon model S BSG [X] = 1 4π Σ d 2 z ∂X∂X + ∂Σ du λ cos X(u) in which X = iX 0 is a field with space-like rather than time-like signature. All spatial coordinates X µ , µ = 0, have been suppressed here since their contribution to the model is not affected by the interaction. The boundary Sine-Gordon theory has been studied rather intensively, see e.g. [10,11,12,13,14]. Let us briefly review some of the most important results. To begin with, we point out that the boundary interaction is exactly marginal so that the theory is conformal for all values of λ. Properties of the model, however, depend crucially on the strength λ of the boundary potential. In fact, it is well known that variations of λ allow to interpolate smoothly between Neumann and Dirichlet boundary conditions. The former appear for all integer values of λ (in particular for λ = 0, of course) while the latter are reached when λ ∈ 1 2 + Z. At these points, the theory describes a one-dimensional infinite lattice of D0 branes. From the geometric pictures we can infer that the spectra of boundary excitations must also depend drastically on the parameter λ. In fact, at the points λ ∈ Z with Neumann boundary conditions, the open string spectrum is continuous. If we now start tuning λ away from these special values, the spectrum develops band gaps which become wider and wider until we reach the Dirichlet points λ ∈ 1 2 + Z at which the spectrum is discrete. The first computation of the spectrum for generic values of λ can be found in [11] (see also [14] for a much more elegant derivation). Despite of these significant insights into the structure of the boundary Sine-Gordon model, there are several important quantities that remain unknown. This applies in particular to the boundary 2-and 3-point functions and the bulk-boundary operator product expansions. In the string theoretic context, these missing data determine the time-dependence of open string couplings on a decaying brane and the back-reaction to the bulk geometry. Our desire to understand such important quantities is one of the main motivations to study the following, closely related world-sheet theory [15], We have written the interaction term for general parameters b and also added a similar interaction in the bulk, mainly to emphasize the relation with boundary Liouville theory. S T BL [X] = 1 4π Σ d 2 z η µν ∂X µ∂ X ν + ∂Σ du µ B exp X 0 (u) . What makes this relation so valuable for us is the fact that boundary Liouville theory has been solved over the last years [17,18,19,20,21,22,23,24,25,26]. Needless to stress that the solution includes explicit formulas for the bulk-boundary structure constants [24] and the boundary 3-point couplings [25]. There is one crucial difference between Liouville theory and the model we are interested in: whereas the usual Liouville model is defined for real b, our application requires to set b = i. A continuation of results in Liouville theory from c = 13 + 6b 2 + 6b −2 ≥ 25 to c = 1 (i.e. b = i) might appear to be a rather daring project, even more so as a naive inspection of the world-sheet action would suggest the c = 1 model could not possibly be unitary. Nevertheless, we shall show below that the theory is entirely well-defined and unitary. For the pure bulk theory, a similar result was established in [27]. It was shown there that the bulk 3-point couplings of Liouville theory possess a b = i limit which is well-defined for real momenta of the participating closed strings. The b = i theory, however, is no longer analytic in the momenta. Furthermore, the limit turned out to agree with the c = 1 limit of unitary minimal models which was constructed by Runkel and Watts in [28]. Our analysis here extends these findings to the boundary theory. We shall see that the latter is considerably richer than the bulk model. Some qualitative properties of the c = 1 boundary model can be understood using no more than a few general observations. To begin with, let us reconsider the simpler bulk model. Recall from ordinary Liouville theory that it has a trivial dependence on the coupling constant µ. Since any changes in the coupling can be absorbed in a shift of the zero mode, one cannot vary the strength of the interaction. This feature of Liouville theory persists when the parameter b moves away from the real axis into the complex plane. As we reach the point b = i, our model seems to change quite drastically: at this point, the 'Liouville wall' disappears and the potential becomes periodic. Standard intuition therefore suggests that the spectrum of closed string modes develops gaps at b = i. But since the strength of the interaction cannot be tuned in the bulk theory, the band gaps must be point-like. Though our argument here was based on properties of the classical action which we cannot fully trust, the point-like band-gaps are indeed a characteristic property of the c = 1 bulk theory [28]. These band-gaps also explain why the couplings of Liouville theory cease to be analytic when we reach c = 1. For the boundary theory, we can go through a similar argument, but the consequences are more pronounced. In the presence of a boundary, Liouville theory contains a second coupling constant µ B which controls the strength of an exponential interaction on the boundary of the world-sheet and is a real parameter of the model. In fact, the freedom of shifting the zero mode can only be used for one of the couplings µ or µ B . Once more, the boundary potential becomes periodic at b = i and hence the open string spectrum should develop gaps, as in the case of the bulk model. But this time, the width of these gaps can be tuned by changes of the parameter µ B . Hence, we expect the boundary Liouville theory to possess gaps of finite width at c = 1. Our exact analysis will confirm this outcome of the discussion. After these more qualitative remarks on the model we are about to construct, we shall summarize our main results in more detail. In section 3 we shall provide the boundary states for a family of boundary conditions of the c = 1 Liouville theory that is parametrized by one parameter s ∈ R. Formulas for these boundary states are trivially obtained from the corresponding expressions in c ≥ 25 Liouville theory and, as we shall also explain, through particular limits of boundary theories in minimal models. We shall then use both approaches to establish the existence of finite band gaps in the boundary spectrum of the c = 1 boundary theories. More precisely, we show that boundary fields Ψ ss p in the boundary theory with parameter s carry a momentum whose allowed values are taken from the following set S s := {p ∈ R | | sin πp| ≤ | sin πs| } . When s is a half-integer, the set S s fills the entire real line. As we vary s away from such half-integer values, the allowed momenta are restricted to intervals of decreasing width until we reach integer values of s where the spectrum of boundary fields becomes discrete. In this sense the parameter s interpolates between Neumann-type and Dirichlettype boundary conditions. Let us note that the boundary conditions with s ∈ Z are related to those constructed by Runkel and Watts in [28]. All other boundary theories, however, are new. It is worthwhile pointing out how closely the c = 1 limit of Liouville theory mimics the behavior of the boundary Sine-Gordon model. In both theories we can interpolate smoothly between continuous and discrete boundary spectra. Even the geometric interpretation of the Dirichlet points is very similar: in the context of the c = 1 Liouville model they describe a semi-infinite array of point-like branes from which individual branes can be removed through shifts of the boundary parameter s. Let us point out that finite arrays of such point-like branes are described by the c = 1 limit of ZZ branes. 3 Hence, the picture we suggest for the Dirichlet points of the c = 1 FZZT branes provides a nice geometric interpretation for the well-known relation of FZZT and ZZ branes in Liouville theory [29,30]. There are two important quantities that we shall construct for all these new boundary theories. These are the boundary 2-point function and the bulk-boundary 2-point function. For the former we obtain Ψ s 1 s 2 p 1 (u 1 )Ψ s 2 s 1 p 2 (u 2 ) c=1 = N s 1 s 2 (p) |u 1 − u 2 | 2p 2 1 [δ(p 1 + p 2 ) + R s 1 s 2 c=1 (p 1 )δ(p 1 − p 2 )] (1.4) where R s 1 s 2 c=1 (p) = R is 1 is 2 c=25 (1 − p) −1 . (1.5) Detailed formulas for N (see eq. (4.6)) and for the so-called reflection amplitude R are spelled out in section 4. Here, we have expressed the result in terms of the reflection amplitude for the theory at b = 1. It is quite remarkable that the reflection amplitudes of the two models are related in such a simple way. Let us point out, though, that the boundary reflection amplitude for the c = 1 theory is only defined within a subset S s 1 s 2 of momenta p. It is more difficult to write down the bulk-boundary coupling B of the c = 1 model, i.e. the coefficient in the 2-point function of a bulk field Φ with one of the allowed boundary fields, Φ pα (z,z)Ψ ss p β (u) c=1 = B(p α , p β , s) |z −z| 2p 2 α −p 2 β |z − u| 2p 2 β . We shall prove below that this coupling B is given by B c=1 (p α , p β , s) = 4π 2 i(πµ ren ) −pα−p β /2 e −iπs(p β +1) Γ(−2p β ) −1 × (1.6) × e −2πipα(s+1) 1 − e −4πipα g(y 0 ) 2 e − ih(y 0 ) 2π y 2 0 h ′′ (y 0 ) exp ∞ 0 dt t (e −p β t − e 2pαt ) sinh 2 p β t 2 sinh 2 t 2 − (p α → −p α ) where p β ∈ S s and the functions g and h are defined through g(y) = 1 − A 0 1 − A 0 y −pα−p β /2 1 − B 0 1 − B 0 y −p β /2 1 − C 0 1 − C 0 y pα+1/2 y 1+s−p β 2 √ 1 − y (1.7) h(y) = Li 2 (A 0 ) − Li 2 (A 0 y) + Li 2 (B 0 ) − Li 2 (B 0 y) −Li 2 (C 0 ) + Li 2 (C 0 y) − Li 2 (1) + Li 2 (y) + log y log z 0 . (1.8) Here, Li 2 denotes the dilogarithm and we have abbreviated B 0 = exp(−2πip β ), C 0 = exp(−4πip α ), A 0 = B 0 C 0 , z 0 = exp(−2πi(s − p β )) and the parameter y 0 = y 0 (p α , p β , s) is one of the two solutions of the following quadratic equation exp(y h ′ (y)) = 1 . For more details and explicit formulas, see section 5. We also show that the coefficients in the bulk-boundary operator product expansion vanish whenever the open string momentum p β lies within the band gaps, i.e. when p β ∈ S s . This provides another non-trivial consistency check for the couplings we propose. Let us add that correlators with boundary insertions in a similar model where also discussed recently in [31]. The techniques used there, however, only allowed to determine such correlations functions for a discrete set of boundary momenta p β . The c = 1 limit of the bulk theory In this section we shall mainly review some results on the c = 1 limit of bulk Liouville theory [27] (see also [32]). Our derivation of the 3-point couplings, however, is different from the one given in [27]. The new construction is simpler and uses some of the same techniques that we shall also employ to analyze the boundary model later on. To begin with, let us recall a few standard facts about the usual c ≥ 25 Liouville theory. As in any bulk conformal field theory, the exact solution of the Liouville model is entirely determined by the structure constants of the 3-point functions for the (normalizable) primary fields Φ α (z,z) ∼ e 2αX(z,z) with α = Q b 2 + ip , p ≥ 0 (2.1) where Q b = b + b −1 , b ∈ R. These fields are primaries with conformal weight h α = h α = α(Q b − α) under the action of the two Virasoro algebras whose central charge is c = 1 + 6Q 2 b . For real values of b, the couplings of three such fields are given by the following expression C b (α 1 , α 2 , α 3 ) = πµγ(b 2 )b 2−2b 2 (Q b −2α)/b Υ ′ (0|b) Υ(2α − Q b |b) 3 j=1 Υ(2α j |b) Υ(2α j |b) (2.2) whereα andα j are linear combinations of α j , α = 1 2 (α 1 + α 2 + α 3 )α j =α − α j , γ(y) = Γ(y)/Γ(1 − y) is a quotient of ordinary Γ-functions and the function Υ is defined in terms of Barnes' double Gamma function Γ 2 (see appendix A.2) by Υ(α|b) := Γ 2 (α|b) −1 Γ 2 (Q b − α|b) −1 . (2.3) The solution (2.2) was first proposed several years ago by H. Dorn and H.J. Otto [17] and by A. and Al. Zamolodchikov [18]. Crossing symmetry of the conjectured 3-point function was then checked analytically in two steps by Ponsot and Teschner [19] and by Teschner [20,21]. It is well known that Barnes' double Gamma function is analytic for Re b = 0. In the limit b = i, however, the function becomes singular. Nevertheless, the combinations of double Gamma functions that appear in the various couplings of Liouville theory turn out to be well defined, though they are no longer analytic. Most of our analysis of the b = i limit is based on the following relation between double Gamma functions with parameter b and −ib, Γ 2 (ip + n Q b 2 |b) = (−q; q 2 ) ∞ e iπ 4 (ip+(n−1) Q b 2 ) 2 (e 2πb −1 p (−1) n q n ; q 2 ) ∞ Γ −1 2 (−ib − p + in Q b 2 | − ib) (2.4) which holds for Im b 2 > 0. Here we have introduced q = exp(−iπ/b 2 ) and the so- Γ 2 (ip + n Q b 2 |b) ∼ √ 2 e −i π 4 p 2 2| sin πp|e −πi(p−[p]−1/2) −(p+1−n)/2 × (2.5) × e 1 ǫ (Li2(e −2πip )−Li 2 (1)) Γ −1 2 (1 − p|1) 1 + o(ǫ 0 ) with ǫ = 2πib −1 Q b and [p] denoting the integer part of p. Note that the double Gamma function diverges as we send b → i (or ǫ → 0) with a divergent factor involving the Dilogarithm Li 2 . In the expression for closed string couplings, Barnes' double Gamma functions appears only through Υ, and using a standard dilogarithm identity (see appendix A.3) one may show that Υ(ip + n Q b 2 |b) b→i ∼ 1 2 e − 1 ǫ λ(p)−3Li 2 (1) e iπp 2 /2 e −iπ(p−n+1)(p−[p]−1/2) Υ(1 − p|1) −1 . (2.6) Here, λ(p) is a continuous periodic function which is quadratic on each interval [n, n + 1] for n ∈ Z, λ(p) = 2π 2 (p − [p] − 1/2) 2 . Observe that the divergent factor in Υ is much simpler than for the double Gamma functions it is built from. In the function Υ, the divergence comes from a product of q-Pochhammer symbols. The latter is closely related to Jacobi's function ϑ 1 and its divergence may be controlled more directly using modular properties of ϑ-functions (see [27]). Using once more the formula (2.4) and some simple facts about Barnes' double Gamma function is is not difficult to show that Υ ′ (0|b) b→i ∼ 2π 2 ǫ Υ(1|1) −1 (1 + o(ǫ 0 )) . (2.7) With the help of the two asymptotic expressions (2.6) and (2.7), we find Υ ′ (0|b) Υ(2α − Q b |b) 3 j=1 Υ(2α j |b) Υ(2α j |b) = Υ ′ (0|b) Υ(Q b /2 + 2ip|b) 3 j=1 Υ(Q b + 2ip j |b) Υ(Q b /2 + 2ip j |b) = e − 1 ǫ F (p 1 ,p 2 ,p 3 ) 4π 2 ǫ e 2πip(p−[2p]−1/2) 3 j=1 e πip j e −πi(2p j −1)(p j −[2p j ]−1/2) e −πi2p j (p j −[2p j ]−1/2) × Υ(1 − 2p|1) Υ(1|1) 3 j=1 Υ(1 − 2p j |1) Υ(1 − 2p j |1) (1 + o(ǫ 0 )) (2.8) where F (p 1 , p 2 , p 3 ) = 3 j=1 λ(2p j ) − 3 j=1 λ(2p j ) − λ(2p) + 3Li 2 (1) . (2.9) It can be shown that F (p 1 , p 2 , p 3 ) ≥ 0, so that the exponential is never diverging, but it can suppress the whole three-point coupling. We write 2p j = [2p j ] + f j and find that F is zero if either [2p 1 ] + [2p 2 ] + [2p 3 ] even, and |f 1 − f 2 | ≤ f 3 ≤ min{f 1 + f 2 , 2 − f 1 − f 2 } , or [2p 2 ] + [2p 2 ] + [2p 3 ] odd, and |f 1 − f 2 | ≤ 1 − f 3 ≤ min{f 1 + f 2 , 2 − f 1 − f 2 } , otherwise F is strictly greater than zero and the three-point coupling vanishes for ǫ → 0. In more formal terms, the function P (2p 1 , 2p 2 , 2p 3 ) := lim ǫ→0 e − 1 ǫ F (p 1 ,p 2 ,p 3 ) assumes the value P = 1 when one of the above conditions is fulfilled and it vanishes otherwise. With the new function P being introduced, we can use eq. (2.8) to recast the 3-point couplings in the form C c=1 (p 1 , p 2 , p 3 ) b→i ∼ (πµ ren ) −2p 4π 2 i ǫ 3 j=1 e −πi[2p j ] P (2p 1 , 2p 2 , 2p 3 ) × × Υ(1 − 2p|1) Υ(1|1) 3 j=1 Υ(1 − 2p j |1) Υ(1 − 2p j |1) . (2.10) In our conventions, Q b ∼ ǫ/2π, so up to a factor i we reproduce precisely the result of [27] (see eqs. (4.8) and (5.3) of that article). To obtain a finite limit, the bulk fields should be renormalized by a factor ǫ and correlators on the sphere receive an additional factor 1 ǫ 2 due to a rescaling of the vacuum (see section 4.2 in [28] for a related discussion). This prescription produces a finite 3-point function and it does not affect the 2-point function. At first, the form of the couplings, and in particular the non-analytic factor P , may seem a bit surprising. It is therefore reassuring that exactly the same couplings emerged several years ago through a c = 1 limit of unitary minimal models [28]. Runkel Boundary states of FZZT branes at c = 1 It is well known that for real b, Liouville theory admits a 1-parameter family of boundary conditions which correspond to branes stretching out along the real line. The boundary state of this theory was found in [22], Φ α (z,z) s = Γ(1 + 2ibp) Γ(1 + 2ip/b) cos(2πsp) (πµγ(b 2 )) −ip/b 2πi2 1/4 p 1 |z −z| 2hα (3.1) where µ 2 B sin πb 2 = µ cosh 2 πsb and α = Q b /2 + ip, as usual. Note that the coupling on the right hand side is analytic in b and hence there is no problem to extend the FZZT brane boundary state into the c = 1 theory, Φ α (z,z) c=1 s = (πµ ren ) −p i2 1/4 cos 2πsp sin 2πp 1 |z −z| 2p 2 . (3.2) This boundary state is related to its Lorentzian analogue [33,15] by analytic continuation. It is, however, not entirely obvious that eq. (3.2) really provides the one-point function of a consistent boundary conformal field theory. We shall argue below that this is the case, at least as long as the parameter s remains real. Let us now see how such a boundary state of the c = 1 model can arise from a limit of boundary minimal models. To begin with, we need to set up a few notations. We shall label minimal models by some integer m = 3, 4, 5, . . . so that c = 1 − 6/m(m + 1). For representations of the corresponding Virasoro algebra we use the Kac labels (r, r ′ ) with 1 ≤ r ≤ m−1 and 1 ≤ r ′ ≤ m along with the usual identification (r, r ′ ) ∼ (m−r, m+1−r ′ ). Recall that the boundary conditions of the theory are in one to one correspondence with the representations, i.e. that they are also labeled by Kac labels (ρ, σ). The associated one-point functions are given by Φ r,r ′ (z,z) (ρ,σ) = 2 3/4 (m(m + 1)) 1/4 sin πρ(r/t − r ′ ) sin πσ(r − r ′ t) sin(r/t − r ′ ) sin π(r − r ′ t) 1 |z −z| 2h r,r ′ (3.3) where t = m/(m + 1) and h r,r ′ is the conformal dimension of the primary field with Kac labels (r, r ′ ), h r,r ′ = ((mr ′ − (m + 1)r) 2 − 1) 4m(m + 1) =: p 2 r,r ′ . Here, we have introduced the new quantity p r,r ′ that can be regarded as a discrete analogue of the continuous momentum p when c < 1. In taking the limit m → ∞, the bulk fields Φ r,r ′ of minimal models approach the fields Φ p in the continuum theory. The momentum variable p is related to the Kac-labels by 2p = 2p r,r ′ ∼ r − r ′ + 1 2m + 2 (r + r ′ ) + O(1/m 2 ) . In other words, the integer part of the quantity 2p is determined by the distance of the Kac label (r, r ′ ) from the diagonal r = r ′ , whereas the fractional part 2p−[2p] corresponds to the (rescaled) position along the diagonal. For the A-series of minimal models, the bulk spectrum is obtained by filling every point of the Kac table above the diagonal. When we send m to infinity, approximately the same number of fields appears at any given distance from the diagonal of the Kac table and these are distributed almost homogeneously along the diagonal r ′ = r. Hence, the spectrum of p in the c = 1 limit is homogeneous. Only integer values of 2p are not part of this spectrum since the Kac labels satisfy r, r ′ ≥ 1. This leads to the point-like band-gaps in the spectrum of bulk fields that we also saw emerging from Liouville theory. To obtain a one-parameter family of branes with 1-point function given by eq. (3.2) we propose the following Ansatz, 3) of minimal models, a short analysis similar to the one carried out in [28] gives (ρ, σ) = ( [f s m] + [s] + 1 , [f s m] + 1 ) .(3.+ 1)(r/t − r ′ ) sin π([f s m] + 1)(r − r ′ t) m→∞ ∼ cos 2πps − cos 4πp([f s m] + 1 + [s]/2) + . . . with p ∼ (r − r ′ t)/2Φ p (z,z) s = 2 −1/4 cos 2πsp | sin 2πp| 1 |z −z| 2p 2 . The result agrees with the 1-point function (3.2) of the c = 1 Liouville model up to some prefactors which may be absorbed by an appropriate renormalization of the bulk fields. If, on the other hand, the label s is an integer, then our limit of boundary conditions in minimal models becomes Φ p (z,z) MM s = cos 2πsp − cos 2π(s + 2)p 2 1/4 | sin 2πp| |z −z| 2p 2 ∼ Φ p (z,z) s − Φ p (z,z) s+2 . (3.5) The last identification with the c = 1 limit of boundary states in Liouville theory holds up to some factors that are due to the slightly different normalizations of bulk fields in minimal models and Liouville theory (see [27]). Our result (3.5) is very reminiscent of a similar relation between localized and extended branes in Liouville theory [29,30] and we shall comment on the precise connection and its geometric interpretation in the c = 1 model below. Now that we have identified the construction of the boundary state from minimal models, it is instructive to study the associated spectrum of boundary fields. According to the usual rules [34], the boundary spectrum of the brane (ρ, σ) contains fields with Kac labels r = 1, 3, . . . , min(2ρ − 1, 2m − 2ρ − 1) and r ′ = 1, 3, . . . , min(2σ − 1, 2m − 2σ + 1). Using the identification (r, r ′ ) ∼ (m − r, m + 1 − r ′ ) we can assume that r ≥ r ′ . Hence, the Kac labels of boundary fields fill every second lattice point within two triangles in the Kac table (see Figure 1). In the first triangle with corner at (1, 1), the difference r − r ′ of Kac labels is even. For Kac labels in the second triangle, on the other hand, r − r ′ is an odd integer since the reflection (r, r ′ ) → (m − r, m + 1 − r ′ ) shifts the difference of Kac labels by one unit. Recall that the quantity r − r ′ measures the distance of a point in the Kac table from the diagonal and thereby determines the integer part of 2p in the c = 1 theory. We conclude that the two triangles contribute boundary fields with [2p] being even or odd integer, respectively. Furthermore, points from the first triangle are uniformly distributed in the direction along the diagonal up to a maximal value 0 < (min(2ρ − 1, 2m − 2ρ − 1) + min(2σ − 1, 2m − 2σ + 1))/2(m + 1) < 1. A similar observation holds true for the second triangle. After we have sent m to infinity, this implies that the spectrum of p is given by S s = { p ∈ R | − min(f s , 1 − f s ) < p < min(f s , 1 − f s ) mod 1 } . Hence, the momentum p has bands of width 2 min(f s , 1 − f s ) centered around integer momentum p. The gaps between these bands widen while we vary s between a halfinteger and an integer value. Once we reach a point s ∈ Z, the boundary spectrum becomes discrete. ZZ and FZZT branes at the Dirichlet points Branes in the c = 1 limit of minimal models with a discrete open string spectrum have also been constructed by Runkel and Watts (see also [35]). Actually, the branes that we obtained through the limit of minimal models when s ∈ Z are identical to the limit of (s + 1, 1) branes in minimal models and hence to the discrete boundary conditions that were found by Runkel and Watts. We wish to point out that these branes may also be obtained from the so-called ZZ branes of Liouville theory. In general, the ZZ branes are labeled by a pair (m, n) of positive integers and they possess the following 1-point functions, to read off the following geometric interpretation of the ZZ brane with label (m, n) in the c = 1 theory: it corresponds to an array of min(m, n) point-like branes that are distributed equidistantly such that the rightmost brane is located at m+n. This interpretation agrees with the geometry of branes that emerges from the coset construction of minimal models (see [36]). Φ α (z,z) (m,n) = − 2 3/4 2πip Γ(1 − 2ibp)Γ(1 − 2ip/b) sinh 2πmp/b sinh 2πnbp sinh 2πp/b sinh 2πbp (πµγ(b 2 )) −ip/b |z −z| 2hα . Ö ¼ Ö ¾ ¾ ¾ ¾ With this in mind, we shall also be able to interpret our FZZT branes at the Dirichlet points s ∈ Z. Let us recall that the c = 1 limit of FZZT branes with integer label s differs from the limit of minimal model branes. The precise relation between their two boundary states is given in eq. (3.5). On its right hand side we consider the difference of one-point couplings for two FZZT branes whose labels differ by two units. This is claimed to be the same as the one-point coupling to the discrete brane with label (s + 1, 1). Therefore, the For the annulus amplitude between a ZZ brane (m, n) and an FZZT brane at parameter s one has Z (m,n),s (q) = m−1 ′ k=1−m n−1 ′ l=1−n χ (s+i(k/b+lb))/2 (q) (3.7) where χ p (q) = η −1 (q) q p 2 . Here Cardy's rule implies that there appears a single primary with Kac label given by eq. (3.4) in the corresponding spectrum of the m th minimal model. When we send m to infinity, this field approaches a c = 1 primary field with momentum 2p ∼ [f s m] + [s] − [f s m] + 1 2(m + 1) (2[f s m] + [s] + 2) + O(1/m 2 ) m→∞ −→ [s] + f s = s . Hence, there is perfect agreement between the annulus amplitudes in the c = 1 limit of unitary minimal models and of Liouville theory. Let us finally return to the analysis of our FZZT branes at the Dirichlet points. Since we already understand the effect of shifts in s, we can restrict the following discussion to the case s = 0. We would like to probe this FZZT brane with the ZZ brane (1, 1). The amplitude (3.7) then becomes Z (1,1),0 (q) = η −1 (q) = lim r→∞ Z (1,1),(r,r) (q) . In the second equality, we inserted formula (3.6). This result nicely confirms our interpretation of the FZZT brane with label s = 0 as an infinite array of point-like branes. The boundary 2-point function In this section we shall discuss how the boundary 2-point function of FZZT branes in Liouville theory can be continued to c = 1. This will allow us to derive the exact boundary spectrum, in particular the position and width of the expected band gaps, from Liouville theory. Our results will confirm nicely the outcome of our previous discussion in the context of unitary minimal models. Let us recall that the boundary 2-point functions for the FZZT branes in c ≥ 25 Liouville theory are given by Ψ s 1 s 2 β 1 (u 1 )Ψ s 2 s 1 β 2 (u 2 ) = |u 1 − u 2 | −2h β 1 [ δ(Q b − β 1 − β 2 ) + R s 1 s 2 b (β 1 ) δ(β 1 − β 2 )] . (4.1) The coefficient in front of the first term has been set to 1 by an appropriate normalization of the boundary fields. Once this freedom of normalization has been fixed, the coefficient of the second term is entirely determined by the physics. It takes the form Our aim now is to continue these expressions to the c = 1 model (see also [15] There exists a distinguished normalization of the boundary fields in which the reflection amplitude of the boundary theory becomes trivial. Since the usual 'Liouville wall' ceases to exist at the point b = i, the trivialization of the boundary reflection amplitude seems quite well adapted to our physical setup. Explicitly, the new normalization of the boundary fields is given by [25,30],Ψ R s 1 s 2 b (β) = (πµγ(b 2 )b 2−2b 2 ) (Q b −2β)/2b S(β + is|b)S(β − is|b)S(β + is 1 |b)S(β + is 2 |b) Γ 2 (2β − Q b |b) Γ 2 (Q b − 2β|b)s 1 s 2 β (u) = g s 1 s 2 β Ψ s 1 s 2 β (u) , (4.3) with g s 1 s 2 β = πµγ(b 2 )b 2−2b 2 β 2b Γ 2 (Q b − β − is|b) Γ 2 (Q b |b)Γ 2 (Q b − 2β|b)Γ 2 (Q b + is 1 |b)Γ 2 (Q b − is 2 |b) Γ 2 (Q b − β + is 2 |b)Γ 2 (Q b − β + is 1 |b)Γ 2 (Q b − β + is|b) . It is straightforward to rewrite the 2-point function in terms of the fieldsΨ s 1 s 2 β (u). The result is Ψ s 1 s 2 β 1 (u 1 )Ψ s 2 s 1 β 2 (u 2 ) ∼ (πµγ(b 2 )) (Q b /2−β 1 −is)/b |u 1 − u 2 | 2h β 1 C −1 b (β 1 , σ 1 , σ 2 ) × (4.4) × Γ(1 + 2ibp 1 )Γ(2ib −1 p 1 ) Γ(1 − ibs 1 )Γ(−ib −1 s 1 )Γ(1 − ibs 2 )Γ(−ib −1 s 2 ) [ δ(Q b − β 1 − β 2 ) + δ(β 1 − β 2 ) ] . Here we use β = Q b /2 + ip and the term C b in the denominator denotes the bulk 3point couplings of Liouville theory evaluated at the points β 1 and σ i = Q b /2 + is i /2. In writing this expression we have omitted a factor that is constant in the momentum Here, f s = s − [s] denotes the fractional part of the boundary parameter as before. This is exactly the same spectrum we found in our discussion of boundary conditions in the c = 1 limit of unitary minimal models ! Our result for the 2-point functions of boundary fields in the spectrum of the model is given by Ψ s 1 s 2 p 1 (u 1 )Ψ s 2 s 1 p 2 (u 2 ) c=1 ∼ e −Q(iP 1 ,is 1 /2,is 2 /2) |u 1 − u 2 | 2p 2 1 sin πs 1 sin πs 2 sin 2πp 1 [δ(p 1 + p 2 ) + δ(p 1 − p 2 )] where e Q(α 1 ,α 2 ,α 3 ) := Υ(1 + 2iα|1) Υ(1|1) 3 j=1 Υ(1 + 2iα j ) Υ(1 + 2iα j ) . In a final step, we change the normalization of the fieldsΨ in the c = 1 theory again so that they possess unit norm, Ψ s 1 s 2 p (u) :=g s 1 s 2 pΨ s 1 s 2 p (u) , (4.5) with g s 1 s 2 p = (πµ ren ) −p/2 Γ(−2p) Γ 2 (1|1)Γ 2 (1 + 2p|1)Γ 2 (1 − s 1 |1)Γ 2 (1 + s 2 |1)Γ(s 1 )Γ(1 − s 2 ) Γ 2 (1 + p +s|1)Γ 2 (1 + p −s 1 |1)Γ 2 (1 + p −s 2 |1)Γ 2 (1 + p −s|1) . Ψ s 1 s 2 p 1 (u 1 )Ψ s 2 s 1 p 2 (u 2 ) c=1 = N s 1 s 2 (p) |u 1 − u 2 | 2p 2 1 [δ(p 1 + p 2 ) + R s 1 s 2 c=1 (p 1 )δ(p 1 − p 2 )] (4.6) where R s 1 s 2 c=1 (p) = R is 1 is 2 c=25 (1 − p) −1 and N s 1 s 2 (p) = −2ǫp 1 (−1) [2p 1 ]+[s 1 ]+[s 2 ] for all p 1 , p 2 ∈ S s 1 s 2 . Note that we have been able to express the "boundary reflection amplitude" of the c = 1 theory through the corresponding quantity of the c = 25 model. We shall comment on this interesting outcome of our computation in the section 6. The bulk-boundary coupling constants Our final aim is to compute the bulk-boundary couplings of the c = 1 boundary Liouville models. For Liouville theory with real b, expressions for these couplings were provided by Hosomichi in [24]. We shall depart from these formulas to derive the corresponding couplings in the c = 1 model. The analysis turns out to be rather intricate. Nevertheless, the final answer is in some sense simpler than for c ≥ 25. Let us begin by reviewing Hosomichi's formula for the bulk-boundary correlation func- tion Φ α (z,z)Ψ ss β (u) = |z −z| h β −2hα |z − u| −2h βB (α, β, s) of an extended branes labeled by a parameter s. In our normalization (4.3) for the boundary fieldsΨ the couplingsB read B(α, β, s) = 2π(µπγ(b 2 )b 2−2b 2 ) 1 2b (Q b −2α) I(α, β, s, b) × Γ 2 (Q b − β|b)Γ 2 (2Q b − 2α − β|b)Γ 2 (2α − β|b)Γ 2 (Q b + is|b)Γ 2 (Q b − is|b) Γ 2 (Q b − β − is|b)Γ 2 (Q b − β + is|b)Γ 2 (β|b)Γ 2 (2α|b)Γ 2 (Q b − 2α|b) (5.1) where I(α, β, s) is given by the integral I(α, β, s, b) = ∞ −∞ dt e 2πist S( β+2α−Q b 2 + it|b)S( β+2α−Q b 2 − it|b) S( 2α−β+Q b 2 + it|b)S( 2α−β+Q b 2 − it|b) . (5.2) While sending b to b = i, an infinite number of poles and zeroes of the integrand approach the integration path. It turns out to be more convenient to evaluate the integral by Cauchy's theorem. For s > |Im β| we can close the integration contour and rewrite the sum over residues as a combination of basic hypergeometric series (see appendix A.1), I(α, β, s, b) = 2 Φ 1 e 2πib(β+2α−Q b ) , e 2πibβ , e 4πibα ; e 2πib 2 ; e −2πib(β−Q b −is) × 2 Φ 1 e −2πib −1 (β+2α−Q b ) , e −2πib −1 β , e −4πib −1 α ; e −2πib −2 ; e 2πib −1 (β−Q b +is) × e −πs(β+2α−Q b ) S(β|b)S(β + 2α − Q b |b)S(Q b − 2α|b) + (α → Q b − α) . (5.3) Our previous results allow us to estimate the behavior of the double Gamma functions and its close relative S in the vicinity of b = i. But the function 2 Φ 1 (A, B, C;q; z) has not appeared in our discussion before. Hence, in order to continue our analysis, we need to study the basic hypergeometric series in the limit when the parameterq approaches q = 1. We shall begin with a separate treatment of the two factors 2 Φ 1 before we combine the results to address the full couplingB. In dealing with one of the basic hypergeometric series, the basic idea is to approximate the series (we denote the summation parameter by k) through an integral and to evaluate the latter using a saddle-point analysis. We will not give a rigorous derivation but shall content ourselves with a rough sketch of the argument. This will help to understand the main features of our final formulas. Let us introduce the variable y =q k . Using the asymptotic behavior of the q-Pochhammer symbols (see appendix B) we obtain Once we have understood which saddle-points y i contribute to the integral, we end up with an expansion of the form 2 Φ 1 (A, B, C;q; z) ∼ 1 √ −2π logq2 Φ 1 (A, B, C;q, z) ∼ i f (y i ) y i h ′′ (y i ) e 1 logq h(y i ) . The most difficult part, however, is to find the right saddle-points. To begin with, the saddle-point equation (5.6) has two different solutions, at least for generic choice of the momenta. But this is not the full story since the function h is a complex function with branch-cuts. Hence, the description of the saddle point y i is not complete before we have specified on which branch the relevant saddle-point is located. We were not able to solve the problem in full generality, but for the combination of parameters in the problem at hand, we could find the saddle-points through a comparison with numerical studies (Mathematica). Let us state the result for the first q-hypergeometric function appearing in eq. (5.3). Here, the parameterq is e 2πib 2 and thus the prefactor 1/ logq of h in the exponent becomes − 1 ǫ − i 2π , where ǫ = 2πiQ b /b as introduced in section 2. Note that the arguments A, B, C and z depend on b and thus on ǫ and hence they need to be expanded around ǫ = 0. We shall use the symbols A 0 , B 0 , C 0 and z 0 to denote the leading terms, i.e. A 0 = B 0 C 0 , B 0 = e −2πip β , C 0 = e −4πipα , z 0 = e −2πi(s−p β ) , where we have set α = Q b /2 + ip α and β = Q b /2 + ip β . Since the function h in the exponential comes with an extra factor 1/ǫ, the sub-leading term in the ǫ-expansion of h contributes an extra factor which we shall combine with f into a new function g. Before we spell out our results on the saddle points, we also have to specify the branch we use for log z. We found it most convenient to choose log z → 2πi(p β − [p β ] − s + [s]). In the end we obtain, 2 Φ 1 e 2πib(β+2α−Q b ) , e 2πibβ , e 4πibα ; e 2πib 2 ; e −2πib(β−Q b −is) ∼ i=± g(y i ) y i h ′′ (y i ) e − 1 ǫ − i 2π h(y i ) with g(y) = 1 − A 0 1 − A 0 y −pα−p β /2 1 − B 0 1 − B 0 y −p β /2 1 − C 0 1 − C 0 y pα+1/2 (1 − y) −1/2 y 1+s−p β 2 . The two solutions of the saddle-point equation are denoted by y ± , and they are given explicitly by y ± = y ± (p α , p β , s) =                                e πi(2pα+p β ) cos(2πpα) sin(πs) sin(π(p β +s)) 1 ∓ i tan(2πp α ) 1 − sin(πp β ) 2 sin(πs) 2 sin(2πpα) 2 for | sin(πp β )| < | sin(πs) sin(2πp α )| e πi(2pα+p β ) cos(2πpα) sin(πs) sin(π(p β +s)) 1 ∓ sin 2 (πp β )−sin 2 (2πpα) sin 2 (πs) sin 2 (πs) cos 2 (2πpα) for | sin(πs)| ≥ | sin(πp β )| ≥ | sin(πs) sin(2πp α )| e πi(2pα+p β ) cos(2πpα) sin(πs) sin(π(p β +s)) 1 ∓ sin(π(s+2pα)) sin(πs) cos(2πpα) sin 2 (πp β )−sin 2 (2πpα) sin 2 (πs) sin 2 (π(s+2pa)) for | sin(πp β )| ≥ | sin(πs)| The last of these three different cases concerns the band gaps, i.e. momenta p β ∈ S s . The first two cases, on the other hand, apply to the bands so that all bands appear to be split into an inner (first case) and an outer (second case) region, depending on the value of the bulk momentum p α . Analyzing the limit in the outer part of the bands turns out to be the most difficult task because the two saddle-points y ± are of the same order, i.e. Re h(y + ) = Re h(y − ), and there are at least some subsets in momentum space where they both contribute. Within the inner region and the gap, the saddle-point y + dominates so that only the summand i = + remains in our asymptotic expansion. The exact description of our findings for the intermediate region is rather cumbersome. Fortunately, the distinction between inner and outer parts of the band disappears once we combine our two basic hypergeometric series into the bulk-boundary coupling. Since our main goal is to provide a formula for the c = 1 limit ofB, we shall not attempt to present our findings for 2 Φ 1 in the outer region. Instead, let us recall from above that we also need to specify the branch on which the saddle-points sit. The prescription is as follows: If |y i | > 1, then we choose the branch by moving it from the fundamental branch first to |y i | ± i0 + , infinitesimally above/below the real axis depending on whether the fractional part s − [s] is greater or smaller than 1/2. Then we move it on the shortest arc to y i . Having completed the analysis for the first hypergeometric function we now turn to the second for which an analogous investigation gives the following results, h(y) = h(A 0 , B 0 , C 0 , z −1 0 B −2 0 , y) andg (y) = 1 − A 0 1 − A 0 y pα+p β /2 1 − B 0 1 − B 0 y p β /2 1 − C 0 1 − C 0 y −pα+1/2 (1 − y) −1/2 y 1+s+p β 2 . Note that in this case 1 logq is directly given by − 1 ǫ . The relevant saddle-points arẽ y ± = y ± (p α , p β , −s) . Our discussion of the three different cases and the choice of branches carries over from the previous discussion without any significant changes. Before we return to the study ofB it is necessary to make one more important observation which relates the two saddle-points y ± andỹ ± . It is not difficult to see that (A 0ỹ± ) −1 solves the saddle-point equation (5.6) of y ± . Indeed we find for p β ∈ S s that y ± = (A 0ỹ∓ ) −1 . This fact along with some simple formulas in appendix A.3 allows to re-express the final formulas entirely in terms of y + (p α , p β , s). Now we have all ingredients at our disposal to analyze the asymptotic behavior of the bulk-boundary correlator. We must combine the divergences coming from the qhypergeometric functions with the divergences of the prefactors in eq. (5.1). Since we know already that boundary fields with p β ∈ S s do not exist, we can restrict our evaluation ofB to the inner and outer band regions (see discussion above). In the inner region, only one saddle-point contributes and the divergent terms may be determined easily from the formulas we have provided. It then turns out that they cancel each other thanks to a rather non-trivial dilogarithm identity which is attributed to Ray [37]. For convenience we state Ray's identity in appendix A.3 (eq. (A.13)). Hence, we are left with a finite result. After a tedious but elementary computation we obtain lim b→iB (α, β, s) = 4πi(πµ ren ) −pα e −πis(p β +1) sin πs e −2πipα(s+1) 1 − e −4πipα × × g(y + ) 2 e − ih(y + ) 2π y 2 + h ′′ (y + ) Γ 2 (1 + p β + s|1)Γ 2 (1 + p β − s|1)Γ 2 (1 − 2p α |1) 2 Γ 2 (1 − 2p α − p β |1)Γ 2 (1 − 2p α + p β |1)Γ 2 (1 − s|1)Γ 2 (1 + s|1) − (p α → −p α ) where y + = y + (p α , p β , s). In the derivation we used the fact that y + and (A 0ỹ+ ) −1 = y − are the two solutions of the same quadratic saddle-point equation along with a few elementary identities which we list in appendix A.3. Within the outer regions of the bands, the q-hypergeometric functions can have contributions from both saddle-points. It is possible to show that out of the four possible terms in our product of two q-hypergeometric functions, only two appear at any point in momentum space. Moreover, after the divergences of the prefactors have been taken into account, only one of the two terms is free of divergences. The other keeps a rapidly oscillating factor. The cancellation is again due to Ray's identity and the finite contributions come from the combination (y + ,ỹ + ). In the end, the result for the outer region of the band is therefore given by the same formula as for the inner region. It now remains to change the normalization of our boundary fields. We pass from the fieldsΨ to Ψ using the prescription (4.5), as before. Our result then reads B(p α , p β , s) = 4π 2 i(πµ ren ) −pα−p β /2 e −πis(p β +1) Γ −1 (−2p β ) × e −2πipα(s+1) 1 − e −4πipα g(y + ) 2 e − ih(y + ) 2π y 2 + h ′′ (y + ) Γ 2 (1|1)Γ 2 (1 + 2p β |1)Γ 2 (1 − 2p α |1) 2 Γ 2 (1 − 2p α − p β |1)Γ 2 (1 − 2p α + p β |1)Γ 2 (1 + p β |1) 2 − (p α → −p α ) . This is the formula we anticipated in the introduction. For a direct comparison one has to employ the integral representation (A.2) of Barnes' double Gamma function. Before we conclude this section, we would like to comment briefly on the band-gaps. Actually, we can use the analysis of the bulk-boundary couplings to confirm our previous statement that boundary fields Ψ ss p β with p b ∈ S s decouple from the theory. To this end, we shall focus on the coefficients A of the bulk boundary operator product expansion rather than the couplings B. The former are related to latter by multiplication with the boundary 2-point function. Our claim is that the coefficients A b=i vanish within the band-gaps. As we have seen before, the 2-point function contributes a factor that diverges when p β ∈ S s . With the help of our asymptotic expansion formulas in this section it is possible to show that the couplings B always diverge slower. Hence, boundary fields whose momenta lie in the gaps cannot be excited when a bulk field approaches the boundary. Conclusion and open problems In this work we have constructed boundary conditions of the Euclidean c = 1 Liouville model. In particular we argued that there exists a family of boundary conditions that is parametrized by one real parameter s ∈ R. We provided explicit expressions for their boundary states (see eq. (3.2)), the boundary 2-point function (4.6,4.2), and the bulkboundary coupling (1.6). The only quantity that we are missing for a complete solution of the model is the boundary 3-point coupling. In the case of Liouville theory, the corresponding formulas have been found by Ponsot and Teschner in [25]. We believe that the analysis of their c = 1 limit can proceed along the lines of the studies we have presented in section 5, but obviously this program remains to be carried out in detail. A crucial ingredient in our study was the formula (2.6) for the asymptotics of Barnes' double Gamma function. In section 2, the latter enabled us to provide a new derivation of the bulk couplings in the c = 1 Liouville model. Our approach here is simpler and more general than the one that was developed in [27]. With this technical progress it is now also possible to calculate the bulk couplings of non-rational conformal field theories with c < 1. In contrast to the c = 1 case, the models with c < 1 are certainly non-unitary. We will comment on the results and their relation with minimal models elsewhere. Possible applications of such developments include the 2-dimensional cigar background with 0 ≤ k ≤ 2 [38] and similar limits of the associated boundary theories [39]. R s 1 s 2 c=1 (x) = R is 1 is 2 c=25 (1 + ix) −1 . This result does not agree with the proposal in [15]. Note, however, that in the Lorentzian case the correlation functions can depend on the details of how b approaches b = i. Such a behavior might be related to a choice of vacuum 5 . Let us also anticipate that consistency of our expression for the boundary 2-point coupling with the half-brane boundary states [33,15] can be checked by means of the modular bootstrap. A related observation, though with a somewhat problematic domain of open string momenta, has also been made in [41]. Concerning the bulk-boundary coupling, our investigations are not complete yet. But before having worked out the expression in the Lorentzian model, it is worthwhile comparing our formula (1.6) for the bulk-boundary coupling with a corresponding expression in the time-like theory that was suggested recently in [42] (eq. [4.14] of that paper). In fact, the exponential in the second line of our formula is identical to a corresponding factor in the work of Balasubramanian et. al. It will be interesting to determine the other factors through our approach. We shall return to this issue in a forthcoming publication. (1 − aq n ) . Its limit for k → ∞ exists for |q| < 1 and is denoted by a;q ∞ = ∞ n=0 (1 − aq n ) . (A.1) The tilde is used to avoid confusion with our convention for q in the rest of the text. With the help of the the finite q-Pochhammer symbol we can now introduce the basic hypergeometric series 2 Φ 1 . This q-deformation of the hypergeometric function 2 F 1 is defined as 2 Φ 1 (A, B, C;q; z) = ∞ k=0 A;q k B;q k C;q k q;q k z k . The interested reader can find many basic properties of these functions in the literature (see e.g. [44,43]). All the properties we need in the main text are stated and derived there. A.2 Barnes' double Gamma function and related functions Barnes' double Γ-function Γ 2 (x|b) is defined for x ∈ C and complex b with Re b = 0 (see [45]), and can be represented by an integral (for Re x > 0), log Γ 2 (x|b) = − ∞ 0 dt t e −Q b t/2 − e −xt (1 − e −bt )(1 − e −t/b ) + (Q b /2 − x) 2 2 e −t + Q b /2 − x t , (A.2) whenever Re x > 0. Here we have also introduced the symbol Q b = b + b −1 . Throughout the main text, we often use the following special combinations of double Gamma functions, Υ(x|b) = Γ 2 (x|b) −1 Γ 2 (Q b − x|b) −1 (A.3) S(x|b) = Γ 2 (x|b) Γ 2 (Q b − x|b) −1 . (A.4) For the latter, we would also like to spell out an integral representation that easily follows from the formula (A.2) above, log S(x|b) = ∞ 0 dt t sinh(Q b − 2x)t 2 sinh(bt) sinh(t/b) − Q b /2 − x t . This representation is valid for |Re x| < Re Q b . Let us also note that S is unitary in the sense that S(x|b)S(Q b − x|b) = 1 The function S is also closely related to Ruijsenaars' hyperbolic Gamma function Γ h (see [46]), S(x) = Γ h i(x − Q b /2) . Many further properties of double Gamma functions, in particular on the position of their poles and shift properties, can be found in the literature (see e.g. [47]). Here we would like to prove one relation that involves the q-Pochhammer symbols we introduced in the previous subsection. To this end, we depart from the integral representation (A.2). Let us denote the integrand in formula (A. 2) by f (t, x, b), s.t. x, b). The function f has the property log Γ 2 (x|b) = − ∞ 0 dt f (t,if (it, x, b) = −f (t, −ib + ix, −ib) + ( Q b 2 − x) 2 2t (e −it − e −t ) . (A.5) We evaluate the integral by closing the contour in the first quadrant, log Γ 2 (x|b) = −i ∞ 0 dt f (it, x, b) − 2πi Poles tn Res t=tn f (t, x, b) (A.6) = − log Γ 2 (−ib + ix| − ib) − ∞ 0 dt 2t ( Q b 2 − x) 2 (e −it − e −t ) − 2πi Poles tn Res t=tn f (t, x, b) where we have used the formula (A.5) in passing to the second line. The integral in the second term is ∞ 0 dt t (e −it − e −t ) = −i π 2 . (A.7) The poles appearing in the sum over residues are at t n = 2πin/b with n = 1, 2, . . . Hence we find 2πi Poles tn Res t=tn f (t, x, b) = ∞ n=1 1 n e −Q b πin/b − e −2πinx/b 1 − e −2πin/b 2 = ∞ n=1 1 n (−1) n − e πin/b 2 −2πinx/b 2i sin πn/b 2 . Replacing sin πn/b 2 through 1 2i sin πn/b 2 = e −πin/b 2 ∞ m=0 e −2πinm/b 2 and changing the order of summation, we obtain 2πi Poles tn Res t=tn f (t, x, b) = ∞ m=0 log(1 − e −2πim/b 2 −2πix/b ) − log(1 + e −2πi(m+1/2)/b 2 ) = log ∞ m=0 1 − q 2m e −2πix/b 1 + q 2m+1 with q = e −iπ/b 2 . We can now re-express the infinite product with the help of q-Pochhammer symbols (see appendix A.1). Inserting this result and eq. (A.7) into eq. (A.6) we finally arrive at Γ 2 (x|b) = e iπ 4 (Q b /2−x) 2 − q; q 2 ∞ e −2πix/b ; q 2 ∞ Γ 2 (−ib + ix| − ib) −1 . (A.8) This formula is the starting point for our evaluation of the asymptotic behavior of Γ 2 (x|b) near b → i. We shall return to this issue in appendix B. A.3 Dilogarithm There is one more special function that plays an important role in our analysis: Euler's dilogarithm (L. Euler 1768). It is defined by Li 2 (x) := ∞ n=1 x n n 2 , for |x| ≤ 1. At x = 1, we find Li 2 (1) = π 2 6 . The dilogarithm has the integral representation Li 2 (x) = − x 0 log(1 − t) t dt . It can be analytically continued with a branch cut along the real axis from 1 to +∞. At x = 1, the dilogarithm is still continuous, but not differentiable. Properties of Dilogarithms are used frequently to derive the formulas that appear in the main text. Here we list the most relevant properties Li 2 (z 2 ) = 2Li 2 (z) + 2Li 2 (−z) (A.9) Li 2 (z) + Li 2 (1 − z) = Li 2 (1) − log(z) log(1 − z) (A.10) Li 2 (−z) + Li 2 (−1/z) = 2Li 2 (−1) − 1 2 log 2 (z) (A.11) Li 2 (e −ǫ ) = Li 2 (1) − (1 − log ǫ)ǫ + o(ǫ) . (A.12) Many more properties of the dilogarithm can be found in the literature (see e.g. [48]). Of particular relevance for our analysis of the bulk-boundary 2-point function is the following equality 2 j=1 2 l=1 Li 2 (η j x l ) − Li 2 (η j /ζ l ) − Li 2 (ζ j x l ) + Li 2 (ζ j /ζ l ) (A.13) = Li 2 (z) − Li 2 (z η 1 η 2 ζ 1 ζ 2 ) + log z log(x 1 x 2 ζ 1 ζ 2 ) . Here, x 1,2 are the two solutions of the quadratic equation z(1 − η 1 x)(1 − η 2 x) = (1 − ζ 1 x)(1 − ζ 2 x) . (A.14) Equation (A.13) is derived from the usual 5-term relation of the dilogarithm (see e.g. [48]) with the help of the following list of relations that hold for any two solutions x 1 , x 2 of the equation (A.14), 1 − z η 1 η 2 ζ 1 ζ 2 (1 − η i x 1 )(1 − η i x 2 ) = 1 − η i ζ 1 1 − η i ζ 2 (A.15) 1 − z η 1 η 2 ζ 1 ζ 2 (1 − ζ i x 1 )(1 − ζ i x 2 ) = (ζ i − η 1 )(ζ i − η 2 ) ζ i (A.16) x 1 x 2 = 1 − z ζ 1 ζ 2 − zη 1 η 2 . (A.17) We leave the details to the reader. Let us note that the properties (A.15) to (A.17) are also used frequently to simplify the final formula for the bulk-boundary 2-point function. B Asymptotic behavior of q-Pochhammer symbols In order to evaluate the behavior of the double Gamma function near b = i one can start from our equation (A.8). Since the double Gamma function that appears on the right hand side of this equation is analytic at b = i, the main issue is to understand the asymptotic behavior of the q-Pochhammer symbol. This is what we are concerned with here. To begin with, we shall parametrizeq by logq = −ǫ. In general, the limit can depend on the way we send ǫ to zero. We start by taking the logarithm of the definition (A.1), log aq z/2 ;q ∞ = ∞ n=0 log(1 − aq z/2qn ) = 1 2 log(1 − aq z/2 ) + ∞ 0 log(1 − aq z/2+x )dx + ∞ 0 φ 1 (t − [t]) d dx log(1 − aq z/2+x ) x=t dt . Here, we used the Euler-MacLaurin sum formula to re-express the sum as an integral, φ 1 is the first Bernoulli polynomial, φ 1 (t) = 1 2 − t . Note that the branch of the logarithm log(1 − aq z 2 +x ) is chosen such that it approaches 0 when we send x along the real axis toward +∞. The first integral is given by the dilogarithm (see appendix A.3 for the definition and properties), so we obtain aq z/2 ;q ∞ = e − 1 ǫ Li 2 (aq z/2 ) (1 − aq z/2 ) When ǫ goes to zero, the strongly oscillating sign of the Bernoulli polynomial suppresses the integral, so that generically I(ǫ, a, z) vanishes in that limit. The leading contribution comes from the pole of the integrand, and if it approaches the path of integration when sending ǫ to zero, we can get a finite contribution. For |a| < 1, the pole is always outside of the integration region, but for |a| ≥ 1, the pole might come close to the real axis. We can take this effect into account by extracting the pole part from the integrand, 1 a −1 e ǫz/2 e t − 1 = 1 t − log a + ǫz/2 + regular . The regular part will not contribute to the integral in the limit ǫ → 0, so we can rewrite I(ǫ, a, z) as I(ǫ, a, z) = ∞ 0 φ 1 (t − [t]) 1 t − log a ǫ + z 2 + o(ǫ 0 ) . The remaining integral is given by the difference of − log Γ(− log a ǫ + z 2 ) and the first terms of its Stirling series, I(ǫ, a, z) = − log Γ(− log a ǫ + z 2 ) + 1 2 log 2π + − log a ǫ + z 2 log − log a ǫ + z 2 − 1 − 1 2 log − log a ǫ + z 2 + o(ǫ 0 ) . As long as |a| ≤ 1 and a = 1, the limit of I vanishes because the Γ-function approaches its Sterling approximation. If a = 1, we find exp I(ǫ, 1, z) = √ 2π Γ(z/2) This formula together with (B.1) describe the asymptotics of q-Pochhammer symbols in the cases we shall need for our applications. In the following we want to apply the gained insight in a number of special cases. First, let us write down the full asymptotics (B.1) for a = 1, q z/2 ;q ∞ = e − 1 ǫ Li 2 (1) ǫ −(z−1)/2 √ 2π Γ(z/2) (1 + o(ǫ 0 )) where we used (A.12) from appendix A.3 to expand the dilogarithm close to 1. We observe that this expression has zeroes for z = −2, −4, −6, . . . which is consistent with the general definition (A.1). For integer z these asymptotics can alternatively be derived using modular properties of Dedekind's η-function. We are often interested in q-Pochhammer-symbols of the form f (q);q ∞ with some function f (q). In most cases, it is enough to expand f to first order aroundq = 1, and we find Let us finally look at the behavior of a;q ∞ when a is close to 1. Obviously 1;q ∞ = 0, and the derivative is easily obtained as d da a;q ∞ a=1 = − q;q ∞ . (B.4) Similar formulas for the asymptotics of q-Pochhammer symbols have been derived before (see in particular [49]). f (q);q ∞ = f (1)q f ′ (1)/f (1) ;q ∞ (1 + o(ǫ 0 )) . 2 S 2has been named time-like boundary Liouville theory. Since the tachyon vanishes in the far past, the model seems to describe a half-brane, i.e. a brane that dissolves as time progresses. After Wick rotation X = iX 0 , the theory of the time coordinatebecomes 1 BL [X] = 1 4π Σ d 2 z ∂X∂X + µ exp 2bX(z,z) + ∂Σ du µ B exp bX(u) called q-Pochhammer symbols (.; .) ∞ (see appendix A.1). Relation (2.4) follows from a rotation of the contour in the integral representation (A.2) of the double Gamma function. The factor involving the q-Pochhammer symbols assembles contributions from the poles in the integrand of the double Gamma function (see appendix A.2 for details). Using formula (B.3) for the behavior of our q-Pochhammer symbols at b = i we conclude and Watts investigated the consistency of these couplings and demonstrated that crossing symmetry holds when all the half-integer momenta p are removed from the spectrum. As we have argued in the introduction, the appearance of such point-like band-gaps is rather natural from the point of view of Liouville theory.3 The c = 1 limit of the boundary theory Now we are ready to analyze the boundary model. We shall begin with the so-called FZZT boundary conditions of Liouville theory. They describe extended branes with an exponential potential for open strings (see eq. (1.3)). Taking the c = 1 limit of their 1-point function is straightforward and results in an expression with analytic dependence on the closed string momentum. These c = 1 boundary states are parametrized through one real parameter s and for non-integer values of s they can be reproduced by taking an appropriate limit of boundary conditions in unitary minimal models. The latter construction will provide a first derivation of the spectrum of boundary vertex operators, including the precise position and width of the band-gaps.When s ∈ Z, the bands become point-like. At these special values of the parameter s, our two constructions through Liouville theory and minimal models do not produce the same boundary states, though they still give closely related results. From minimal models we obtain the discrete set of boundary conditions that is already contained in the work of Runkel and Watts. These agree with the c = 1 limit of ZZ branes in Liouville theory and they possess a simple geometric interpretation as finite arrays of point-like branes.The relation between ZZ and FZZT branes in Liouville theory then implies that we can interpret FZZT branes with s ∈ Z as half-infinite arrays of point-like branes in the c = 1 model. being the momentum in the c = 1 theory. Provided s ∈ Z, the second term in the previous formula oscillates rapidly when we send m to infinity and hence we can drop it in the limit (recall that all correlation functions should be understood as distributions in momentum space). Together with the other factors in the 1-point function(3. Figure 1 : 1Boundary excitations for the boundary condition (ρ, σ) fill a rectangle, or, after reflection, two triangles in the Kac table. In the large m limit, they form bands with band-gaps around 2p = 1.If we send the parameter b to b = i, these 1-point functions assume the following formΦ p (z,z) c=1 (m,n) = −i(πµ ren ) −p 2 3/4 sin 2πmp sin 2πnp sin 2πp 1 |z −z| 2p 2 .We can rewrite the coupling on the right hand side with the help of the (m + n − 2l − 1) brane configuration that is associated to the FZZT brane with integer label s contains an additional point-like object with transverse position parameter s along the real line.The latter gets removed as we shift from s to s + 2. Our observation suggests to think of the c = 1 FZZT branes with integer s as being built up from discrete objects, or more specifically, as some half-infinite array of point-like objects. They occupy points with even or odd integer position x in target space, depending on whether s is odd or even. Changes of s by one unit should therefore correspond to a simple translation of the entire array in the target space.At least qualitatively, such an interpretation in terms of point-like branes appears consistent with our previous study of brane spectra on FZZT branes. In fact, we noticed before that their bands shrink to points when s reaches integer values. Let us also stress that our geometric picture of the branes with integer s is very similar to the interpretation of the Dirichlet points in the boundary Sine-Gordon model (see introduction).More support for the proposed geometric pictures comes from considering spectra involving the ZZ branes in the c = 1 theory. The annulus amplitude for two ZZ branes with labels (m, n) and (m ′ , n ′ ) was found in[26] to be of the formZ (m,n),(m ′ ,n ′ ) (q) = min(m,m ′ )−1 k=0 min(n,n ′ )−1 l=0 χ m+m ′ −2k−1,n+n ′ −2l−1 (q) where χ m,n (q) = η −1 (q) q −(m/b+nb) 2 /4 − q −(m/b−nb) 2 /4 b=i = η −1 (q) q (m−n) 2 /4 − q (m+n) 2 /4 . Note that in the c = 1 theory, excitations of the ZZ branes have non-negative conformal dimensions. Since χ m,n are characters of irreducible Virasoro representations at c = 1, the expression for the annulus amplitude coincides with what is obtained from the limit of Cardy type branes in unitary minimal models. A particularly simple case arises from setting (m, n) = (1, 1) and (m ′ , n ′ ) = (r, r), Z (1,1),(r,r) (q) = χ r,r (q) = η −1 (q) 1 − q r 2 . (3.6) According to our previous discussion, this amplitude encodes the spectrum of open strings between a single point-like brane and an array of length 2r. , Σ ′ denotes a summation in steps of two. Let us point out that once more, the open string excitations between ZZ and FZZT branes have real, non-negative conformal dimensions when b = i. This is in contrast to the situation for c ≥ 25 where the spectrum contains fields with complex dimensions if n = 1 or m = 1. The spectrum between the (1, 1) branes and an FZZT brane of parameter s contains a single primary field with conformal dimension h = s 2 /4, in perfect agreement with the construction from unitary minimal models. In fact, the construction of the c = 1 brane we have proposed along with 1 + s 2 )/2 ands i =s − s i . The special function S is constructed as a ratio of two Barnes double Gamma functions (see appendix A.2 for details). R is known as the reflection amplitude since it describes the phase shift of wave functions that occurs when open strings spanning between the FZZT branes with parameters s 1 and s 2 are reflected by the Liouville potential. p 1 1and invariant under the exchange of s 1 and s 2 since this can easily be absorbed in a redefinition of the boundary fields.When rewritten in the new normalization, the only nontrivial term in the expression (4.4) is the bulk 3-point function. Hence, we can use the results of[27] (see also section 2) to continue the boundary 2-point function to c = 1. Note, however, that the bulk 3-point coupling needs to be inverted. This is possible whenever it is non-zero. Since the coupling C c=1 (β, σ 1 , σ 2 ) vanishes whenever the factor P (2p, s 1 , s 2 ) does, we conclude that boundary fields Ψ s 1 s 2 β exist for p ∈ S s 1 s 2 ⊂ R where S s 1 s 2 := {p ∈ R |P (2p, s 1 , s 2 ) = 0} . Fields which are labeled by momenta outside this range S s 1 s 2 do not correspond to normalizable states of the model. On a single brane with label s = s 1 = s 2 we obtain in particular S ss = S s = {p ∈ R | − min(f s , 1 − f s ) < p < min(f s , 1 − f s ) mod 1 } . In an abuse of notation, we have denoted the normalized boundary fields of the c = 1 model by the same letter Ψ as in c ≥ 25 Liouville theory. Nevertheless it important to keep in mind that they are not justs limits of the corresponding fields in ordinary Liouville theory. After this final change of normalization, the boundary 2-point functions take the form 4 y) = h(A, B, C, z, y) = Li 2 (A) − Li 2 (Ay) + Li 2 (B) − Li 2 (By) −Li 2 (C) + Li 2 (Cy) − Li 2 (1) + Li 2 (y) + log y log z , and f(y) = f (A, B, C, z, y) = (1 − A)(1 − B)(1 − Cy) (1 − Ay)(1 − By)(1 − C)(1 − y) ofthe integral in eq. (5.4) can be determined by the method of steepest descent. The leading contribution comes from saddle-points y i , i.e. from points satisfying the condition h ′ (y i ) = 0. An elementary computation shows that such saddle-points are obtained as a solution of the quadratic equation z(1 − Ay)(1 − By) = (1 − Cy)(1 − y) . (5.6) All this research, however, was mainly motivated by the desire to construct an exact conformal field theory model for the homogeneous condensation of open string tachyons, such as the tachyon on an unstable D0 brane in type IIB theory. As we explained in the introduction, the underlying world-sheet theory is a Lorentzian version of the model we have considered in this work. But since the couplings of our theory are not analytic in the momenta (with the boundary state being the only exception), this Lorentzian theory cannot be obtained by a simple Wick rotation from the solution we described here.Instead, it was suggested in[27] to perform the Wick rotation before sending b to its limiting value b = i. Such a prescription makes sense because the couplings of Liouville theory are analytic in the momenta as long as b is not purely imaginary. It is certainly far from obvious that the Wick rotated couplings again possess a well-defined b = i limit.Preliminary studies of this issue show that our analysis of the boundary 2-point function extends to the Lorentzian case. In the time-like model, the spectrum of boundary fields is continuous (there are no gaps) and their 2-point function is still given by eq. (4.6) with a reflection amplitude Acknowledgements: We would like to thank V. Balasubramanian, P. Etingof, J. Fröhlich, M.R. Gaberdiel, K. Graham, G. Moore, B. Ponsot, I. Runkel, J. Teschner and G. Watts for very helpful discussions and some crucial remarks. Part of this research has been carried out during a stay of VS with the String Theory group of Rutgers University and of both authors at the Erwin Schroedinger Institute for Mathematical Physics in Vienna. We are grateful for their warm hospitality during these stays. A Some background on special functions In this first appendix we collect a few results on the special functions which appear in the analysis of the c = 1 boundary Liouville model. We shall start with some q-deformed special functions and then introduce Barnes' double Gamma functions and certain closely related special functions. Dilogarithms and some of their properties are finally reviewed in the third subsection.A.1 q-Pochhammer symbols and q-hypergeometric functionsOne of the most basic objects in the theory of q-deformed special functions is the finite q-Pochhammer symbol (see e.g.[43]) As long as |a| < 1, I(ǫ, a, z) is suppressed by logq = −ǫ. On the other hand, if a = 1, then the integral gives a non-trivial contribution. Let us explain that in more detail. o(ǫ 0 ) for z ∈ C \ (−∞, 0] . This relation breaks down if f (1) = 1 and f ′ (1)/f (1) = −n for some n = 0, 1, 2, . . . . Let us look at an example. Setq = e −2πi/b 2 and f (q) = e −2πp/b(q)qz/2 for p ∈ R. Then we obtain log e −2πp/bqz/2 ;q ∞ = − 1 ǫ Li 2 (e 2πip ) + 1 2 (1 − z − p) log(1 − e 2πip ) + o(ǫ 0 ) . (B.2) By carefully choosing the correct branch of the logarithm we can rewrite the result as e −2πp/bqz/2 ;q ∞ = e − 1 ǫ Li 2 (e 2πip ) 2| sin πp|e πi(p−[p]−1/2) (1−z−p)/2 1 + o(ǫ 0 ) , (B.3) where [p] denotes the largest integer smaller or equal p. 4 ) 4Here, s ∈ R is the parameter of the resulting limiting boundary condition.[s] and f s denote the integer and fractional part of s, respectively. The prescription (3.4) instructs us to pick a boundary condition that is represented by a point at distance [s] from the diagonal of the Kac table and to scale its projection to the diagonal with m. Our first aim now is to show that this recipe reproduces the 1-point function(3.2) in the c = 1 Liouville theory. The main idea is to rewrite the product of sin-functions in the numerator of eq.(3.3) as follows 2 sin π([f s m] + [s] ofBarnes' double Gamma function at b ∼ i, it is not hard to see that the boundary reflection amplitude R of Liouville theory vanishes outside a discrete set of open string momenta. This behavior is unacceptable since it does not allow for a sensible physical interpretation. Actually, it does not originate from the physics of the model but rather from an inappropriate choice of the limiting procedure. To see this we recall that the boundary spectrum of the theory is expected to develop gaps of finite width. For such gaps to emerge from Liouville theory, it is necessary that the corresponding states in the Liouville model become non-normalizable. But our normalization of boundary fields was chosen such that all states in the boundary theory have unit norm. Hence, the normalization we have chosen above is clearly not appropriate for the limit we are about to take.for a first attempt in this direction). Using once more the general formula (2.6) for the behavior Let us remark in passing that this model also arises in the context of flux line pinning in planar superconductors, see appendix F of[16] 2 In writing this action we think of the corresponding model as being defined through analytic continuation in the parameter b. This might be relevant for comparisons with previous results in models with exponential interactions. The two integers (m, n) that parametrize the ZZ branes correspond to the transverse position and the length of the finite array. Following the discussion on normalization of the vacuum after eq. (2.10), disc correlators should be rescaled by a factor 1 ǫ so that the limit of the properly renormalized 2-point function is finite. 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[]
[ "Evidence for transient clumps and gas chemical evolution in the CS core of L673", "Evidence for transient clumps and gas chemical evolution in the CS core of L673" ]
[ "Oscar Morata \nDepartment of Physics\nThe Ohio State University\n174 West 18th Avenue43210ColumbusOH\n", "Josep Miquel Girart \nDepartament d'Astronomia i Meteorologia\nUniversitat de Barcelona\nAv. Diagonal 64708028BarcelonaCatalunyaSpain\n\nInstitut de Ciències de l'Espai (CSIC) / IEEC\nGran Capità 208034BarcelonaCatalunyaSpain\n", "Robert Estalella \nDepartament d'Astronomia i Meteorologia\nUniversitat de Barcelona\nAv. Diagonal 64708028BarcelonaCatalunyaSpain\n" ]
[ "Department of Physics\nThe Ohio State University\n174 West 18th Avenue43210ColumbusOH", "Departament d'Astronomia i Meteorologia\nUniversitat de Barcelona\nAv. Diagonal 64708028BarcelonaCatalunyaSpain", "Institut de Ciències de l'Espai (CSIC) / IEEC\nGran Capità 208034BarcelonaCatalunyaSpain", "Departament d'Astronomia i Meteorologia\nUniversitat de Barcelona\nAv. Diagonal 64708028BarcelonaCatalunyaSpain" ]
[]
We present FCRAO maps as well as combined BIMA and FCRAO maps of the high density molecular emission towards the CS core in the L673 region. With the FCRAO telescope we mapped the emission in the CS (2→ 1), C 34 S (2→ 1), HCO + (1→ 0), and H 13 CO + (1→ 0) lines. The high-density molecular emission, which arises from a filamentary structure oriented in the NW-SE direction, shows clear morphological differences for each molecule. We find that HCO + has an extremely high optical depth, and that the H 13 CO + emission is well correlated with submm sources. The BIMA and FCRAO combined maps recover emission from structure previously undetected or marginally detected, and show an overall aspect of a filamentary structure connecting several intense clumps. We found a total 15 clumps in our combined data cube, all of them resolved at our angular resolution, with diameters in the 0.03-0.09 pc range. Their estimated masses range between 0.02 and 0.2 M ⊙ , except for the largest clump, which has a mass of ∼ 1.2 M ⊙ . We find a clear segregation between the northern and southern region of the map: the northern section shows the less chemically evolved gas and less massive but more numerous clumps, while the southern region is dominated by the largest and most massive clump, and contains the more evolved gas, as traced by emission of late-time molecules. We find that the derived clump masses are below the virial mass, and that the clump masses become closer to the virial mass when they get bigger and more massive. This supports the idea that these clumps must be transient, and that only the more massive ones are able to condense into stars. The clumps we detect are probably in an earlier evolutionary stage than the "starless cores" reported recently in the literature. Only the most massive one has properties similar to a "starless core".
10.1051/0004-6361:20041829
[ "https://export.arxiv.org/pdf/astro-ph/0412621v2.pdf" ]
16,017,616
astro-ph/0412621
81adb3210885fc5a7b732d59a77ff50e50cad063
Evidence for transient clumps and gas chemical evolution in the CS core of L673 March 20, 2022 Oscar Morata Department of Physics The Ohio State University 174 West 18th Avenue43210ColumbusOH Josep Miquel Girart Departament d'Astronomia i Meteorologia Universitat de Barcelona Av. Diagonal 64708028BarcelonaCatalunyaSpain Institut de Ciències de l'Espai (CSIC) / IEEC Gran Capità 208034BarcelonaCatalunyaSpain Robert Estalella Departament d'Astronomia i Meteorologia Universitat de Barcelona Av. Diagonal 64708028BarcelonaCatalunyaSpain Evidence for transient clumps and gas chemical evolution in the CS core of L673 March 20, 2022arXiv:astro-ph/0412621v2 24 Jan 2005 Astronomy & Astrophysics manuscript no. morata2004 (DOI: will be inserted by hand later) Received ...; accepted ...ISM: individual objects: L673 -ISM: abundances -ISM: clouds -ISM: molecules -Radio lines: ISM - Stars: formation We present FCRAO maps as well as combined BIMA and FCRAO maps of the high density molecular emission towards the CS core in the L673 region. With the FCRAO telescope we mapped the emission in the CS (2→ 1), C 34 S (2→ 1), HCO + (1→ 0), and H 13 CO + (1→ 0) lines. The high-density molecular emission, which arises from a filamentary structure oriented in the NW-SE direction, shows clear morphological differences for each molecule. We find that HCO + has an extremely high optical depth, and that the H 13 CO + emission is well correlated with submm sources. The BIMA and FCRAO combined maps recover emission from structure previously undetected or marginally detected, and show an overall aspect of a filamentary structure connecting several intense clumps. We found a total 15 clumps in our combined data cube, all of them resolved at our angular resolution, with diameters in the 0.03-0.09 pc range. Their estimated masses range between 0.02 and 0.2 M ⊙ , except for the largest clump, which has a mass of ∼ 1.2 M ⊙ . We find a clear segregation between the northern and southern region of the map: the northern section shows the less chemically evolved gas and less massive but more numerous clumps, while the southern region is dominated by the largest and most massive clump, and contains the more evolved gas, as traced by emission of late-time molecules. We find that the derived clump masses are below the virial mass, and that the clump masses become closer to the virial mass when they get bigger and more massive. This supports the idea that these clumps must be transient, and that only the more massive ones are able to condense into stars. The clumps we detect are probably in an earlier evolutionary stage than the "starless cores" reported recently in the literature. Only the most massive one has properties similar to a "starless core". Introduction Dense cores of molecular clouds have been extensively used to study the star formation process. The emission of several molecules known to be good tracers of high density molecular gas has been used to study their properties, such as mass and density, temperature, and velocity distributions. However, it was apparent from the first surveys (Zhou et al. 1989;Myers et al. 1991) that there existed large discrepancies between the emission of some of these molecules, such as CS, NH 3 or HCO + , which needed to be clarified in order to obtain the actual distribution of the high density gas in these sources. Pastor et al. (1991) and Morata et al. (1997) carried out a systematic comparison between the emission of the CS (J=1→0) and NH 3 (J, K)=(1,1) transitions, under similar conditions of angular resolution, which confirmed these discrepancies, in particu-Send offprint requests to: Oscar Morata; [email protected] lar the systematic difference in size and separation between the emission peaks of both molecules. To explain these results, Taylor et al. (1996Taylor et al. ( , 1998 developed a chemical model in which high density condensations, or dense cores, are formed by clumps of different masses, ages, sizes, and densities but with sizes of < ∼ 0.1 pc. These clumps would be unresolved in single-dish angular resolution observations such as the ones listed in the previous paragraph. Most of the clumps would disperse before NH 3 abundances build up to significant levels, though these clumps would contain substantial CS, so its emission should be observable. A few clumps, those sufficiently long-lived, or in a more advanced stage of physical and chemical evolution because of being denser or more massive, would form a significant amount of NH 3 , while the CS abundance decreases with time. These clumps would possibly continue their evolution to eventually form stars. Moreover, Taylor et al. (1996Taylor et al. ( , 1998 found that a classification could be made in the model between "earlytime" molecules (showing extended emission like CS) and "late-time" molecules (showing more compact emission like NH 3 ), according to the time at which these molecules reached their peak fractional abundance. In order to test the Taylor et al. chemical model, and in particular to study the small-size structure of the dense cores, we carried out high angular resolution observations with the BIMA telescope towards the L673 molecular cloud of several transitions corresponding to molecules of both families of species (Morata, Girart & Estalella 2003;hereafter MGE03). The region selected had been previously mapped at an angular resolution of ∼ 1 ′ in CS (2→ 1) (Morata et al. 1997) and NH 3 (J, K)=(1,1) (Sepúlveda et al. in preparation). It has no signs of star formation and is located ∼ 9. ′ 5 to the southeast of IRAS 19180+1116. The distance to the L673 cloud is ∼ 300 pc (Herbig & Jones 1983) MGE03 detect emission in the CS (J=2→1), N 2 H + (J=1→0), HCO + (J=1→0) and SO (J K = 2 3 → 1 2 ) lines. The high angular resolution interferometric observations reveal a much clumpier medium than the lower resolution single-dish observations. Several clumps of size < ∼ 0.08 pc are detected distributed all over the region encompassed by the single-dish observations. However, the BIMA interferometer only detects 9-12% of the CS (2→1) emission. Modelling the filtering effect of BIMA, MGE03 show that a clumpy and heterogeneous medium could explain this effect. MGE03 find that the BIMA observations can be fitted by a model with a density at which collapse is halted of n f = 5 × 10 5 cm −3 . A tentative classification of the studied clumps according to the stage of chemical evolution indicated by the molecular abundances is proposed. However, the model cannot explain the observed low HCO + abundances with respect to the predicted values obtained at two positions, but this is possibly due to strong self-absorption in the HCO + (1→ 0) line. The results of MGE03 show that although the interferometric observations lose information on the total emission of the region, they are in agreement with the results obtained with single-dish telescopes, suggesting that interferometric observations are useful for studying with high angular resolution the chemical evolution of the inner structures of starless cores. However, in order to avoid the filtering effect of interferometers, array observations should be combined with single dish observations. This would give high resolution maps of the full emission. In this paper we present the study of the spatial distribution of the molecular emission in L673 from the maps obtained with the 14-m FCRAO telescope, and from the maps obtained by combining the BIMA data of MGE03 and the FCRAO maps presented in this paper. In Sect. 2 we describe the FCRAO observations and the BIMA and FCRAO combination procedure. In Sect. 3 we describe the morphological properties of the molecular emission. In Sect. 4 we analyze the properties of the molecular cloud observed. Observations FCRAO Observations The observations were carried out in 2003 January with the 14-m Five College Radio Astronomy Observatory (FCRAO) telescope using the SEQUOIA (Second Quabbin Observatory Imaging Array) 16 beam array receiver. We used the dual channel correlator (DCC), which provided a bandwidth of 25 MHz and a spectral resolution of 24.4 kHz. The DCC was configured to observe the 4 molecular transitions using 3 different frequency setups: one for HCO + , one for CS, and one for H 13 CO + and C 34 S, which were observed simultaneously. The mainbeam efficiency at the frequencies observed is η mb ≃ 0.50. At 89 and 98 GHz the half-power beam width is 58 ′′ and 52 ′′ , respectively. The weather conditions during the observations were excellent, providing averaged system temperatures of ∼120 K. The observations were performed using the On-The-Fly (OTF) mapping technique, covering a region of 8 ′ × 8 ′ for HCO + and CS and of 6 ′ × 6 ′ for H 13 CO + and C 34 S. The center of the maps was α(J2000) = 19 h 20 m 52. s 20; δ(J2000) = 11 • 14 ′ 45. ′′ 5. The selected reference position was 30 ′ east of the center position. A summary of the observed molecular line transitions is given in Table 1. The OTF maps were regridded to have a cell size of 20 ′′ . The line intensities are given in main-beam brightness temperature, T mb = T * A /η mb . Combination procedure for the FCRAO and BIMA data The BIMA interferometer detects only 9-12% of the CS (2→ 1) emission. Therefore, in order to recover the missing flux of the interferometer data we combined the CS (2→ 1) and HCO + (1→ 0) data obtained with the FCRAO presented in this paper with the BIMA data from MGE03. The first step to do this is to convert the FCRAO antenna temperature to a flux scale. We used a 43 Jy K −1 conversion factor for the two transitions (M. Heyer, private communication). There are basically two single-dish and interferometric data combination techniques that work well (Helfer et al. 2003). One is to add synthetic short-spacing visibilities in the (u, v) plane (e. g., Vogel et al. 1984;Girart et al. 1999). The other is a linear combination procedure of the single-dish and interferometric maps (e. g., Ye, Turtle & Kennicutt 1991;Stanimirovic et al. 1999). We tested these two techniques. In order to fill the BIMA visibility hole with synthetic FCRAO visibilities, we first deconvolved the FCRAO maps from the FCRAO beam pattern using the CLEAN algorithm. The beam was modeled as a Gaussian with the FWHM given in Table 1. The deconvolved FCRAO map was then multiplied by the interferometer primary-beam response, taken as a Gaussian, and centered on the two position observed with BIMA (see MGE03). From these two resulting maps, a randomly distributed sample of visibilities was generated in the (u, v) plane of 0-20 light-ns. The number of visibilities generated was chosen to have a visibility density similar to the BIMA data for the inner part of the (u, v)-plane. To check the relative flux calibration of the FCRAO data with the BIMA data, FCRAO visibilities were generated in the range of 21 to 30 light-ns at the same location as the BIMA visibilities. The two data sets were compared by vector-averaging the visibilities in this range over concentric annuli. The values obtained from the two instruments agreed to within 15%. The linear combination of the maps consists of first generating mosaicked BIMA dirty maps, M bima ; as well as the BIMA dirty beam, B bima ; then, the "dirty" combined map and beam (M comb , B comb ) are generated as follows (Stanimirovic et al. 1999): M comb = M bima + f α M fcrao 1 + α (1) B comb = B bima + α B fcrao 1 + α (2) M fcrao is the FCRAO map regridded to match the M bima cell size and map size and weighted with the primary gain of the mosaicked BIMA maps; B fcrao is modeled as a Gaussian with the FWHM given in Table 1; f is the calibration factor, taken as 1.0 given the agreement of the relative flux calibration between the two instruments; and α is the ratio of the beam areas for the FCRAO and BIMA. We adopted a value of α = 0.050 and 0.042 for the CS and HCO + data, respectively. The different M comb were deconvolved from B comb using the CLEAN algorithm. Note that modelling the FCRAO beam as a Gaussian is a good approximation: typical sidelobes in the beam pattern of a single-dish radio telescope are of the order of −20 dB, which will give an error in the combined beam pattern of less than 10 −3 , well below the dynamical range of the map. As pointed out by Helfer et al. (2003), the method of adding synthetic single-dish visibilities in the (u, v) plane is parameterdependent, in particular on the number and weighting of the synthetic visibilities. Choosing the right combination of pa- rameters, we found results almost identical to those of the linear method technique. Yet the linear method is much faster and simpler, so the combined maps presented here were obtained using the linear method. Details of the combined maps are shown in Table 2. Results and Analysis 3.1. Spatial distribution of large scale molecular gas Figure 1 shows the integrated emission maps of the four species observed with the FCRAO. The maps show that the highdensity molecular emission arises from a filamentary structure oriented in the NW-SE direction. Despite sharing the same filamentary structure, the four maps show clear morphological differences. First, the CS is more spread out than HCO + : the FWHM along the axis perpendicular to the filament major axis is about 2 ′ -3 ′ for HCO + , whereas is about 3 ′ -5 ′ for CS. Second, the HCO + and H 13 CO + relative intensity peaks do not coincide. In particular, at the center of the map they are anti- correlated, i. e., the H 13 CO + maximum is located in a relative minimum of the HCO + . Indeed the H 13 CO + and HCO + intensities are comparable at this position. This suggests a very high optical depth. In order to estimate quantitatively the effect of the opacity on the emission of CS and HCO + we estimated their optical depths from the ratio of the integrated emission maps of H 13 CO + and HCO + and from C 34 S and CS, using the isotope abundance ratios shown in Table 4. Figure 2 shows the grey scale image of the HCO + and CS optical depth overlaid with the contour map of their integrated emission. This figure shows clearly that the HCO + is highly optically thick, with τ > ∼ 7 for most of the emitting region and with a maximum of τ = 90 at the position of L673 SMM 5. On the other hand, the CS emission is moderately optically thick (τ ≃ 1-4). The extremely high optical depths of the HCO + (1→ 0) emission imply that this line is not a good tracer of the distribution of the HCO + molecule in L673. Thus, we confirm the hypothesis pointed out by MGE03 that the high optical depth of HCO + (1→ 0) is responsible for the lack of agreement between the observed BIMA HCO + column densities and the theoretically predicted column densities. Indeed, the maximum HCO + optical depth measured is located within the region observed with BIMA. Strong absorption features and very high optical depths of the HCO + (1→ 0) emission are also observed in other molecular clouds (e. g., Langer et al. 1978;Girart et al. 2000). From an inspection to the IRAS catalogues within the region shown in Fig. 1 we found four infrared sources. IRAS 19182+1112 and IRAS 19183+1109 are detected only at 12 and 25 µm. On the other hand, the other two sources, IRAS 19186+1106 and IRAS 19187+1107 are detected only at 100 µm. There are also four weak submillimeter sources (Visser, Richer & Chandler 2002). Only the strongest submm source, L673 SMM8, is associated with an infrared source, IRAS 19187+1107. All the submm sources are located in strong-molecular-line-emitting regions. Indeed, they are well correlated with the relative intensity peaks of H 13 CO + (1→ 0). Visser et al. (2002) classify these submm sources, except SMM3, as starless cores. SMM3 is classified as a candidate protostar (where there is some evidence of high-velocity CO emission). id. T mb T mb V a ∆V b Area τ c N d (CS) (C 34 S) (CS) (CS) (CS) (CS) (CS) (K) (K) (km s −1 ) (km s −1 ) (K km s −1 ) (10 13 cm −2 ) a 1.34±0.14 < 0.14 e +7.0 0.7 1.03 ± 0. id. T mb T mb V a ∆V b Area τ g N d N (HCO + ) (H 13 CO + ) (H 13 CO + ) (H 13 CO + ) (H 13 CO + ) (H 13 CO + ) (H 13 CO + ) (HCO + ) (K) (K) (km s −1 ) (km s −1 ) (K km s −1 )(.1 km s −1 . b The error in the line width is lower than 0.1 km s −1 . c Using an abundance ratio CS/C 34 S=20 and T ex = 4 K. d Beam averaged column density calculated from the formula in Table 3 of MGE03. e Upper limit taken as 3σ, where σ is the sensitivity per channel. f Range of values for τ ≪ 1 and τ = 1.72. g using an abundance ratio HCO + /H 13 CO + = 62 (Langer & Penzias 1993) and T ex = 4 K. h We adopt a value of τ = 4 as a lower limit. By analyzing the kinematical properties of the CS and H 13 CO + emission, we found a smooth velocity gradient along the major axis of the filamentary molecular structure: from NW to SE the v LSR goes from ∼ 7.0 to 7.4 km s −1 . In addition, east of the center of the map the CS emission is redshifted to ∼ 7.7 km s −1 . Comparison of the emission of the CS (1→ 0) and (2→ 1) lines We have also compared the CS (1→ 0) line measured with the Yebes telescope (Morata et al. 1997) with the CS (2→ 1) line obtained with the FCRAO telescope, after convolving the spectra of the (2→ 1) line with the 1.9 ′ beam of the Yebes telescope. We compared the resulting spectra at the position of the CS (1→ 0) peak, corresponding to the North field of the BIMA map, and the position of the peak NH 3 intensity, corresponding to the South field. The derived T ex at the North position is the same for both lines, 4.2 K. In the South position, where we did not observe the C 34 S (1→ 0) transition, the excitation temperature derived from the (2→ 1) line, 4.4 K, is very similar to the value found in the northern region. Column densities As in MGE03, we selected several positions in the region mapped with the FCRAO observations in order to obtain the physical parameters of the gas and to compare them with the chemical model. Table 3 shows the coordinates for each position and their map counterparts, if any. These positions were selected mainly to coincide with the 4 positions we selected in MGE03, and with the SMM and IRAS sources of the region. We also selected two positions not related to any object in order to see if they traced "younger" gas. Table 4 shows the line parameters for the CS (2→ 1) and H 13 CO + lines, and the physical parameters obtained for each position. We adopted a value of T ex = 4 K for our calculations, based on the result we had obtained from the comparison between the CS (1→ 0) and (2→ 1) lines (Morata et al. 1997 and MGE03, respectively). Table 5 shows the ratio between the derived column densities for CS and HCO + obtained for each position, which can be compared to the results of our chemical model (MGE03). We observe that there is a similar trend to the results we obtained in MGE03. In the positions where there are no SMM or IRAS sources (positions b, d, and e) the CS column density is higher than the HCO + column density, which would indicate that the clumps are chemically young. In the positions where there is a SMM or IRAS source, the column density fo HCO + is higher than for CS. This is what we expect for later times, when the clumps are more chemically evolved, HCO + is more abundant, and CS starts to suffer freeze-out. We found the highest ratio for SMM5 (position g), which we previously classified with the BIMA observations as one of the two chemically more evolved points, and where the N 2 H + emission is more intense. Thus, we confirm our previous classification. Visser et al. (2002) calculate the masses of the SMM sources in L673 from the total flux measured in an area of radius r. They also calculated the H 2 column density over this area. We have convolved the spectra of the CS (2→ 1) and C 34 S (2→ 1) emission with a Gaussian with FWHM equal to the r derived for each SMM source, in order to estimate the CS column density for each SMM position. Table 6 shows the averaged CS abundance obtained from the H 2 and CS column densities. The derived values are in the range 2-10 × 10 −9 cm −2 . We found the lowest value near the IRAS source at the SE part of the large scale maps, and the highest abundance at the position of SMM6, a submm source that seems to be less centrally peaked than other submm sources (Visser et al. 2002), which would indicate that it is in a less evolved stage. CS abundances 3.2. Spatial distribution at high angular resolution Figure 3 shows the composite of the combination process of the BIMA and FCRAO observations for the CS (2→ 1) and HCO + (1→ 0) emission. The HCO + combined map shows a clear filamentary structure connecting several clumps, although some of them, particularly to the SE, are far from the phase center of the map, which makes it difficult to distinguish them from another filament. However, as indicated by Fig. 2, the region observed has a very high optical depth, so the emission shown in the combined map cannot be used to describe the true morphological and kinematical behaviour of the HCO + emission. On the other hand, the combined map of the CS emission, although it has a moderately high opacity (2-4), can be used to trace the CS distribution. When we recover all the flux, we find that all the most prominent clumps found in the BIMA observations are also detected but with more intensity. We believe that these clumps, which lie near the phase center of the map, are real. We also recover emission from a great deal of other structure which was not, or only marginally, detected previously. This confirms the predicted effect of the BIMA filtering in MGE03. The overall aspect is also of a filamentary structure connecting several intense clumps. One of these clumps coincides with the nominal position of the source SMM 5. The clumps and the filamentary structure of the emission are more clearly revealed in Fig. 4, which shows the channel maps of the combined BIMA and FCRAO maps for the CS (2→ 1) line. Here we find a very complex structure, with several clumps with sizes of ∼ 13 ′′ -25 ′′ , appearing in two or three of the channel maps. Some of these clumps might coincide with clumps found in the HCO + map. Clump Analysis Since the combined high-angular-resolution maps show a highly clumpy CS spatial and velocity structure, we carried out the clump analysis technique developed by Williams, de Geus & Blitz (1994) in our combined BIMA and FCRAO CS (2→ 1) data cube. This technique consists mainly of two algorithms, CLFIND and CLSTATS. CLFIND, which does not assume any clump profile, works by contouring the data at a multiple of the rms noise of the data cube, then searches for peaks of emission to locate the clumps, and then follows them down to lower intensities. CLSTATS is a task that calculates integrated intensities, peak positions, sizes, velocity dispersion, etc, for the clumps found by CLFIND. A very detailed explanation of this analysis technique is found in Williams et al. (1994), who also found that the CLFIND algorithm obtains the best results when the contour interval and the lower contour are set to 2σ. However, they recommend using a higher value of the lower contour if there is background emission. This is the case for the combined L673 CS (2→ 1) maps, where there is some lowlevel (∼ 2σ) extended emission. Thus, we run the CLFIND and CLSTATS algorithms for the L673 CS channel maps with the contour interval set to 0.48 Jy beam −1 (2σ) and the lowest contour set at 0.96 Jy beam −1 (4σ). A total of 15 clumps were found, all of them with a peak intensity of at least ∼ 4σ. Table 7 lists the physical parameters derived for the 15 clumps found by the algorithm. All the clumps are clearly resolved at our angular resolution, and have deconvolved diameters in the 0.03-0.09 pc range. The velocity dispersion of the clumps (i. e., the FWHM corrected for the spectral resolution) is not very different for the different clumps, 0.31-0.65 km s −1 . The clumps have masses between 0.02 and 0.36 M ⊙ . The estimated clump densities range between 0.6 and 4.6 × 10 4 cm −3 . This value of the averaged density of the clumps implies that, since the CS (2→ 1) critical density is 5 × 10 5 cm −3 , they are subthermalized. It also explains why the derived excitation temperature, 4 K, is significantly below the kinetic temperature expected in the cloud, 10 K. Figure 5 shows the location of the clumps derived by the algorithm as circles. The areas of the circles are proportional to the masses of the clumps. This figure shows a segregation of clumps between the northern and southern section of the map. The clumps with peak intensities in the 7.2 and 7.5 km s −1 channels arise from the northern region, whereas the clumps that have a peak in the 6.9 km s −1 channel are located in the southern region. In addition, the southern part contains fewer clumps than the northern part, but it is dominated by the largest and most massive clump. Interestingly, this clump is associated with the two N 2 H + clumps found by MGE03. This confirms the idea that both N 2 H + and NH 3 trace the more evolved, massive clumps. Discussion 4.1. The large-scale structure of L673. Correlation of HCO + and N 2 H + emission with dust As we have pointed out, the H 13 CO + seems to be well correlated with the dust as detected by the submillimeter observations. This suggests that H 13 CO + traces the more massive or more evolved clumps. At the same time, Fig 1 shows that the peak H 13 CO + emission is also coincident with the position of the BIMA peak of N 2 H + , and a SMM source (SMM 5). In our chemical models, HCO + and N 2 H + are late-time molecules. Thus, the apparent discrepancy found in MGE03 between the column densities observed with BIMA and the expected values from the chemical model was due to the extremely high opacity of the HCO + emission. H 13 CO + is found where N 2 H + is also found, although a little more extended, as can be expected from the models, because it is formed a little earlier. Moreover, the emission of H 13 CO + , a late-time molecule, coincides with the position of the dusty submm sources, which also agrees with the theoretical predictions (Taylor et al. 1998). The small-scale structure of L673 4.2.1. CS clump distribution properties Figure 6 shows the mass and size distribution of the derived clumps. It is clear that with only 15 clumps we cannot pursue a statistical analysis of the mass distribution of the clumps. However, with the data available, the mass clump distribution is compatible with the low-mass end of the mass distribution observed in low-mass star formation clouds, such as in ρ Oph (Motte et al. 1998;Johnstone et al. 2000), where the slope of dN/dM is not so steep (see Fig. 6). The derived clump masses are significantly below the virial mass (see Table 7). This strongly supports the idea that these clumps must be transient (Taylor et al. 1996(Taylor et al. , 1998. Recently, a detailed study of the illuminated clumps ahead of the HH 2 object also suggested that clumps in molecular clouds are transient (Girart et al. 2002;Viti et al. 2003). Fig. 7 also shows that the mass of the clumps becomes closer to the virial mass when they get bigger and more massive, which is when they have a higher chance to form stars. However, the mass calculated for the clumps is derived assuming constant CS abundance. If, as it seems, the CS abundance is lower for more evolved clumps, the slope of the relation shown in Fig. 7 would be steeper, and the clumps would probably have a mass closer to the virial mass. Figure 8 shows the comparison of two channel maps for the combined CS (1→ 2) and N 2 H + (1→ 0) emission. We can Fig. 4). The thick circles are centered at the position and at the channel at which each clump shows its highest intensity, as found by the CLFIND algorithm of Williams et al. (1994) and described in Sect. 3.3. The areas of the circles are proportional to the masses of the clumps. The numbers correspond to the identifier in the first column of Table 7. clearly see that the peak of the CS emission is shifted with respect to the peak of N 2 H + . Moreover, the CS emission is more spread than N 2 H + , so we are detecting chemical differentiation inside the clump. Largest clump and N 2 H + clumps properties The largest CS clump is not only the clump with the highest M clump /M virial ratio (and thus with the highest probability of undergoing star formation) but it is also the only one associated with N 2 H + emission (see Sect. 3.3). This confirms the Taylor et al. (1996Taylor et al. ( , 1998 scenario. Interestingly, the density of this clump, as derived from the CS emission, is similar to that of the other clumps. This may be explained if the emission of the N 2 H + traces the highest densities of the clump, high enough for CS depletion to occur, as has been observed in several dense starless cores (e. g., Tafalla et al. 2002Tafalla et al. , 2004. We have calculated from the BIMA data the mass traced by the N 2 H + for this clump. Assuming a N 2 H + fractional abundance of 8 × 10 −11 , which was obtained using the H 2 column density determination from the submm sources of Visser et al. (2002), we could derive a total mass for the biggest clump of 1.2 M ⊙ . This mass is significantly higher than those of the other clumps. In addition, M clump /M vir is 0.52, so it is close to collapse. On the small scale structure of L673 The previous results, along with the results from the lowerresolution analysis in Sect. 3.1.2, show at least the differences between the two regions within the BIMA observations. a Optical depth derived at the peak position of the clump from the intensity ratio of the CS (2→ 1) and C 34 S (2→ 1) FCRAO maps. b FWHM of the line corrected for channel resolution as described in Appendix A of Williams et al. (1994) c Equivalent circular radius corrected for beam size as described in Appendix A of Williams et al. (1994) d Clump mass derived from the integrated emission obtained with CLSTATS corrected for the CS (2→ 1) optical depth, τ, and assuming a constant excitation temperature of T ex = 4 K. An abundance ratio of X[CS/H 2 ] = 10 −8 is adopted. e Virial mass obtained from M vir = 5 ∆R σ 2 v /α G with σ v = ∆v/2.355 and assuming an inverse-square power law density profile (α=5/3). f Average volume density obtained assuming a sphere of mass M cl and radius ∆R. The northern region shows a gas less chemically evolved, and smaller and more numerous clumps. The southern region contains the more evolved gas and the biggest clump. The clumps detected in our observations seem to be of a different kind than the ones usually studied in surveys of "starless cores" (Tafalla et al. 2002;Bergin et al. 2001). Those are usually older cores, which have already achieved the more evolved and more massive stage of evolution, as is clearly indicated by the detection of late-time molecules, as N 2 H + and NH 3 , and by the high level of depletion of some molecules onto dust grains. We find one such clump in our region, which might be just arriving at that evolved phase. This clump is at the same time the most massive, shows emission of N 2 H + , and different emission peaks for CS and N 2 H + , which could point to depletion. However, we detect many other clumps, less massive, that show almost no emission of late-time molecules, and which probably will disperse before they can achieve the "starless core" stage. We face the question of what the origin of these transient clumps could be. They are probably caused by interstellar turbulence (see e.g. Padoan 1995;Ballesteros-Paredes et al. 1999). In support of this idea, we observe that the derived masses for our clumps are around 0.1M ⊙ , which is also the massscale of turbulent fragmentation in the ISM (Padoan 1995). This small scale turbulence is probably related to large-scale flows in the diffuse Galactic interstellar medium, which several studies (Hartmann et al. 2001;Bergin et al. 2004) propose as a mechanism that can lead to the fast formation and dispersal of molecular clouds, which could explain the apparent rapid star formation and short cloud lifetimes observed in the regions of the solar neighborhood. On a smaller scale, several authors postulate the possible existence of transient clumps. Falle & Hartquist (2002) propose that slow-mode MHD waves could create, under suitable conditions, large density enhancements, of the order of a factor of 30, which could last on the order of 1 Myr, before dispersing back to the original density. At the same time, Vázquez-Semadeni et al. (2005 and Ballesteros-Paredes et al. (2003) also argue that cores within molecular clouds may not be in hydrostatical equilibrium, and would be transient features generated by the dynamical flow in the cloud, not necessarily requiring strong magnetic fields. These cores must then either collapse, if they have an excess of gravitational energy, or re-expand and merge back into the surrounding molecular cloud. The re-expansion time is expected to be longer than the compression time because of the retarding action of self-gravity, and is estimated to be of the order of a few free-fall times. This would be consistent with the result that typically there are more starless than star-forming cores in molecular clouds and that, as in our case, most of the cores do not appear to be gravitationally bound. Only sufficiently massive cores would eventually undergo local collapse. In any case, such models would have consequences for the physical and chemical evolution of these regions . Firstly, the chemistry would be mainly "young", as the molecules would be destroyed when the gas returns to a more diffuse state. Secondly, this mechanism would halt the process of freeze-out of molecules onto dust, as the frozen-out molecules would also return to the gas phase as the density of the gas goes back to the diffuse state. Thus, these regions would not be completely depleted of molecules in the gas phase, as is apparently shown by some observations (Gibb & Little 1998). Finally, the mass distribution of these transient clumps probably determines the low-mass star formation rate. However, an alternative possibility is that the more massive clumps were formed by aggregation of smaller clumps into a massive one, which could then be closer to collapse. The analysis of our clumps allows us to make some rough estimates to see if this can happen since transient clumps are expected to survive for ∼ 1 Myr (Falle & Harquist 2002;Vázquez-Semadeni et al. 2004). Two limiting cases can be studied. One is that of a 1-dimensional structure, that is, the clumps are assumed to move inside a filamentary structure, with relative velocities (between clumps) of 0.3 km s −1 . This is the typical difference in the radial velocities of the clumps, which should roughly give a reasonable lower limit for the true relative velocities. In the course of ∼ 1 Myr any of the observed clumps would have travelled ∼ 63000 AU or ∼ 3. ′ 5, which is the size of the field of view of our maps. The other case is that of a 3dimensional (3-D) structure with similar scales in the three dimensions. In this case the projected area of this volume in any particular direction is A proj ≃ V 2/3 tot , where A proj and V tot are the area and volume, respectively. Let us assume that in a volume V tot there are n identical spherical clumps of radius R distributed randomly, with a random velocity distribution with an average velocity v 3D . The 3-D filling factor of the clumps for this volume is f 3D = nV cl /V tot = nV cl /A 3/2 proj , where V cl is the clump volume. The volume swept up by a clump in a time t is V sw (t) = v 3D t πR 2 . For a given clump, the time scale needed to cross another clump, t X , is such that f 3D V sw (t X ) = V cl . From the previous equations, the average crossing scale time can be written as t X = A 3/2 proj /(n v 3D πR 2 ). Since what we measure is the radial velocity, v obs and v 3D = 2 v obs , then t X = A 3/2 proj /(2 n v obs πR 2 ). For A proj equal to our field of view (∼ 3. ′ 5 × 2. ′ 0), which includes 15 clumps, and using a radius equal to the average value for the measured clumps (from Table 7), R ≃ 18 ′′ , we found that t X ≃ 6×10 5 yr. Then it could be possible that star formation was the result not of the gravitational collapse of just one clump, but of a "merging" effect. In this case, the geometry of dense cores would be far from spherical symmetry, even the density distribution would not follow n(r) ∝ r −2 . Tafalla et al. (2004) find that their starless cores are not spherical, but that clear inhomogeneities are found in them. Conclusions We have made a multitransitional study of the spatial distribution of the molecular emission with the 14 m FCRAO telescope in the starless core found in the L673 region, which had been previously observed at lower angular resolution (Morata et al. 1997) and with the BIMA array (MGE03). The main goal of these observations was to combine them with the array ob- Fig. 8. Comparison of the channel maps of the combined CS (2→ 1) emission (solid contours) for the 7.16 (left) and 6.87 km s −1 (right) channels; and the BIMA N 2 H + (1→ 0) emission (contours and grey scale) for the −0.8 (left) and −1.1 km s −1 (right) channels. The contour levels for the CS maps are the same as in Fig. 4. The contour levels for the N 2 H + maps are 3, 3.5, 4, 5, 6, and 7 times 0.20 Jy beam −1 (the rms noise of the maps). The symbols are the same as those in Fig. 1. servations in order to obtain high resolution maps of the full emission, and thus be able to confirm that the clumpy structure detected with the BIMA telescope was real. The main results were: 1. With the FCRAO we detected emission in the CS (2→ 1), C 34 S (2→ 1), HCO + (1→ 0), and H 13 CO + (1→ 0) lines. Although every map shows clear morphological differences, we found that the high density molecular emission in the four molecules arises from a filamentary structure oriented in the NW-SE direction. The ratio between HCO + and H 13 CO + also confirms that the apparent lack of HCO + (1→ 0) emission in the BIMA maps was due to the extremely high optical depth in this line. We also found that several submm sources detected in this region are located in strong molecular line emitting regions, and that they seem to be well correlated with the relative intensity peaks of H 13 CO + . 2. We selected several positions to study the local physical parameters. We confirm the results of MGE03: the column density of CS is higher than that of HCO + for positions were the gas is chemically younger, i.e. not associated with IRAS or submm sources. In the positions where there is a submm or IRAS source, the HCO + column density is higher than that of CS. 3. The BIMA and FCRAO combined maps of the CS emission show that all the prominent clumps found in the BIMA observations are also detected but with more intensity. We believe that these clumps are real. We also recover emission from other structure which was previously undetected or only marginally detected. The overall aspect is also of a filamentary structure connecting several intense clumps. 4. We found a total of 15 clumps in our combined data cube. All of them are resolved at our angular resolution, with diameters in the 0.03−0.09 pc range. Their estimated masses range between 0.02 and 0.36 M ⊙ , and their densities between 0.6 and 4.6×10 4 cm −3 , which indicates that the CS (2→ 1) emission is clearly subthermalized. 5. There is a clear segregation in the properties of the detected clumps between the northern and southern regions of the map. The northern region shows the less chemically evolved gas, and less massive but more numerous clumps. On the other hand, the southern region contains the more evolved gas and the most massive clump. This clump is also the only one associated with BIMA N 2 H + emission. Moreover the H 13 CO + emission correlates well with the dust emission traced by the submillimeter observations, and its emission peak also coincides with the BIMA N 2 H + peak and the SMM 5 source. All these results support the predictions of our chemical model, which proposed that late-time molecules, such as N 2 H + and HCO + , should be found in the same regions. 6. The clump masses derived are below the virial mass, which means that these clumps must be transient, as only the more massive ones are able to condense into stars. 7. The properties of the clumps we detect seem to be different from those of the "starless cores" recently found in the literature. Although we might have detected one such core, the rest are in an early evolutionary stage, after which they will most likely disperse. We find that the simple chemical model proposed in MGE03 to explain the properties of the CS core of L673 is confirmed by the combination of single-dish and array observations. However, there is still a lot of work to be done on the formation mechanisms of these clumps and on the possibility that the merging of smaller clumps can lead to more massive and gravitationally unstable clumps. Further observing and modelling of the chemistry of these clumps and of the interclump gas will be needed in order to test the transient nature of the clumps. Fig. 1 . 1Contour maps of the integrated emission over the v LSR range 6.05-8.45 km s −1 of (from top to bottom) HCO + (1→ 0), H 13 CO + (1→ 0), CS (2→ 1), and C 34 S (2→ 1). The HCO + contour map is overlaid on the J band image from the 2MASS All-Sky Survey. The filled triangles show the submm sources reported by Visser, Richer & Chandler 2002 (see labels in the bottom panel). The crosses show the positions of infrared sources from the IRAS Serendipitous Survey Catalog. From west to east:IRAS 19182+1112, IRAS 19183+1109, IRAS 19186+1106 and IRAS 19187+1107. Contour levels are 2, 3, 4, . . . 8 times 0.24 K km s −1 for CS and HCO + and 1, 2, 3, . . . 6 times 0.12 and 0.086 K km s −1 for C 34 S and H 13 CO + , respectively. Fig. 2 . 2Grey scale image of the optical depth of HCO + (1→ 0) (top panel) and CS (2→ 1) (bottom panel) derived from the integrated maps of Fig. 1. The scale levels are shown on the right side of each panel. The overlaid contour maps show the integrated emission of HCO + and CS. The dashed line shows the gain at the 0.5 level of the BIMA primary beam response. The symbols are the same as those in Fig. 1. Fig. 3 . 3Composite of the combination process for CS (2→ 1) (bottom panels) and HCO + (1→ 0) (top panels). All maps show the integrated emission over the v LSR range 6.05-8.45 km s −1 . Left panels: FCRAO maps. Middle-left panels: FCRAO maps corrected for the primary beam of the mosaicked BIMA maps. Middle-right panels: mosaicked BIMA maps. Right panels: Combined BIMA and FCRAO maps. For the FCRAO maps, the contour levels are 3, 4, 5, . . . 12 times 3.17 and 2.80 Jy beam −1 km s −1 for CS (2→ 1) and HCO + (1→ 0), respectively. For the BIMA and combined maps, the contour levels are 2, 3, 4, . . . 9 times 0.27 and 0.24 Jy beam −1 km s −1 for CS (2→ 1) and HCO + (1→ 0), respectively. The inner and outer dashed lines show the 0.5 and 0.25 level of the BIMA primary beam response. The symbols are the same as those inFig. 1. Fig. 4 . 4Channel maps of the CS (2→ 1) emission. The contour levels are -3, 3, 4.5, 6, . . . 12 times 0.24 Jy beam −1 (the rms noise of the maps). The inner and outer dashed lines show the 0.5 and 0.25 level of the BIMA primary beam response. The symbols are the same as inFig. 1. Fig. 5 . 5Combined BIMA and FCRAO CS (2→ 1) contour maps for the three channels with the strongest emission (from Fig. 6 . 6Top Mass distribution of the L673 CS clumps within the primary beam of the BIMA maps. The lines show the clump mass spectrum for a power-law distribution, N ∝ M −0.27 . Bottom Size (equivalent circular radius) distribution of the L673 CS clumps. The dotted line shows the radius of the CS (2→ 1) synthesized beam. Note that all the clumps are well resolved at our angular resolution. Fig. 7 . 7Relation between the clump to virial mass ratio with respect to the clump mass (top) and the equivalent circular radius (bottom). The open circle represents the most massive clump when the mass derived from the N 2 H + emission is considered. Table 1 . 1Lines Observed with the 14-m FCRAO telescopeν FWHM ∆v ∆T mb Molecule Transition (GHz) Beam (km/s) (K) H 13 CO + (1→ 0) 86.75434 59 ′′ 0.17 0.04 HCO + (1→ 0) 89.18852 58 ′′ 0.16 0.11 C 34 S (2→ 1) 96.41295 53 ′′ 0.15 0.07 CS (2→ 1) 97.98097 52 ′′ 0.15 0.14 Table 2 . 2Parameters of the FCRAO and BIMA Combined MapsRms Molecular Synthesized Beam ∆v Noise Transition HPBW P.A. (km/s) (Jy Beam −1 ) HCO + (1→ 0) 13. ′′ 3 × 10. ′′ 3 −16 • 0.33 0.22 CS (2→ 1) 13. ′′ 1 × 10. ′′ 3 −17 • 0.30 0.25 Table 3 . 3Selected positionsid. Position map counterpart a α (J2000) δ (J2000) a 19 h 20 m 34.3 s 11 • 18 ′ 05.5 ′′ IRAS 19182+1112 b 19 h 20 m 43.8 s 11 • 16 ′ 05.5 ′′ c 19 h 20 m 47.9 s 11 • 16 ′ 05.5 ′′ SMM 6 d 19 h 20 m 54.4 s 11 • 15 ′ 25.5 ′′ N e 19 h 20 m 54.7 s 11 • 14 ′ 45.5 ′′ f 19 h 20 m 47.9 s 11 • 14 ′ 05.5 ′′ SMM 3 g 19 h 20 m 52.0 s 11 • 13 ′ 45.5 ′′ SMM 5 / S h 19 h 21 m 02.9 s 11 • 11 ′ 45.5 ′′ SMM 8 a Positions N and S correspond to positions selected for study in MGE03. Table 4 . 4Physical parameters The error in the line velocity is lower than 010 11 cm −2 ) (10 13 cm −2 ) a 1.49 ± 0.23 0.46 ± 0.12 +7.0 0.4 0.22 ± 0.03 0.37 3.86 ± 0.59 2.40 ± 0.36 b 1.53 ± 0.19 0.42 ± 0.06 +7.1 0.5 0.23 ± 0.03 0.32 3.91 ± 0.44 2.42 ± 0.27 c 0.95 ± 0.18 0.49 ± 0.07 +7.2 0.7 0.37 ± 0.02 0.73 7.16 ± 0.47 4.44 ± 0.29 d 0.70 ± 0.16 0.34 ± 0.05 +7.4 0.7 0.25 ± 0.03 0.66 4.77 ± 0.48 2.95 ± 0.30 e 0.54 ± 0.15 0.21 ± 0.07 +7.5 0.5 0.12 ± 0.02 0.48 2.12 ± 0.38 1.31 ± 0.24 f 1.17 ± 0.19 0.78 ± 0.04 +7.3 0.5 0.38 ± 0.02 1.10 8.12 ± 0.43 5.03 ± 0.26 g 0.81 ± 0.17 0.89 ± 0.05 +7.3 0.5 0.45 ± 0.02 > 4 h > 18.2 ± 0.83 > 11.3 ± 0.51 h 2.80 ± 0.25 0.42 ± 0.97 +7.4 0.6 0.25 ± 0.03 0.16 4.10 ± 0.48 2.55 ± 0.30 a Table 5 . 5Ratio of column densitiesid. N(HCO+)/N(CS) a 1.6-2.5 a b 0.67 ± 0.08 c 1.54 ± 0.11 d 0.99 ± 0.10 e 0.44 ± 0.08 f 1.29 ± 0.08 g > 2.96 ± 0.19 h 1.63 ± 0.21 a The value 1.6 is obtained for τ = 1.72. The value 2.5 is obtained when τ ≪ 1. Table 6 . 6CalculatedCS abundances at the positions of the SMM sources Beam N(H 2 ) a N(CS) (arcsec) (cm −2 ) T C 34 S /T CS τ (cm −2 ) X(CS) SMM3 60 4.3×10 21 0.15±0.02 3.2 2.74×10 13 6.4×10 −9 SMM5 80 4.0×10 21 0.18±0.02 3.8 2.66×10 13 6.7×10 −9 SMM6 100 2.5×10 21 0.15±0.02 3.2 2.56×10 13 1.0×10 −8 SMM8 110 3.8×10 21 0.11±0.02 2.0 1.08×10 13 2.8×10 −9 a From Visser et al. 2002 Table 7 . 7Clumps within the BIMA primary beamN T peak τ a CS ∆v b ∆R c M d cl M e vir n(H 2 ) f (K) (km s −1 ) (pc) (M ⊙ ) (M ⊙ )(10 4 cm −3 ) 1 3.5 3.2 0.49 0.045 0.36 2.28 1.9 2 3.0 3.2 0.39 0.037 0.14 1.18 1.3 3 2.8 2.7 0.43 0.023 0.12 0.89 4.6 4 2.5 2.8 0.55 0.026 0.11 1.65 3.0 5 2.5 2.7 0.51 0.020 0.07 1.08 4.1 6 2.2 3.1 0.38 0.036 0.15 1.10 1.6 7 2.1 2.6 0.37 0.025 0.04 0.73 1.2 8 2.0 3.5 0.49 0.022 0.08 1.12 3.4 9 2.0 2.4 0.31 0.033 0.08 0.64 1.2 10 2.0 3.0 0.56 0.021 0.09 1.39 4.4 11 1.9 2.7 0.44 0.020 0.04 0.80 2.3 12 1.9 1.9 0.57 0.027 0.03 1.83 0.6 13 1.6 4.0 0.39 0.015 0.02 0.47 2.5 14 1.4 3.0 0.65 0.022 0.07 1.98 3.3 15 1.4 2.0 0.39 0.021 0.02 0.67 1.2 Acknowledgements. 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[ "Chiral dynamics in few-nucleon systems", "Chiral dynamics in few-nucleon systems" ]
[ "Evgeny Epelbaum [email protected] \nInstitut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany\n", "Ulf-G Meißner \nInstitut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany\n", "Walter Glöckle \nInstitut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany\n", "C Elster \nInstitut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany\n", "H Kamada \nInstitut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany\n", "A Nogga \nInstitut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany\n", "H Witala \nInstitut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany\n" ]
[ "Institut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany", "Institut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany", "Institut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany", "Institut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany", "Institut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany", "Institut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany", "Institut für Kernphysik (Th)\nForschungszentrum Jûlich\nD-52425JülichGermany" ]
[]
We report on recent progress achieved in calculating various few-nucleon low-energy observables from effective field theory. Our discussion includes scattering and bound states in the 2N, 3N and 4N systems and isospin violating effects in the 2N system. We also establish a link between the nucleon-nucleon potential derived in chiral effective field theory and various modern high-precision potentials.
10.1016/j.nuclphysa.2008.02.265
[ "https://export.arxiv.org/pdf/nucl-th/0109065v1.pdf" ]
14,205,713
nucl-th/0109065
9d9fe519ad951b1d82301a23ae9ebbeef2067d39
Chiral dynamics in few-nucleon systems 21 Sep 2001 Evgeny Epelbaum [email protected] Institut für Kernphysik (Th) Forschungszentrum Jûlich D-52425JülichGermany Ulf-G Meißner Institut für Kernphysik (Th) Forschungszentrum Jûlich D-52425JülichGermany Walter Glöckle Institut für Kernphysik (Th) Forschungszentrum Jûlich D-52425JülichGermany C Elster Institut für Kernphysik (Th) Forschungszentrum Jûlich D-52425JülichGermany H Kamada Institut für Kernphysik (Th) Forschungszentrum Jûlich D-52425JülichGermany A Nogga Institut für Kernphysik (Th) Forschungszentrum Jûlich D-52425JülichGermany H Witala Institut für Kernphysik (Th) Forschungszentrum Jûlich D-52425JülichGermany Chiral dynamics in few-nucleon systems 21 Sep 2001 We report on recent progress achieved in calculating various few-nucleon low-energy observables from effective field theory. Our discussion includes scattering and bound states in the 2N, 3N and 4N systems and isospin violating effects in the 2N system. We also establish a link between the nucleon-nucleon potential derived in chiral effective field theory and various modern high-precision potentials. INTRODUCTION Chiral effective field theory (EFT) offers a systematic and controlled method to study the dynamics of few-nucleon systems. In the approach proposed by Weinberg [1,2], one starts from an effective Lagrangian for nucleon and pion fields as well as external sources, in harmony with chiral and gauge invariance. The effective nucleon-nucleon (NN) potential is constructed using a unitary transformation to avoid any energydependence and applying systematic power counting, as described in ref. [3]. To leading order, one has the one-pion exchange (OPE) together with two contact terms accompanied by low-energy constants (LECs). At next-to-leading order (NLO), renormalizations of the OPE, the leading two-pion exchange (TPE) diagrams and seven more contact operators appear. At NNLO, one has to include the subleading TPE with one insertion of dimension two pion-nucleon vertices (the corresponding coupling constants are denoted by c 1,3,4 ). The potential is then used in a properly regularized Lippmann-Schwinger equation (e.g. by a sharp or exponential momentum cut-off) to generate the bound and scattering states, as detailed in ref. [4]. The iteration of the potential leads to a non-perturbative treatment of the pion exchange which is of major importance to properly describe the NN tensor force. The nine LECs are to be determined from a fit to the low np partial waves. Here, we discuss implications of the uncertainties in the values of the c i 's to various properties of chiral forces. Apart from the already published NNLO potential with numerically large values of the c i 's, taken from ref. [5], we construct the NNLO* version with (numerically) reduced and fine tuned values of c 3 , c 4 . We also discuss physical mechanisms, which can explain such smaller values of these lowenergy constants. In contrast to the NNLO version, the NNLO* chiral potential does not lead to deeply lying bound states. This makes it more suitable for many-body applications. Both versions reproduce np phase shifts in most partial waves fairly well up to E lab ∼ 200 MeV. In the same framework, one can also include charge symmetry breaking and charge dependence of the nuclear force by including the light quark mass difference and elecromagnetic corrections [6]. This allows to predict differences between some phase shifts in pp and nn systems. The extension to three-and four-nucleon systems has also been started using the NLO potential [7]. At this order, no 3N forces (3NF) appear and one can make parameter-free predictions. We also present some first NNLO* results including only two-nucleon forces. We further show that the numerical values of the LECs can be understood on the basis of phenomenological one-boson-exchange (OBE) models [8]. We also extract these values from various modern high accuracy NN potentials and demonstrate their consistency and remarkable agreement with the values in the chiral effective field theory approach. This paves a way for estimating the lowenergy constants of operators with more nucleon fields and/or external probes. FEW-NUCLEON FORCES IN CHIRAL EFT Chiral EFT is a powerful tool, which allows to calculate low-energy observables performing an expansion in powers of the low-energy scale Q associated with momenta of external particles. 1 The pion mass M π is treated on the same footing as Q. To select the relevant diagrams contributing to the S-matrix at a certain order, one makes use of power counting. This consists of a set of rules to calculate the power ν of the low-energy scale Q for any given diagram. The precise meaning of the power counting scheme depends on the system one is investigating. In what follows, we will concentrate on various low-energy processes between few (2, 3 and 4) nucleons. The essential complication in that case compared to pion-pion or pion-nucleon scattering is given by the fact, that the nucleon-nucleon interaction is nonperturbative even at very low energies. To deal with this problem Weinberg proposed to use time-dependent ("old-fashioned") perturbation theory instead of the covariant one. The expansion of the S-matrix, obtained using this formalism, has the form of a Lippmann-Schwinger equation with an effective potential, defined as the sum of all diagrams without pure nucleonic intermediate states. Such states would lead in "old-fashioned" perturbation theory to energy denominators, which are by a factor of Q/m smaller than those from the states with pions and which destroy the power counting. Here and below, m denotes the nucleon mass. The effective potential is free from such small energy denominators and can, in principle, be calculated perturbatively to any given precision. This strategy has been followed in the pioneering work by Ordóñez et al. [9]. The effective NN Hamiltonian derived in this formalism possesses some unpleasant properties: it depends, in general, explicitly on the energy of incoming nucleons and is not Hermitean. This complicates its application to few-nucleon systems. In order to avoid these problems we construct an effective NN Hamiltonian using the method of unitary transformation (projection formalism), as described in ref. [3]. For that we modify Weinberg's power counting in an appropriate way. All details are given in [3]. In what follows, we will show the qualitative and the quantitative results for the effective potential in the first few orders of the chiral expansion. Let us begin with the power counting. As pointed out above, any (time-ordered) diagram T contributing to the few-nucleon scattering process scales as: T ∼ Q/Λ χ ν ,(1) where Λ χ ∼ 1 GeV is the typical scale of chiral symmetry breaking. For any diagram with E n nucleons, L loops and V i vertices of type i one has (we only consider connected diagrams): ν = −4 + 2E n + 2L + ∑ i V i ∆ i ,(2) where each vertex carries the index ∆ i (also called chiral dimension) given by ∆ i = d i + 1 2 n i − 2 .(3) Here, n i is the number of nucleon field operators and d i is the number of derivatives (or pion mass insertions). Due to spontaneously broken chiral symmetry, the index ∆ i cannot be negative. As a consequence, the power ν of the low-energy scale Q is bounded from below and a systematic and perturbative expansion for an effective Hamiltonian becomes possible. Let us now concentrate on the first few orders in the chiral expansion. • LO (ν = 0) The power ν of the low-energy scale Q takes its minimal value ν = 0 for tree diagrams with two nucleons (E n = 2) and with all interactions of dimension ∆ i = 0. Thus, the potential at LO is given by OPE and contact interactions without derivatives, see fig. 1. There are no 3N and 4N forces at LO. • NLO (ν = 2) The first corrections to the 2N potential are given by tree diagrams with one insertion of ∆ i = 2 interaction (seven independent contact interactions with two derivatives) as well as by one-loop graphs with all leading vertices 2 (leading chiral TPE). Nominally, one has also three-nucleon forces given by tree diagrams with ∆ i = 0 vertices. In the projection formalism it turnes out, that the total contribution from those diagrams vanishes. If "old-fashioned" perturbation theory is used to derive the effective potential, the contribution of the corresponding 3N force cancels against energy dependent part of once iterated LO potential [1,10]. Thus, there are no 3N and 4N forces at this order. • N 2 LO (ν = 3) The N 2 LO 2N potential is given by the subleading chiral TPE with one ππNN vertex of dimension ∆ i = 1. There are three independent structures in the Lagrangian, which contribute to NN scattering. The corresponding LECs are denoted by c 1,3,4 [11]. Note that no additional contact interactions appear at that order. There are first nonvanishing three-nucleon forces. • N 3 LO (ν = 4) The chiral potential at this order has not yet been worked out completly. Some calculations were performed by Kaiser [12]. At this order one has first 4N forces as well as a large number of various 3N and 2N interactions (including two-and three-pion exchange graphs). Note that one can easily write down the contributions to the effective potential within "old-fashioned" perturbation theory, which correspond to the diagrams shown in fig. 1. The contributions in the projection formalism are, however, not easily recoverable without knowledge of the explicit operator expressions. The diagrams of fig. 1 should therefore serve only as a guidance. For the precise numerical prefactors as well as the energy denominators corresponding to the graphs shown the reader is refered to ref. [3]. We would like to point out one of the most important qualitative findings of chiral EFT applied to few-nucleon systems [2], [10]. As shown in fig. 1, the chiral power counting eq. (2) suggests, that 3N forces are weaker than 2N ones, 4N forces are weaker than 3N ones etc.. APPLICATIONS In what follows, we will discuss application of the described formalism to 2N, 3N and 4N systems, consider some isospin violating effects and establish connection between chiral NN forces and OBE models. Two nucleons Let us now concentrate on the parameters entering the NN potential. The two unknown LECs at LO associated with the contact interactions without derivatives have to be fixed by a fit to S-wave phase shifts at low energies. The leading chiral TPE at NLO is parameter-free. At this order one has in addition seven unknown LECs, which corresponds to contact terms with two derivatives. Those constants as well as the two LO LECs are fixed from fit to phase shifts in S-and P-waves and to the mixing angle ε 1 . Thus, one has nine unknown constants at this order, see ref. [4] for more details. As already stressed before, the subleading TPE at NNLO depends on the LECs c 1,3,4 , which corresponds to ππNN vertices of dimension ∆ i = 1. Precise numerical values of these constants are crucial for some properties of the effective potential. Clearly, the subleading ππNN verices represent an important link between NN scattering and other processes, such as πN scattering. Ideally, one would like, therefore, to take their values from the analysis of the πN system. Various calculations for πN scattering have been performed and are published. From the Q 2 analyses [13] one gets: c 1 = −0.64 , c 3 = −3.90 , c 4 = 2.25 . Here all values are given in GeV −1 . From various Q 3 calculations [13], [14], [15], [16], [5] one gets the following bands for the c i 's: c 1 = −0.81 . . . − 1.53 , c 3 = −4.70 . . . − 6.19 , c 4 = 3.25 . . .4.12 .(4) Recently, the results from the Q 4 analysis have become available [17]. At this order the S-matrix is sensitive to 14 LECs (including c 1,3,4 ), which have been fixed from a fit to πN phase shifts. It turnes out that different available phase shift analyses (PSA) from refs. [18], [19] and [20] lead to sizable variation in the actual values of the LECs. In particular, the dimension two LECs acquire a quark mass renormalization. The corresponding shifts are proportional to M 2 π . The renormalized c i 's are denoted bỹ c i . A typical fit based on the phases of ref. [19] leads to: c 1 = −0.27 ± 0.01 ,c 3 = −1.44 ± 0.03 ,c 4 = 3.53 ± 0.08 .(5) However, using the older Karlsruhe or the VPI phases as input, one finds sizeable variation in thec i . Alternatively, one can also keep the c i at their third order values and fit the fourth order corrections separately, see [17]. Due to the fitting range chosen and uncertainties in the isoscalar amplitudes, these pieces are not very well determined. In principle, these fits could be improved by including the scattering lengths determined from pionic hydrogen/deuterium. It should also be stressed that numbers consistent with the band given in eq.(4) have been obtained in [23] using IR regularized baryon chiral perturbation theory and dispersion relations. Comparable values of c 1,3,4 have also been obtained by Rentmeester et al. from analysis of the pp data [21]. 3 A few comments are in order. First of all, the numerical values of some of the c i 's appear to be quite large. Indeed, from the naive dimensional analysis one would expect, for example, c 3 to scale like: c 3 ∼ ℓ/2m ∼ ℓ/2Λ χ ,(6) where ℓ is some number of order one. Taking the value c 3 = −4.70 from ref. [5] and Λ = M ρ = 770 MeV we end up with ℓ ∼ −7.5 . Such a large value can be partially explained by the fact that the c 3,4 are to a large extent saturated by the ∆-excitation. This implies that the different and smaller scale, namely m ∆ − m ∼ 293 MeV, enters the values of these constants, see [14]. More work on pion-nucleon scattering (dispersive versus chiral representation), new dispersive analyses and more precise low-energy data are needed to pin down these LECs to the precision required here. Applied to NN scattering, the large numerical values of the c i 's might lead to a slow convergence of the low-momentum expansion. Another consequence of the large c i 's for the NN system is the appearance of spurious deeply bound states in low partial waves, which can be traced back to a very strong attractive central potential related to the subleading TPE. The spurious states do not influence low-energy observables, as explained in [4]. The important consequence is, however, that strong 3NF are needed 4 As a consequence, no unphysical deeply bound states appear This is partially motivated by the fact that the ∆ is not included as an explicit degree of freedom in existing OBE models leading to very good quantitative description of observables. Some steps along a deeper understanding of the surprisingly modest role of the ∆ in the NN system within boson exchange models have been undertaken in the framework of the Bonn potential [24,25]. It was pointed out that there are strong cancellations between the TPE, whose dominant part is given by diagrams with intermediate ∆-excitations, with the πρexchange. Such a cancellation should ultimately also be observed in EFT studies, but that would require a consistent power counting scheme including vector mesons. Such a scheme has not yet been constructed. Let us finally point out that we do not know at the moment, whether the NNLO or NNLO* versions are closer to reality. Further studies of different processes as well as going to higher orders in the NN system may shed some light on to this question. In what follows, we will use the NNLO* potential in our 3N and 4N calculations as well as for comparison with conventional models. 5 We are now in the position to give quantitative results. In fig.2 we show the two np S-wave phase shift 1 S 0 and 3 S 1 and the 3 S 1 − 3 D 1 mixing parameter at NLO (left panel) and NNLO* (right panel) in comparison to the Nijmegen PSA. To regularize the LS equation, we have used an exponential regulator f R ( p) = exp(−p 4 /Λ 4 ) (for details, see [4]). The two lines correspond to cut-offs Λ = 500 and 600 MeV. We note that the description of the phases improves when going from NLO to NNLO* and that also the cut-off dependence gets weaker (especially at low energies). This is to be expected from a converging EFT and we emphasize again that this is not the result of an increasing number of free parameters. For the other phase shifts and a more detailed discussion, see [22]. Isospin violation It is well known that charge symmetry (CS) and charge independence (CI) of the nuclear force are violated. In the Standard Model, these are manifestations of isospin violation (IV). There are two distinct sources of IV. In pure QCD, the light quark mass difference m u − m d is the only source of IV. This can easily be incorporated in the EFT by means of an external scalar source. Since the charges of the quarks are also different, there is an additional electromagnetic contribution to IV. The electromagnetic effects due to hard photons are represented by local contact interactions. Soft photons appear in loop diagrams and also generate the long-range Coulomb potential. In [6], these effects have been studied in some detail, extending earlier work of van Kolck and collaborators [26,27]. The power counting is extended to include the electromagnetic interactions with the fine structure constant α serving as the additional small parameter. With this, one can construct the various contributions to the NN potential. It consists of two distinct pieces, the strong (nuclear) potential including isospin violating effects and the Coulomb potential. The nuclear potential consists of one-and two-pion exchange graphs (with different pion and nucleon masses), πγ exchange diagrams and a set of local four-nucleon operators (some of which are isospin symmetric, some depend on the quark charges and some on the quark mass difference). Since the nuclear effective potential is naturally formulated in momentum space, the matching procedure developed in ref. [28] to incorporate the correct asymptotical Coulomb states was employed in [6]. More precisely, the classification of the IV contributions to the NN potential is a as follows: To leading order one has to consider the pion mass difference in the OPE and the Coulomb potential. NLO IV corrections stem from the pion mass difference in the TPE, from πγ exchange and from two four-nucleon contact interactions with no derivatives which have the generic structure: O CSB ∼ (N † τ 3 N)(N † N), O CIB ∼ (N † τ 3 N) 2 .(7) These operators parameterize non-pionic CS breaking and CI breaking effects. The low-energy constants accompanying the contact interactions and the cut-off Λ were determined in [6] by a simultaneous best fit to the S-and P-waves of the Nijmegen phase shift analysis in the np and the pp systems for laboratory energies below 50 MeV. This allows to predict these partial waves at higher energies and all higher partial waves. Most physical observables come out independent of the sharp cut-off for Λ between 300 and 500 MeV. The upper limit on this range is determined by the pp interactions. In fig.3, we show the predictions for two P-waves in comparison to the Nijmegen PSA and the CD-Bonn potential. The range expansion for the np and the pp system was also studied in [6]. For the range of cut-offs, the pp scattering length varies modestly with Λ due to the scheme-dependent separation of the nuclear and the Coulomb part, |δa pp /a pp | ≃ 0.6 fm/17 fm ≃ 3%. This shows that the arbitrariness in separating the Coulomb and the nuclear part to a pp is strongly reduced within EFT because only a certain range of cut-offs allows to simultaneously describe np and pp scattering. Finally, it is worth to stress that CSB is largely driven by a contact interaction. The corresponding LEC can be understood to some extent in terms of ρ − ω mixing, see [27]. Three and four nucleons The NLO np potential was applied to systems of three and four nucleons in [7]. At this order, one has no 3NF and thus obtains parameter-free predictions. One also gets a first indication about the theoretical accuracy of this approach. Consider first the binding energy. For changing the cut-off between 540 and 600 MeV, the 3 H and 4 He binding energies vary between −7.55 . . . − 8.28 and −24.0 . . . − 28.1 MeV, respectively, showing the level of accuracy of the NLO approximation for these observables. As discussed in detail in [7], the chiral predictions for most observables like the differential cross section or the tensor analyzing powers T i j for elastic nd scattering as well as break-up observables are in agreement with what is found for high precision potentials like CD-Bonn. One also observes a clear improvement in the analyzing power A y for low and moderate energies, compare the left and the right panels in fig.4. The cutoff dependence for the canonical range of Λ is moderate, as indicated by the band in fig.4. These calculations have now been extended to NNLO, more precisely, using the NNLO* potential and neglecting all 3NFs, which appear at that order. For a detailed discussion of the results we refer to [22]. Here, we only remark that the cut-off sensitivity of the 3N and 4N binding energies is sizeably reduced at NNLO*, one finds e.g. −29.96 . . . − 27.87 MeV for E B ( 4 He) for even extended cut-off range between 500 and 600 MeV (the overbinding is partly due to the fact that only an np force is used). Also, the description of A y at 3 and 10 MeV is worse than at NLO, but it is still somewhat better than for the high-precision potentials and, more importantly, the cut-off dependence is much weaker as compared to NLO. Another crucial observation is that at NNLO* we are able to go to much higher energies than at NLO: our NNLO* prediction for A y at 65 MeV is in excellent agreemnet with the data. Of course, final conclusions can only be drawn when the 3NFs have been included. Work along these lines is underway. For some first steps in this direction see ref [29]. Connection to "realistic" potentials Finally, we wish to provide a bridge between the EFT and traditional nuclear physics approaches. In the latter case, one constructs (semi)phenomenological potentials such that one can describe the NN data very precisely. One particular class are the boson exchange potentials, which besides OPE have heavy meson exchanges (σ, ρ, ω, . . .) to generate the intermediate range attraction, short range repulsion and so on. In most modern high-precision potentials (which lead to fits with a χ 2 /datum ∼ 1) the shortdistance physics is parametized in different ways, say by boundary conditions, r-space fit functions or partial wave dependent boson exchanges. In [8] it was demonstrated that existing one-boson-exchange (or more phenomenologically constructed) models of NN force allow to explain the LECs in the chiral EFT potential in terms of resonance parameters. To be specific, consider a heavy meson exchange graph in a generic OBE potential. In the limit of large meson masses M R , keeping the ratio of coupling constant g to mass fixed, one can interpret such exchange diagrams as a sum of local operators with increasing number of derivatives (momentum insertions). In a highly symbolic relativistic notation, this reads, (NP i N) g 2 δ i j M 2 R − t (NP j N) = g 2 M 2 R (NP i N)(NP i N) + g 2 t M 4 R (NP i N)(NP i N) + . . . ,(8) where the P i are projectors on the appropriate quantum numbers for a given meson exchange (including also Dirac matrices if needed). Clearly, it is easy to express the dimension zero and two LECs, corresponding to the two terms of the r.h.s. of eq.(8), in terms of meson masses, coupling constants (and form factor scales, if required by the model). In the chiral EFT, one has to expand the TPE in a similar way and adjust its contribution to the various LECs accordingly. One can also repeat this procedure for the high-precision potentials, this has to be done numerically for the various partial waves. In fig.5 we show a comparison between the 4 S-and the 5 P-wave LECs obtained at NLO (leftmost bands) and NNLO* (central bands) and extracted from various OBE and high-precision potentials (see the inset in the figure). The agreement is rather stunning since one would have expected higher dimension operators to play a more prominent role in the phenomenological potentials (related e.g. to the cut-off sensitivity of the πN form factor like in the Bonn potential). Note also that the theoretical uncertainties determined in EFT are a) small compared to their average values and b) are smaller than the band spanned by the potential models (even if one only includes the high-precision ones). Note that the comparison has been performed for the partial wave projected set of coupling constants. Let us now discuss the important issue of naturalness of the various LECs related to contact interactions. For that is turns out to be more appropriate to work directly with the coupling constants C S , C T , C 1...7 , which enter the chiral Lagrangian, see [8]. Here, the constants C S and C T correspond to operators without derivatives, whereas the remaining ones to contact terms with two derivatives. Dimensional arguments suggest the following scaling properties of these LECs: C C,S ∼ l S,C /( f 2 π ) , C 1...7 ∼ l 1...7 /( f 2 π Λ 2 χ ), where the l's are some numbers of order one. These scaling properties of the LECs are crucial for the convergence of the low-momentum expansion. Note that one has to take into account numerical prefactors which accompany the various terms of the Lagrangian, see ref. [8] for more details. Taking Λ χ = 1 GeV it turnes out that the values of the l's fluctuate between 0.3 and 3.5, i.e. the values found for these LECs are indeed natural, with the notable exception of f 2 π C T , which is much smaller than one: f 2 π C T = −0.002 . . .0.147 at NLO and f 2 π C T = 0.002 . . .0.040 at NNLO*. These unnaturally small numbers can be traced back to the SU(4) symmetry of the NN interaction, proposed about 65 years ago by Wigner [30]. For the recent discussion on that subject within the EFT approach see ref. [31]. SUMMARY AND OUTLOOK Despite the original scepticism by its creator [2], chiral effective field theory not only offers qualitative but also quantitative insight into the dynamics of few-nucleon systems, as should have become clear from the discussed topics. In addition, it is the only framework at present in which one can address the question of the size of three (four) nucleon forces in a truely systematic manner. It is therefore evident that a vast effort has to be undertaken to pin down the chiral 3NF and apply it not only to few-but also many-body systems. Exciting times are ahead of us. FIGURE 1 . 1First orders in the chiral expansion for few-nucleon forces, as explained in the text. Solid (dashed) lines denote nucleons (pions). The heavy dots, filled square and filled diamond denote the ∆ i = 1, ∆ i = 2 and ∆ i = 4 vertices, respectively. . Note that despite this very different scenario from what is expected in conventional nuclear physics (small 3N forces), the individual contributions of the 2N and 3N forces to observables can, in principle, not be observed experimentally. Since the actual values of the c i 's may possibly change in future analyses of the πN system (higher orders, more precise PSA, etc.), we constructed the NNLO* version of the NN potential, in which we basically subtracted the ∆-contributions to these LECs and allowed for some fine tuning. This results in numerically reduced values of the c 3,4 : c 3 = −1.15 GeV −1 , c 4 = 1.20 GeV −1 . FIGURE 2 . 2Phase shifts at NLO (left panel) and NNLO* (right panel) versus the lab energy (in GeV) in comparison to the Nijmegen PSA. The solid (dashed) line corresponds to Λ = 500 (Λ = 600) MeV. FIGURE 3 . 3Phase shift difference δ pp − δ np for the 3 P 0 and 3 P 2 partial waves in comparison to the Nijmegen PSA (dashed lines) and results from the CD-Bonn potential (filled circles). FIGURE 4 . 4Analyzing power A y for elastic nd scattering, for E lab = 3, 10, 65 MeV (top to bottom). Results at NLO (left panel) and NNLO* (middle panel). The band corresponds to the range Λ = 500 to 600 MeV. Results based on the high-precision potentials (CD-Bonn, AV-18, Nijm-93, Nijm-I,II) are shown in the right panel. Here the band refers to the uncertainty using the various potentials. FIGURE 5 . 5LECs from phenomenological models and chiral EFT. The leftmost band refers to NLO (the length reflects the variation with the cut-off), the middle bar is NNLO*, and the symbols correspond to the indicated potentials (see inset). To be more precise, Q is associated with the 4-momenta of external pions and 3-momenta of external nucleons. Only irreducible topologies have to be taken into account in "old-fashioned" perturbation theory. On the contrary, in the projection formalism one has also to include reducible topologies, as explained below. Note, however, that it is not possible to fix all three constants in this prosess. For that reason the constant c 1 has been fixed at the value c 1 = −0.77 GeV −1 .4 Calculated with only 2N forces, the triton is underbound by about 4 MeV[22].5 The application of the NNLO potential to few-nucleon systems is technically more complicated. ACKNOWLEDGMENTSE.E. thanks the organizers for the invitation and support. . S Weinberg, Phys. Lett. 288S. Weinberg, Phys. Lett. B251 (1990) 288. . S Weinberg, Nucl. 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R A Said On-Line Program, Arndt, SAID on-line program, R.A. Arndt et al., see website http://gwdac.phys.gwu.edu/. . M C M Rentmeester, R G E Timmermans, J L Friar, J J De Swart, Phys. Rev. Lett. 824992M.C.M. Rentmeester, R.G.E. Timmermans, J.L. Friar, and J.J. de Swart, Phys. Rev. Lett. 82 (1999) 4992. . E Epelbaum, in preparationE. Epelbaum et al., in preparation. . T Becher, H Leutwyler, JHEP. 010617T. Becher and H. Leutwyler, JHEP 0106 (2001) 017. . R Machleidt, K Holinde, Ch Elster, Phys. Rep. 1491R. Machleidt, K. Holinde, and Ch. Elster, Phys. Rep. 149 (1987) 1. . G Janssen, K Holinde, J Speth, Phys. Rev. 542218G. Janssen, K. Holinde, and J. Speth, Phys. Rev. C54 (1996) 2218. Few-Body Systems Suppl. U Van Kolck, 9444U. van Kolck, Few-Body Systems Suppl. 9 1995 444 . U Van Kolck, J L Friar, T Goldman, Phys. Lett. 371169U. van Kolck, J.L. Friar and T. Goldman, Phys. Lett. B371 (1996) 169. . C M Vincent, S C Phatak, Phys. Rev. 10391C.M. Vincent and S.C. Phatak, Phys. Rev. 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[ "An Algebraic Theory of Complexity for Discrete Optimisation *", "An Algebraic Theory of Complexity for Discrete Optimisation *" ]
[ "David A Cohen [email protected] ", "Martin C Cooper [email protected] ", "Páidí Creed [email protected] ", "Peter G Jeavons [email protected] ", "\nDepartment of Computer Science Royal Holloway\nUniversity of London\nUK\n", "\nSchool of Mathematical Sciences Queen Mary\nIRIT University of Toulouse III\nFrance\n", "\nDepartment of Computer Science\nUniversity of London\nUK\n", "\nUniversity of Oxford\nUK\n", "\nStanislavŽivný University College University of Oxford\nUK\n" ]
[ "Department of Computer Science Royal Holloway\nUniversity of London\nUK", "School of Mathematical Sciences Queen Mary\nIRIT University of Toulouse III\nFrance", "Department of Computer Science\nUniversity of London\nUK", "University of Oxford\nUK", "StanislavŽivný University College University of Oxford\nUK" ]
[]
Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimisation problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised.These results provide a new approach to studying the complexity of discrete optimisation. We use this approach to identify certain maximal tractable subproblems of the general problem, and hence derive a complete classification of complexity for the Boolean case.
10.1137/130906398
[ "https://arxiv.org/pdf/1207.6692v1.pdf" ]
14,403,432
1207.6692
afa188bf0f16e14c56e54e0bc27cb59e1f036d71
An Algebraic Theory of Complexity for Discrete Optimisation * 28 Jul 2012 David A Cohen [email protected] Martin C Cooper [email protected] Páidí Creed [email protected] Peter G Jeavons [email protected] Department of Computer Science Royal Holloway University of London UK School of Mathematical Sciences Queen Mary IRIT University of Toulouse III France Department of Computer Science University of London UK University of Oxford UK StanislavŽivný University College University of Oxford UK An Algebraic Theory of Complexity for Discrete Optimisation * 28 Jul 2012Galois connectionvalued constraint satisfaction problemsconstraint optimisationweighted polymorphismsweighted clonescomplexity Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimisation problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised.These results provide a new approach to studying the complexity of discrete optimisation. We use this approach to identify certain maximal tractable subproblems of the general problem, and hence derive a complete classification of complexity for the Boolean case. Introduction Discrete optimisation problems arise in many different areas and are studied under many different names, including Min-Sum Problems, Gibbs energy minimisation, Markov Random Fields, Conditional Random Fields, 0/1 integer programming, pseudo-Boolean function minimisation, constraint optimisation and valued constraint satisfaction [31,10,40,39,17,16,7,6]. Here we adopt a very general framework where each problem instance is specified by a set of variables, a set of possible values for those variables, and a set of constraints. Each combination of values allowed by each constraint has an associated cost, and the goal is to find an assignment with minimal total cost. This simple abstract mathematical framework can be used to express discrete optimisation problems arising in a wide variety of fields, including operational research (scheduling, resource utilisation, transportation), computer vision (region segmentation, object recognition, image enhancement), automated reasoning (Max SAT, Min ONES), graph theory (Min-Cut, Maximum Independent Set), and many others. In the special case when all costs are zero, the problem we are studying collapses to the standard constraint satisfaction problem (CSP). The general CSP is NP-hard, and so is unlikely to have a polynomialtime algorithm. However, there has been much success in finding tractable fragments of the CSP by restricting the types of relation allowed in the constraints. A set of allowed relations has been called a constraint language [27]. For some constraint languages the associated constraint satisfaction problems with constraints chosen from that language are solvable in polynomial-time, whilst for other constraint languages this class of problems is NP-hard [28,27,22]; these two cases are referred to as tractable languages and NP-hard languages, respectively. Dichotomy theorems, which classify each possible constraint language as either tractable or NP-hard, have been established for languages over 2-element domains [34], 3-element domains [13], for conservative languages [15,1], and maximal languages [11,12]. The more general framework we consider here, which allows non-zero costs, is also NP-hard, but again we can try to identify tractable fragments by restricting the types of allowed constraints. Each type of constraint is specified by a rational-valued function defined on a set of tuples, which specifies the cost associated with each allowed tuple of values. Such a function is called a weighted relation, and a set of weighted relations will be called a valued constraint language [16]. Much less is known about the complexity of the optimisation problems associated with different valued constraint languages, although some results have been obtained for certain special cases. In particular, a complete characterisation of complexity has been obtained for valued constraint languages over a 2-element domain [16]. This result generalises a number of earlier results for particular optimisation problems such as MAX-SAT [18] and MIN-ONES [19]. A complete classification has also been obtained for valued constraint languages containing all unary {0, 1}valued weighted relations (such languages are called conservative) [30]. This result generalises a number of earlier results such as GRAPH MIN-COST-HOM [25] and DIGRAPH MIN-COST-HOM [37]. One class of weighted relations that has been extensively studied and shown to be tractable is the class of submodular functions [19,16,29,21,30,41]. In the classical CSP framework it has been shown that the complexity of any constraint language over any finite domain is determined by certain algebraic properties known as polymorphisms [28,27]. This result has reduced the problem of the identification of tractable constraint languages to that of the identification of suitable sets of polymorphisms. In other words, it has been shown to be enough to study just those constraint languages which are characterised by having a given set of polymorphisms. Using this algebraic approach, considerable progress has now been made towards a complete characterisation of the complexity of constraint languages over finite domains of arbitrary size [22,14,4,2,3,5]. In this paper, we introduce a new algebraic construct which we call a weighted polymorphism. We are able to show that the weighted polymorphisms of a valued constraint language are sufficient to determine the complexity of that language. In addition, we are able to define a Galois connection between valued constraint languages and sets of weighted polymorphisms, and characterise the closed sets on both sides. The Galois connection we establish here can be used in the search for tractable valued constraint languages in a very similar way to the use of polymorphisms in the search for tractable constraint languages in the classical CSP. First, we need only consider valued constraint languages characterised by weighted polymorphisms. This greatly simplifies the search for a characterisation of all tractable valued constraint languages. Second, we will show below that any tractable valued constraint language must have an associated non-trivial weighted polymorphism. Hence the results of this paper provide a powerful new set of tools for analysing the complexity of finite-domain discrete optimisation problems. In fact, a recent result on the power of linear programming for valued constraint languages builds on weighted polymorphisms [38]. The structure of the paper is as follows. In Section 2 we introduce the general framework of the Valued Constraint Satisfaction Problem (VCSP) and define the notion of expressibility. In Section 3 we focus on the classical Constraint Satisfaction Problem and show how it fits in the VCSP framework as a special case. We briefly recall the notion of polymorphism, and the Galois connection that has been so fruitful in the study of the complexity of the classical Constraint Satisfaction Problem. In Sections 4 and 5 we introduce weighted relational clones (valued constraint languages closed under expressibility and certain other operations) and the corresponding closed sets of weighted polymorphisms, which we call weighted clones. We then state our main result: weighted relational clones are in 1-to-1 correspondence with weighted clones. In Section 6 we give proofs of the theorems establishing this Galois connection. In Section 7 we use the algebraic theory to establish necessary conditions that must be satisfied by any tractable valued constraint language. Using these results, we obtain a complete classification for the Boolean case in Section 8. Finally, in Section 9, we state some conclusions and outline directions for future work. The Valued Constraint Satisfaction Problem For any set D, the set of tuples of length r over D is denoted by D r , and a subset of D r is called a relation over D of arity r. A (partial) function ̺ from D r to Q associates a rational 1 weight with each of the tuples in some subset of D r , and so will be called a weighted relation on D of arity r. The idea of a weighted relation is very general, and can be used to define a wide variety of discrete optimisation problems. The general framework we shall use for such problems is defined as follows. Definition 2.1. An instance of the valued constraint satisfaction problem, (VCSP), is a triple P = V, D, C where: V is a finite set of variables; D is a finite set of possible values; C is a finite multiset of constraints. Each element of C is a pair c = σ, ̺ where σ is a tuple of variables called the scope of c, and ̺ is a weighted relation on D of arity |σ|. An assignment for P is a mapping s : V → D. The cost of an assignment s, denoted Cost P (s), is given by the sum of the weights assigned to the restrictions of s onto each constraint scope, that is, Cost P (s) def = v 1 ,v 2 ,...,vm ,̺ ∈C ̺(s(v 1 ), s(v 2 ), . . . , s(v m )). If ̺(s(v 1 ), s(v 2 ), . . . , s(v m )) is undefined for some v 1 , v 2 , . . . , v m , ̺ ∈ C, then the assignment s is said to be infeasible and Cost P (s) is undefined. A solution to P is a feasible assignment with minimal cost. Example 2.2 (MIN-CUT). In graph theory a cut of a graph is a partition of the vertices into two disjoint sets. The size of a cut is the number of edges of the graph that intersect both sides of this partition. The MIN-CUT problem for a graph is to find a cut with the smallest possible size. This problem can be solved in polynomial-time [32]. The MIN-CUT problem for the graph (V, E) can be expressed as an instance V, {0, 1}, C of VCSP by setting C = { e, ̺ = | e ∈ E}, where ̺ = is the binary weighted relation on the set {0, 1} defined by: ̺ = (x, y) def = 0 if x = y 1 otherwise . Any assignment for this VCSP instance partitions the elements of V into those assigned the value 0 and those assigned the value 1. The cost of the assignment is equal to the size of the corresponding cut. Example 2.3 (MAX-CUT). The MAX-CUT problem for a graph is to find a cut with the largest possible size. This problem is NP-hard [23]. The MAX-CUT problem for the graph (V, E) can be expressed as an instance V, {0, 1}, C of VCSP by setting C = { e, ̺ = | e ∈ E}, where ̺ = is the binary weighted relation on the set {0, 1} defined by: ̺ = (x, y) def = 0 if x = y 1 otherwise . Example 2.4 (DIGRAPH MIN-COST-HOM). Given two directed graphs (digraphs) G = (V G , E G ) and H = (V H , E H ), a homomorphism from G to H is a mapping f : V G → V H that preserves edges, that is, (u, v) ∈ E G implies (f (u), f (v)) ∈ E H . Assume that for any u ∈ V G and v ∈ V H a rational cost c u (v) is given. The cost of a homomorphism f from G to H is then defined to be u∈V G c u (f (u)). The DIGRAPH MIN-COST-HOM problem is to find a homomorphism from G to H of minimum cost [25,37]. Given a fixed digraph H = (V H , E H ), we denote by ̺ H the binary weighted relation on the set V H defined by: ̺ H (x, y) def = 0 if (x, y) ∈ E H ∞ otherwise . The DIGRAPH MIN-COST-HOM problem for input graph G = (V G , E G ) and fixed target graph H = (V H , E H ) can be expressed as an instance V G , V H , C of VCSP by setting C = { e, ̺ H | e ∈ E G } ∪ { u, c u | u ∈ V G }. A valued constraint language is any set Γ of weighted relations on some fixed set D. We define VCSP(Γ) to be the set of all VCSP instances in which all weighted relations in all constraints belong to Γ. A valued constraint language Γ is called tractable if, for every finite subset Γ ′ ⊆ Γ, there exists an algorithm solving any instance P ∈ VCSP(Γ ′ ) in polynomial time. Conversely, Γ is called NP-hard if there is some finite subset Γ ′ ⊆ Γ for which VCSP(Γ ′ ) is NP-hard. Example 2.2 shows that the valued constraint language {̺ = } is tractable, and Example 2.3 shows that the valued constraint language {̺ = } is NP-hard. We now define a closure operator on weighted relations, which adds to a given set of weighted relations all other weighted relations which can be expressed using that set, in the sense defined below. We say that a weighted relation ̺ is expressible over a valued constraint language Γ if there exists a VCSP instance P ∈ VCSP(Γ) and a list L of variables of P such that π L (P) = ̺. We call the pair P, L a gadget for expressing ̺ over Γ. We define Express(Γ) to be the expressive power of Γ; that is, the set of all weighted relations expressible over Γ. Note that the list of variables L in a gadget may contain repeated entries, the sum over an empty set is zero, and the minimum over an empty set is undefined. Example 2.6. Let P be the VCSP instance with a single variable v and no constraints, and let L = v, v . Then, by Definition 2.5, π L (P)(x, y) = 0 if x = y undef ined otherwise . Hence for any valued constraint language Γ, over any set D, Express(Γ) contains this binary weighted relation, which will be called the weighted equality relation. π L (P)(x, y) = 0 if x = y 1 otherwise . Hence π L (P) = ̺ = , so we have that ̺ = ∈ Express({̺ = }). However, using the results of this paper, we will be able to show, for example, that ̺ = ∈ Express({̺ = }) (see Example 8.4). The next result shows that expressibility preserves tractability. Theorem 2.8. A valued constraint language Γ is tractable if and only if Express(Γ) is tractable; similarly, Γ is NP-hard if and only if Express(Γ) is NP-hard. Proof. By the definition of a tractable valued constraint language, it is sufficient to show that for any finite subset Γ ′ of Express(Γ) there exists a polynomial-time reduction from VCSP(Γ ′ ) to VCSP(Γ ′′ ), where Γ ′′ is a finite subset of Γ. Let Γ ′ be a finite subset of Express(Γ) and let P ′ be any instance of VCSP(Γ ′ ). By Definition 2.5, any weighted relation ̺ ′ ∈ Express(Γ) can be constructed by using some gadget P ̺ ′ , L where P ̺ ′ is an instance of VCSP(Γ). Hence we can simply replace each constraint in P ′ which has a weighted relation ̺ ′ not already in Γ with the corresponding gadget to obtain an instance P of VCSP(Γ) which is equivalent to P ′ . The maximum size of any of the gadgets used is a constant determined by the finite set Γ ′ , so this construction can be carried out in polynomial time in the size of P ′ . This result shows that, when trying to identify tractable valued constraint languages, it is sufficient to consider only languages of the form Express(Γ). In the following sections, we will show that such languages can be characterised using certain algebraic properties. Classical Constraint Satisfaction In this section we consider the special case when the weights are all zero. There is a one-to-one correspondence between the set of zero-valued weighted relations R D and the set of all relations over D. In this correspondence each weighted relation ̺ in R D is associated with the relation R(̺) containing precisely those tuples on which ̺ is defined. Similarly, each zero-valued weighted relation in R D is associated with the predicate which is true for precisely those tuples where the weighted relation is defined. Subsets of R D are sometimes referred to as crisp constraint languages [16] and VCSP(R D ) is equivalent to the classical constraint satisfaction problem, or CSP, where each assignment is either allowed (cost 0) or disallowed (infeasible, or cost undefined). Definition 3.2. A weighted relation ̺ of arity r can be obtained by addition from the weighted relation ̺ 1 of arity s and the weighted relation ̺ 2 of arity t if ̺ satisfies the identity ̺(x 1 , . . . , x r ) = ̺ 1 (y 1 , . . . , y s ) + ̺ 2 (z 1 , . . . , z t ), for some (fixed) choice of y 1 , . . . , y s and z 1 , . . . , z t from amongst the x 1 , . . . , x r . For zero-valued weighted relations this notion of addition corresponds to performing a relational join operation on the associated relations R(̺ 1 ) and R(̺ 2 ) [26]. It also corresponds to taking a conjunction of the associated predicates [14]. Moreover, minimising a weighted relation ̺ ∈ R D over one of its arguments corresponds to taking a relational projection of R(̺) onto its remaining co-ordinates. It also corresponds to existential quantification of the associated predicate over that argument. For each Γ ⊆ R D we define RelClone(Γ) to be the smallest relational clone containing Γ. It is a straightforward consequence of Definitions 2.5 and 3.3 that the expressive power of a crisp constraint language is given by the smallest relational clone containing it, as the next result indicates. This alternative characterisation for the expressive power of a crisp constraint language was first observed in [27], and used to study the complexity of such languages using tools from universal algebra. We now give a brief summary of this algebraic approach. For any set D, a function f : D k → D is called a k-ary operation on D. We can extend k-ary operations to operate on tuples in a natural way, as follows. Let x 1 , . . . , x k be tuples of length r over a set D, where each x i = x i,1 , x i,2 , . . . , x i,r . We can obtain another element of D r by applying f to the tuples x i co-ordinatewise, as follows: For any set D, the mappings Pol and Inv form a Galois connection between O D and R D [9]. A characterisation of this Galois connection for finite sets D is given by the following two theorems, originally obtained for sets of relations [24,8]. As with any Galois connection [9], this means that there is a one-to-one correspondence between clones and relational clones. Together with Proposition 3.4, this result shows that the expressive power of any crisp constraint language Γ on a finite set D corresponds to a particular clone of operations on D. Hence, by Theorem 2.8, the search for tractable crisp constraint languages corresponds to a search for suitable clones of operations [27,14]. This key observation paved the way for applying deep results from universal algebra in the search for tractable constraint languages [11,15,12,13,4,2,3,5]. f (x 1 , . . . , x k ) def = f (x 1,1 , . . . , x k,1 ), f (x 1,2 , . . . , x k,2 ), . . . , f (x 1,r , . . . , x k,r ) . Definition 3.9. Let ̺ be a weighted relation of arity r on a set D and let f ∈ O (k) D . We say that f is a polymorphism of ̺ if, for any x 1 , x 2 , . . . , x k ∈ D r such that ̺(x i ) is defined for i = 1, . . . , k, we have that ̺(f (x 1 , x 2 , . . . , x k )) is also defined. If f is a polymorphism of ̺ we say ̺ is invariant under f . Weighted Relational Clones In this section we return to the general case of weighted relations taking arbitrary values in Q in order to define the notion of a weighted relational clone. We now define a closure operator on weighted relations, which adds to a set of weighted relations all other weighted relations which can be obtained from the original set by non-negative scaling and addition of a constant. Definition 4.2. A weighted relation ̺ ′ ∈ Φ D can be obtained from a weighted relation ̺ ∈ Φ D by non-negative scaling and addition of constants if there exist α, β ∈ Q with α ≥ 0 such that ̺ ′ ≡ α̺ + β . We denote by Γ ∼ the smallest set of weighted relations containing Γ which is closed under non-negative scaling and addition of constants. The next result shows that adding weighted relations that can be obtained by non-negative scaling and addition of constants preserves tractability. Theorem 4.3. A valued constraint language Γ is tractable if and only if Γ ∼ is tractable; similarly, Γ is NP-hard if and only if Γ ∼ is NP-hard. Proof. By the definition of tractable valued constraint language, it is sufficient to show that for any finite subset Γ ′ of Γ ∼ there exists a polynomial-time reduction from VCSP(Γ ′ ) to VCSP(Γ ′′ ), where Γ ′′ is a finite subset of Γ. Let Γ ′ be a finite subset of Γ ∼ and let P ′ be any instance of VCSP(Γ ′ ). By Definition 4.2, any weighted relation ̺ ′ ∈ Γ ∼ can be obtained by non-negative scaling and addition of constants from some weighted relation ̺ ∈ Γ. Hence we can replace each of the constraints σ, ̺ ′ in P ′ with a new constraint σ, ̺ , where ̺ ∈ Γ and ̺ ′ = α̺ + β for some non-negative rational value α and some arbitrary rational constant β, to obtain an instance P of VCSP(Γ). The constant β is added to the cost of all assignments and so does not affect the choice of solution. Since α is a non-negative rational value, it can be expressed as p/q for some non-negative integer p and positive integer q. If p is non-zero, then the effect of the scale factor p/q can be simulated by taking p copies of the new constraint in P and q copies of all other constraints in P. The maximum values of p, q are constants determined by the finite set Γ ′ , so this construction can be carried out in polynomial time in the size of P ′ . It only remains to deal with the cases where p is zero. Assume that P ′ contains k constraints of the form σ, ̺ ′ where ̺ ′ = 0̺ + β, and replace each of these with the corresponding constraint σ, ̺ to obtain a new instance, P. Let M be the maximum weight assigned by any weighted relation in the finite set Γ ′ , and let m be the minimum difference between any two distinct weights assigned by weighted relations in Γ ′ . The cost of any feasible assignment for P is greater by at most kM than the cost of the same assignment for P ′ . Hence if we also take ⌈ M k m + 1⌉ copies of all the remaining constraints of P ′ , then we obtain an instance of VCSP(Γ) with the same solutions as P ′ . Since M and m are constants determined by the finite set Γ ′ , this construction can again be carried out in polynomial time in the size of P ′ . For each Γ ⊆ Φ D we define wRelClone(Γ) to be the smallest weighted relational clone containing Γ. It is a straightforward consequence of Definitions 2.5, 4.2 and 4.4 that, for any valued constraint language Γ ⊆ Φ D , the set of weighted relations that can be expressed using weighted relations obtained from Γ by non-negative scaling and addition of constants, is given by the smallest weighted relational clone containing Γ, as the next result indicates. Proposition 4.5. For any Γ ⊆ Φ D , Express(Γ ∼ ) = wRelClone(Γ). Hence, by Theorem 2.8 and Theorem 4.3, the search for tractable valued constraint languages corresponds to a search for suitable weighted relational clones. In the next section we establish an alternative characterisation for weighted relational clones which facilitates this search. Weighted Clones To obtain a suitable alternative characterisation for weighted relational clones we now generalise the notion of a clone of operations, introduced in Definition 3.8, by introducing the notion of a weighted clone. Recall that a clone of operations, C, is a set of operations on some fixed set D that contains all projections and is closed under superposition. The k-ary operations in a clone C will be denoted C (k) . Definition 5.1. We define a k-ary weighting of a clone C to be a function ω : C (k) → Q such that ω(f ) < 0 only if f is a projection and f ∈C (k) ω(f ) = 0 . We denote by W C the set of all possible weightings of C and by W For any weighting ω, we denote by dom(ω) the set of operations on which ω is defined. We denote by ar(ω) the arity of ω. Since a weighting is simply a rational-valued function satisfying certain inequalities it can be scaled by any non-negative rational to obtain a new weighting. Similarly, any two weightings of the same clone of the same arity can be added to obtain a new weighting of that clone. The notion of superposition from Definition 3.7 can also be extended to weightings in a natural way, by forming a superposition with each argument of the weighting, as follows. Definition 5.2. For any clone C, any ω ∈ W (k) C and any g 1 , g 2 , . . . , g k ∈ C (ℓ) , we define the superposition of ω and g 1 , . . . , g k , to be the function ω[g 1 , . . . , g k ] : C (ℓ) → Q defined by ω[g 1 , . . . , g k ](f ′ ) def = f ∈C (k) f [g 1 ,...,g k ]=f ′ ω(f ) . (1) Example 5.3. Let D be a totally ordered set D, and let C = Clone({max}) where max is the binary maximum operation on D. Note that C (2) contains just three binary operations: e 1 , e 2 and max. Let ω be the 2-ary weighting of C given by ω(f ) def =      −1 if f = e (2) 1 +1 if f = e (2) 2 0 if f = max and let g 1 , g 2 = e (2) 2 , max . Note that e 1 [g 1 , g 2 ] = g 1 = e(2) 2 and e 2 [g 1 , g 2 ] = g 2 = max, so, applying Definition 5.2, we have ω[g 1 , g 2 ](f ) =      0 if f = e (2) 1 −1 if f = e (2) 2 +1 if f = max .(2) Note that ω[g 1 , g 2 ] satisfies the conditions of Definition 5.1 and hence is a 2-ary weighting of C. Example 5.4. Let C be a clone on some totally ordered set D and let max and min be binary maximum and minimum operations which are contained in C. Note that C (4) contains operations such as max[e (4) i , e j ] which returns the maximum of the ith and jth argument values. Operations of this form will be denoted max(x i , x j ). Let ω be the 4-ary weighting of C given by ω(f ) def =    −1 if f is a projection, that is, f ∈ {e (4) 1 , e(4) 2 , e 3 , e 4 } +1 if f ∈ {max(x 1 , x 2 ), min(x 1 , x 2 ), max(x 3 , x 4 ), min(x 3 , x 4 )} 0 otherwise and let g 1 , g 2 , g 3 , g 4 = e(4) 1 , e 2 , e 3 , max(x 1 , x 2 ) .(3) Then, by Definition 5.2 we have ω[g 1 , g 2 , g 3 , g 4 ](f ) =    −1 if f is a projection, that is, f ∈ {e (3) 1 , e(3) 2 , e 3 } +1 if f ∈ {max(x 1 , x 2 , x 3 ), min(x 1 , x 2 ), min(x 3 , max(x 1 , x 2 ))} 0 otherwise .(3) Note that ω[g 1 , g 2 , g 3 , g 4 ] satisfies the conditions of Definition 5.1 and hence is a 3-ary weighting of C. Example 5.5. Let C and ω be the same as in Example 5.4 but now consider g ′ 1 , g ′ 2 , g ′ 3 , g ′ 4 = e(4)1 , max(x 2 , x 3 ), min(x 2 , x 3 ), e(4) 4 . By Definition 5.2 we have ω[g ′ 1 , g ′ 2 , g ′ 3 , g ′ 4 ](f ) =        −1 if f ∈ {e (4) 1 , max(x 2 , x 3 ), min(x 2 , x 3 ), e(4)4 } +1 if f ∈ max(x 1 , x 2 , x 3 ), min(x 1 , max(x 2 , x 3 )), max(min(x 2 , x 3 ), x 4 ), min(x 2 , x 3 , x 4 ) 0 otherwise . Note that ω[g ′ 1 , g ′ 2 , g ′ 3 , g ′ 4 ] does not satisfy the conditions of Definition 5.1 because, for example, we have that ω[g ′ 1 , g ′ 2 , g ′ 3 , g ′ 4 ](f ) < 0 when f = max(x 2 , x 3 ), which is not a projection. Hence ω[g ′ 1 , g ′ 2 , g ′ 3 , g ′ 4 ] is not a valid weighting of C. It follows immediately from Definition 3.7 that the sum of the weights in any superposition ω[g 1 , . . . , g k ] is equal to the sum of the weights in ω, which is zero, by Definition 5.1. However, as we have seen in Example 5.5, it is not always the case that an arbitrary superposition satisfies the other condition in Definition 5.1, that negative weights are only assigned to projections. Hence we make the following definition: Definition 5.6. If the result of a superposition is a valid weighting, then that superposition will be called a proper superposition. We are now ready to define weighted clones. C containing all zero-valued weightings of C is a weighted clone with support C. Note that W 0 C contains exactly one weighting of each possibly arity, which assigns the value 0 to all operations in C of that arity. We now establish a link between weightings and weighted relations, which will allow us to link weighted clones and weighted relational clones. Definition 5.11. Let ̺ be a weighted relation of arity r on some set D and let ω be a k-ary weighting of some clone of operations C on the set D. We say that ω is a weighted polymorphism of ̺ if, for any x 1 , x 2 , . . . , x k ∈ D r such that ̺(x i ) is defined for i = 1, . . . , k, we have that ̺(f (x 1 , x 2 , . . . , x k )) is defined for all f ∈ C (k) , and f ∈C (k) ω(f )̺(f (x 1 , x 2 , . . . , x k )) ≤ 0 .(2) If ω is a weighted polymorphism of ̺ we say ̺ is improved by ω. Note that, by Definition 3.9, if ̺ is improved by the weighting ω ∈ W (k) C , then every element of C (k) must be a polymorphism of ̺. Example 5.12. Consider the class of submodular functions [32]. These are precisely the functions ̺ satisfying ̺(min(x 1 , x 2 )) + ̺(max(x 1 , x 2 )) − ̺(x 1 ) − ̺(x 2 ) ≤ 0 . In other words, the set of submodular functions are the set of weighted relations with a 2-ary weighted polymorphism ω sub , defined by: ω sub (f ) def =    −1 if f ∈ {e(2) 1 , e 2 } +1 if f ∈ {min(x 1 , x 2 ), max(x 1 , x 2 )} 0 otherwise .(2) Submodular function minimisation is known to be tractable [32]. To define a mapping in the other direction, we need to consider the union of the sets W C over all clones C on some fixed set D, which will be denoted W D . If we have a set W ⊆ W D which may contain weightings of different clones over D, then we can extend each of these weightings with zeros, as necessary, so that they are weightings of the same clone C, given by C = Clone( ω∈W dom(ω)). This set of extended weightings obtained from W will be denoted W . For any set W ⊆ W D , we define wClone(W ) to be the smallest weighted clone containing W . Definition 5.14. For any W ⊆ W D , we denote by Imp(W ) the set of all weighted relations in Φ D which are improved by all weightings ω ∈ W . The set of r-ary weighted relations in Imp(W ) will be denoted Imp (r) (W ). It follows immediately from the definition of a Galois connection [9] that, for any set D, the mappings wPol and Imp form a Galois connection between W D and Φ D , as illustrated in Figure 1. A characterisation of this Galois connection for finite sets D is given by the following two theorems, which are proved in Section 6. As with any Galois connection [9], this means that there is a one-to-one correspondence between weighted clones and weighted relational clones. Hence, by Proposition 4.5, Theorem 2.8, and Theorem 4.3, the search for tractable valued constraint languages over a finite set corresponds to a search for suitable weighted clones of operations. Proofs of Theorems 5.15 and 5.16 Our proofs of Theorems 5.15 and 5.16 will both use the following result, which is a variant of the wellknown Farkas' Lemma used in linear programming [32,35]. Farkas 1894). Let S and T be finite sets of indices, where T is the disjoint union of two subsets, T ≥ and T = . For all i ∈ S, and all j ∈ T , let a i,j and b j be rational numbers. Exactly one of the following holds: • Either there exists a set of non-negative rational numbers {x i | i ∈ S} and a rational number C such that for each j ∈ T ≥ , i∈S a i,j x i ≥ b j + C, and, for each j ∈ T = , i∈S a i,j x i = b j + C. • Or else there exists a set of integers {y j | j ∈ T } such that j∈T y j = 0 and: for each j ∈ T ≥ , y j ≥ 0, for each i ∈ S, j∈T y j a i,j ≤ 0, and j∈T y j b j > 0. Such a set is called a certificate of unsolvability. We note that there is an effective procedure to decide which of the cases mentioned in Lemma 6.1 holds for any instance, and to calculate the values of the corresponding coefficients x i or y j [35]. We will prove Theorem 5.15 in two parts. First, we show in Proposition 6.2 that the set of all weighted relations improved by any given set of weightings is always a weighted relational clone. Then we show that for any finite set Γ the set of weighted relations improved by all weightings in wPol(Γ) is precisely the weighted relational clone wRelClone(Γ). Proof. Certainly Imp(W ) contains the weighted equality relation, ̺ = , since ̺ = satisfies inequality (2) in Definition 5.11 for all x i such that ̺ = (x i ) is defined. Similarly, Imp(W ) is closed under non-negative scaling, addition of constants, addition and rearrangement of arguments, since all of these operations preserve inequality (2). Hence, to show Imp(W ) is a weighted relational clone we only need to show Imp(W ) is closed under minimisation. Let ̺ ∈ Imp (r) (W ) and assume that ̺ ′ is obtained from ̺ by minimising over the last argument. In other words, ̺ ′ (x 1 , x 2 , . . . , x r−1 ) = min xr (̺(x 1 , x 2 , . . . , x r )). We will now show that ̺ ′ ∈ Imp(W ). Let ω ∈ W be a k-ary weighting of a clone C. Since ̺ ∈ Imp(W ), we know that ̺ and ω satisfy inequality (2) for all x 1 , x 2 , . . . , x k such that ̺(x i ) is defined. Now consider any x ′ 1 , x ′ 2 , . . . , x ′ k for which each ̺ ′ (x ′ i ) is defined. Extend each x ′ i to a tuple x ′′ i of arity r in such a way that ̺(x ′′ i ) is minimised. Since all negative values of ω are associated with projections, we have f ∈C (k) ω(f )̺ ′ (f (x ′ 1 , x ′ 2 , . . . , x ′ k ) ≤ f ∈C (k) ω(f )̺(f (x ′′ 1 , x ′′ 2 , . . . , x ′′ k ) ≤ 0 . We now prove Theorem 5.15, which states that that for any finite set D, and any finite Γ ⊂ Φ D , Imp(wPol(Γ)) = wRelClone(Γ). Proof. We first establish that for any Γ ⊂ Φ D we have the inclusion wRelClone(Γ) ⊆ Imp(wPol(Γ)). To see this, observe that Γ ⊆ Imp(wPol(Γ)) and, hence, wRelClone(Γ) ⊆ wRelClone(Imp(wPol(Γ))) which is equal to Imp(wPol(Γ)) by Proposition 6.2. We will prove the reverse inclusion, Imp(wPol(Γ)) ⊆ wRelClone(Γ), as follows. Given a weighted relation ̺ of arity r, we will show that either there exists a weighted operation ω ∈ wPol(Γ) such that ̺ ∈ Imp({ω}) or else ̺ plus some constant can be obtained by minimisation from a non-negative weighted sum of weighted relations in Γ, and hence ̺ ∈ wRelClone(Γ). We now give the details of this argument. Let k be the number of r-tuples for which ̺ is defined and fix an arbitrary order, x 1 , . . . , x k , for these tuples. This list of tuples can be viewed as (the rows of) a matrix with k rows and r columns, which we will call S ̺ . By Proposition 4.5, ̺ ∈ wRelClone(Γ) if and only if it can be expressed using weighted relations from Γ ∼ . By Definition 2.5, a weighted relation ̺ ′ is expressible over Γ ∼ if and only if there exists an instance P ∈ VCSP(Γ ∼ ) and a list L of variables of P such that π L (P) = ̺ ′ . We consider instances P with |D k | variables, where each variable is associated with a distinct tuple from D k . Each constraint of P is a pair S, γ for some γ ∈ Γ ∼ and some list of variables S. Each such S can be viewed as a list of k-tuples over D, and hence as a matrix over D, whose columns are these k-tuples. Since we are using P to express the defined values of ̺, it is sufficient to consider only matrices S with rows t 1 , . . . , t k such that γ(t i ) is defined for i = 1, . . . , k. For any γ ∈ Γ, a pair S, γ with this property will be called a k-match to Γ. Each assignment to the variables of P can be seen as a mapping from k-tuples over D to D, and hence associated with an operation f ∈ O (k) D . For any list of variables S of P, we will write f (S) to denote the assignment to those variables obtained by applying f to the columns of S, viewed as a matrix. With this notation, we have that π L (P) = ̺ ′ with L = S ̺ if we can find non-negative rationals x S,γ for all k-matches to Γ, and a constant c ∈ Q, such that the following system of inequalities and equations is satisfied: Moreover, if f ∈ Pol (k) (Γ) then the left-hand-side of the corresponding inequality will be undefined, by Definition 3.9, so it is sufficient to consider only f ∈ Pol (k) (Γ). This gives us a system of inequalities and equations with rational coefficients. For each f ∈ O (k) D , If this system has a solution then ̺ ∈ wRelClone(Γ). On the other hand, if this system of equations and inequalities has no solution, then we appeal to Lemma 6.1, to get a certificate of unsolvability. That is, in this case we know that there exists a set of integers {y f | f ∈ Pol (k) (Γ)}, such that f ∈Pol (k) (Γ) y f = 0, y f < 0 only if f is a projection, and: for each k-matching S, γ of Γ, f ∈Pol (k) (Γ) y f γ(f (S)) ≤ 0, and(3)f ∈Pol (k) (Γ) y f ̺(f (S ̺ )) > 0(4) Now, consider the k-ary weighting ω of the clone Pol(Γ) defined by ω(f ) = y f for each f ∈ Pol (k) (Γ). From (3), we can see that ω is a weighted polymorphism of every γ ∈ Γ. On the other hand, (4) shows that ω is not a weighted polymorphism of ̺. Remark 6.3. The proof of Theorem 5.15 demonstrates the decidability of the following question: for any finite Γ ⊆ Φ D and any weighted relation ̺ defined on D, does ̺ belong to wRelClone(Γ)? We will prove Theorem 5.16 in two parts. First, we show in Proposition 6.4 that the set of weighted polymorphisms of any given set of weighted relations is always a weighted clone. Then we show that for any finite set W the set of weightings that improve all weighted relations in Imp(W ) is precisely the weighted clone wClone(W ). Let ω ∈ wPol (k) (Γ) and suppose ω ′ = ω[g 1 , . . . , g k ] is a proper superposition of ω, where g 1 , g 2 , . . . , g k ∈ Pol (ℓ) (Γ). We will now show that ω ′ ∈ wPol (ℓ) (Γ). Suppose ̺ is a weighted relation of arity r satisfying ω ∈ wPol({̺}), i.e., ̺ and ω satisfy inequality (2) for all x 1 , x 2 , . . . , x k such that each ̺(x i ) is defined. Given any x ′ 1 , x ′ 2 , . . . , x ′ ℓ for which each ̺(x ′ i ) is defined, set x i = g i (x ′ 1 , x ′ 2 , . . . , x ′ ℓ ) for i = 1, 2, . . . , k. Then, if we set f ′ = f [g 1 , . . . , g k ], we have f ′ (x ′ 1 , x ′ 2 , . . . , x ′ ℓ ) = f (x 1 , x 2 , . . . , x k ) , for any f ∈ Pol (k) (Γ). Hence, by Definition 5.2, we have f ′ ∈Pol (ℓ) (Γ) ω ′ (f ′ )̺(f ′ (x ′ 1 , x ′ 2 , . . . , x ′ k )) = f ∈Pol (ℓ) (Γ) ω(f )̺(f (x 1 , x 2 , . . . , x k )) ≤ 0 . We will make use of the following technical lemma, which shows that any weighted sum of arbitrary superpositions of a pair of weightings ω 1 and ω 2 can be obtained by taking a weighted sum of superpositions of ω 1 and ω 2 with projection operations, and then taking a superposition of the result. This result implies that any weighting which can be expressed as a weighted sum of arbitrary superpositions can also be expressed as a superposition of a weighted sum of proper superpositions. Lemma 6.5. Let C be a clone, and let ω 1 and ω 2 be weightings of C, of arity k and ℓ respectively. For any g 1 , . . . , g k ∈ C (m) and any g ′ 1 , . . . , g ′ ℓ ∈ C (m) , c 1 ω 1 [g 1 , . . . , g k ] + c 2 ω 2 [g ′ 1 , . . . , g ′ ℓ ] = ω[g 1 , . . . , g k , g ′ 1 , . . . , g ′ ℓ ] ,(5) where ω = c 1 ω 1 [e Proof. For any f ∈ C (m) , the result of applying the right-hand side expression in equation (5) to f is: f ′ ∈C (k+ℓ) f ′ [g 1 ,...,g k ,g ′ 1 ,...,g ′ ℓ ]=f       h ′ ∈C (k) h ′ [e (k+ℓ) 1 ,...,e (k+ℓ) k ]=f ′ c 1 ω 1 (h ′ ) + h ′ ∈C (ℓ) h ′ [e (k+ℓ) k+1 ,...,e (k+ℓ) k+ℓ ]=f ′ c 2 ω 2 (h ′ )       . Replacing each f ′ by the equivalent superposition of h ′ with projections, we obtain: h ′ ∈C (k) h ′ [g 1 ,...,g k ]=f c 1 ω 1 (h ′ ) + h ′ ∈C (ℓ) h ′ [g ′ 1 ,...,g ′ ℓ ]=f c 2 ω 2 (h ′ ) , which is the result of applying the left-hand-side of Equation 5 to f . We now prove Theorem 5.16, which states that for any finite set D, and any finite W ⊂ W D , wPol(Imp(W )) = wClone(W ). Proof. We first establish that for any W ⊂ W D we have the inclusion wClone(W ) ⊆ wPol(Imp(W )). To see this, observe that every operation in C = Clone( ω∈W dom(ω)) is a polymorphism of Imp(W ), by Definition 3.9, so W ⊆ wPol(Imp(W )). Hence, wClone(W ) ⊆ wClone(wPol(Imp(W ))) which is equal to wPol(Imp(W )) by Proposition 6.4. We will prove the reverse inclusion, wPol(Imp(W )) ⊆ wClone(W ), as follows. Given any weighting ω 0 ∈ W D , we will show that either there exists a weighted relation ̺ ∈ Imp(W ) such that ω 0 ∈ wPol({̺}) or else ω 0 is equal to a non-negative weighted sum of superpositions of weightings in W , and hence ω 0 ∈ wClone(W ). We now give the details of this argument. Let k be the arity of ω 0 , and let M = |D| k . We first observe that it is sufficient to consider weighted relations of arity M in Imp(W ). To see this, suppose there exists a weighted relation ̺ ∈ Imp(W ) with arity N > M such that ω 0 ∈ wPol({̺}) and let x 1 , . . . , x k ∈ D N be any set of tuples for which inequality (2) fails to hold for ω 0 and ̺. Let X be the k × N matrix whose rows are the tuples x 1 , . . . , x k . Since N > M it follows that some of the columns in this matrix must be equal. Moreover, if the i-th and j-th column of X are equal, then so will be the i-th and j-th entry of the tuple f (x 1 , . . . , x k ) obtained by applying any f ∈ O (k) D to these k tuples. Now let ̺ ′ be the weighted relation of arity ≤ M that depends only on the first of each of these repeated columns, and takes the same values as ̺ takes on arguments with the appropriate entries repeated. Let X ′ be the reduced form of X (with repeated columns deleted). By this approach, we can construct ̺ ′ so that ̺ ′ ∈ Imp(W ), but X ′ gives a certificate to show that ω 0 ∈ wPol({̺}), i.e., the rows of X ′ form a list of tuples for which (2) fails to hold for ω 0 and ̺ ′ . Moreover, if we have a weighted relation ̺ ∈ Imp(W ) with arity N < M such that ω 0 ∈ wPol({̺}), then ̺ can be extended to a weighted relation ̺ ′ of arity M that does not depend on the M − N added inputs, and, hence, is also contained in Imp(W ) and is such that ω 0 ∈ wPol({̺ ′ }). By the argument given above, there exists a weighted relation ̺ ∈ Imp(W ) such that ω 0 ∈ wPol({̺}) if and only if there exists a weighted relation ̺ M of arity M in Imp(W ) such that ω 0 ∈ wPol({̺ M }). Furthermore, by reordering the arguments of ̺ M if necessary, we can assume that ̺ M and ω 0 violate (2) on the particular list of tuples x 1 , . . . , x k given by taking the rows of a matrix, X M , whose columns are precisely the tuples in D k , ordered lexicographically. By Definition 5.11, such a weighted relation ̺ M exists if and only if the following system of inequalities can be satisfied, for all ω ∈ W and all t 1 , . . . , t ar(ω) ∈ D M such that ̺ M (t i ) is defined for i = 1, . . . , ar(ω), g∈dom(ω) ω(g) ̺ M (g(t 1 , . . . , t ar(ω) )) ≤ 0 , and, for the tuples x 1 , . . . , x k forming the rows of X M , ̺ M (x i ) is defined for i = 1, . . . , k and f ∈dom(ω 0 ) ω 0 (f ) ̺ M (f (x 1 , . . . , x k )) ≤ 0 .(7) There is a one-to-one correspondence between operations g : D k → D and tuples t g ∈ D M , where the tuple t g contains the list of values returned by the operation g applied to the columns of X M . Set C = Clone( ω∈W dom(ω)). We observe that, to satisfy inequality (6), for any ω ∈ W , if ̺ M (t i ) is defined for i = 1, . . . , ar(ω), then ̺ M (g(t 1 , . . . , t ar(ω) )) must be defined for all g ∈ dom(ω). To achieve this, it is sufficient to ensure that, for all g ∈ C (k) , ̺ M (t g ) is defined. All other values of ̺ M can be left undefined, as this just reduces the number of inequalities in the system. Using superposition (Definition 5.2), we can rewrite inequalities (6) to obtain the following equivalent system: for all ω ∈ W , and all g 1 , . . . , g ar(ω) ∈ C (k) , f ∈dom(ω[g 1 ,...,g ar(ω) ]) ω[g 1 , . . . , g ar(ω) ](f ) ̺ M (f (x 1 , . . . , x k )) ≤ 0 .(8) Now, by applying Lemma 6.1 to the resulting system of inequalities, we conclude that either a solution ̺ M exists, in which case ω 0 ∈ wPol(Imp(W )), or else there exists a set of non-negative rational numbers {c ω[g 1 ,...,g ar(ω) ] | ω ∈ W , g 1 , . . . , g ar(ω) ∈ C (k) } such that for every f ∈ C (k) , ω∈W g 1 ,...,g ar(ω) g i ∈C (k) c ω[g 1 ,...,g ar(ω) ] ω[g 1 , . . . , g ar(ω) ](f ) ≥ ω 0 (f ) .(9) By Definition 5.1, adding the left-hand side of these inequalities over all f gives 0, and so does adding the right hand sides, so each inequality must actually be an equality. In other words, ω 0 is equal to a nonnegative weighted sum of superpositions of weightings in W . Hence, by Lemma 6.5 and Remark 5.7, ω 0 is equal to a proper superposition of some element ω ′ 0 ∈ wClone(W ), so ω 0 ∈ wClone(W ). Remark 6.6. The proof of Theorem 5.16 demonstrates the decidability of the following question: for any finite W ⊆ W D and any weighting ω defined on D, does ω belong to wClone(W )? Necessary Conditions For Tractability In this section, we will start to investigate the structure of weighted clones, and hence establish some necessary conditions for any valued constraint language to be tractable. Note that, by Definition 3.8, the smallest possible clone of operations over a fixed set D is the set of all projection operations on D, which is denoted J D . Proof. Let W be a weighted clone with support J D . If the weights assigned by every weighting ω ∈ W are all zero, then W is the zero-valued weighted clone W 0 J D described in Example 5.10. Otherwise, there is some ω ∈ W (of arity k) such that ω assigns positive weight to some k-ary projections and negative weights to some of the others (the sum of the weights is zero, by Definition 5.1). If we form the superposition of ω with the sequence of projections g 1 , g 2 , . . . , g k , where g i = e By adding appropriate multiples of such functions for each successive pair of indices a and b, we can obtain any desired weighting of J D . Hence, in this case W contains all possible weightings of J D , so W = W J D . Any weighting ω which is defined only for projection operations will be called a trivial weighting, Proposition 7.2. For any set of trivial weightings W , Imp(W ) is NP-hard. Proof. By Theorems 5.15 and 5.16, we have that Imp(wClone(W )) = wRelClone(Imp(W )), so by by Proposition 4.5, Theorem 2.8, and Theorem 4.3 we have that Imp(W ) is NP-hard if and only if Imp(wClone(W )) is NP-hard. If W contains only trivial weightings, then wClone(W ) has support J D , so it is equal to W J D or W 0 J D by Proposition 7.1. Every weighting in W 0 J D is a weighted polymorphism of any possible weighted relation, by Definition 5.11. Hence Imp(W 0 J D ) = Φ D . The weighted relations that are improved by all weightings are precisely those which take at most one value. Hence Imp(W J D ) = (R D ) ∼ . In both cases the resulting valued constraint language is NP-hard. Now consider weightings whose values are all 0. Proposition 7.3. For any set of zero-valued weightings W , Imp(W ) is NP-hard. Proof. By Definition 5.11, a zero-valued weighting will be a weighted polymorphism of any weighted relation which is a total function (i.e., any weighted relation where all assignments are feasible). Some valued constraint languages containing only total functions are NP-hard [16]. For example, consider the valued constraint language consisting of the following total function: ̺ = (x, y) = 0 if x = y 1 otherwise . We observed in Example 2.3 that on the domain {0, 1} the problem VCSP({̺ = }) corresponds to the MAX-CUT problem which is known to be NP-hard. Over domains of size k > 2 this problem corresponds to the problem MAX-k-CUT, which is also known to be NP-hard. Using the Galois connection developed in the previous sections, these two results tell us that any valued constraint language that is not NP-hard must have a weighted polymorphism which is non-trivial and assigns at least some non-zero weights. A weighting which assigns positive weight to at least one operation that is not a projection will be called a positive weighting. Corollary 7.4. For any finite set D, and any Γ ⊂ Φ D , either Γ is NP-hard, or else wPol(Γ) is a weighted clone containing some positive weightings. Proof. By Proposition 6.4, in all cases wPol(Γ) is a weighted clone. By Theorem 5.15, for any finite Γ ′ ⊆ Γ, Imp(wPol(Γ ′ )) = wRelClone(Γ ′ ). By Proposition 4.5, Theorem 2.8, and Theorem 4.3, if Γ ′ is NP-hard, then wRelClone(Γ ′ ) is also NP-hard, so Imp(wPol(Γ ′ )) must be NP-hard. Conversely, if Γ ′ is not NP-hard, then the same argument shows that Imp(wPol(Γ ′ )) is not NP-hard, so by Propositions 7.2 and 7.3, wPol(Γ ′ ) must contain some weightings that are non-trivial and some weightings that are not zero-valued. Choose a weighting ω ∈ wPol(Γ) that is not zero-valued, and a weighting ω ′ that is non-trivial (but may be zero-valued). If ω assigns positive weight to any non-projection, then it is a positive weighting and we are done. Otherwise, we have that ω assigns positive weight to some projections and negative weight to some other projections. Let f be an operation on which ω ′ is defined that is not a projection, and let k be the arity of f . If we form the superposition of ω with the sequence of functions g 1 , g 2 , . . . , g ar(ω) , where g i = f if ω(e (ar(ω)) i ) is positive, and g i = e (k) 1 otherwise, then we obtain a new weighting ω[g 1 , g 2 , . . . , g ar(ω) ] which assigns positive weight to f (see Example 5.3). Assuming that P = N P , this result tells us that tractable valued constraint languages are associated with certain kinds of weighted clones. To obtain further information about the weighted clones associated with tractable valued constraint languages, we now consider some special kinds of operations. For any k ≥ 2, a k-ary operation f is called sharp if f is not a projection, but the operation obtained by equating any two inputs in f is a projection [20]. In other words, f is sharp if for all i, j ∈ {1, . . . , k−1} with i = j, there exists an index m ∈ {1, . . . , k−1} such that f satisfies the identity: Each of these weightings has arity k − 1, so, by the choice of ω, must not assign positive weight to any operation except (possibly) projections. Hence all non-projection operations assigned positive weight by ω are sharp. f (x 1 , x 2 , . . . , x j−1 , x i , x j , x j+1 , . . . , x k−1 ) = x m . We can obtain further details about these weighted clones by considering the possible types of sharp operations. First, we observe that all sharp operations must satisfy the identity f (x, x, . . . , x) = x; such operations are called idempotent. Ternary sharp operations may be classified according to their values on tuples of the form (x, x, y), (x, y, x) and (y, x, x), which must be equal to either x or y. There are precisely 8 possibilities, as listed in Table 1. Input 1 2 3 4 5 6 7 8 (x,x,y) x x x x y y y y (x,y,x) x x y y x x y y (y,x,x) x y x y x y x y Table 1: Sharp ternary operations The first column in Table 1 corresponds to operations that satisfy the identities f (x, x, y) = f (x, y, x) = f (y, x, x) = x for all x, y ∈ D; such operations are called majority operations. The last column in the table corresponds to operations that satisfy the identities f (x, x, y) = f (x, y, x) = f (y, x, x) = y for all x, y ∈ D; such operations are called minority operations. Columns 4, 6 and 7 in Table 1 correspond to operations that satisfy the identities f (y, y, x) = f (x, y, x) = f (y, x, x) = y for all x, y ∈ D (up to permutations of inputs); such operations are called Pixley operations [33]. For any k ≥ 3, a k-ary operation f is called a semiprojection if it is not a projection, but there is an index i ∈ {1, . . . , k} such that f (x 1 , . . . , x k ) = e (k) i for all x 1 , . . . , x k ∈ D such that x 1 , . . . , x k are not pairwise distinct. In other words, a semiprojection is a particular form of sharp operation where the operation obtained by equating any two inputs is always the same projection. Columns 2,3 and 5 in Table 1 correspond to semiprojections. The following lemma shows that the only sharp operations of arity k ≥ 4 are semiprojections. Lemma 7.6 (Świerczkowski's Lemma [36]). Given an operation of arity ≥ 4, if every operation arising from the identification of two variables is a projection, then these projections coincide. Hence we may refine Theorem 7.5 to obtain the following corollary. Classification of Boolean Valued Constraint Languages In this section, we consider the special case of valued constraint languages over a 2-valued domain, such as the Boolean domain D = {0, 1}. There are only four unary operations on the Boolean domain, and one of these is the projection operation e (1) 1 , which is the identity operation. The remaining three unary operations are the operations given by f (x) = 0, f (x) = 1, and f (x) = 1 − x. These will be referred to as constant 0, constant 1, and inversion. There are only two binary idempotent operations on the Boolean domain that are not projections: the operations min and max. The only sharp ternary operations are the unique majority operation (which we will call Mjrty), the unique minority operation (which we will call Mnrty), and three Pixley operations. There are no semiprojections. Hence we can refine Corollary 7.7 even further in the special case of the Boolean domain, to limit the possibilities for weighted clones associated with tractable valued constraint languages to just nine cases. Proof. Let C be the support of W , and let ω be a positive weighting in W with minimal possible arity. Since there are no semiprojections on the Boolean domain, Corollary 7.7 tells us that ω is either unary, binary or ternary. Consider first the case when ω is unary. Since there are just three unary operations on the Boolean domain that are not projections, scale ω so it assigns weight -1 to the projection e (1) 1 , weight a to the constant 0 operation f 0 , weight b to the constant 1 operation f 1 , and weight c to the inversion operation f ¬ . If c = 1, then ω assigns positive weight only to f ¬ and we are done. Otherwise, if f ¬ ∈ C, and hence c is defined, we consider the weighting ω ′ = 1 c+1 ω + c c+1 ω[f ¬ ]. It is straightforward to check that ω ′ assigns weight c − 1 to e (1) 1 , weight a to f 0 , weight b to f 1 and weight 0 to f ¬ . By Lemma 6.5, ω ′ belongs to W . If a = 1, then ω ′ assigns positive weight only to f 0 and we are done. Otherwise, if f 0 ∈ C, and hence a is defined, we consider the weighting ω ′′ = ω ′ + a 1−a ω ′ [f 0 ]. It is straightforward to check that ω ′′ assigns positive weight only to f 1 . By Lemma 6.5, ω ′′ belongs to W . Next consider the case when ω is binary. By Corollary 7.7 and our observations above about the possible binary idempotent operations on the Boolean domain, we know that ω assigns positive weight only to one or both of the operations min and max. If either of these weights is undefined (because the corresponding function does not belong to C), or zero, then we are done, so assume that ω assigns positive weight to both min and max. By taking the weighting ω + ω[e (2) 2 , e 1 ], with a suitable scaling, we can obtain a weighting ω a ∈ W that assigns weight -1 to both binary projections, weight a to min and weight 2 − a to max, for some 0 < a < 2. If a < 1, then the weighting ω a + a 1−a ω a [min, max] assigns positive weight only to max. If a > 1, then the weighting ω a + 2−a a−1 ω a [min, max] assigns positive weight only to min. If a = 1, then ω assigns equal weight to min and max. Finally, we consider the case when ω is ternary. By Corollary 7.7 and our observations above about the possible ternary sharp operations on the Boolean domain, we know that ω assigns positive weight to some subset of Mjrty, Mnrty and the three Boolean Pixley operations f 1 , f 2 and f 3 (corresponding to the fourth, sixth and seventh columns of Table 1). We note that f 1 [e (3) 2 , e(3) 3 , e 1 ] = f 3 , f 2 [e (3) 2 , e(3) 3 , e 1 ] = f 1 and f 3 [e (3) 2 , e(3) 3 , e 1 ] = f 2 . Hence, if ω assigns positive weight to any Pixley operation, then we have that W also contains the weighting ω ′ = ω + ω[e (3) 2 , e 3 , e 1 ] + ω[e (3) 3 , e(3) 1 , e 2 ] which assigns equal negative weight to each projection, and equal positive weight to each Pixley operation. By a suitable scaling we shall assume that ω ′ assigns weight -1 to each projection. Suppose first that ω ′ assigns positive weight to at least one of Mjrty and Mnrty, and assigns weight 0 < w < 1 to the three Pixley operations. We note that f i [f 1 , f 2 , f 3 ] = e = ω ′ + wω ′ [f 1 , f 2 , f 3 ] is non-zero, assigns weight 0 to each Pixley operation and equal negative weight to all projections. By Lemma 6.5, ω ′′ ∈ W . Assume that ω ′′ assigns positive weight to both Mnrty and Mjrty. By taking a suitable scaling, we can obtain a weighting ω a ∈ W that assigns weight -1 to all three projections, weight a to Mnrty and weight 3 − a to Mjrty, for some 0 < a < 3. If a < 1, then the weighting ω a + a 1−a ω a [Mjrty, Mjrty, Mnrty] assigns positive weight only to Mjrty. If a > 1, then the weighting ω a + 3−a a−1 ω a [Mjrty, Mjrty, Mnrty] assigns positive weight only to Mnrty. If a = 1, then ω a assigns positive weight to both Mjrty and Mnrty, in the ratio 2:1. The only remaining case is when ω ′ assigns positive weight 1 just to the three Pixley operations. In this case we note that f 1 [e 2 , f 1 ] = e 2 . Thus the function µ 1 = ω[e 1 , e 2 , f 1 ] assigns weight −1 to f 1 , +1 to Mjrty, and 0 otherwise. For i = 2, 3, we can obtain in a similar way a function µ i , which assigns weight −1 to f i and +1 to Mjrty. Then the weighting ω + µ 1 + µ 2 + µ 3 will assign positive weight only to Mjrty. Again, by Lemma 6.5, ω ′′ ∈ W . Each of the nine types of weightings mentioned in Theorem 8.1 can be supported by a different clone, so these nine types of weightings can each generate different weighted clones. Using the Galois connection developed above, this result tells us that any tractable valued constraint language over the Boolean domain must have as a weighted polymorphism one of nine specific kinds of weightings. Eight of these can be been shown to be sufficient to ensure tractability using the results of [16]. The only remaining case is the unary weighting that assigns positive weight to the inversion operation only. Our next result shows that having a weighted polymorphism of this kind is not a sufficient condition for tractability on its own, but if a language has any additional positive weightings as weighted polymorphisms which are not implied by this one, then it will be tractable. Proof. By Theorem 8.1, W must contain either a weighting ω ¬ that assigns positive weight to the inversion operation only, or at least one of the eight other kinds of weighting listed in Theorem 8.1 (or both). If W = wClone({ω ¬ } ∪ W 0 C ), for some unary weighting ω ¬ that assigns positive weight to the inversion operation only and C is the support of W , then we are in case (1). In this case, the weighted relation ̺ = defined in Example 2.3 is an element of Imp(W ), so Imp(W ) is NP-hard (see the proof of Proposition 7.3). If W contains a suitable weighting ω ¬ , but W = wClone({ω ¬ } ∪ W 0 C ), then W must also contain a non-zero weighting ω of minimal possible arity such that ω ∈ wClone({ω ¬ } ∪ W 0 C ). If ω is unary, then we can use the same argument as in the proof of Theorem 8.1 to show that W must contain a unary operation that assigns positive weight to the constant 1 operation only or the constant 0 operation only. If ω is not unary, then ω[e 1 , . . . , e 1 ] must lie in wClone(ω ¬ ), so ω assigns positive weights only to operations f such that f (x, . . . , x) = x or f (x, . . . , x) = 1 − x. If ω(f ) = a > 0 for some f such that f (x, . . . , x) = 1 − x, then we consider the weighting ω ′ = ω + aω ¬ [f ], and note that ω ′ (f ) = 0. By repeating this process we obtain a weighting ω ′′ which assigns positive weight only to operations f such that f (x, . . . , x) = x. Since ω has minimal arity, these must be sharp operations, so we can proceed as in the proof of Theorem 8.1 to show that case (2) holds. Each of the eight types of weightings in case (2) is sufficient to ensure the tractability of Imp(W ), by the results of [16]. The corresponding classification for valued constraint languages over the Boolean domain was obtained in [16] using a more intricate argument involving the explicit construction of gadgets to express particular weighted relations. Here we have considered only the properties of weighted clones. Hence the valued constraint language Γ = {̺ = } is tractable, and remains tractable if we add to Γ any other weighted relations that have this weighting as a weighted polymorphism. For example, we may add unary weighted relations with a single allowed value, which allow us to fix individual variables to a desired value, and still retain tractability. It follows that ̺ = ∈ Express({̺ = }). Conclusions We have presented an algebraic theory of valued constraint languages that generalizes and extends the algebraic theory developed over the past few years to study the complexity of the classical constraint satisfaction problem. We have shown that the complexity of any valued constraint language over a finite domain with rational-valued costs is determined by certain algebraic properties which we have called weighted polymorphisms. When the weights are all zero, the optimisation problem we are considering collapses to the classical constraint satisfaction problem, CSP. In previous work [28,27] it has been shown that every tractable constraint language for the CSP can be characterised by an associated clone of operations. That work initiated the use of algebraic properties in the search for tractable constraint languages, an area that has seen considerable activity in recent years; see, for instance, [11,15,14,13,29,21,4,2,3,5]. The results in this paper show that a similar result holds for the valued constraint satisfaction problem: every tractable valued constraint language is characterised by an associated weighted clone. We therefore hope that our results here will provide a similar impetus for the investigation of a much broader class of discrete optimisation problems. For example, a recent result on the power of linear programming for valued constraint languages [38] provides a characterisation of languages solvable by a standard LP relaxation in terms of weighted polymorphisms. Many questions about the complexity of discrete optimisation problems over finite domains can now be translated into questions about the structure of weighted clones. This provides a new approach to tackling such questions by investigating the algebraic properties of weighted clones. Example 2.2 shows that VCSP({̺ = }) includes the MIN-CUT problem, and Example 2.3 shows that VCSP({̺ = }) includes the MAX-CUT problem. In fact it is easy to show that VCSP({̺ = }) corresponds to the MIN-CUT problem in the sense that not only does VCSP({̺ = }) include the MIN-CUT problem, but also any instance from VCSP({̺ = }) can be reduced to an instance of the MIN-CUT problem. Similarly, VCSP({̺ = }) corresponds to the MAX-CUT problem. The problem VCSP({̺ = , ̺ = }) includes both MIN-CUT and MAX-CUT, as well as many other problems which can be expressed using these two types of constraints. Definition 2. 5 . 5For any VCSP instance P = V, D, C , and any list L = v 1 , . . . , v r of variables of P, the projection of P onto L, denoted π L (P), is the weighted relation on D of arity r defined as follows: π L (P)(x 1 , . . . , x r ) def = min {s:V →D | s(v 1 ),...,s(vr) = x 1 ,...,xr } Cost P (s) . Example 2. 7 . 7Let P be the VCSP instance with domain {0, 1}, variables v 1 , v 2 , v 3 , and constraints v 1 , v 2 , ̺ = and v 2 , v 3 , ̺ = , and let L = v 1 , v 3 . Then, by Definition 2.5, Definition 3. 1 . 1We denote by R D the set of all zero-valued weighted relations on a set D. Definition 3.3. A set Γ ⊆ R D is called a relational clone if it contains the weighted equality relation and is closed under addition and minimisation over arbitrary arguments. Proposition 3. 4 . 4For any Γ ⊆ R D , Express(Γ) = RelClone(Γ). Definition 3. 5 . 5We denote by O D the set of all finitary operations on D and by O (k) D the k-ary operations in O D . Definition 3. 6 . 6The k-ary projections on D are the operations e (k) i : D k → D , (a 1 , . . . , a k ) → a i . Definition 3.7. Let f ∈ O (k) D and g 1 , . . . , g k ∈ O (ℓ) D . The superposition of f and g 1 , . . . , g k is the ℓ-ary operation f [g 1 , . . . , g k ] : D ℓ → D , (x 1 , . . . , x ℓ ) → f (g 1 (x 1 , . . . , x ℓ ), . . . , g k (x 1 . . . , x ℓ )) . Definition 3.8. A set F ⊆ O D is called a clone of operations if it contains all the projections on D and is closed under superposition. For each F ⊆ O D we define Clone(F ) to be the smallest clone containing F . Definition 3. 10 . 10For any valued constraint language Γ over a set D, we denote by Pol(Γ) the set of all operations on D which are polymorphisms of all weighted relations ̺ ∈ Γ and by Pol (k) (Γ) the k-ary operations in Pol(Γ).Definition 3.11. For any F ⊆ O D , we denote by Inv(F ) the set of all weighted relations in R D that are invariant under all operations f ∈ F . Theorem 3 . 12 . 312For any finite set D, and any finite Γ ⊆ R D , Inv(Pol(Γ)) = RelClone(Γ). Theorem 3.13. For any finite set D, and any finite F ⊆ O D , Pol(Inv(F )) = Clone(F ). Definition 4. 1 . 1We denote by Φ D the set of all weighted relations on D taking values in Q and by Φ (r) D the weighted relations in Φ D of arity r. Definition 4.4.A set Γ ⊆ Φ D is a weighted relational clone if it contains the weighted equality relation and is closed under non-negative scaling and addition of constants, addition, and minimisation over arbitrary arguments. C the set of k-ary weightings of C. Remark 5. 7 . 7The superposition of a projection operation and other projection operations is always a projection operation. So, by Definition 5.2, for any clone C and any ω ∈ W (k) C , if g 1 , . . . , g k ∈ C (ℓ) are projections, then the function ω[g 1 , . . . , g k ] can take negative values only on projections, and hence is a valid weighting. This means that a superposition with any list of projections is always a proper superposition. Definition 5. 8 . 8A weighted clone, W , is a non-empty set of weightings of some fixed clone C which is closed under non-negative scaling, addition of weightings of equal arity, and proper superposition with operations from C. The clone C is called the support of W . Example 5.9. For any clone, C, the set W C containing all possible weightings of C is a weighted clone with support C. Example 5.10. For any clone, C, the set W 0 Definition 5. 13 . 13For any Γ ⊆ Φ D , we denote by wPol(Γ) the set of all weightings of Pol(Γ) which are weighted polymorphisms of all weighted relations ̺ ∈ Γ. The set of k-ary weightings in wPol(Γ) will be denoted wPol (k) (Γ). Theorem 5. 15 . 15For any finite set D, and any finite Γ ⊆ Φ D , Imp(wPol(Γ)) = wRelClone(Γ). Theorem 5.16. For any finite set D, and any finite W ⊆ W D , wPol(Imp(W )) = wClone(W ). Figure 1 : 1Galois connection between Φ D and W D . Proposition 6. 2 . 2For any finite set D, and any W ⊂ W D , Imp(W ) is a weighted relational clone. γ∈Γ {all k-matches S, γ } x S,γ γ(f (S)) ≥ ̺(f (S ̺ )) + c For each projection e ∈ O (k) D , γ∈Γ {all k-matches S, γ }x S,γ γ(e(S)) = ̺(e(S ̺ )) + c Proposition 6. 4 . 4For any finite set D, and any Γ ⊂ Φ D , wPol(Γ) is a weighted clone. Proof. By Definition 5.13, wPol(Γ) is a set of weightings of Pol(Γ). Similarly, wPol(Γ) is closed under addition and non-negative scaling, since both of these operations preserve inequality (2). Hence, to show wPol(Γ) is a weighted clone we only need to show wPol(Γ) is closed under proper superposition by members of Pol(Γ). Proposition 7. 1 . 1For any finite set D, there are precisely two weighted clones with support J D . These are W J D and W 0 J D . is positive, and g i = e (k) b otherwise, then we obtain a new weighting ω[g 1 , g 2 , . . . , g k ] of J D which assigns some positive weight w to e (k) a and −w to e (k) b . Theorem 7. 5 . 5Any weighted clone W containing positive weightings must contain a weighting that assigns positive weight to either: 1. A set of unary operations that are not projections; or 2. A set of sharp operations. Proof. Let ω be a positive weighting in W with the smallest possible arity, k. If k = 1, then we are done. Otherwise, we consider the weightings ω[e for all i, j ∈ {1, . . . , k} with i = j. Theorem 8. 1 . 1Any weighted clone W on the Boolean domain that contains positive weightings must contain a weighting ω that assigns positive weight to either: 1. Exactly one of the unary operations constant 0, constant 1, or inversion; 2. Exactly one of the binary operations min and max, or both of them equally; 3. Exactly one of the ternary operations Mjrty and Mnrty, or both of them with ω(Mjrty) = 2ω(Mnrty). each i = 1, 2, 3. Moreover, Mjrty[f 1 , f 2 , f 3 ] = Mnrty and Mnrty[f 1 , f 2 , f 3 ] = Mjrty. Thus, the weighting ω ′′ f 1 ] = Mjrty, f 2 [e f 1 ] = e 1 and f 3 [e Corollary 8. 2 . 2Any weighted clone W on the Boolean domain that contains positive weightings, satisfies exactly one of the following:1. W = wClone({ω ¬ } ∪ W 0 C ),for some unary weighting ω ¬ that assigns positive weight to the inversion operation only, where C is the support of W ; in this case Imp(W ) is NP-hard. 2. 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[]
[ "Malign Overfitting: Interpolation Can Provably Preclude Invariance", "Malign Overfitting: Interpolation Can Provably Preclude Invariance" ]
[ "Yoav Wald ", "Gal Yona ", "Uri Shalit ", "Yair Carmon " ]
[]
[]
Learned classifiers should often possess certain invariance properties meant to encourage fairness, robustness, or out-of-distribution generalization. However, multiple recent works empirically demonstrate that common invariance-inducing regularizers are ineffective in the over-parameterized regime, in which classifiers perfectly fit (i.e. interpolate) the training data. This suggests that the phenomenon of "benign overfitting," in which models generalize well despite interpolating, might not favorably extend to settings in which robustness or fairness are desirable.In this work we provide a theoretical justification for these observations. We prove that-even in the simplest of settings-any interpolating learning rule (with arbitrarily small margin) will not satisfy these invariance properties. We then propose and analyze an algorithm that-in the same setting-successfully learns a non-interpolating classifier that is provably invariant. We validate our theoretical observations on simulated data and the Waterbirds dataset. c are uniformly distributed on unit spheres a subspace of dimension d − N .Proof. Recalling the notation z i = y i x i , note that {z i } i∈S are sufficient statistics for µ s , µ c given the training data, i.e., P(µ s , µ c | {z i } i∈S ) = P(µ s , µ c | {x i , y i } i∈S ). Furthermore, since the joint distribution of µ s , µ c , {z i } i∈S is rotationally invariant, we havefor any orthogonal matrix R ∈ R d×d . Focusing on matrices R that presereve that data, i.e., satisfying Rz i = z i for all i ∈ [N ], we have P(µ s , µ c | {z i } i∈S ) = P(Rµ s , Rµ c | {z i } i∈S ).
10.48550/arxiv.2211.15724
[ "https://export.arxiv.org/pdf/2211.15724v1.pdf" ]
254,069,432
2211.15724
205fcf07feb4e09e07ba765045244ce3da20c500
Malign Overfitting: Interpolation Can Provably Preclude Invariance Yoav Wald Gal Yona Uri Shalit Yair Carmon Malign Overfitting: Interpolation Can Provably Preclude Invariance Learned classifiers should often possess certain invariance properties meant to encourage fairness, robustness, or out-of-distribution generalization. However, multiple recent works empirically demonstrate that common invariance-inducing regularizers are ineffective in the over-parameterized regime, in which classifiers perfectly fit (i.e. interpolate) the training data. This suggests that the phenomenon of "benign overfitting," in which models generalize well despite interpolating, might not favorably extend to settings in which robustness or fairness are desirable.In this work we provide a theoretical justification for these observations. We prove that-even in the simplest of settings-any interpolating learning rule (with arbitrarily small margin) will not satisfy these invariance properties. We then propose and analyze an algorithm that-in the same setting-successfully learns a non-interpolating classifier that is provably invariant. We validate our theoretical observations on simulated data and the Waterbirds dataset. c are uniformly distributed on unit spheres a subspace of dimension d − N .Proof. Recalling the notation z i = y i x i , note that {z i } i∈S are sufficient statistics for µ s , µ c given the training data, i.e., P(µ s , µ c | {z i } i∈S ) = P(µ s , µ c | {x i , y i } i∈S ). Furthermore, since the joint distribution of µ s , µ c , {z i } i∈S is rotationally invariant, we havefor any orthogonal matrix R ∈ R d×d . Focusing on matrices R that presereve that data, i.e., satisfying Rz i = z i for all i ∈ [N ], we have P(µ s , µ c | {z i } i∈S ) = P(Rµ s , Rµ c | {z i } i∈S ). Introduction Modern machine learning applications often call for models which are not only accurate, but which are also robust to distribution shifts or satisfy fairness constraints. For example, we might wish to avoid using hospital-specific traces in X-ray images [16,65], as they rely on spurious correlations that will not generalize to a new hospital, or we might seek "Equal Opportunity" models attaining similar error rates across protected demographic groups, e.g., in the context of loan applications [5,22]. A developing paradigm for fulfilling such requirements is learning models that satisfy some notion of invariance [40, 41] across environments or sub-populations. For example, in the X-ray case, spurious correlations can be formalized as relationships between a feature and a label which vary across hospitals [65]. Equal Opportunity [22] can be expressed as a statistical constraint on the outputs of the model, where the false negative rate is invariant to membership in a protected group. Many techniques for learning invariant models have been proposed including penalties that encourage invariance [3,28,57,60,43,34,44,25], data re-weighting [49, 63,24], causal graph analysis [54,55], and more [1]. While the invariance paradigm holds promise for delivering robust and fair models, many current invarianceinducing methods often fail to improve over naive approaches. This is especially noticeable when these methods are used with overparameterized deep models capable of interpolating, i.e., perfectly fitting the training data [20,18,21,68,36,58,12]. Existing theory explains why overparameterization hurts invariance for standard interpolating learning rules, such as empirical risk minimization and max-margin classification [50,37,15], and also why reweighting and some types of distributionally robust optimization face challenges when used with overparameterized models [6,49]. In contrast, training overparameterized models to interpolate the training data typically results in good in-distribution generalization, and such "benign overfitting" [26,63] is considered a key characteristic of modern deep learning [8,62,52]. Consequently, a number of works attempt to extend benign overfitting to robust or fair generalization by designing new interpolating learning rules [7,26,63,33]. In this paper, we demonstrate that such attempts face a fundamental obstacle, because all interpolating learning rules (and not just maximum-margin classifiers) fail to produce invariant models in certain high-dimensional settings where invariant learning (without interpolation) is possible. This does not occur because there are no invariant models that separate the data, but because interpolating learning rules cannot find them. In other words, beyond identically-distributed test sets, overfitting is no longer benign. More concretely, we consider linear classification in a basic overparameterized Gaussian mixture model with invariant "core" features as well as environment-dependent "spurious" features, similar to models used in previous work to gain insight into robustness and invariance [51,45,50]. We show that any learning rule producing a classifier that separates the data with non-zero margin must necessarily rely on the spurious features in the data, and therefore cannot be invariant. Moreover, in the same setting we analyze a simple two-stage algorithm that can find accurate and nearly invariant linear classifiers, i.e., with almost no dependence on the spurious feature. Thus, we establish a separation between the level of invariance attained by interpolating and noninterpolating learning rules. We believe that learning rules which fail in the simple over-parameterized linear classification setting we consider are not likely to succeed in more complicated, real-world settings. Therefore, our analysis provides useful guidance for future research into robust and fair machine learning models, as well as theoretical support for the recent success of non-interpolating robust learning schemes [46,58,27,36,29,67,24,10]. Paper organization. The next section formally states our full result (Theorem 1). In Section 3 we outline the arguments leading to the negative part of Theorem 1, i.e., the failure of interpolating classifiers to be invariant in our model. In Section 4 we establish the positive part Theorem 1, by providing and analyzing a non-interpolating algorithm that, in our model, achieves low robust error. We validate our theoretical findings with simulations and experiments on the Waterbirds dataset in Section 5, and conclude with a discussion of additional related results and directions for future research in Section 6. Statement of Main Result Preliminaries Data model. Our analysis focuses on learning linear models over covariates x distributed as a mixture of two Gaussian distributions corresponding to the label y. Definition 1. An environment is a distribution parameterized by (µ c , µ s , d, σ, θ) where θ ∈ [−1, 1] and µ c , µ s ∈ R d satisfy µ c ⊥ µ s and with samples generated according to: P θ (y) = Unif{−1, 1}, P θ (x|y) = N (yµ c + yθµ s , σ 2 I). (1) Our goal is to find a (linear) classifier that predicts y from x and is robust to the value of θ (we discuss the specific robustness metric below). To do so, the classifier will need to have significant inner product with the "core" signal component µ c and be approximately orthogonal to the "spurious" component µ s . We focus on learning problems where we are given access to samples from two environments that share all their parameters other than θ, as we define next. We illustrate our setting with Figure 3 in Appendix A. Definition 2 (Linear Two Environment Problem). In a Linear Two Environment Problem we have datasets S 1 = {x (1) i , y (1) i } N1 i=1 and S 2 = {x (2) i , y (2) i } N2 i=1 of sizes N 1 , N 2 drawn from P θ1 and P θ2 respectively. A learning algorithm is a (possibly randomized) mapping from the tuple (S 1 , S 2 ) to a linear classifier w ∈ R d . We let S = {x i , y i } N i=1 denote that dataset pooled from S 1 and S 2 where N = N 1 + N 2 . Finally we let r c := µ c and r s := µ s . We study settings where θ 1 , θ 2 are fixed and d is large compared to N , i.e. the overparameterized regime. We refer to the two distributions P θe for e ∈ {1, 2} as "training environments", following [40,3]. In the context of Out-of-Distribution (OOD) generalization, environments correspond to different experimental conditions, e.g., collection of medical data in two hospitals. In a fairness context, we may think of these distributions as subpopulations (e.g., demographic groups). 1 While these are different applications that require specialized methods, the underlying formalism of solutions is often similar [see, e.g., 14, Table 1], where we wish to learn a classifier that in one way or another is invariant to the environment variable. Robust performance metric. An advantage of the simple model defined above is that many of the common invariance criteria all boil down to the same mathematical constraint: learning a classifier that is orthogonal to µ s , which induces a spurious correlation between the environment and the label. These include Equalized Odds [22], conditional distribution matching [31], calibration on multiple subsets of the data [23,60], Risk Extrapolation [28] and CVaR fairness [64]. In terms of predictive accuracy, the goal of learning a linear model that aligns with µ c (the invariant part of the data generating process for the label) and is orthogonal to µ s coincides with providing guarantees on the robust error, i.e. the error when data is generated with values of θ that are different from the θ 1 , θ 2 used to generate training data. 2 Definition 3 (Robust error). The robust error of a linear classifier w ∈ R d is: max θ∈[−1,1] θ (w), where θ (w) := E x,y∼P θ [sign( w, x ) = y].(2) Normalized margin. We study is whether algorithms that perfectly fit (i.e. interpolate) their training data can learn models with low robust error. Ideally, we would like to give a result on all classifiers that attain training error zero in terms of the 0-1 loss. However, the inherent discontinuity of this loss would make any such statement sensitive to instabilities and pathologies. For instance, if we do not limit the capacity of our models, we can turn any classifier into an interpolating one by adding "special cases" for the training points, yet intuitively this is not the type of interpolation that we would like to study. To avoid such issues, we replace the 0-1 loss with a common continuous surrogate, the normalize margin, and require it to be strictly positive. Definition 4 (Normalized margin). Let γ > 0, we say a classifier w ∈ R d separates the set S = {x i , y i } N i=1 with normalized margin γ if for every (x, y) ∈ S y i w, x i w > γ √ σ 2 d. The √ σ 2 d scaling of γ is roughly proportional to x under our data model in Equation (1), and keeps the value of γ comparable across growing values of d. Main Result Equipped with the necessary definitions, we now state and discuss our main result. Theorem 1. For any sample sizes N 1 , N 2 > 65, margin lower bound γ ≤ 1 4 √ N , target robust error > 0, and coefficients θ 1 = 1, θ 2 > − N1γ √ 288N2 , there exist parameters r c , r s > 0, d > N , and σ > 0 such that the following holds for the Linear Two Environment Problem (Definition 2) with these parameters. 1. Invariance is attainable. Algorithm 1 maps (S 1 , S 2 ) to a linear classifier w such that with probability at least 99 100 (over the draw S), the robust error of w is less than . 2. Interpolation is attainable. With probability at least 99 100 , the estimator w mean = N −1 i∈[N ] y i x i separates S with normalized margin (Definition 4) greater than 1 4 √ N . 3. Interpolation precludes invariance. Given µ c uniformly distributed on the sphere of radius r c and µ s uniformly distributed on a sphere of radius r s in the subspace orthogonal to µ c , let w be any classifier learned from (S 1 , S 2 ) as per Definition 2. If w separates S with normalized margin γ, then with probability at least 99 100 (over the draw of µ c , µ s , and the sample), the robust error of w is at least 1 2 . Theorem 1 shows that if a learning algorithm for overparameterized linear classifiers always separates its training data, then there exist natural settings for which the algorithm completely fails to learn a robust classifier, and will therefore fail on multiple other invariance and fairness objectives. Furthermore, in the same setting this failure is avoidable, as there exists an algorithm (that necessarily does not always separate its training data) which successfully learns an invariant classifier. This result has deep implications for theoreticians attempting to prove finite-sample invariant learning guarantees: it shows that-in the fundamental setting of linear classification-no interpolating algorithm can have guarantees as strong as non-interpolating algorithms such as Algorithm 1. Importantly, Theorem 1 requires interpolating invariant classifiers to exist-and shows that these classifiers are information-theoretically impossible to learn. In particular, the first part of the theorem implies that the Bayes optimal invariant classifier w = µ c has robust test error at most . Therefore, for all < 1 100N we have that µ c interpolates S with probability > 99 100 . Furthermore, a short calculation (see Appendix C.1) shows that (for r c , r s , d and σ satisfying Theorem 1) the normalized margin of µ c is Ω((N + √ N 2 /γ) − 1 2 ). However, we prove that-due to the high-dimensional nature of the problem-no algorithm can use (S 1 , S 2 ) to reliably distinguish the invariant interpolator from other interpolators with similar or larger margin. This learnability barrier strongly leverages our random choice of µ c , µ s , without which the (fixed) vector µ c would be a valid learning output. We establish Theorem 1 with three propositions, each corresponding to an enumerated claim in the theorem: (1) Proposition 2 (in §4) establishes that invariance is attainable, (2) Proposition 3 (Appendix C) establishes that interpolation is attainable, and (3) Proposition 1 (in §3) establishes that interpolation precludes invariance. We choose to begin with the latter proposition since it is the main conceptual and technical contribution of our paper. Conversely, Proposition 3 is an easy byproduct of the developments leading up to Proposition 1, and we defer it to the appendix. With Propositions 1, 2 and 3 in hand, the proof of Theorem 1 simply consists of choosing the free parameters in Theorem 1 (r c , r s , d and σ) based on these propositions such that all the claims in the theorem hold simultaneously. For convenience we take σ 2 = 1/d. Then (ignoring constant factors) we pick r 2 s ∝ 1 N and r 2 c ∝ r 2 s /(1+ √ N2 N1γ ) in order to satisfy requirements in Propositions 1 and 3. Finally, we take d to be sufficiently large so as to satisfy the remaining requirements, resulting in d ∝ max N 2 , N γ 2 N 2 1 r 2 c , (Q −1 ( )) 2 Nminr 4 c , 1 N 2 min r 4 c , where N min = min{N 1 , N 2 } and Q is the Gaussian tail function (see Appendix E for the full proof). We conclude this section with remarks on the range of parameters under which Theorem 1 holds. The impossibility results in Theorem 1 are strongest when N 2 is smaller than N 2 1 γ 2 . In particular, when N 2 ≤ N 2 1 γ 2 /288, our result holds for all θ 2 ∈ [−1, 1] and moreover the core and spurious signal strengths r c and r s can be chosen to be of the same order. The ratio N 2 /(N 2 1 γ 2 ) is small either when one group is under-represented (i.e., N 2 N 1 ) or when considering large margin classifiers (i.e., γ of the order 1/ √ N ). Moreover, unlike prior work on barriers to robustness [e.g., 50, 37], our result continue to hold even for balanced data and arbitrarily low margin, provided θ 2 is close to 0 and the core signal is sufficiently weaker than the spurious signal. Notably, the normalized margin γ can be arbitrarily small while the maximum achievable margin is always at least of the order of 1 √ N . Therefore, we believe that Theorem 1 essentially precludes any interpolating learning rule from being consistently invariant. Interpolating Models Cannot Be Invariant In this section we prove the third claim in Theorem 1: for essentially any nonzero value of the normalized margin γ, there are instances of the Linear Two Environment Problem (Definition 2) where with high probability, learning algorithms that return linear classifiers attaining normalized margin at least γ must incur a large robust error. The following proposition formalizes the claim; we sketch the proof below and provide a full derivation in Appendix B.3. Proposition 1. For σ = 1/ √ d, θ 1 = 1, there are universal constants c r ∈ (0, 1) and C d , C r ∈ (1, ∞), such that, for any target normalized γ, θ 2 > −N 1 γ/ √ 288N 2 , and failure probability δ ∈ (0, 1), if max{r 2 s , r 2 c } ≤ c r N , r 2 s r 2 c ≥ C r 1 + √ N 2 N 1 γ and (3) d ≥ C d N γ 2 N 2 1 r 2 c log 1 δ ,(4) then with probability at least 1 − δ over the drawing of µ c , µ s and (S 1 , S 2 ) as described in Theorem 1, anŷ w ∈ R d that is a measurable function of (S 1 , S 2 ) and separates the data with normalized margin larger than γ has robust error at least 0.5. Proof sketch. We begin by noting that for any fixed θ, the error of a linear classifier w is θ (w) = Q w, µ c + θ w, µ s σ w = Q w, µ c σ w 1 + θ w, µ s w, µ c ,(5) where Q(t) := P(N (0; 1) > t) is the Gaussian tail function. Consequently, when w, µ s / w, µ c ≥ 1 it is easy to see that θ (w) = 1/2 for some θ ∈ [−1, 1] and therefore the robust error is at least 1 2 ; we prove that w, µ s / w, µ c ≥ 1 indeed holds with high probability under the proposition's assumptions. Our proof has two key parts: (a) restricting the set of classifiers to the linear span of the data and (b) lower bounding the minimum value of w, µ s / w, µ c for classifier in that linear span. For the first part of the proof we use the spherical distribution of µ c and µ s and concentration of measure to show that (with high probability) any component of w chosen outside the linear span of {x i } i∈[N ] will have negligible effect on the predictions of the classifier. To explain this fact, let P ⊥ denote the projection operator to the orthogonal complement of the data, so that P ⊥ w is the component of w orthogonal to the data and P ⊥ w, µ c = w, P ⊥ µc P ⊥ µc P ⊥ µ c . Conditional on (S 1 , S 2 ) and the learning rule's random seed, the vector P ⊥ µ c / P ⊥ µ c is uniformly distributed on a unit sphere of dimension d − N while the vector w is deterministic. Assuming without loss of generality that w = 1, concentration of measure on the sphere implies that | w, P ⊥ µc P ⊥ µc | is (with high probability) bounded by roughly 1/ √ d, and therefore | P ⊥ w, µ c | is roughly of the order r c / √ d. For sufficiently large d (as required by the proposition), this inner product would be negligible, meaning that w, µ c is roughly the same as (I − P ⊥ )w, µ c , and (I − P ⊥ )w is in the span of the data. The same argument applies to µ s as well. In the second part of the proof, we consider classifiers of the form w = i∈[N ] β i y i x i (which parameterizes the linear span of the data) and minimize w, µ s / w, µ c over β ∈ R N subject to the constraint that w has normalize margin of at least γ. To do so, we first use concentration of measure to argue that it is sufficient to lower bound i∈[N1] β i subject to the margin constraint and w 2 ≤ 1, which is convex in β-we obtain this lower bound by analyzing the Lagrange dual of the problem of minimizing i∈[N1] β i subject to these constraints. Overall, we show a high-probability lower bound on w,µs w,µc that (for sufficiently high dimensions) scales roughly as r 2 s N1γ r 2 c √ N2 . For parameters satisfying Equation (3) we thus obtain w,µs w,µc ≥ 1, completing the proof. Implication for invariance-inducing algorithms. Our proof implies that any interpolating algorithm should fail at learning invariant classifiers. This alone does not necessarily imply that specific algorithms proposed in the literature for learning invariant classifiers fail, as they may not be interpolating. Yet our simulations in Section 5 show that several popular algorithms proposed for eliminating spurious features are indeed interpolating in the overparameterized regime. We also give a formal statement in Appendix G regarding the IRMv1 penalty [3], showing that it is biased toward large margins when applied to separable datasets. Our results may seem discouraging for the development of invariance-inducing techniques using overparameterized models. It is natural to ask what type of methods can provably learn such models, which is the topic of the next section. Stage 2: Return f v (x) = v 1 · w 1 + v 2 · w 2 , x that solves maximize (x,y)∈S post yf v (x) subject to v ∞ = 1 and f v ∈ F(S fine 1 , S fine 2 ) A Provably Invariant Overparameterized Estimator We now turn to propose and analyze an algorithm (Algorithm 1) that provably learns an overparametrized linear model with good robust accuracy in our setup. Our approach is a two-staged learning procedure that is conceptually similar to some recently proposed methods [46,58,27,36,29,67]. In Section 5 we validate our algorithm on simulations and on the Waterbirds dataset [49], but we leave a thorough empirical evaluation of the techniques described here to future work. Let us describe the operation of Algorithm 1. First, we evenly 3 split the data from each environment into the sets S train Crucially, the invariance penalty is only used in the second stage, in which we are no longer in the overparamterized regime since we are only fitting a two-dimensional classifier. In this way we overcome the negative result from Section 3. While our approach is general and can handle a variety of invariance notions (we discuss some of them in Appendix F), we analyze the algorithm under the Equal Opportunity (EOpp) criterion [22]. Namely, for a model f : R d → R we write: F(S fine 1 , S fine 2 ) = f :T 1 (f ) =T 2 (f ) , whereT e (f ) := 4 N e (x,y)∈S fine e :y=1 f (x). This is the empirical version of the constraint E P θ 1 [f (x)|y = 1] = E P θ 2 [f (x)|y = 1] . From a fairness perspective (e.g., thinking of a loan application), this constraint ensures that the "qualified" members (i.e., those with y = 1) of each group receive similar predictions, on average over the entire group. We now turn to providing conditions under which Algorithm 1 successfully learns an invariant predictor. The full proof for the following proposition can be found in section D.1 of the appendix. While we do not consider the following proposition very surprising, the fact that it gives a finite sample learning guarantee means it does not directly follow from existing work (discussed in §6 below) that mostly assume inifinite sample size. Robust Accuracy Robust Accuracy vs. d Proposition 2. Consider the Linear Two Environment Problem (Definition 2), and further suppose that |θ 1 − θ 2 | > 0.1. 4 There exist universal constants C p , C c , C s ∈ (1, ∞) such that the following holds for every target robust error > 0 and failure probability δ ∈ (0, 1). If N min := min{N 1 , N 2 } ≥ C p log(4/δ) for some C p ∈ (1, ∞), 5 r 2 s ≥ C s log 68 δ σ 2 √ d N min , r 2 c ≥ C c σ 2 log 68 δ max Q −1 ( ) d N min , √ d N min , r 2 s N min r 2 c ,(6) and d ≥ log 68 δ then, with probability at least 1 − δ over the draw of the training data and the split of the data between the two stages of learning, the robust error of the model returned by Algorithm 1 does not exceed . Proof sketch. Writing down the error of f v = v 1 · w 1 + v 2 · w 2 under P θ , it can be shown that to obtain the desired bound on the robust error of the classifier returned by Algorithm 1, we must upper bound the ratio (v 1 θ 1 + v 2 θ 2 ) µ s 2 + µ s , v 1n1 + v 2n2 (v 1 + v 2 ) µ c 2 + µ c , v 1n 1 + v 2n 2 , whenn e is the mean of Gaussian noise vectors, and v 1 and v 2 are the solutions to the optimization problem in Stage 2 of Algorithm 1. The terms involving inner-products with the noise terms are zero-mean and can be bounded using standard Gaussian concentration arguments. Therefore, the main effort of the proof is upper bounding v 1 θ 1 + v 2 θ 2 v 1 + v 2 · µ s 2 µ c 2 . To this end, we leverage the EOpp constraint. The population version of this constraint (corresponding to infinite N 1 and N 2 ) implies that v 1 θ 1 + v 2 θ 2 = 0. For finite sample sizes, we use standard Gaussian concentration and the Hanson-Wright inequality to show that the empirical EOpp constraint implies that |v 1 θ 1 + v 2 θ 2 | goes to zero as the sample sizes increase. Furthermore, we argue that |v 1 + v 2 | ≥ |θ 1 − θ 2 |/2, implying that-for appropriately large sample sizes-the above ratio indeed goes to zero. Empirical Validation The empirical observations that motivated this work can be found across the literature. We therefore focus our simulations on validating the theoretical results in our simplified model. We also evaluate Algorithm 1 on the Waterbirds dataset, where the goal is not to show state-of-the-art results, but rather to observe whether our claims hold beyond the Linear Two Environment Problem. Simluations Setup. We generate data as described in Theorem 1 with two environments where θ 1 = 1, θ 2 = 0 (see Figure 4 in the appendix for results of the same simulation when θ = − 1 2 ). We further fix r c = 1 and r c = 2, while N 1 = 800 and N 2 = 100. We then take growing values of d, while adjusting σ so that (r c /σ) 2 ∝ d/N . 6 For each value of d we train linear models with IRMv1 [3], VREx [28], MMD [31], CORAL [56], GroupDRO [49], implemented in the Domainbed package [20]. We also train a classifier with the logistic loss to minimize empirical error (ERM), and apply Algorithm 1 where the "post-processing" stage trains a linear model over the two-dimensional representation using the VREx penalty to induce invariance. We repeat this for 15 random seeds for drawing µ c , µ s and the training set. Evaluation and results. We compare the robust accuracy and the train set accuracy of the learned classifiers as d grows. First, we observe that all methods except for Algorithm 1 attain perfect accuracy for large enough d, i.e., they interpolate. We further note that while invariance-inducing methods give a desirable effect in low dimensions (the non-interpolating regime)-significantly improving the robust error over ERMthey become aligned with ERM in terms of robust accuracy as they go deeper into the interpolation regime (indeed, IRM essentially coincides with ERM for larger d). This is an expected outcome considering our findings in section 3, as we set here N 1 to be considerably larger than N 2 . Waterbirds Dataset We evaluate Algorithm 1 on the Waterbirds dataset [49], which has been previously used to evaluate the fairness and robustness of deep learning models. Setup. Waterbirds is a synthetically created dataset containing images of water-and land-birds overlaid on water and land background. Most of the waterbirds (landbirds) appear in water (land) backgrounds, with a smaller minority of waterbirds (landbirds) appearing on land (water) backgrounds. We set up the problem following previous work [50,58], where a logistic regression model is trained over random features extract from a fixed pre-trained ResNet-18. Please see Appendix H for details. Fairness. We use the image background type (water or land) as the sensitive feature, denoted A, and consider the fairness desiderata of Equal Opportunity [22], i.e., the false negative rate (FNR) should be similar for both groups. Towards this, we use the MinDiff penalty term [42]. The Evaluation. We compare the following methods: (1) Baseline: Learning a linear classifier w by minimizing L p + λ · L M , where L p is the standard binary cross entropy loss and L M is the MinDiff penalty; (2) Algorithm 1: In the first stage, we learn group-specific linear classifiers w 0 , w 1 by minimizing L p on the examples from A = 0 and A = 1, respectively. In the second stage we learn v ∈ R 2 by minimizing L p +λ·L M on examples the entire dataset, where the new representation of the data isX = [ w 1 , X , w 2 , X ] ∈ R 2 . 7 Results. Our main objective is to understand the effect of the fairness penalty. Toward this, for each method we compare both the test error and the test FNR gap when using either λ = 0 (no regularization) or λ = 5. The results are summarized in Figure 2. We can see that for the baseline approach, the fairness penalty successfully reduces the FNR gap when the classifier is not interpolating. However, as our negative result predicts and as previously reported in [58], the fairness penalty becomes ineffective in the interpolating regime (d ≥ 1000). On the other hand, for our two-phased algorithm, the addition of the fairness penalty reduces does reduce the FNR gap with an average relative improvement of 20%; crucially, this improvement is independent of d. Discussion and Additional Related Work In terms of formal results, most existing guarantees about invariant learning algorithms rely on the assumption that infinite training data is available [3,60,57,43,45,17]. Wang et al. [61], Chen et al. [11] analyze algorithms that bear resemblance to Algorithm 1 as they first project the data to a lower dimension and then fit a classifier. While these algorithms deal with more general assumptions in terms of the number of environments, number of spurious features, and noise distribution, the fact that their guarantees assume infinite data prevents them from being directly applicable to Algorithm 1. A few works with results on finite data are Ahuja et al. [2], Parulekar et al. [38] (and also Efroni et al. [19] who work on related problems in the context of sequential decision making) that characterize the sample complexity of methods that learn invariant classifiers. However, they do not analyze the overparameterized cases we are concerned with. Negative results about learning overparameterized robust classifiers have been shown for methods based on importance weighting [66] and max-margin classifiers [50]. Our result is more general, applying to any learning algorithm that separates the data with arbitrarily small margins, instead of focusing on maxmargin classifiers or specific algorithms. While we focus on the linear case, we believe it is instructive, as any reasonable method is expected to succeed in that case. Nonetheless, we believe our results can be extended to non-linear classifiers, and we leave this to future work. One take-away from our result is that while low training loss is generally desirable, overfitting to the point of interpolation can significantly hinder invariance-inducing objectives. This means one cannot assume a typical deep learning model with an added invariance penalty will indeed achieve any form of invariance; this fact also motivates using held-out data for imposing invariance, as in our Algorithm 1 as well as several other two-stage approaches mentioned above. Our work focuses theory underlying a wide array of algorithms, and there are natural follow-up topics to explore. One is to conduct a comprehensive empirical comparison of two-stage methods along with other methods that avoid interpolation, e.g., by subsampling data [24,10]. Another interesting topic is whether there are other model properties that are incompatible with interpolation. For instance, recent work [9] connects the generalization gap and calibration error on the training distribution. We also note that our focus in this paper was not on types of invariance that are satisfiable by using clever data augmentation techniques (e.g. invariance to image translation), or the design of special architectures (e.g. [13,30,35]). These methods carefully incorporate a-priori known invariances, and their empirical success when applied to large models may suggest that there are lessons to be learned for the type of invariant learning considered in our paper. These connections seem like an exciting avenue for future research. x > e = 1 e = 2 Y = 1 Y = 1 2 1 0 1 2 < u 2 , x > < u 1 , x > Projections of datapoints on c , s and Random Directions u 1 , u 2 for e {1, 2} Figure 3: Example of datasets sampled from two training envrionments, where we set θ 1 = 1, θ 2 = 0, N 1 = 800, N 2 = 100, r s = 2, r c = 1. Left and right plots show projections of training points on µ c , µ s ∈ R d , and on u 1 , u 2 drawn uniformly from the d-dimensional unit sphere, respectively. As we increase d there are many hyperplanes w ∈ R d that separate the data, for some w, µ c is much higher than w, µ s (i.e. their predictions are invariant) and for some the opposite may hold. We ask whether interpolating learning rules can find the former. A Setting and Helper Lemmas A.1 Notation U ∼ U(O(d)) µ c = U 1 · r c , µ s = U 2 · r s y i = Rad( 1 2 ), n i ∼ N (0, σ 2 I d ) ∀i ∈ [N ] x i = y i µ c + y i θ e µ s + n i ∀e, i ∈S e . The vectors E 1 , E 2 ∈ {0, 1} N are binary vectors where [E e ] i = 1 for i ∈S e and e ∈ {1, 2}, while 1 is the vector of length N whose entries equal 1. We also denote z i = x i y i for i ∈S and Z = [z 1 , . . . , z N ] ∈ R N ×d the matrix that stacks all these vectors. The i-th column of a matrix M is denoted by M i , s min (M), s max (M) are its smallest and largest singular values accordingly. The unit matrix of size n is denoted by I n and for convenience we denote the direction of any vector v asv := v v . Finally, for some vector of coefficients β ∈ R N , we will use the formŵ = i∈S β i y i x i + w ⊥ where w ⊥ is in the orthogonal complement of span({x i } i∈S ), to write any linear model (here normalized to unit norm). For convenience we will write our proofs for the case where θ 1 = 1 and σ 2 = d −1 , extensions to different settings of these parameters are straightforward but result in a more cumbersome notation. A.2 Operator Norms of Wishart Matrices We begin with stating the required events for our results and their occurrence with high-probability: Lemma 1. Consider the matrix G = Z − 1µ c − (E 1 + θ 2 E 2 )µ s . For any t > 0, with probability at least 1 − 6 exp(−t 2 /2) the following hold simultaneously: 1 − N d − t √ d ≤ s min (G ) ≤ s max (G ) ≤ 1 + N d + t √ d (8) Gµ c ≤ t N d µ c (9) Gµ s ≤ t N d µ s(10) Proof. G is a random Gaussian matrix with G i,j ∼ N (0, d −1 I N ). By concentration results for random Gaussian matrices [59,Cor. 5.35] we obtain that with probability at least 1 − 2 exp(−t 2 /2) Equation (8) holds. Next we note that Gµ c ∼ N (0, d −1 µ c 2 I N ) and similarly for Gµ s . The norm of a Gaussian random vector can be bounded for any t 2 > 0: P [ Gµ c ≥ t 2 ] ≤ 2 exp − dt 2 2 2N µ c 2 Setting t 2 = t N d µ c we get that with probability at least 1 − 2 exp(−t 2 /2) Equation (9) holds. Repeating the analogous derivation for Equation (10) and taking a union bound over the 3 events, we arrive at the desired result. Lemma 2. Conditioned on the events in Lemma 1 with parameter t ≥ 0, if √ N + t √ d + √ N ( µ c + µ s ) ≤ 1 2 ,(11) then ZZ − E[ZZ ] op ≤ 3 √ N + t √ d and 1 2 I N ZZ 2I N . We note that we already assume d N and µ c N −1/2 , hence the additional assumption introduced in the conditions of this lemma is regarding the size of µ s √ N 1 . Proof. Since GG ∼ W (d −1 I N , d) we have that E[GG ] = I N . Then from Equation (8) we can also obtain (1 − N d − t √ d ) 2 I n GG (1 + N d + t √ d ) 2 I n , which leads to: GG − E[GG ] op ≤ 1 + N d + t √ d 2 − 1. Combining this with Equation (9) and Equation (10) ZZ − E ZZ op ≤ GG − E GG op + Gµ c 1 op + Gµ s (E 1 + θ 2 E 2 ) op ≤ N d 2 √ N + t √ N + ( √ N + t) 2 √ N d + t √ N ( µ c + µ s ) ≤ √ N + t √ d 2 + √ N + t √ d + t √ N + t √ N ( µ c + µ s ) ≤ √ N + t √ d · 2.5, where the last transition follows from substituting Equation (11). To obtain the spectral bound on ZZ we have that Z = G + 1µ c + (E 1 + θ 2 E 2 )µ s . From Weyl's inequality for singular values: |s min (G + µ c 1 +µ s (E 1 + θ 2 E 2 ) ) − s min (G )| ≤ s max (µ c 1 + µ s (E 1 + θ 2 E 2 ) ) ≤ µ c √ N + µ s √ N . Taken together with Equation (8) and the assumption in Equation (11) we get: s min (Z ) ≥ s min (G ) − µ c √ N − µ s √ N ≥ 1 − 1 √ d √ N + t − µ c √ N − µ s √ N ≥ 1 2 . To prove that ZZ 2 we simply need to follow the same steps while taking notice that Weyl's inequality also holds for s max (G ). This will give us s max (Z ) ≤ 3/2 ≤ 2 from which the upper bound follows. A.3 Sufficiency of Linear Classifiers Spanned by Data Points We wish to bound ŵ ⊥ , µ c = r c ŵ ⊥ , U 1 . To this end let us take an orthonormal basis {v 1 , . . . , v N } and let these vectors form the columns of the orthogonal matrix V ∈ R d×N . Let P V be the orthogonal projection matrix on the columns of V . We first claim that conditioned on the data, the component of the mean vectors that is not spanned by the data is distributed uniformly. We may also write this equality as P(P V µ s , P V µ c , (I − P V )µ s , (I − P V )µ c | {z i } i∈S ) = P(P V Rµ s , P V Rµ c , (I − P V )Rµ s , (I − P V )Rµ c | {z i } i∈S ). The fact that R preserves {z i } i∈S implies that P V R = P V = RP V and therefore P(P V µ s , P V µ c , µ ⊥ s , µ ⊥ c | {z i } i∈S ) = P(P V µ s , P V µ c , Rµ ⊥ s , Rµ ⊥ c | {z i } i∈S ). Marginalizing P V µ s , P V µ c , we obtain that, conditional on the training data, the distribution of µ ⊥ s , µ ⊥ c , is invariant to rotations that preserve the training data. Therefore, the unit vectors in the directions of µ ⊥ s and µ ⊥ c must each be uniformly distributed on the sphere orthogonal to the training data, which has dimension d − N . Now we simply need to derive a bound on w ⊥ , µ s : Corollary 1. For any t > 0 as in Lemma 1, with probability at least 1 − 10 exp(−t 2 /2), all the events in Lemma 1 hold and additionally | w ⊥ , µ s | < µ s √ d − N t and | w ⊥ , µ c | < µ c √ d − N t.(12) Proof. Note that | w ⊥ , µ s | = w ⊥ , µ ⊥ s = µ ⊥ s w ⊥ w ⊥ w ⊥ , µ ⊥ s µ ⊥ s ≤ µ s w ⊥ w ⊥ , µ ⊥ s µ ⊥ s . Conditional on the training data and the algorithm's randomness, w ⊥ w ⊥ is a fixed unit vector in the subspace orthogonal to the training data (of dimension d − N ), while µ ⊥ s µ ⊥ s is a spherically uniform unit vector in that subspace. Therefore, standard concentration bounds [4, Lemma 2.2] imply that, for any t 2 > 0 P w ⊥ w ⊥ , µ ⊥ s µ ⊥ s ≥ t 2 ≤ 2 exp(−(d − N )t 2 2 /2). The claimed result follows by taking t 2 = t/ √ d − N , applying the same argument for µ c and taking a union bound. B Proofs of Main Result In this section, we provide the proof of Proposition 1, our main theoretical finding highlighting a fundamental limitation to the robustness of any interpolating classifier. Following the notation of Appendix A, we write a general unit-vector classifier asŵ = i∈S β i z i + w ⊥ , where z i = y i x i . As explained in the proof sketch at Section 3, in order to show a lower bound on robust accuracy, we show a lower bound on the spurious-to-core ratio w,µs w,µc or equivalently upper bound w,µs w,µc , which we can write as w, µ c w, µ s = ŵ, µ c ŵ, µ s = µ c 2 µ s 2 · 1 β + 1 µc 2 i∈S β i n i , µ c + w ⊥ , µ c (E 1 + θ 2 E 2 ) β + 1 µs 2 i∈S β i n i , µ s + w ⊥ , µ s .(13) We develop the lower bound -and prove Proposition 1 -in three steps, each corresponsding to a subsection below. First, we give a lower bound on (E 1 + θ 2 E 2 ) β using Lagrange duality (Lemma 4). Second, in Lemma 5, we bound the residual terms of the form 1 µ 2 i∈S β i n i , µ + w ⊥ , µ (for µ ∈ {µ c , µ s }) using concentration of measure arguments from Appendix A. Finally, we combine these two results with the conditions of Proposition 1 to conclude its proof. B.1 Lower bounding (E 1 + θ 2 E 2 ) β The crux of our proof is showing that the term (E 1 + θ 2 E 2 ) β, i.e., the sum of the contributions of elements from the first environment to w, must grow roughly as N 1 γ for any interpolating classifier. This will in turn imply a large spurious component in the classifier via manipulation of Equation (13). Lemma 4. Conditional on the events in Corollary 1 (with parameter t > 0), if Equation (11) holds and w has normalized margin at least γ, we have that (E 1 + θ 2 E 2 ) β ≥ 1 2 (N 1 + [θ 2 ] + N 2 )γ − 2N 2 N 1 µ c 2 − √ 18N · √ N + t √ d − 8N 2 [−θ 2 ] + ,(14) where [z] + = max{x, 0} denotes the positive part of x. Proof of Lemma 4. Our strategy for bounding (E 1 + θ 2 E 2 ) β begins with writing down the smallest value it can reach for any unit-norm classifierŵ with normalized margin at least γ. Recalling thatŵ = Z β + w ⊥ (for w ⊥ such that Zw ⊥ = 0), the smallest possible value of E 1 β is the solution to the following optimization problem: min β∈R N ,w ⊥ ∈ker(Z) (E 1 + θ 2 E 2 ) β (15) subject to Z β + w ⊥ , y i x i ≥ γ ∀i ∈ [N ] Z β + w ⊥ = 1. Since z i = y i x i and Zw ⊥ = 0, the first constraint is equivalent to the vector inequality ZZ β ≥ γ1, and the second constraint is equivalent to β ZZ β = 1 − w ⊥ 2 . Relaxing the second constraint, the smallest value of (E 1 + θ 2 E 2 ) β is bounded from below by the solution to: min β∈R N β (E 1 + θ 2 E 2 )(16) subject to ZZ β ≥ γ1 β ZZ β ≤ 1. We now treat separately the two cases where θ 2 ≤ 0 and θ 2 > 0. The case where θ 2 ≤ 0. Take Lagrange multipliers λ ∈ R N + and ν ≥ 0, from strong duality the above equals: max λ∈R N + ,ν≥0 min β∈R N β (E 1 + θ 2 E 2 ) + λ (1γ − ZZ β) + 1 2 ν(β ZZ β − 1) Optimizing the quadratic form over β, the above becomes: max λ∈R N + ,ν≥0 λ 1γ − 1 2 ν − 1 2 E 1 + θ 2 E 2 − ZZ λ νZZ −1 E 1 + θ 2 E 2 − ZZ λ Maximizing over ν this becomes: max λ∈R N + λ 1γ − (E 1 + θ 2 E 2 − ZZ λ) (ZZ ) −1 (E 1 + θ 2 E 2 − ZZ λ) := max λ∈R N + L(λ)(17) Thus, (E 1 +θ 2 E 2 ) β is lower bounded by L(λ), for any λ ∈ R N + . Taking λ = αE 1 for α = 1 + µ c 2 + µ s 2 N 1 −1 , we obtain: L(λ) = N 1 γα − (E 1 + θ 2 E 2 − αZZ E 1 ) (ZZ ) −1 (E 1 + θ 2 E 2 − αZZ E 1 ) ≥ N 1 γα − √ 2 E 1 + θ 2 E 2 − αZZ E 1 ≥ N 1 γα − √ 2 E 1 − αZZ E 1 − 2N 2 |θ 2 | = N 1 γα − √ 2 I N − α E ZZ + ZZ − E ZZ E 1 − 2N 2 |θ 2 | ≥ N 1 γα − √ 2 I N − αE ZZ E 1 − √ 2 α ZZ − E ZZ E 1 − 2N 2 |θ 2 | Here, the first inequality is from our assumption that Equation (11) L(λ) ≥N 1 γα − √ 2 I N − αE ZZ E 1 − α − α 18N 1 · √ N + t √ d − 2N 2 |θ 2 | ≥N 1 γα − √ 2 I N − αE ZZ E 1 − α − α √ 18N · √ N + t √ d − 2N 2 |θ 2 |. Let us break down the second term in the bound above: I N − αE ZZ E 1 = 1 − α − αN 1 µ s 2 E 1 − αN 1 µ c 2 1 = 1 − α − αN 1 µ s 2 E 1 − αN 1 µ c 2 (E 1 + E 2 ) = (1 − α (1 + N 1 ( µ s 2 + µ c 2 )) 2 N 1 + α 2 N 2 1 µ c 4 N 2 = αN 1 µ c 2 N 2 , where the final equality used α 1 + N 1 ( µ s 2 + µ c 2 = 1. Overall, we get: β (E 1 + θ 2 E 2 ) ≥ L(λ) ≥ α N 1 γ − 2N 2 N 1 µ c 2 − √ 18N · √ N + t √ d − 2N 2 |θ 2 |.(18) The proof is complete by noting that α ≥ 1/2 due to Equation (11). The case where θ 2 > 0. Before introducing Lagrange multipliers, we revisit Equation (16) and this time we first bound the optimum from below as min β∈C(Z,γ) β (E 1 + θ 2 E 2 ) ≥ θ 2 min β∈C(Z,γ) β (E 1 + E 2 ) + (1 − θ 2 ) · min β∈C(Z,γ) β E 1 , where we used C(Z, γ) as a shorthand for the constraints in Equation (16). By our derivation for the case of θ 2 ≤ 0, the second term on the right hand side is readily bounded by the term in Equation (18) with θ 2 = 0. We are left with bounding the first term, which turns out to be simpler since E 1 + E 2 = 1. We repeat the process of taking the Lagrangian up until Equation (17), and now choose λ = α1 for α = 1 + µ c 2 + µ s 2 N −1 . For completeness, let us rewrite the lower bound on the Lagrangian with these slight changes: L(λ) = N γα − (1 − αZZ 1) (ZZ ) −1 (1 − αZZ 1) ≥ N γα − √ 2 1 − αZZ 1 = N γα − √ 2 I N − α E ZZ + ZZ − E ZZ 1 ≥ N γα − √ 2 I N − αE ZZ 1 − √ 2 α ZZ − E ZZ 1 Using again the bound on ZZ − E ZZ op , the above bound becomes L(λ) ≥ N γα − √ 2 I N − αE ZZ 1 − α − α √ 18N · √ N + t √ d . This time the chosen value for α makes the second term vanish, since I N − αE ZZ 1 = 1 − α − αN µ s 2 + µ c 2 1 = 0. Equation (11) again tells us that α > 1/2 which leads us to the bound: β (E 1 + θ 2 E 2 ) ≥ 1 2 γ(θ 2 N + (1 − θ 2 )N 1 ) − (1 − θ 2 ) 2N 2 N 1 µ c 2 − √ 18 θ 2 √ N + (1 − θ 2 ) N 1 · √ N + t √ d ≥ 1 2 γ(θ 2 N 2 + N 1 ) − 2N 2 N 1 µ c 2 − √ 18N · √ N + t √ d . Combining the two cases for negative and positive θ 2 , we arrive at the desired bound in Equation (14). B.2 Controlling residual terms We now provide a bound on the terms in Equation (13) associated with quantities that vanish as the problem dimension grows. Lemma 5. Conditioned on all the events in Corollary 1 with parameter t > 0 (which happen with probability at least 1 − 10 exp(−t 2 /2)) and the additional condition of Lemma 2, we have for µ ∈ {µ c , µ s }: 1 µ 2 i∈S β i n i , µ + w ⊥ , µ ≤ 3t µ N d − N(19) Proof. We prove the claim for µ s ; the proof for µ c is analogous. Recall the random matrix G = Z − 1µ c − E 1 µ s ∈ R N ×d from Lemma 1. From Equation (10) we get that Gµ s ≤ t N d µ s and then: i∈S β i n i , µ s = β Gµ s ≤ β Gµ s ≤ t β N d µ s . To eliminate β from this bound, we use ZZ 1 2 I N due to Lemma 2 to write 1 √ 2 β ≤ β ZZ β ≤ β Z Zβ + w ⊥ 2 = ŵ = 1. Finally, we use Equation (12) from Corollary 1 to bound | w ⊥ , µ |. B.3 Proof of Proposition 1 Proposition 1. There are universal constants c r ∈ (0, 1) and C d , C r ∈ (1, ∞), such that, for any target normalized γ, θ 2 such that θ 2 > −N 1 γ/ √ 288N 2 , and failure probability δ ∈ (0, 1), if max{r 2 s , r 2 c } ≤ c r N , r 2 s r 2 c ≥ C r 1 + √ N 2 (N 1 + [θ 2 ] + N 2 )γ and d ≥ C d N γ 2 N 2 1 r 2 c log 1 δ , then with probability at least 1 − δ over the drawing of µ c , µ s and (S 1 , S 2 ) as described in Theorem 1, anŷ w ∈ R d that is a measurable function of (S 1 , S 2 ) and separates the data with normalized margin larger than γ has robust error at least 0.5. Proof of Proposition 1. Let t 10 log 10 δ ≥ 2 log 10 δ , so that the events described in the previous lemmas and corollaries all hold with probability at least 1 − δ. Note that for c r ≤ 1/64 we have (20) and (since γ ≤ 1 √ N ( µ c + µ s ) ≤ 1 44 √ N ) d ≥ C d 10 1 γ 2 N t 2 N 2 1 µ c 2 ≥ C d 10c r N t 2 N 1 γ 2 ≥ 16C d 10c r N 2 t 2 N 1 N ≥ 6 4 C d N t 2 . Consequently, for C d ≥ 1 √ N + t √ d ≤ 2 1 64C d ≤ 1 4 .(21) Combining Equations (20) and (21), we see that the condition in Equation (11) holds. Therefore, we may apply Lemma 4; we now argue that under the assumptions of Proposition 1 the lower bound on (E 1 + θ 2 E 2 ) β simplifies to a constant multiple of N 1 γ. First, taking c r ≤ 1/9 and C r ≥ 1, we have 2N 2 N 1 µ c 2 ≤ √ 2N 2 N 1 µ s 2 C r 1 + √ N2 N1γ ≤ N 1 γ √ 2N 1 µ s 2 C r ≤ N 1 γ √ 2c r C r ≤ 1 6 N 1 γ. Second, using again c r ≤ 1/64 and taking C d ≥ 180, √ 18 [θ 2 ] + √ N + (1 − [θ 2 ] + ) N 1 √ N + t √ d ≤ N 1 γ √ 18 C d /10 √ N + t t √ N √ N µ c ≤ 1 6 N 1 γ. Finally, due to our condition on θ 2 in the proposition statement, we have √ 8N 2 [−θ 2 ] + ≤ N 1 γ/6. Substituting all these into Equation (14), we conclude that under our assumptions (E 1 + θ 2 E 2 ) β ≥ 1 4 (N 1 + [θ 2 ] + N 2 )γ. Next, we combine the lower bound on (E 1 + θ 2 E 2 ) β with Lemma 5 to handle the denominator and numerator in the RHS of Equation (13). Beginning with the numerator, we have 1 β + 1 r 2 c   i∈S β i n i , µ c + w ⊥ , µ c   ≤ (E 1 + θ 2 E 2 ) β + (1 − θ 2 ) E 2 β + 3t r c N d − N . As argued in the proof of Lemma 5, we have β ≤ √ 2 and therefore (1−θ 2 ) E 2 β ≤ √ 8N 2 . Substituting again our assumptions d (which imply d > 2N ), and taking C d ≥ 64 · 180, we have 3t r c N d − N ≤ √ 18t µ c √ d ≤ N 1 γ 180 C d ≤ 1 8 N 1 γ. For the denominator, noting µ c ≤ µ s by our assumption, we may similarly write (E 1 + θ 2 E 2 ) β + 1 µ s 2   i∈S β i n i , µ s + w ⊥ , µ s   ≥ (E 1 + θ 2 E 2 ) β − 1 8 N 1 γ. Consequently (since (E 1 +θ 2 E 2 ) β ≥ 1 4 N 1 γ), we have that the denominator is nonnegative. (If the numerator is not positive, w will have error greater than 1/2 for θ = 0). Substituting back to Equation (13) and using the lower bound ( E 1 + θ 2 E 2 ) β ≥ 1 4 N 1 γ, we get w, µ c w, µ s µ s 2 µ c 2 ≤ (E 1 + θ 2 E 2 ) β + √ 8N 2 + 1 8 N 1 γ (E 1 + θ 2 E 2 ) β − 1 8 N 1 γ ≤ 1 4 (N 1 + [θ 2 ] + N 2 ) γ + √ 8N 2 + 1 8 N 1 γ 1 4 (N 1 + [θ 2 ] + N 2 ) γ − 1 8 N 1 γ ≤ 3 + √ 512N 2 (N 1 + [θ 2 ] + N 2 ) γ . Therefore, for C r ≥ 32 we have w,µs w,µc ≥ 1 as required. Since the error of classifier w in environment with parameter θ is Q w, µ c σ w 1 + θ w, µ s w, µ c , (where Q(t) := P(N (0; 1) > t) is the Gaussian tail function), the fact that w,µs w,µc ≥ 1 implies that there exists θ ∈ [−1, 1] for which the error is Q(0) = 0.5, implying the stated bound on the robust error. C Lower Bounds On the Achievable Margin We now argue that, in our model, a simple signed-sample-mean estimator interpolates the data with normalized margin scaling as 1/ √ N . This fact establishes the first part of Theorem 1. Proposition 3. There exist universal constants c r , C d > 0 such that, in the Linear Two Environment Problem with parameters N 1 , N 2 , d > 0, µ c , µ s ∈ R d , θ 1 , θ 2 ∈ [−1, 1] and σ 2 = 1/d, for any δ ∈ (0, 1/2) if max{r c , r s } ≤ c r N , θ 1 N 1 ≥ −θ 2 N 2 and d ≥ C d N 2 log 1 δ then with probability at least 1 − δ, the signed-sample-mean estimator w mean = 1 N N i=1 y i x i obtains normal- ized margin of at least 1 √ 8N . Proof. Using the notation defined in the beginning of Appendix A, we note that w mean = 1 N Z 1 and (for σ 2 d = 1) its normalized margin is min i∈[N ] y i x i , w mean w mean = min i∈[N ] [Zw mean ] i w mean = min i∈[N ] [ZZ 1] i Z 1 . Substituting the assumed bounds on d and µ c , µ s into Lemma 2 (with t = 8 log 1 δ ≥ 2 log 6 δ ), it is easy to verify that for sufficiently small c r and sufficiently large C d , the condition in Equation (11) holds, and therefore ZZ − EZZ op ≤ 3 √ N + t √ d ≤ 1 √ 4N , with the final inequality following by choosing C d sufficiently large. Lemma 2 then also implies that ZZ 2I N . Noting that EZZ = I N + µ c 2 11 + µ s 2 (θ 1 E 1 + θ 2 E 2 )(θ 1 E 1 + θ 2 E 2 ) , we have that, for all i ∈ [N ], [ZZ 1] i ≥ [EZZ 1] i − ZZ − EZZ op 1 ≥ 1 − 1 √ 4N 1 = 1 2 . Moreover, ZZ 2I N implies that Z 1 = √ 1 ZZ 1 ≤ 2 1 = 2 √ N . Combining the above two displays yields the claimed margin bound. C.1 Margin for invariant classifiers We now lower bound the margin achieved by the invariant classifier w = µ c . To that end, note that y i µ c , x i / µ c ∼ N ( µ c ; σ 2 ) for all i ∈S. Therefore, taking σ 2 = 1/d, with probability at least 1 − δ we have, for all i ∈S form the proof of Theorem 1 (see Appendix E), we obtain that, with probability at least 1−δ, the normalized margin of µ c is y i µ c , x i µ c √ σ 2 d ≥ µ c − 1 √ d Q −1 δ N ≥ µ c − 2 log N δ d .Ω( µ c ) = Ω     1 N 1 + √ N2 N1γ     = Ω min 1 √ N , γ N 2 . In addition, letting n i ∼ N (0; σ 2 I d ), we may also consider the invariant classifier w = µ c + 1 N N i=1 n i = 1 N e∈{1,2} (x,y)∈Si (y i x i − θ e µ s ). The proof of Proposition 3 (with θ 1 = θ 2 = 0) shows that, under the assumptions of that proposition, the classifier w defined above attains margin of 1 4 √ N with high probability. We emphasize once more that, while the discussion above shows the existence of invariant classifiers with good margin, our main results proves that these classifiers may be unlearnable from the finite samples S 1 and S 2 . To show this existence numerically too, we run our simulations from Section 5 and add a model that is fitted on the features where µ s is removed (i.e. the new datapoints are x i − θ e µ s for each point x i ∈ S e ). We observe in Figure 5 that for a sufficiently large dimension the model is both interpolating and has high robust accuracy, demonstrating the existence discussed above. We emphasize again that this model cannot necessarily be learned by an algorithm that receives the original (S 1 , S 2 ) (before the removal of µ s ). D Analysis of Algorithm 1 The proof that Algorithm 1 indeed achieves a non-trivial robust error will require some definitions and more mild assumptions which we now turn to describe. Definitions. Denote the first-stage training set indices by S, where |S| = N and second stage "fine-tuning" set by |D| = M . Let us denote: n e = 1 N e i∈Se n i ,m e = 1 M e i∈De n i ,m e,1 = 1 M e,1 i∈De,1 n i . • Return the solution: v ∈ {v pos , v neg } which has the higher score, where the score is: v * ∈ arg max v∈{vpos,vneg} i∈D v 1 w 1 + v 2 w 2 , y i x i We first analyze the two possible solution v pos and v neg and show that their coordinates cannot be negations of each other. Intuitively, in an ideal scenario with infinite data, the EOpp constraint will enforce v 1 θ 1 = −v 2 θ 2 . Then v 1 = −v 2 is only possible if θ 1 = θ 2 , which we assume is not the case (if it is, we cannot identify the spurious correlation from data). The assumption of a fixed ∆ > 0, will let us show that indeed with high probability v 1 = −v 2 does not occur. Corollary 3 will show that they occur with the desired probability in our statement. Let us incorporate these events into the EOpp constraint. We group the items multiplied by v 1 and those multiplied by v 2 : −v * 1 θ 1 µ s 2 ∆ + n 1 , µ s ∆ + m 1,1 −m 2,1 , µ c + θ 1 µ s +n 1 = v * 2 θ 2 µ s 2 ∆ + n 2 , µ s ∆ + m 2,1 −m 1,1 , µ c + θ 2 µ s +n 2 Let us denote for convenience (where we drop the dependence on parameters in the notation): a = µ s −2 ∆ n 1 , µ s + ∆ −1 m 1,1 −m 2,1 , µ c + θ 1 µ s +n 1 b = µ s −2 ∆ n 2 , µ s + ∆ −1 m 2,1 −m 1,1 , µ c + θ 2 µ s +n 2 Now the EOpp constraint can be written as −v * 1 µ s 2 ∆ (θ 1 + a) = v * 2 µ s 2 ∆ (θ 2 + b). Plugging in Equation (23) and Equation (24), we see that max{|a|, |b|} ≤ r. Assume that |θ 1 +b| ≥ |θ 2 +a|, and note that since v * ∞ = 1 we have that |v * 1 | = 1 (the proof for the other case is analogous). 8 We note that by definition ∆ ≤ 2θ max , hence if v * 2 = 0 we have |v * 1 + v * 2 | = 1 ≥ ∆ 2θmax and our claim holds. Otherwise, we can write: |v * 1 + v * 2 | = 1 − θ 2 + b θ 1 + a = ∆ + a − b θ 1 + a ≥ ∆ − 2r θ max + r = ∆ − 2 ∆θmax ∆+4θmax θ max + ∆θmax ∆+4θmax = ∆ (∆ + 4θ max − 2θ max ) θ max (∆ + 4θ max + ∆) = ∆ 2θ max ≥ ∆ 2 The result above will be useful for proving the rest of our claims towards the performance guarantees of the algorithm. We first show that the retrieved solution is the one that is positively aligned with µ c . Lemma 8. With probability at least 1 − 34 exp(−t 2 ) , between the two solutions considered at the second stage of our algorithm, the one with v 1 + v 2 ≥ 0 achieves a higher score. Proof. Let's write down the score on environment e ∈ {1, 2} in detail: i∈De w x i y i =(v 1 + v 2 ) µ c 2 + µ c , v 1n1 + v 2n2 +(25)(v 1 θ 1 + v 2 θ 2 )θ e µ s 2 + µ s , θ e (v 1n1 + v 2n2 ) + m e , (v 1 + v 2 )µ c + (θ 1 v 1 + θ 2 v 2 )µ s + v 1n1 + v 2n2 8 In the case where |θ 2 + a| ≥ |θ 1 + b| then |v * 2 | = 1 would hold. We will bound all the items other than (v 1 + v 2 ) µ s 2 with concentration inequalities, and for the second line also use the EOpp constraint. Regrouping items in Equation (22) we have: (v 1 θ 1 + v 2 θ 2 ) µ s 2 + µ s , v 1n1 + v 2n2 · ∆ = | 2 (v) − 1 (v)| In Corollary 3 we will prove that with probability at least 1 − 34 exp(−t 2 ) , it holds that | 2 (v) − 1 (v)| ≤ ∆ 6 |v 1 + v 2 | · µ c 2 . Combined with |θ e | < 1, we get that the magnitude of the terms in the second line of Equation (25) is bounded by 1 6 |v 1 + v 2 | · µ c 2 . We will also show in Corollary 3 that the other two terms in Equation (25) besides (v 1 + v 2 ) µ c 2 , are bounded by 1 6 |v 1 + v 2 | · µ c 2 . Hence we have for some b such that |b| ≤ 1 2 |(v 1 + v 2 )| · µ c 2 that: i∈De w x i y i = (v 1 + v 2 ) µ c 2 + b We note that the score in the algorithm is a weighted average of the scores over the training environments, yet the derivation above holds regardless of e. That is, θ e did not play a role in the derivation other than the assumption that its magnitude is smaller than 1. Hence it is clear that the solution v * = v pos will be chosen over v neg . Once we have characterized our returned solution, it is left to show its guaranteed performance over all environments θ ∈ [−1, 1]. We can draw a similar argument to Lemma 8 to reason about the expected score obtained in each environment. Lemma 9. Let t > 0 and consider the retrieved solution v * . With probability at least 1 − 34 exp(−t 2 ) , the expected score of v * over any environment corresponding to θ ∈ [−1, 1] is larger than ∆ 3 µ c 2 . Proof. The expected score can be written same as in Equation (25), except we can drop the last item since it has expected value 0. We let θ ∈ [−1, 1] and write: E x,y∼P θ w xy =(v * 1 + v * 2 ) µ c 2 + µ c , v * 1n1 + v * 2n2 + (v * 1 θ 1 + v * 2 θ 2 )θ µ s 2 + µ s , θ (v * 1n1 + v * 2n2 ) ≥ 2 3 (v * 1 + v * 2 ) µ c 2 . The inequality follows from the arguments already stated in Lemma 8, where the second and third items in the above expression have magnitude at most 1 6 (v * 1 + v * 2 ) µ c 2 . Now it is left to conclude that (v * 1 + v * 2 ) ≥ ∆ 2 , which is a direct consequence of Lemma 7 and Lemma 8. D.1 Proof of Proposition 2 Now we are in place to prove the guarantee given in the main paper on the robust error of the model returned by the algorithm. We will restate it here with compatible notation to the earlier parts of this section which slightly differ from those in the main paper (e.g. by incorporating ∆). We also note that to obtain the statement in the main paper we should eliminate the dependence of Assumption 1 on M e,1 . We do this by assuming that our algorithm draws M e as half of the original dataset for environment e. Then we have that P(M e,1 ≤ N min /8) is bounded by the cumulative probability of a Binomial variable with k = N min /8 successes and at least N min trials. This may be bounded with a Hoeffding bound by 1 − 2 exp( 1 2 N min ) and with a union bound over the two environments. To absorb this into our failure probability we require N min > c eo log(1/δ), leading to this added constraint in the main paper. Q w, µ c + θµ s σ w The nominator of this expression is simply the expected score from Lemma 9, which we already proved is at least ∆ 3 µ c 2 . Then we need to bound w from above to get a bound on the robust accuracy. According to Corollary 3, if we denote N min = min{N 1 , N 2 }, this upper bound can be taken as 5t σ 2 d/N min . We plug this in to get: w, µ c + θµ s σ w ≥ ∆ 15t µ c 2 1 σ 2 N min d Since Q is a monotonically decreasing function, if µ c 2 ≥ tQ −1 ( ) 15 ∆ σ 2 d Nmin our model achieves the desired performance. D.2 Required Concentration Bounds To conclude the proof we now show all the concentration results used in the above derivation. Note that v * is determined by all the other random factors in the problem, hence we should be careful when using them in our bounds. We will only use the fact that v * ∞ = 1 and hence v * 1 ≤ 2. To bound the inner product of noise vectors, we use [48, Theorem 1.1]: Theorem 2. (Hanson-Wright inequality). Let X = (X 1 , . . . , X n ) ∈ R n be a random vector with independent components X i which satisfy EX i = 0 and X i ψ2 ≤ K. Let A be an n × n matrix. Then, for every t ≥ 0, P X AX − E[X AX | > t ≤ 2 exp −c min t 2 K 4 A 2 HS , t K 2 A We can apply this theorem to get the following result. Proof. We take X as the concatenation ofn e andm e , then A is set such that X AX = n e ,m e (e.g. Setting t 2 = r∆ 4 µ s 2 or t 2 = ∆ 2 288 µ c 2 we will get that: P | n e ,m e ,1 | ≥ min r∆ 4 µ s 2 , ∆ 2 288 µ c 2 ≤ 2 exp −c M 2 e ,1 σ 4 d min (r∆) 2 16 µ s 4 , ∆ 4 288 2 µ c 4 Hence we require µ c 2 ≥ t · c · (M e ,1 ∆ 2 ) −1 · (288σ 2 √ d) and µ s 2 ≥ t · c · (M e ,1 r∆) −1 · (4σ 2 √ d) for Equation (31) to hold. For Equation (33) we can get in a similar manner that it holds in case that µ c 2 ≥ t · c · (M e ∆) −1 (48σ 2 √ d). The probability for all the events listed so far to occur is at last 1 − 32 exp −t 2 /2 . Finally, for Equation (34) we simply use the bound on a norm of Gaussian vector: P ( n e ≥ t 2 ) ≤ 2 exp − t 2 2 N e 2σ 2 d Plugging in t 2σ 2 d Ne we arrive at the desired result with a final union bound that give the overall probability of at least 1 − 34 exp −t 2 /2 . We now use the bounds above to write down the specific bounds on expressions that we used during proof. Corollary 3. Conditioned on all the events in Lemma 10, we have for e ∈ {1, 2} that: These are the events required for Lemma 7, hence from now on we can now assume that: ∆ 6 |v 1 + v 2 | · µ c 2 ≥ | 2 (v) − 1 (v)|(35)1 6 |v 1 + v 2 | · µ c 2 ≥ | µ c , v 1n1 + v 2n2 |(36)1 6 |v 1 + v 2 | · µ c 2 ≥ | m e , (v 1 + v 2 )µ c + (θ 1 v 1 + θ 2 v 2 )µ s + v 1n1 + v 2n2 |(|v 1 + v 2 | ≥ ∆ 2 = ∆ 4 · 2 ≥ ∆ 4 v 1 Now we can combine with Equation (28) to prove Equation (36): µ c , v 1n1 + v 2n2 ≤ e |v e | · | µ c ,n e | ≤ v 1 ∆ 24 µ c 2 ≤ 1 6 |v 1 + v 2 | · µ c 2 Next we prove Equation (37) in a similar manner using Equation (32) and Equation (33): | m e ,(v 1 + v 2 )µ c + (θ 1 v 1 + θ 2 v 2 )µ s + v 1n1 + v 2n2 | ≤ e |v e | · (| m e , µ c + θ e µ s | + | m e ,n e |) ≤ v 1 · 2 · 1 48 ∆ µ c 2 ≤ 1 6 |v 1 + v 2 | · µ c 2 E Proof of Theorem 1 Proof of Theorem 1. Our proof simply consists of choosing the free parameters in Theorem 1 (r c , r s , d and σ) based on Propositions 1, 2 and 3 such that all the claims in the theorem hold simultaneously. Keeping in line with the setting of Propositions 1 and 3, we take σ 2 = 1/d. Next, our strategy is to pick r s and r c so as to satisfy the requirements of Propositions 1 and 3, and then pick a sufficiently large d so that the requirements of Proposition 2 hold as well. Throughout, we set δ = 99/100 so as to meet the failure probability requirement stated in the theorem; it is straightforward to adjust the proof to guarantee lower error probabilities. Starting with the value of r s , we let r 2 s = min{c n , c n } N where the parameters c n , c m and c n are as given by Propositions 1 and 3, respectively. Next, we pick r c to be r 2 c = r 2 s C r 1 + √ N2 N1γ = min{c n , c n } C r N 1 + √ N2 N1γ with C r from Proposition 1 (this setting guarantees r c ≤ r s as C r ≥ 1). Thus, we have satisfied the requirements in Equation (3) Using r s ≥ r c and r 2 s ≤ 1 Nmin , the above two displays simplify to d ≥ max{C c , C s } 2 log 1 δ N min r 4 c max (Q −1 ( )) 2 , 1 N min . Therefore, taking d = max{C d , C d , C 2 s , C 2 c } max N 2 , N γ 2 N 2 1 r 2 c , (Q −1 ( )) 2 N min r 4 c , 1 N 2 min r 4 c log 1 δ fulfills all the requirements and completes the proof. F Definitions of Invariance and Their Manifestation In Our Model In section 4 we show that the Equalized Odds principle in our setting reduces to the demand that w, µ s = 0. Here we provide short derivations that show this is also the case for some other invariance principles from the literature. We will show this in the population setting, that is in expectation over the training data. We also assume that θ 1 = θ 2 . Calibration over environments [60] Assume σ( w, x ) is a probabilistic classifier with some invertible function σ : R → [0, 1] such as a sigmoid, that maps the output of the linear function to a probability that y = 1. Calibration can be written as the condition that: P θ (y = 1 | σ( w, x − b) =p) =p ∀p ∈ [0, 1]. Calibration on training environments in our setting then requires that this holds simultaneously for P θ1 and P θ2 . We can write the conditional probability of y on the prediction (when the prior over y is uniform) as: P θe (y = 1 | w, x − b = α) = exp (α− w,µc+θ1µs +b) 2 2σ 2 w 2 exp (α− w,µc+θ1µs +b) 2 2σ 2 w 2 + exp (α+ w,µc+θ1µs +b) 2 2σ 2 w 2 Now it is easy to see that if the classifier is calibrated across environments, we must have equality in the log-odds ratio for the above with e = 1 and e = 2 and all α ∈ R: (α − w, µ c + θ 1 µ s + b) 2 2σ 2 w 2 − (α + w, µ c + θ 1 µ s + b) 2 2σ 2 w 2 = (α − w, µ c + θ 2 µ s + b) 2 2σ 2 w 2 − (α + w, µ c + θ 2 µ s + b) 2 2σ 2 w 2 . After dropping all the terms that cancel out in the subtractions we arrive at: w, µ c + θ 1 µ s = w, µ c + θ 2 µ s . Clearly this holds if and only if w, µ s = 0, hence calibration on both environments entails invariance in the context of the data generating process of Definition 2. Conditional Feature Matching [31, 57] Treating the environment index as a random variable, the conditional independence relation w, x ⊥ ⊥ e | y is a popular invariance criterion in the literature. Other works besides the ones mentioned in the title of this paragraph have used this, like the Equalized Odds criterion [22]. This independence is usually enforced w.r.t available training distributions, hence in our case w.r.t P θ1 , P θ2 . Writing this down we can see that: P θe ( w, x | y = 1) = N ( w, µ c + θ e µ s , w 2 σ 2 I). Hence requiring conditional independence in the sense of P θ1 ( w, x | y = 1) = P θ2 ( w, x | y = 1) means we need to have equality of the expectations, i.e. w, µ c + θ 1 µ s = w, µ c + θ 2 µ s which happens only if w, µ s = 0. Other notions of invariance. It is easy to see that even without conditioning on y, the independence relation w, x ⊥ ⊥ e used in Veitch et al. [57] among many others will also require that w, µ s = 0. For the last invariance principle we discuss here, we note that VREx and CVaR Fairness essentially require equality in distribution of losses [64,28] under both environments. Examining the expression for the error of w under our setting (Equation (5)) reveals immediately that these conditions will also impose w, µ s = 0. G Invariant Risk Minimization and Maximum Margin In the main paper we note that the IRMv1 penalty of Arjovsky et al. [3] can be shown to prefer large margins when applied with linear models to separable datasets. This can be shown when we apply the IRMv1 principle with exponentially decaying losses such as the logistic or the exponential loss. We characterize the condition on the losses below and then give the result using a technique similar to [47] who prove that exponentially decaying losses maximize margins under separable datasets. For generality we do not assume anything about the data generating process (specifically, we do not assume the data is Gaussian as we do in the main paper) and also allow for more than two training environments. We do assume for simplicity that the datasets for each environment are of the same size, yet the proof can be easily adjusted to account for varying sizes. Let S e = {(x e i , y e i )} m i=1 be datasets for each environment e ∈ E train , with X = R d , Y = {−1, 1}. Assume the pooled dataset S = {(x e i , y e i )} i,e is linearly separable and we are learning with an l 2 regularized IRM, that is L(w; S) + λ 1 e ∇ v:v=1 L(v · w; S e ) + λ 2 w 2 2 .(41) Here we defined the average loss over a dataset as L(w;S) = 1 |S| (x,y)∈S l(w xy). Our result holds for losses that satisfy the following conditions for any > 0: lim t→∞ l(t · [1 − ]) l(t) = ∞,(42)lim t→∞ ∇ s:s=t·[1− ] l(s) ∇ s=t l(s) = ∞.(43) Relation to Previous Formal Results. Previous works [68,32] have noted that the IRMv1 penalty cannot distinguish between solutions that achieve 0 loss. That is, they prove that the solution of the IRMv1 problem is not unique when 0 loss is achievable and that the set of possible solutions coincides with the set of possible solutions for ERM. Yet here we are interested in the implicit bias of learning algorithms, hence we are interested in a more specific characterization of the IRMv1 solutions that is not provided by prior works. We ask whether out of the hyperplanes that separate S, will the IRMv1 principle find one that attains a margin that is considerably smaller than the attainable margin (hence our negative result would not imply its failure), or instead it finds a large-margin separator and hence our theory predicts its failure in learning a robust classifier? While ideally we would like to give a characterization of the solutions towards SGD will converge, as in e.g. Soudry et al. [53], the techniques used to gain such results are inapplicable to non-convex losses such as the IRMv1 penalty. Hence we turn to prove a different type of result, concerning the convergence of solutions for an 2 regularized problem, as the regularization term vanishes. This has been used in previous work to gain intuition on the type of losses that lead to margin-maximizing solutions [47]. Claim 1. Letŵ(λ 2 ) be a minimizer of Equation (41) with λ 2 > 0, where we assume that the empirical loss l : R → (0, ∞) is monotone non-increasing, and satisfies Equation (42) and Equation (43). Any convergence point ofŵ (λ2) ŵ(λ 2) 2 as λ 2 → 0 is a maximum margin classifier on S. Proof. We prove the claim in three steps. Showing that lim λ2→0 ŵ(λ 2 ) 2 2 = ∞. To this end we note that the loss is strictly positive and due to Equation (42) it approaches 0 if and only if the margin approaches ∞. The margin can only grow unboundedly large if the weights also do. Therefore if a sequenceŵ(λ 2 ) approaches loss 0 as λ 2 → 0 then also ŵ(λ 2 ) → ∞. Now we will show that this must happen, thus concluding this part of the proof. Due to Equation (43) we can also observe that the IRMv1 regularizer approaches 0 as the margin approaches ∞. This holds since for any datasetS, ∇ v:v=1 L v · w;S = 1 |S| (xi,yi)∈S w, x i · y i · ∇ s:s= w,xi·yi (s) . Invoking Equation (43), we may gather that for any > 0 it holds that Due to non-increasing monotonicity of the loss, we see that t · ∇ s:s=t l(s) < 0 any t > t 0 for some t 0 and approaches 0 as t → ∞. 10 This means that if the margin attained by some series of hyperplanes approaches ∞, then the value of the IRMv1 regularizer also approaches 0. Now to see that ŵ(λ 2 ) → ∞ as λ 2 → 0, we observe that the objective of this series as given by Equation (41) must approach 0. This holds since there is a series of hyperplanes whose objective approaches 0 (e.g. any series whose norm grows sub-linearly with λ −1 2 and separates the dataset with margin that grows to infinity). Hence any series whose objective does not approach 0 cannot be a series of minimizers. As mentioned before, the loss can only approach 0 if ŵ(λ 2 ) approaches ∞. Maximum margin separators approach 0 faster than others. Let w 1 and w 2 be vectors on the unit sphere that define separating hyperplanes, and assume w 1 achieves a larger margin than w 2 . That is, if we define for k ∈ {1, 2} s (k) i,e = w k x e i y e i , s (k) e = min i s (k) i,e , and s (k) = min e s (k) e , then our assumption on the margins is that s (1) > s (2) . We will show that the direction ofŵ(λ 2 ) cannot approach w 2 as λ 2 → 0. To this end, note that for some t 0 it holds that for all t > t 0 we have simultaneously that: t · s (1) ∇ s:s=t·s (1) l(s) < (m 2 |E train |) −1 t · s (2) ∇ s:s=t·s (2) l(s) t · s (1) i,e ∇ s:s=t·s (1) i,e l(s) ≥ t · s (1) ∇ s:s=t·s (1) l(s) ∀i, e This is true since s (1) > s (2) , s i,e ≥ s (1) for all i, e, and Equation (43) holds. We will now show that the value of the IRMv1 regularizer under tw 1 is lower than that under tw 2 : e ∇ v:v=1 L(v · w 1 ; S e ) = e 1 m m i=1 t · s (1) i,e ∇ s:s=t·s (1) i,e l(s) 2 ≤ |E train | · (t · s (1) ∇ s:s=t·s (1) l(s)) 2 < (m −1 · t · s (2) ∇ s:s=t·s (2) l(s)) 2 ≤ e 1 m m i=1 t · s (2) i,e ∇ s:s=t·s (2) i,e l(s) 2 = e ∇ v:v=1 L(v · w 2 ; S e ) . The third inequality is true since for a separating hyperplane, all the summands are negative, thus if we only add summands then the square of the entire sum becomes larger. The second one is due to Equation (44), while the first inequality is due to Equation (45) when we again use the negativity of the summands. Following the proof of [47], we can gather that the above also holds when we replace the multiplication between gradients of the loss and the margin, simply to the value the loss. Taken together, this means that tw 1 achieves a lower loss than tw 2 for all t > t 0 for some t 0 . Showing that only max-margin classifiers can be limit points. To finish the proof we simply note that if w 1 is a convergence point ofŵ(λ 2 )/ ŵ(λ 2 ) and it achieves margin γ 1 , while w 2 is another vector on the unit sphere that attains margin γ 2 > γ 1 . Then there is also a neighborhood around w 1 , N w1 = {w : w = 1, w − w 1 ≤ δ} with sufficiently small δ, such that each w ∈ N w1 attains margin at most γ 2 − > γ 1 . Therefore by our previous paragraph we know that for some t 0 and all t > t 0 then tw 2 attains a lower loss than tw for all w ∈ N w1 , which means w 1 cannot be a convergence point of w(λ 2 )/ ŵ(λ 2 ) (since the items of this series are kept out of a neighborhood N w1 around w 1 ). This is a contradiction to w 1 being a convergence point, meaning w 2 cannot attain a larger margin than w 1 . Our simulations indicate that indeed, when we apply unregularized IRMv1 to the synthetic setting we study, it finds a separator that is very close to the max-margin solution. This also holds for some other penalties such as VREx and we believe that the proof above can be adapted to this loss and several other ones, which we leave for future work. We illustrate this in ref by plotting the cosine similarity of the learned hyperplane with the hyperplane found by an SVM classifier (trained with a hinge loss using LinearSVC from sklearn [39] with C = 1000) as d grows and we approach the interpolating regime. We note that while some other methods such as CORAL and MMD do not maximize margin, they still separate the data with non-vanishing margin and indeed fail achieve worst robust accuracy as the dimension grows. H Experimental Details for Waterbirds Dataset Here we elaborate on experimental details in our Waterbirds experiment that were left out from the main paper due to lack of space. The dataset is split into training, validation and test sets with 4795, 1199 and 5794 images in each set, respectively. We follow previous work [50,58] in defining a binary task in which waterbirds is the positive class and landbirds are the negative class, and using the following random features setup: for every image, a fixed pre-trained ResNet-18 model is used to extract a d rep -dimensional feature vector x (d rep = 512). This feature vector is then converted into an d-dimensional feature vector x = ReLU(U x ), where U ∈ R d×drep is a random matrix with Gaussian entries. Finally, a logistic regression classifier is trained on x. The extent of over-parameterization in this setup is controlled by varying d, the dimensionality of x. In our experiments we vary d from 50 to 2500, with interpolation empirically observed at d = 1000 (which we refer to as the interpolation threshold). For all the experiments we use the Adam optimizer, a batch size of 128 and a learning rate schedule with initial rate of 0.01 and a decay factor of 10 for every 10,000 gradient steps. Every experiment is repeated 25 times and results are reported over all runs. For the baseline model we train for a total of 30,000 gradient steps whereas for our two-phased algorithm we use 15,000 gradient steps for each model in Phase A and an additional 250 steps for Phase B. = S e \ S trn e for e ∈ {1, 2} and S post = S e ∈ {1, 2}. The two stages of the algorithm operate on different splits of the data as follows. 1. "Training" stage: We use {S train e } to fit overparameterized, interpolating classifiers {w e } separately for each environment e ∈ {1, 2}. 2. "Post-processing" stage: We use the second portion of the data S post 1 , S post 2 to learn an invariant linear classifier over a new representation, which concatenates the outputs of the classifiers in the first stage. In particular, we learn this classifier by maximizing a score (i.e., minimizing an empirical loss), subject to an empirical version of an invariance constraint. For generality we denote this constraint by membership in some set of functions F( Figure 1 : 1Numerical validation of our theoretical claims. Invariance inducing methods improve robust accuracy compared to ERM in low values of d, but their ability to do so is diminished as d grows (top plot) and they enter the interpolation regime, as seen on the bottom plot for d > 10 2 . Algorithm 1 learns robust predictors as d grows and does not interpolate. Figure 2 : 2Results for the Waterbirds dataset [49]. Top row: Train error (left) and test error (right). The train error is used to identify the interpolation threshold for the baseline method (approximately d = 1000). Bottom row: Comparing the FNR gap on the test set (left), with zoomed-in versions on the right. Let U(O(d)) be the uniform distribution over d × d orthogonal matrices, Rad(α) the Rademacher distribution with parameter α, and N (µ, Σ) the Gaussian and multivariate normal distribution with mean µ and covariance Σ (the dimension will be clear from context) and W (Σ, d) the Wishart distribution with scale matrix Σ and d degrees of freedom. For the dataset S = {x i , y i } N i=1 we denote the indices of examples with setS = [N ], and recalling that S comprises two datasets S 1 , S 2 , we denote the indices of their respective examples within S byS 1 ,S 2 ⊆S where |S e | = N e for e ∈ {1, 2}. Our generative process is then: Figure 4 : 4Results of the simulation described in Section 5.1 with θ 2 = − 1 2 (all other parameters are kept at the same value as in Section 5.1). Lemma 3 . 3Let µ ⊥ c := (I − P V )µ c and µ ⊥ s := (I − P V )µ c . Conditional on the training set {x i , y i } i∈S , Figure 5 : 5Simulation from Section 5 with an added model trained after removing the spurious feature. This demonstrated the existence of an invariant interpolator, yet our theoretical results suggest that this type of model cannot be learned by an interpolating Lemma 7 . 7Let t > 0 and consider the solutions v neg , v pos that the algorithm may return. With probability at least 1 − 34 exp(−t 2 ) , the solutions satisfy |v 1 + v 2 | ≥ ∆ 2 . Proof. Assume that for e ∈ {1, 2} the following events occur:| n e , µ s | ≤ µ c + θ e µ s +n e | ≤ r∆ µ s 2 Proposition 4 . 4Under Assumption 1, let > 0 be the target maximum error of the model and t > 0. Ifµ c 2 ≥ tQ −1 ( ) 15 ∆ σ 2 dNmin , then with probability at least 1 − 34 exp(−t 2 ) the robust accuracy error of the model is at most .Proof. The error of the model in the environment defined by θ ∈ [−1, 1] is given by the Gaussian tail function: Corollary 2 . 2for some universal constant c > 0 (when we assume w.l.o.g that M e ≤ N e ): P {| n e ,m e | > t} ≤ 2 exp −c min A i,i+d = 1 for 1 ≤ i ≤ d and 0 elsewhere). Then A 2 HS = d and A = √ d. Since entries inn e ,m e are distributed as N (0, σ 2 Ne ), N (0, σ 2 Me ) respectively, we have K ≤ C σ √ min {Ne,M e } (assume w.l.o.g that M e < N e ) for some universal constant C which we can incorporate into the constant c in the theorem. This gives: P {| n e ,m e | > t} ≤ 2 exp −c min The next statement collects all of the concentration results we require for the other parts of the proof. Lemma 10. Define r := ∆θmax ∆+4θmax where θ max := arg max e∈{1,2} {|θ e |}, denote by v * the solution retrieved by the algorithm, and let t > 0. When Assumption 1 holds, then with probability at least 1 − 34 exp(−t 2 ) we 37)r∆ µ s 2 ≥ | m 1,1 −m 2,1 , µ c + θ e µ s +n e | (38)r µ s 2 ≥ | n e , µ s | (39) 5t σ 2 d min e N e ≥ w(40)Proof. Equation(39)is just Equation(27)restated for convenience. Equation (38) is a combination of Equation (29) and Equation (31): | m 1,1 −m 2,1 , µ c + θ e µ s +n e | ≤ e | m e ,1 , µ c + θ e µ s | + | m e ,1 ,n e | ≤ r∆ µ s 2 6 6in Proposition 1, as well as the requirement max{r c , r s } ≤ c n N in Proposition 3; it remains to choose d so that the remaining requirements hold. Proposition 1 requires the dimension to satisfy d ≥ C Proposition 3 requires d ≥ C d N 2 log 1 δ . Substituting our choices of σ 2 = 1/d, r s and r c above, let us rewrite the requirements of Proposition 2 as lower bounds on d. The requirement in Equation ( lim t→∞ t · [1 − ] · ∇ s:s=t·[1− ] l(s) t · ∇ s:s=t l(s) = (1 − ) · lim t→∞ ∇ s:s=t·[1− ] l(s) ∇ s:s=t l(s) = ∞. 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Thus, attaining guarantees on the robust error also has consequences on invariance of the model, as defined by these criteria. We discuss this further in section F of the appendix. The even split is used here for simplicity of exposition, and our full proof does not assume it. In practice, allocating more data to the first-stage split would likely perform better. Intuitively, if |θ 1 − θ 2 | = 0 then the two training environments are indistinguishable and we cannot hope to identify that the correlation induced by µs is spurious. Otherwise, we expect |θ 1 − θ 2 | to have a quantifiable effect on our ability to generalize robustly. For simplicity of this exposition we assume that the gap is bounded away from zero; the full result in the Appendix is stated in terms of |θ 1 − θ 2 |.5 This assumption makes sure we have some positive labels in each environment. This is to keep our parameters within the regime where benign overfitting occurs. 7 This is basically Algorithm 1 with the following minor modifications: (1) The we's are computed via ERM, rather than simply taken to be the mean estimators; (2) Since the FNR gap penalty is already computed w.r.t. a small number of samples, we avoid splitting the data and use the entire training set for both phases; (3) we convert the constrained optimization problem into an unconstrained problem with a penalty term. we should also note that the derivative of the loss does not approach −∞ since it is bounded below by 0, hence it must indeed be the case that the denominator approaches 0. Models will be defined by:w e := 1 N e i∈Se y i x i = µ c + θ e µ s +n e , e ∈ {1, 2},The Equalized Opportunity (EOpp) constraint is:T e (f v ; D, S) = 1 M e,1 i∈De,1Additional Assumptions We assume w.l.o.g θ 2 > θ 1 , define ∆ := θ 2 − θ 1 > 0 and r µ = µs µc > 1. We consider r µ , ∆ as fixed numbers. That is, they do not depend on N, d and other parameters of the problem. Also define r := ∆θmax ∆+4θmax , where θ max := argmax{|θ 1 |, |θ 2 |} ≤ 1. The following additional assumptions will be required for our concentration bounds. Assumption 1. Let t > 0 be a fixed user specified value, which we define later and will control the success probability of the algorithm. We will assume that for each e ∈ {1, 2} and some universal constants c c , c s > 0:Analyzing the EOpp constraint. Writing the terms defined above in more detailed form gives:So the EOpp constraint is:We will consider v pos as the solution that satisfies v pos,1 + v pos,2 > 0.Proof. Is it easy to see that the EOpp constraint is a linear equation in v 1 , v 2 and with probability 1 the coefficients in this linear equations are nonzero. Therefore the solutions to this equation form a line in R 2 that passes through the origin. Consequently, this line intersects the l ∞ unit ball at two points, that we denote v pos , v neg , which are negations of one another.The proposed algorithm. Now we can restate our algorithm in terms of v pos and v neg and analyze its retrieved solution.• Calculate w 1 and w 2 according to their definitions. | n e ,m e ,1 | ≤ minProof. We first treat Equation(27)with a tail bound for Gaussian variables:Hence as long as µ s 2 ≥ t 2σ 2 r 2 Ne , Equation(27)holds with probability at least 1 − 4 exp{−t 2 } (since we take a union bound on the two environments). Following the same inequality and taking a union bound, Equation (28) also hold with probability at Similarly with t 2 = 1 48 ∆ · µ c 2 :Taking the required union bounds we get that with probability at least 1 − 24 exp −t 2 /2 Equation(29) Invariant risk minimization games. Kartik Ahuja, Karthikeyan Shanmugam, Kush Varshney, Amit Dhurandhar, International Conference on Machine Learning. 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[]
[ "Design and performance of a Martian autonomous navigation system based on a smallsat constellation", "Design and performance of a Martian autonomous navigation system based on a smallsat constellation" ]
[ "S Molli \nDepartment of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly\n", "D Durante \nDepartment of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly\n", "G Boscagli \nCenter of Aerospace Research Sapienza\nSapienza University of Rome\nItaly\n", "G Cascioli \nCenter for Space Sciences and Technology\nUniversity of Maryland\nBaltimore County\nBaltimoreMDUSA\n\nNASA Goddard Space Flight Center\nGreenbeltMDUSA\n\nCenter for Research and Exploration in Space Science and Technology (CRESST) II\nNASA/GSFC\nGreenbeltMDUSA\n", "P Racioppa \nDepartment of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly\n", "E M Alessi \nInstitute for Applied Mathematics and Information Technologies \"E. Magenes\"\nNational Research Council (IMATI-CNR)\nItaly\n", "S Simonetti \nArgotec S.p.A\nTorinoItaly\n", "L Vigna \nArgotec S.p.A\nTorinoItaly\n", "L Iess \nDepartment of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly\n" ]
[ "Department of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly", "Department of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly", "Center of Aerospace Research Sapienza\nSapienza University of Rome\nItaly", "Center for Space Sciences and Technology\nUniversity of Maryland\nBaltimore County\nBaltimoreMDUSA", "NASA Goddard Space Flight Center\nGreenbeltMDUSA", "Center for Research and Exploration in Space Science and Technology (CRESST) II\nNASA/GSFC\nGreenbeltMDUSA", "Department of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly", "Institute for Applied Mathematics and Information Technologies \"E. Magenes\"\nNational Research Council (IMATI-CNR)\nItaly", "Argotec S.p.A\nTorinoItaly", "Argotec S.p.A\nTorinoItaly", "Department of Mechanical and Aerospace Engineering\nSapienza University of Rome\nItaly" ]
[ "Acta Astronautica" ]
A B S T R A C TDeciphering the genesis and evolution of the Martian polar caps can provide crucial understanding of Mars' climate system and will be a big step forward for comparative climatology of the terrestrial planets. The growing scientific interest for the exploration of Mars at high latitudes, together with the need of minimizing the resources onboard landers and rovers, motivates the need for an adequate navigation support from orbit. In the context of the ARES4SC study, we propose a novel concept based on a constellation that can support autonomous navigation of different kind of users devoted to scientific investigations of those regions. We study two constellations, that differ mainly for the semi-major axis and the inclination of the orbits, composed of 5 small satellites (based on the SmallSats design being developed in Argotec), offering dedicated coverage of the Mars polar regions. We focus on the architecture of the inter-satellite links (ISL), the key elements providing both ephemerides and time synchronization for the broadcasting of the navigation message. Our concept is based on suitably configured coherent links, able to suppress the adverse effects of on-board clock instabilities and to provide excellent range-rate accuracies between the constellation's nodes. The data quality allows attaining good positioning performance for both constellations with a largely autonomous system. Indeed, we show that ground support can be heavily reduced by employing an ISL communication architecture. Periodic synchronization of the clocks on-board the constellation nodes with terrestrial time (TT) is enabled through the main spacecraft (the mother-craft), the only element of the constellation enabling radio communication with the Earth. We report on the results of numerical simulations in different operational scenarios and show that a very high-quality orbit reconstruction can be obtained for the constellation nodes using a batch-sequential filter or a batch filter with overlapping arcs, that could be implemented on board the mother-craft, thus enabling a high level of navigation autonomy. The assessment of the achievable positioning accuracy with this concept is fundamental to evaluate the feasibility of a future positioning system providing a global coverage of the red planet.
10.1016/j.actaastro.2022.11.041
[ "https://export.arxiv.org/pdf/2212.01647v1.pdf" ]
254,005,186
2212.01647
8e271727483722e69078a5cc573f8ba260f9022f
Design and performance of a Martian autonomous navigation system based on a smallsat constellation 2023 S Molli Department of Mechanical and Aerospace Engineering Sapienza University of Rome Italy D Durante Department of Mechanical and Aerospace Engineering Sapienza University of Rome Italy G Boscagli Center of Aerospace Research Sapienza Sapienza University of Rome Italy G Cascioli Center for Space Sciences and Technology University of Maryland Baltimore County BaltimoreMDUSA NASA Goddard Space Flight Center GreenbeltMDUSA Center for Research and Exploration in Space Science and Technology (CRESST) II NASA/GSFC GreenbeltMDUSA P Racioppa Department of Mechanical and Aerospace Engineering Sapienza University of Rome Italy E M Alessi Institute for Applied Mathematics and Information Technologies "E. Magenes" National Research Council (IMATI-CNR) Italy S Simonetti Argotec S.p.A TorinoItaly L Vigna Argotec S.p.A TorinoItaly L Iess Department of Mechanical and Aerospace Engineering Sapienza University of Rome Italy Design and performance of a Martian autonomous navigation system based on a smallsat constellation Acta Astronautica 203202310.1016/j.actaastro.2022.11.0410094-5765/ A R T I C L E I N F OInter-satellite link Mars autonomous navigation Constellation SmallSat Orbit determination A B S T R A C TDeciphering the genesis and evolution of the Martian polar caps can provide crucial understanding of Mars' climate system and will be a big step forward for comparative climatology of the terrestrial planets. The growing scientific interest for the exploration of Mars at high latitudes, together with the need of minimizing the resources onboard landers and rovers, motivates the need for an adequate navigation support from orbit. In the context of the ARES4SC study, we propose a novel concept based on a constellation that can support autonomous navigation of different kind of users devoted to scientific investigations of those regions. We study two constellations, that differ mainly for the semi-major axis and the inclination of the orbits, composed of 5 small satellites (based on the SmallSats design being developed in Argotec), offering dedicated coverage of the Mars polar regions. We focus on the architecture of the inter-satellite links (ISL), the key elements providing both ephemerides and time synchronization for the broadcasting of the navigation message. Our concept is based on suitably configured coherent links, able to suppress the adverse effects of on-board clock instabilities and to provide excellent range-rate accuracies between the constellation's nodes. The data quality allows attaining good positioning performance for both constellations with a largely autonomous system. Indeed, we show that ground support can be heavily reduced by employing an ISL communication architecture. Periodic synchronization of the clocks on-board the constellation nodes with terrestrial time (TT) is enabled through the main spacecraft (the mother-craft), the only element of the constellation enabling radio communication with the Earth. We report on the results of numerical simulations in different operational scenarios and show that a very high-quality orbit reconstruction can be obtained for the constellation nodes using a batch-sequential filter or a batch filter with overlapping arcs, that could be implemented on board the mother-craft, thus enabling a high level of navigation autonomy. The assessment of the achievable positioning accuracy with this concept is fundamental to evaluate the feasibility of a future positioning system providing a global coverage of the red planet. Introduction The growing interest in the exploration of celestial bodies of the Solar System motivates the attention towards new navigation systems, capable of effectively supporting the new scientific and commercial missions planned in the coming years. The objective of this work is to study a navigation system consisting of a constellation of small satellites that can handle, in almost complete autonomy, the relative and absolute positioning of its nodes around planetary bodies. This system must be able to provide the necessary tools to support the navigation of other probes (e.g., objects in Entry, Descent and Landing (EDL) phase, landers, rovers, or orbiters). Using small satellites leads to low development and launch costs, short development times, and offer good flexibility in mission implementation since multiple systems can be launched simultaneously with dedicated launches or as secondary payloads of larger missions. The navigation system is based on radio observables obtained from the inter-satellite link (ISL) between the constellation satellites, heavily reducing the needs for ground support. The use of ISL technologies is widely recognized for its extraordinary results in terrestrial and planetary geodesy (GRACE and GRAIL missions) [1,2]. We demonstrate that the same concept of ISL, in the innovative configuration recently proposed [3,4] can provide observable quantities of high quality, enabling precise orbital determination, if applied to constellations with spacecraft on different orbital planes. Since the ISL provides information only on the relative motion between the spacecraft, the accurate knowledge of the Martian gravity field and rotational state acquired with previous mapping missions like MRO, MGS and Mars Odyssey [5,6], is fundamental, because it allows the conversion to absolute positioning. We examine two different quasi-autonomous constellations of 5 SmallSats, in terms of orbital configuration and system requirements, and we compare their accuracy in terms of satellite positioning. Both constellations have a local coverage of the Mars polar regions, but this concept could be expanded to a global navigation system of the planet. The focus on the Martian polar caps origins from the need for a deep understanding of the connection between the polar deposits and the Martian climate to understand the Martian climate system [7]. The orbital period of the constellation has been selected equal to half of Mars' rotation period for the first constellation, and equal to a quarter of Mars rotation period for the second one, to have repeating ground tracks. The orbits are quasi-circular, resulting in an altitude of around 9500 km and of 4740 km, for the first and second constellation, respectively. The orbits are designed to provide navigation support during the EDL phase of a probe at high latitudes and to allow the positioning of rovers for the exploration of the Martian polar regions. This study allows the demonstration of communication architectures potentially useful for subsequent applications, such as a future positioning system that provides global coverage of the planet. This paper is organized as follows. In Sec. 2, we describe the design of the two constellations in terms of orbital configurations. In Sec. 3, we introduce the navigation system design based on the novel inter-satellite radio-tracking technique advised to precisely determine the orbits of the satellites, and then we focus on its implementation on the spacecraft platform. Starting from the architectural choices, in Sec. 4, we present the radio-tracking system in terms of error budgets, preparatory to the computation of the positioning performance for the two constellations. Then, in Sec. 5 we explain the orbit determination procedure, and in Sec. 6 we describe the numerical simulations in different mission scenarios for both constellations in orbit about Mars. Lastly, in Sec. 7 and 8, we show the results of the numerical simulations in autonomous and quasiautonomous configurations to demonstrate that excellent orbital accuracies can be obtained for a constellation of SmallSat on Mars. Sec 9 gives concluding remarks. Constellation design This study is aimed at realizing a low-cost mission to offer a local communication and navigation system. The constellations' orbits have been designed to optimize the coverage on Mars polar regions, which are selected for scientific purposes. However, this architecture is scalable and could be expanded to realize a global navigation system or could be a part of a more complex mission concept, like MOSAIC [8]. The two constellations presented in this work are both composed of 5 spacecraft deployed on three quasi-circular, high-altitude polar orbits. The main spacecraft, the mother-craft (MSC), is placed on the central orbit, whereas the other four spacecraft, the daughter-craft (DSC), are symmetrically located on the two inclined orbits, as depicted in Fig. 1. The orbital configuration for this navigation system arises from the initial geometric constraints: • Orbital period as an integer submultiple of Mars' rotation period (e. g., T/2, T/4, T/8) to have repeating ground-tracks. • Maximum Line of Sight (LOS) distance between MSC and DSCs less than 8000 km: this result from a preliminary link budget and navigation accuracy assessments. It is a value that can provide sufficient Signal to Noise (SNR) values, while offering good orbital geometry (large distances among nodes are preferred for accurate user triangulation). • Angular separation of two couples of DSCs with respect to the MSC less than 40 • : this is the assumed value for the MSC antenna's HPBW (Half Power Beam Width) for relative navigation. If the two DSCs preceding or following the MSC are always within this field of view, only two antennas are required by the MSC to perform all relative navigation links. Consequently, the relative inclination between the MSC and DSCs orbital planes must remain in the 10-20 • range, to maintain a sufficiently close distance and mitigate losses on the relative MSC-DSC links. Orbital configuration The major differences between the two configurations are the semimajor axis and the inclination of the orbits. The selected semimajor axes, 12868.6 and 8106.7 km respectively, entail an orbital period equal to half Mars' rotation period for the first constellation, hereby referred to as constellation A, and equal to a quarter of Mars rotation period for the second one, referred to as constellation B. The orbital parameters of the spacecraft of the two constellations are reported in Tables 1 and 2 tracks, which repeat every two days and one day, respectively. The maximum distance between the DSCs and the MSC is ~4000 km for the constellation A and ~7000 km for the constellation B. The constellation B, differently from constellation A, is shifted in inclination by 15 • with respect to the pole. This asymmetry enables a more robust position determination, leveraging the effect of the different gravity gradients. For the same principle, the spacecraft of this constellation have a bigger baseline, given by the larger values of the mean anomaly. To avoid collision between the satellites when their orbital planes intersect, in both constellations, the DSC are not phased equally with respect to the MSC. Navigation system design Deep-space ISL applications have been proposed for a variety of scenarios, spanning from the interplanetary range (e.g. Refs. [9,10]) to the atmospheric probing [11], or gravity field determination [12]. The first operative application of ISL outside of the Earth orbit has been accomplished with the GRAIL mission [2], where an inter-satellite link has been established between two spacecraft flying in precision formation around the Moon. The GRAIL concept was derived from the GRACE Earth mission and utilized a modified GRACE payload, i.e., the Lunar Gravity Ranging System (LGRS). GRAIL has obtained extraordinary results, showing that, with this architecture, the estimate of the lunar gravity field can be performed up until degree and order 1200, demonstrating robustness against both dynamic and kinematic errors [13]. These radio tracking systems were based on dual-one-way-ranging (DOWR) measurements in Ka-band to determine the relative motion between the two orbiters [14] and required identical elements onboard the satellites. The drawbacks of this architecture lie in the large complexity and high design and development costs. Moreover, the presence of an ultra-stable oscillator (USO) on both spacecraft leads to the necessity of a precise synchronization between the two clocks, that GRACE accomplished using GPS receivers, taking advantage of being in orbit around the Earth. On the contrary, the GRAIL spacecraft required an S-band time transfer unit and a gravity recovery processor assembly (GPA) to process Ka-band and S-band radio signals that were generated through USOs. Furthermore, a daily calibration of the USO frequency was required, which was accomplished using a Radio Science Beacon (RSB) to transmit radio links at X-and S-band between the orbiters and NASA's DSN stations. The accuracy achieved in terms of range-rate was of 0.03-0.06 μm/s at 5 s integration time [15]. An alternative radio science system architecture based on range-rate measurements is presented in this work. This configuration, shown in Ref. [3], is based on a simplified scheme that leads to significant mass and power savings. In particular, in the ARES4SC study, we chose to establish cross-links only between the MSC and the DSCs, as shown in Fig. 3, realizing a "star" configuration. Recall that for n nodes in a given constellation, the theoretical maximum number of radio links between two nodes at a time is (n 2 − n) 2 . Such a configuration would provide the maximum number of observables and a wider range of relative orbital geometries, which could grant higher accuracies in the determination. of the state of the constellation. However, its implementation may result in significant technical complications, and complexity in managing multiple frequencies simultaneously. For such reasons, this much simpler solution is analysed, demonstrating that it can still provide excellent positioning accuracies, while significantly containing the system complexity. To establish the ISL between MSC and each DSC only three components are necessary (see Fig. 4): i. An Ultra Stable Oscillator (USO) onboard the MSC, which generates the reference signal. ii. A transceiver onboard the MSC, which transmit simultaneously the uplink signal to all the DSCs at frequency f 1 and then receives the downlink signal coherently from DSCs at frequency f 2 , allowing measurement of range and range-rate. iii. A transponder onboard the DSCs, which receives the signal from the MSC and retransmits it back preserving phase coherency. The selected star-like configuration, in which the observables quantities are generated onboard the MSC instead of the DSCs, minimizes the number of required USOs, allowing either the selection, for the same total cost, of a high-performance USO (in terms of Allan deviation) or the reduction of the total cost itself. To allow the use of the same frequency for all the nodes of the constellation, the use of the Code Division Multiplexing (CDM) is proposed. This choice greatly simplifies the use of the frequency spectrum, allowing an easy and simultaneous measurement of the range and range-rate, at the cost of a higher complexity of the equipment on board MSC. The ISL is the crucial component of the autonomous navigation system, but it provides information only on the relative motion between the spacecraft. The knowledge about the absolute positioning can be possible exploiting the knowledge of the asymmetric gravitational field of the central body. If its effect is sufficiently strong, it can allow to reference the spacecraft positions with respect to the body fixed reference frame. Conversely, when the intensity of the gravitational attraction is reduced (when the orbits of the constellation are at a high altitude), an additional ground-tracking system should be provided, as it will be explained later in Section 9, allowing to reference the orbits to the Earth's surface, enabling the absolute positioning. It must be noted that a radio link from the MSC to the Earth is anyhow required to enable Telemetry, Tracking & Command (TT&C) capabilities and to synchronize the time of the constellation with a terrestrial time scale, such as TAI or UTC. DSCs do not have such capability: the reference timescale, as well as commands from ground, are received via the MSC, which effectively acts as a relay satellite. Such a configuration simplifies the radio infrastructure on the DSCs, while retaining system complexity only on the MSC. To further simplify the overall system and reduce the complexity of the radio instrumentation, we select the X-band for ISL. This allows us to utilize the very same band for both the ISL and the link to ground, which is established in the commonly used X-band. Spacecraft platform The navigation system design of both MSC and DSC is based on the Hawk platform, which has been selected for its combination of reliability and radiation hardness that allows to deal with the harsh space environment. The Hawk platform's development, manufacturing, and qualification are carried out entirely by Argotec. Fig. 5 illustrates Argotec's HAWK-6U platform, already used for the current missions ArgoMoon and LiciaCube [16,17]. Due to the different system configurations explained later in this Section, the DSC have been designed to be smaller platforms (6U CubeSat) with respect to MSC (12U CubeSat). Only the 12U CubeSat can take the MSC role in the selected constellation system, leading to a simpler radio link configuration. The constellation can be designed to add redundancy to the whole system considering that some (or all) DSC could be replaced by 12U CubeSat with the possibility that any of these satellites can take the MSC role. The MSC is equipped with a navigation payload that is responsible for the provision of the autonomous navigation service. It is composed by: • Ultra-Stable Oscillator (USO): the Ultra-miniature OCXO 9700 from Microsemi [18] has been preliminarily selected. It has stability in terms of Allan Deviation (ADEV) of 5 • 10 − 12 approximately flat from 1 s to 1000 s, an acceptable value to obtain good accuracies in the orbit determination and time synchronization process, (approximately, two order of magnitude better at 100 s w.r.t. the typical ADEV of a X-band transponder with its internal oscillator (OCXO) in non-coherent mode), and low mass and volume requirements. [19]. The transceiver shall be compliant with CCSDS Standard and SFCG Recommendation for communication in the Mars Region [21]. The minimum RF power (4 W) is derived from a preliminary link budget analysis, and it is limited by the platform power availability. Moreover, to transmit the same signal to all the DSCs nodes, a power splitter is needed to redirect the signal to both Tx antennas. A compatibility analysis and a trade-off have been performed to select the transceiver. A lack of availability of commercial telecommunication payload, for CubeSat, supporting CDMA scheme has been identified. A Software Defined Radio has been preliminarily considered; it is partially compliant with the requirements, and it is well suited for a future reconfiguration to support CDMA communications and generation of navigation signal. The DSC is equipped with a navigation payload composed by: • Tx/Rx Antenna: it provides the RF signal reception and transmission from/to the MSC via one couple of patch antennas onboard the DSCs (which are the same as the MSC) pointed toward the mother node. • X-Band Transponder: it provides the coherent retransmission of the ranging signal sent by the MSC. To be fully compliant with the CCSDS standard, the signal shall be regenerated meaning that the transponder onboard the DSC shall support CDMA technique too. For this reason, in this preliminary selection of the transponder, a high priority is given to mass and volume requirements and reconfigurability. The Iris V2 [22], a miniaturized transponder of limited mass, has been successfully used on board the MarCO SmallSats at Mars [23], and it's used for the current missions ArgoMoon [16] and LICIACube [17]. Future reconfiguration to support CDMA communications must be considered also in this case. With the chosen antennas, the ISL budget, at the maximum distance of 7000 km shows, for the forward link, a SNR of 30 dB-Hz, with a command data rate in the range of 200 bps; for the return link a SNR of 35 dB-Hz, with telemetry data rate in the range of 600 bps; for the ranging measurement a rms value of 1.25 m at integration time of 1 s. To communicate with ground (downlink), high-gain antennas onboard the satellite are needed. Assuming an X-Band communication with 35 m antennas on the ground, it results that to ensure a downlink data rate ranging from 250 bps (2.65 AU) to 8 kbps (0.5 AU) in a nominal scenario, an Effective Isotropic Radiated Power (EIRP) greater than 35 dBW is needed. For this reason, the platform configuration foresees a reflect array antenna with an aperture size of 0.60 × 0.33 m, a gain of 29.2 dBi and an HPBW of 4.9 deg [24,25]; a transmission RF power of 6 dBW is needed to satisfy the requirement. The allocation of the reflect array on the MSC has been made possible by exploiting an integrated reflect array and solar array panels which is a technology demonstrated in both ISARA [26] and OMERA [25] missions. The link from ground to the MSC (uplink) is possible with a mediumgain antenna (2 × 2 patch array) having a gain of 12 dBi and an HPBW of 40 deg. For constellation redundancy purposes, also the DSCs can communicate with Earth at a low data rate only in case the MSC is lost. The same [2] and the novel (bottom) radio system for intersatellite tracking [3]. DSC transponder, used for navigation, can be used for navigation with Earth. Two antennas have been mounted on one side of the spacecraft for TT&C. The Rx antenna is an X-Band 2 × 2 patch array antenna, instead, the Tx antenna is an X-Band 4 × 8 patch array antenna with a gain of 22 dBi. In addition to the Navigation payload and TT&C, both MSC and DSC are composed of the following subsystems: Structure, Thermal Control, Electrical Power (i.e. battery, Solar Panel Array and Power Conversion and Distribution Unit), Attitude Determination and Control (to determine and control where the satellite should be pointed and orient it to establish Inter-Satellite Link, link with Earth, to perform sun-pointing for battery charging, and reaction wheels desaturation), On-Board Computer and Data Handling (the selected one is the OBC&DH Mk I produced by Argotec with heritage from both the ArgoMoon and the LICIACube deep-space CubeSats), On-Board Software, Propulsion and Harness. Fig. 6 and Fig. 7 show the layout of the MSC and DSC platforms, respectively, including the main external equipment location, namely the navigation antennas, the antennas for the link with Earth for TT&C, and the Solar Panel Array. Radio-tracking system This section describes the error budgets for both Earth link and ISL Doppler observables. Space-to-ground The key parameter determining the achievable accuracy of the Doppler tracking system is the frequency stability of the GS-SC-GS loop. Typical performance of deep space Doppler tracking systems employing Doppler measurements (here reported at 10 s integration time) is of the order of 0.2 mm/s for X-band radio links [27] and 0.05 mm/s at Ka-band [28], reaching levels up to 0.015 mm/s for state-of-the-art systems such as the one of BepiColombo [29]. Non-signal disturbances in a Doppler link are due to instrumental noises (random errors introduced by ground or spacecraft themselves), propagation noises (random frequency/phase fluctuations caused by refractive index fluctuations along the line of sight), or systematic errors. However, analyses and experience have shown that at timescales ~1-1000 s systematic contributions are negligible. The radio tracking data at X-band are mainly perturbed by interplanetary plasma. A plasma model proposed by Iess et al. [30] provides the Allan deviation as a function of SEP angle, which is strictly valid for missions in the outer solar system, showing that the level of plasma noise increases approaching superior solar conjunctions. The noise caused by fluctuations of the tropospheric refractive index is another major contributor. At microwave frequencies, tropospheric refractive index fluctuations are nondispersive and dominated by water vapor fluctuations [31]. Up to ~90% of the tropospheric path delay is due to the dry, or homogenous, part of the troposphere. The induced Doppler signal in ~6 h of deep space tracking amounts to ~1 mm/s, but can be efficiently resolved with suitable elevation models and ground readings of meteorological data. On the other hand, the wet path delay amounts to only a few centimeters, but its fluctuations are hardly predictable because of the non-homogenous distribution of the water vapor. Finite SNR of electronic devices on the radio links, on both the ground and spacecraft equipment, induce phase instabilities on the signal carrier. Noise contributions due to the frequency standard and electronics of the ground station are usually negligible. The main contribution from electronic systems comes generally from the onboard deep space transponders (DSTs), unless special design is adopted (as for Juno and BepiColombo Ka/Ka radio science transponder). These devices usually exhibit a white phase noise (S(y) ∝ f 2 ) at frequencies larger than 10 − 4 ÷ 10 − 3 Hz. Transponders based on digital architectures generally have poorer stability, like Rosetta's DST, with Allan deviations σ y ≈ 4.7 ⋅ 10 − 14 , or 0.015 mm/s range rate accuracy, at 60 s integration time. For comparison, Cassini's analog DST performs better (σ y ≈ 1.8 ⋅ 10 − 14 or 0.006 mm/s at the same integration time) [30]. Thus, noise contributions due to the last generation of digital transponders are negligible too. A significant contribution between instrumental noises to the overall Doppler error budget comes from thermal and mechanical deformations of the ground antennas. These deformations are caused by wind and gravitational loading, and time-varying thermal gradients across the antenna structure. An analysis performed by ESA showed that antenna mechanical noise at ESTRACK antennas is at level of σ y ~1.6 ⋅ 10 − 14 (0.05 mm/s) at 60 s integration time [32]. Ground antennas' thermo-mechanical stability is well below other noise sources, but not negligible. Consequently, the error budget for space-to-ground Doppler measurements indicates an accuracy in range rate measurements of 0.04 mm/s @ 60 s in X-band. Inter-satellite link The main factors that determine the Doppler accuracy are: • the error caused by thermal noise of the receiver; • the stability of the transponder on board the nodes; • the stability in frequency (ADEV) of USO. As stated in Section 2, the maximum distance between the nodes was selected as to have adequate power levels, limiting the contribution of the thermal noise, that could be expressed in terms of ADEV as follows [33]: σ y (τ) = 1 2π f τ ̅̅̅̅̅̅̅̅̅ 3B l SNR √ Assuming the noise bandwidth of the receiver (B l ) equal to 0.1 Hz, considering the worst-case scenario with an SNR equal to 30 dB-Hz, the thermal noise contribution at X-band (f = 8.4 GHz) is 3.3 × 10 − 16 at τ = 1000 s and scales with τ − 1 . The latest generation transponders for deep space applications showed ADEV of the order of 3-4 x 10 − 16 at 1000 s integration time and about 1-2 x 10 − 15 at 60 s. In this study we will assume ADEV equal to 1 × 10 − 14 , comparable with that of Iris V2. In terms of range-rate error (2-way) this frequency stability corresponds to 3 μm/s at 60 s of integration time [22]. The frequency stability of the USO is the fundamental element to ensure the accuracy of range rate measurements. As demonstrated in Ref. [3], in a two-way Doppler link, the limited spatial separation, therefore the short round-trip light time (RTLT), of the order of 10-20 ms at most, implies a strong suppression of the noise of the USO. In fact, since the same reference frequency is used in both uplink and downlink, any instability that appears at time scales larger than the RTLT is strongly reduced. This cancellation effect can be appreciated considering that the clock noise transfer function, in a 2-way link, consists of two anticorrelated Dirac delta (δ (t)) functions separated by a RTLT. Let us call y c the random noise process associated with the relative frequency fluctuations introduced by the local oscillator used to generate the reference signal. The instability in the relative frequency shift y(t) = δf f between the received signal and the reference signal can be written as [3]: y(t) = y c δ ( t − 2L c ) − y c δ(t) = yc δ(t − RTLT) − y c δ(t) where L is the spatial separation between the two nodes. The anticorrelation at one RTLT leads to strong noise suppression when the integration time T is much greater than the RTLT. In fact, for low frequencies, the corresponding noise power spectrum S c (f) of the USO becomes: S y (f ) = 4S c (f )sin 2 ( πfL c ) ≈ 4π 2 S c (f ) ( fL c ) 2 , πfL c ≪1 By integrating over a time T (typically 60 s), the spectral power at frequencies higher than 1 T is attenuated by a factor proportional to: 4 ( fL c ) 2 ≤ ( 2L Tc ) 2 = ( RTLT T ) 2 For the ADEV, the suppression factor is equal to its square root: σ y (T) ∝ σ yc (T) RTLT T π RTLT T ≪1 Where σ y is the ADEV of the link measured by the transceiver if only the noise of USO were present and σ yc is the ADEV of the USO. Considering the worst case with 7000 km of inter-satellite distance and T = 60 s, the noise of USO is therefore attenuated by a factor of approximately 800. With the chosen USO with ADEV equal to 5 × 10 − 12 , the residual clock noise in the Doppler measurement would be equal to 6 × 10 − 15 , corresponding to an error in range rate equal to 2 μm/s. This configuration, therefore, allows to obtain excellent range-rate measurements with a very cheap and light USO. The error budget indicates a worst accuracy in range rate measurements of 3.9 and 3.6 μm/s at 60 s integration time for constellation B and A, respectively, as shown in Fig. 8. As a comparison, Ka-band range-rate (KBRR) measurements from GRAIL mission [2], directly derived from the KBR data, have an accuracy of 0.03-0.06 μm ⋅ s − 1 at 5 s integration time but with a very sophisticated system, highly demanding in terms of mass, power, and volume requirements. Orbit determination procedure To investigate the performance of the mission concept ARES4SC, the first step of the orbit determination (OD) procedure is the generation of realistic data from the s/c, called synthetic observables. These are collected using a simulator that obtains a reference trajectory propagating the orbits starting from the chosen initial conditions, using a provided dynamical model. Then, an observational model and a realistic noise process are implemented to simulate the observables. The key part of the OD procedure is the estimation, in which the differential corrections procedure is applied from the available observation to solve for the parameters we are interested in, called state vector [34]. This can be solved with two class of estimators: batch and sequential. The key difference is that batch estimators wait for all observables to be collected and use all of them in a single inversion to estimate the state vector at the desired reference epoch, while the sequential algorithm process one observation (or a few) at a time. One of the most used batch estimators is based on a linear least squares formulation. For this work, we adopt a filter based on optimally weighted least square (WLS hereafter, see 5.1 for a discussion on the weighting procedure) with a-priori information. Due to the high non-linearity of the OD problem, the inversion process shall be iterated a few times to reach convergence. Nevertheless, the solution obtained with this method could not be sufficient when the dynamics of the spacecraft is quite complicated, because the mathematical model and/or the numerical integrator may not be accurate enough for very long data arc, leading to an integrated trajectory that is not consistent with the observations. This is mainly related with difficulty in modelling some of the forces, as in the case of Solar Radiation Pressure (SRP) and thruster firings for trajectory and attitude controls. Furthermore, in several space geodesy experiments the number of solving-for parameters is quite large. This problem is in many cases severe, to the point that a single batch estimation becomes impractical. In such case, the trajectory could be subdivided into different non-overlapping arcs (batches). At this point, different batches can be combined to obtain a unique, and better, estimate of a common subset of parameters, or consecutive batches can be combined sequentially to update the state estimate for the same state vector. In the former approach, called "multiarc" [35], the common "global" parameters (e.g., gravity field coefficients of a body) are uniquely solved, while the non-common "local" parameters (usually the initial position and velocity components) are replicated independently for each arc. The second approach allows performing a near real-time OD of the constellation with the more robust and accurate batch method. In this case, batch filtering (with few iterations) of a generalized state vector for the whole constellation is solved in each arc. The different estimates may be used in combination with different approaches depending on the problem. Batch methods are, in general, computationally more demanding (both for the iterations and the size of involved vectors and matrices) than sequential methods and are not well suited for real-time applications. The latter is the typical area of applicability for sequential estimation methods, providing a state estimate at the observation epoch that is continuously updated after each (or a few) new observation is collected. Kalman Filter (KF) is an example of Sequential filter, linear as well, computationally light (mainly because of the lower memory requirements), but it is not stable enough for OD applications. Extended Kalman Filter (EKF), or Unscented Kalman Filter (UKF) are better options (although computationally more demanding), for real-time OD. In this work, we investigate the first segment of the ARES4SC mission concept: the numerical warm-up period necessary for the constellation to assess its ephemeris, analysing the obtainable accuracies to understand if such a concept is feasible for a future navigation system. Therefore, real-time application is not required for this purpose, leading us to the choice of batch filters, mainly for its robustness. We propose two different procedures for this approach: a first one, called batch sequential filter, in which the components of the spacecraft position and velocity vector, called propagated parameters, and their covariance, estimated at the beginning of the arc, are propagated until the beginning of the next arc, and used as a priori information, optionally using a conservative scaling factor. It's also possible to set constant parameters, such as the gravity field coefficients or the central body's state vector at a given reference epoch, as local parameters. In the second procedure, called batch with overlap filter, the covariance of the propagated parameters is not mapped, since a priori values for each parameter are fixed at the beginning of the process for all the arcs to non-constraining values. Furthermore, to have a smoother solution, the data batches are overlapped. A simplified scheme of the implementation of both filters is provided in Fig. 9. In both approaches, each data arc is solved with WLS filter, which is the building block of the POD algorithms. Future works will assess if the same methods could be used also during the user navigation phase of this mission or if a sequential filter will be needed. This consideration will depend mainly on the quality of the ephemeris ageing during the length of the arc. Weighted least squares with a priori information The OD problem is heavily non-linear as both the state vector X and the observation vector Y are described respectively by a set of n, the number of parameters to estimate, first order non-linear equations F(x, t), and m, the number of observables in the batch, non-linear equations G (x,t). In order to apply the least-square principle, that is a linear estimator, the problem shall be linearized expanding around a reference trajectory x ref , and truncating at the first order. The problem is then reformulated in terms of the state deviation vector x relative to the reference trajectory, and the observation deviation vector y describing the difference between the observed observables and the computed observables on the reference trajectory. This produce the differential corrections estimation method where the governing equation are given by: Where: ẋ(t) = A(x, t)x(t)y(t) =H(x, t)x(t) + ε(t)A(x, t) = ∂F(x, t) ∂x(t) ⃒ ⃒ ⃒ ⃒ x=xrefH (x, t) = ∂G(x, t) ∂x(t) ⃒ ⃒ ⃒ ⃒ x=xref ε(t) ∼ N(0, R) Here x(t) is the unknown state deviation at the observation time t. There is a different state deviation for each set of observables at time t. Instead, ε(t) is a random vector of size m accounting for the observation errors due to measurement noise, assumed to be gaussian with zero mean and covariance matrix R. The matrices A and H contains partial derivatives computed along the reference trajectory for the dynamical and the observations model respectively. Note that H (x, t)x(t) is the correction term accounting for all the modelling and deterministic observation errors. Indeed, we are explicitly assuming the noise is zero mean, so that the estimate is unbiased. This means that any bias in the collected observations shall be calibrated a priori or estimated within the state vector. The key step in a batch filter is to relate all the y(t) to the same state deviation x 0 at time t 0. For this, the state transition matrix Φ(t i , t 0 ) is used to multiply to produce each row of the design matrix H so that a single observation deviation equation can be written for the full batch of observables (the vector y and ε are now not referred to a specific time): y = Hx 0 + ε Where: x(t) = Φ(t, t 0 )x(t 0 ) H = [H(x, t 1 )Φ(t 1 , t 0 )⋮H(x, t m )Φ(t m , t 0 ) ] The state transition matrix is obtained by solving the linear differential equation Φ (t, t 0 ) = A(x, t)Φ(t, t 0 ) with the initial condition Φ(t 0 , t 0 ) = I. The obtained equation is also known as the batch data equation (a linear vector equation of size m). We look for x 0 , the differential correction to the state vector relative to the reference trajectory of the current iteration, by minimizing the sum of squares of the residuals ε(t), so that we can correct for the wrong values in the reference state vector at time t 0 and obtain post-fit residual containing only the unavoidable observation noise. The a priori information are included considering an equivalent data equation that relates the a priori knowledge on the state vector deviation relative to the same reference trajectory as above. That is a data equation were H--I and the associated noise covariance is the a-priori state covariance matrix P: x = x 0 + ε(t) Where: ν(t) ∼ N(0, P) Here x is the a priori state deviation relative to the reference trajectory at the estimation epoch t 0. It is usually zero for the first iteration, when the same reference trajectory and model used to linearize the problem is also that providing the a priori information. On subsequent iterations instead, this it will be different from zero as we will use the updated trajectory as new reference point, but a priori information does not change. If we call x k the estimated state deviation at iteration k, the next a priori state deviation will be: x k+1 = − ∑ k i=1x i Introducing the m-by-m weighting matrix W, the least squares solution is obtained by minimizing the scalar cost function C, given by the sum of squares of the observed and the a priori residuals: C = (y − Hx) T W(y − Hx) + (x − x) T P − 1 (x − x) The solution for the estimated state deviation x and estimated state covariance P is: Fig. 9. Batch-sequential scheme (top) and batch filter with overlap (bottom). x =P ( H T Wy + P − 1 x )P = ( H T WH + P − 1 ) − 1 Numerical simulations In our numerical simulation, we have created synthetic observables by adding a realistic measurement noise for ISL and ground-tracking based on the error budgets shown in Section 4. As shown, the ISL radio link accuracy depends on the inter-satellite distance, while the Earth-link accuracy have been set to 0.04 mm/s @ 60s, all in X-band. The error budget of the pseudolites links has not been investigated in this study, where we conservatively assumed the same accuracy of the Earth link. To estimate the positioning performances of the constellation nodes, we exploited POD algorithms based on the two filters described in Section 5. The dynamic equations of the simulated trajectories account for the Mars gravity field expressed in spherical harmonics to degree and order 120 (MRO120D [5]), the gravitational attraction from Solar system planets and the Martian satellites in a relativistic context, and the non-gravitational acceleration caused by Mars albedo, its infrared radiation, and SRP on the satellites. The spacecraft shape and thermo-optical properties are based on the SmallSat design being developed in Argotec and described in Section 3. Different mission scenarios were investigated to compare orbital solutions obtained through the following analysis: 1. Intersatellite radio only, simulated 24/24 h; 2. Alternating combination of ISL and deep-space radio link; 3. Alternating combination of ISL and pseudolites link. Numerical simulations of these nominal scenarios were carried out with ESA Godot software to investigate the expected positioning accuracies. We performed a perturbation analysis for both constellations, comparing the positioning performance achievable with the two estimation filters we selected. The choice of the batch duration (arc length) is particularly important for a fast convergence and a good accuracy of the final orbital solutions. Its length is a trade-off between the complexity of the estimation procedure (amount of data to be processed in a batch) and the estimation accuracy (the more data processed, the better accuracies). The analysis was carried out for a mission duration of 24 days, with an arc length of 4 days for the batch-sequential filter, while for the batch filter we considered batches of 7 days with 4 days of overlap. We perturbed the dynamical model of the estimation process by limiting the spherical harmonics expansion of Mars gravity field up until degree and order 20 (to limit the computational cost on-board the satellite) and modified the thermo-optical coefficients of the spacecraft shape elements as to introduce a ~10% mismodeling error on both the SRP and the Mars albedo accelerations. Hence, we added a set of whitenoise stochastic accelerations along the Radial, Transverse and Normal (RTN) directions as estimated parameters (local parameters) to absorb the errors. We selected a time update of 3 h after an accurate analysis of the positioning performance obtained considering a wide spectrum of values. Autonomous system results We report in Figs. 10 and 11, the estimation error, i.e., the difference between the estimated and simulated trajectories, the 1-σ and 3-σ uncertainties for the case with ISL alone, of the R, T and N components for the two constellations using both filters. This analysis shows, in the case of ISL-only and constellation A, that the trajectories cannot be recovered with confidence (the error in the normal component is larger than formal uncertainty). As shown in The large discrepancy on the normal component can be explained by considering that the ISLs between the nodes of the constellation provide information only on the relative motion between the spacecraft, hence a translation of the entire constellation in a direction orthogonal to the radial does not change remarkably the relative Doppler and range between the nodes. Furthermore, in case of ISL-only, the orbit determination process relies on gravity gradients, which are well known from the already acquired knowledge of the Mars gravity field and rotational state. Considering that at large distances from Mars the effect of the gravity gradients is negligible, this constellation requires ground support to determine its orbits. Consequently, in this case, the covariance matrix is underestimated; an effect that is more influent with the use of the batch-sequential filter, where the covariance is mapped to extract the priori values of the subsequent arcs. The constellation B, on the contrary, can autonomously reconstruct its position with higher stability, thanks to the dominance of orbit accelerations coming from Mars' gravity field over the solar radiation pressure and other perturbation effects, due to the lower altitude of the orbits. As shown in Fig. 10, using the batch-sequential filter, after a warmup period of about 15 days, we obtain a positioning accuracy of ~10 m, gradually improving as additional Doppler data are analysed with the filter. On the contrary, using the simple batch filter with overlaps, the 3σ accuracy do not improve over time, since the filter is reinitialized without information from previous data arcs. On the normal direction, the accuracy remains steadily around 40 m. For both estimation filters, the radial component is the one better determined, with accuracies in the order of 4 cm, stable over time. Also, the estimation error (solid lines) always remains inside the quoted uncertainty envelopes, indicating the goodness of the dynamical model (and thus of the selection of empirical accelerations). For what concerns the absolute velocity, the Radial and Transverse directions are very well determined (below 0.02 mm/s), while the Normal direction proves again to be the weak direction, with an uncertainty of 5 mm/s and 10 mm/s for the two different estimation filters. If ground support is included (see next paragraph), performances can be increased by more than an order of magnitude. Fig. 11 reports the estimation error and corresponding uncertainty for the relative position and relative velocity between the MSC and one of the DSC projected on the line-of-sight. The relative positioning accuracy is in the order of 5 m and 1 mm/s on average between the nodes. To conclude, by using a batch-sequential filter, constellation B can autonomously reconstruct its trajectory with high accuracy and stability, thanks to the dominance of orbit accelerations coming from Mars' gravity field over the solar radiation pressure, due to the low altitude of the orbits. Mismodelling in Mars gravity and non-gravitational accelerations are correctly accounted for by means of the set of empirical accelerations that have been included in the dynamical model. This can be deduced by the estimation error being always lower than the 3-σ uncertainties. Quasi-autonomous system results In this Section, the orbit determination is performed in a case of a quasi-autonomous system, realized alternating the ISL with groundtracking or with the use of pseudolites on the Martian surface. Positioning performance with alternating combination of ISL and ground-tracking To assess the duration of the Earth-link required by the constellation A to determine the orbits of the nodes, we performed the perturbation analysis simulating an increasing time of ground tracking support, established alternatively with the ISL. We chose the batch-sequential filter for its better performances with respect to the batch filter with overlap, as demonstrated in Section 6.1. In Fig. 12, we illustrate the position error and its relative formal uncertainty obtained after 24 days of simulations in different ground-tracking scenarios. Even 2 h of Earth tracking every 6 days can substantially improve the positioning performance of the constellations, reaching precisions of the order of 10 m. Note that augmenting the ground link duration improves the knowledge on the normal direction and grants an absolute reference point to the constellations, but at the same time decreases the amount of ISL data (the links cannot be simultaneous). This diminution, after a certain threshold, is disadvantageous because the ISL is far more accurate than the Earth-link. So, for this constellation, the best compromise is to alternate the. ISL with ground link in a range between 1 h and 4 h every 3 days. The same analysis has been carried out for the constellation B, showing performance of the order of ~10 cm with 2 h every 6 days of ground-tracking. The best positioning performances are achieved with 4 h every 3 days of ground-tracking, reaching ~10 cm of accuracy. Positioning performance with alternating combination of ISL and pseudolites Implementing the capability of establishing a radio-link from wellknown positions of the Mars surface would allow to drastically augment the positioning performances of the constellations. Pseudolites can provide reference points for the orbit determination, serving a similar role as Earth ground stations. The pseudolite is equipped with a transponder that receives and coherently re-transmits back a Doppler tracking signal. This scheme allows establishing two-way coherent links between the MSC and the pseudolites on the Martian surface. After an initial step in which the positions of the pseudolites have been determined to high accuracy (e.g., with radio links to ground), the location of the pseudolites can be considered as well known and used as fixed points on Mars surface to aid the navigation of the constellation. The pseudolites shall be deployed on the Mars surface either as independent units requiring a dedicated mission or, leveraging the low mass and power requirements, as a piggyback of other missions (e.g., commercial/scientific landers or rovers, surface hubs). Having pseudolites on the Mars surface is surely a more expensive solution than establishing an Earth-link, but it's important to assess the beneficial effect on the orbital determination performances. In these simulations, we considered a link between the MSC and three pseudolites equally spaced on the equator of Mars. For both constellations, the accuracy of the positioning reaches the level of tens of cm, improving of one order of magnitude for constellation A, while remaining approximately at the same level for constellation B, which can be determined, anyhow, more precisely, as shown in Fig. 13. Conclusions We proposed a satellite navigation system able to determine, in almost complete or full autonomy, depending on the altitude of the orbits, the ephemeris of the constellation and we analysed the positioning performance in terms of satellites absolute positioning with respect to Mars. This system exploits a Martian constellation of 5 small satellites on three quasi-circular high-altitude polar orbits, connected via microwave ISL that generate high accuracy Doppler observables. This navigation system is capable of effectively supporting the new missions planned or proposed in the coming years to explore the Martian polar caps. It could be used in the future for supporting the navigation of other probes, such as rovers or landers during the EDL phase. In fact, although this work does not address yet the positional accuracy of the end user, our analysis shows that the pillar of every satellite navigation system, i.e., precise OD, can be attained by means of a relatively simple radio system configuration, fully compatible with the deployment of a SmallSat constellation. Chronization requirements, techniques and performance with a terrestrial time and within the constellation are presented in [53]. Future works will assess the performance achievable with this concept for spacecraft time synchronization. The constellations orbits have been chosen to grant coverage of the polar caps and ground-tracks repetitiveness of two days and one day, respectively. Numerical simulations of the OD process were carried out to investigate the navigation performances of this novel concept. A perturbation analysis, carried out for an autonomous ISL-only case by introducing a 10% mismodeling error on the largest non-gravitational acceleration, shows a positioning accuracy of ~10 m after a warm-up period of about 15 days, if the constellation is sufficiently close to Mars, as our simulations show in the case of constellation B. With higher semi-major axes, a limited but periodic ground-tracking is required. We show that just 2 h every 6 days can be sufficient to obtain ~10 m positioning accuracy. The best performances are achieved by the lower constellation with 4 h every 3 days of ground-tracking, with accuracies down to ~10 cm. To obtain the same results with the higher constellation, we propose a configuration of 3 pseudolites equally spaced on the Mars equator. This solution, while quite more expensive, would greatly strengthen every orbital solution and provide a strong tie between the constellation and the Mars-centered, Mars-fixed reference frame. The pseudolites would also serve a future, global navigation system. The numerical simulations reported in this study were conducted by analysing deep-space, Mars pseudolites and intersatellite data both with a batch-sequential least-squares filter and a batch filter with overlap, showing that the first one is the preferred method in terms of achievable positioning performances. The main strength of this novel technology with respect to a GRACE/ GRAIL radio link configuration is a significant reduction of the required spacecraft mass and power, enabling the use of small satellites, while still providing highly accurate radio tracking measurements. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig. 1 . 1Orbits of the constellations A (top) and B (bottom) in the IAU body-fixed frame and spacecraft relative positions. Fig. 2 . 2Spacecraft ground-track constellation A (top) and B (bottom). Fig. 3 . 3Schematic configuration of the inter-satellite links, for a five-node constellation. Sporadic communications with ground occur through the MSC. • Two Tx/Rx Antennas: given the orbital configuration, two antennas, facing opposite sides, are needed to allow simultaneous communication with all the DSC. An ISL budget and coverage analyses have been performed to select the antennas. A couple of TX/RX patch antennas have been identified with a gain at boresight >11.5 dBi between 8 GHz and 8.4 GHz and an HPBW of 40 deg (which is in line with the maximum angular separation of two DSC). • X-Band Transceiver: The transceiver shall support, for the simultaneous communication with all the four satellites of the constellation, a spread spectrum (SS) multichannel architecture based in transmission on a CDM-M scheme and in reception on a classical parallel SS receiver (CDMA solution) Fig. 4 . 4Block diagram of GRAIL (top) Fig. 5 . 5Argotec's HAWK-6U platform. Fig. 6 . 6MSC -12U CubeSat layout showing the main external equipment. Fig. 7 . 7DSC -6U CubeSat layout showing the main external equipment. Fig. 8 . 8Error budget for the ISL. The USO noise is represented at 7000 km and 4000 km (maximum distances for constellation B and A, respectively), and thermal noise at SNR = 30 dB-Hz (reference value from link budget). Fig. 10 . 10Estimation error (solid lines) with 1-σ and 3-σ uncertainties (shaded area) of RTN components, in case of ISL-only, for constellation B, with a batch sequential filter (a) and with a batch filter with overlap (b). Fig. 10 , 10with both filters, we obtain a positioning error of ~500 m on the normal direction at the first arc, and it's not improved by the following additional Doppler data. Fig. 11 . 11Estimation errors (solid lines) with 1-σ and 3-σ uncertainties (shaded areas) along the line of sight (top: position, bottom: velocity). Results are obtained in the case of ISL-only, with a batch sequential filter (top) and with a batch filter with overlap (bottom). Fig. 12 . 12Estimation error (black) and formal uncertainty (blue) after 24 days of simulation with batch-sequential filter for different duration of Earth link, in case of constellation A (top) and constellation B (bottom). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) Fig. 13 . 13Estimation error (black) and formal uncertainty (blue) after 24 days of simulation with batch-sequential filter for different duration of pseudolites link, in case of constellation A (top) and B (bottom). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) , whileFig. 2shows the spacecraft ground-Acronyms/abbreviations ADEV Allan Deviation AOCS Attitude & Orbit Control System ARG Argotec CDM Code Division Multiplexing CDMA Code Division Multiple Access CCSDS Consultative Committee for Space Data Systems DOWR Dual-One-Way Ranging DSC daughter-craft DSN Deep Space Network EDL Entry, Descent and Landing EIRP Equivalent Isotropic Radiated Power GRACE Gravity Recovery and Climate Experiment GRAIL Gravity Recovery and Interior Laboratory GPS Global Positioning System HPBW Half Power Beam Width ISL Inter-Satellite Link LOS Line of Sight KF Kalman Filter MSC mother-craft OD Orbit Determination POD Precise Orbit Determination RTLT Round Trip Light Time SNR Signal to Noise Ratio TT&C Telemetry, Tracking & Command SRP Solar Radiation Pressure SS Spread Spectrum USO Ultra Stable Oscillator WLS Weighted Least Squares Table 1 1Orbital parameters of constellation A.M D1 D2 D3 D4 a (km) 12868.6 e 0.007 i (deg) 90 75 75 105 105 Ω (deg) − 32 − 32 − 32 − 32 − 32 ω (deg) 270 270 270 270 270 M0 (deg) 0 − 8 8 12 − 12 Table 2 2Orbital parameters of constellation B.M D1 D2 D3 D4 a (km) 8106.7 e 0.007 i (deg) 75 60 60 90 90 Ω (deg) − 32 − 32 − 32 − 32 − 32 ω (deg) 270 270 270 270 270 M0 (deg) 0 − 42 42 48 − 48 AcknowledgmentsPart of this work was supported by a scientific agreement among University of Rome Sapienza, Italian National Research Council (CNR), and Argotec s. r.l. for a Ph.D. doctorate in aerospace engineering. 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S Molli, G Boscagli, M Di Benedetto, D Durante, L Vigna, L Iess, 9th International Workshop on Tracking, Telemetry and Command Systems for Space Applications (TTC) (2022). In pressMolli, S., Boscagli, G., Di Benedetto, M., Durante, D., Vigna, L., Iess, L., Time transfer and orbit determination for a Martian navigation system based on smallsats, 9th International Workshop on Tracking, Telemetry and Command Systems for Space Applications (TTC) (2022). In press.
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[ "E quilibrium and dynam ical properties of the A N N N I chain at the m ultiphase point", "E quilibrium and dynam ical properties of the A N N N I chain at the m ultiphase point" ]
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A bhi shek D har,B .Sri ram Shastry and C handan D asgupta Physics D epartm ent, Indian Institute ofScience,B angal ore 560012,India.(January 11,2022) W estudy theequi l i bri um and dynam i calproperti esoftheA N N N I(axi alnext-nearest-nei ghborIsi ng)chai n atthem ul ti phase poi nt. A n i nteresti ng property ofthe system i sthe m acroscopi c degeneracy ofthe ground state l eadi ng to ni te zero-tem perature entropy. In our equi l i bri um study w e consi der the e ect of softeni ng the spi ns. W e show that the degeneracy of the ground state i s l i fted and there i s a qual i tati ve change i n the l ow tem perature behavi our of the system w i th a w el l de ned l ow tem perature peak of the speci c heat that carri es the therm odynam i c \w ei ght" of the ground state entropy. In our study of the dynam i cal properti es, the stochasti c K aw asaki dynam i cs i s consi dered. T he Fokker-Pl anck operator for the process corresponds to a quantum spi n H am i l toni an si m i l ar to the H ei senberg ferrom agnet butw i th constrai ntson al l ow ed states. T hi s l eads to a num ber ofdi erences i n i ts properti es w hi ch are obtai ned through exact num eri caldi agonal i zati on,si m ul ati ons and by obtai ni ng vari ous anal yti c bounds. PA C S num bers: 05. 50. + q,02. 50. Ey,05. 40. -a I. IN T R O D U C T IO N Pl anck operator and quantum chai ns has often been ex-
10.1103/physreve.62.1592
[ "https://export.arxiv.org/pdf/cond-mat/0004207v1.pdf" ]
23,040,373
cond-mat/0004207
db8bd167b803eb760b61ed3f1988671d3495875b
E quilibrium and dynam ical properties of the A N N N I chain at the m ultiphase point 13 Apr 2000 E quilibrium and dynam ical properties of the A N N N I chain at the m ultiphase point 13 Apr 2000 A bhi shek D har,B .Sri ram Shastry and C handan D asgupta Physics D epartm ent, Indian Institute ofScience,B angal ore 560012,India.(January 11,2022) W estudy theequi l i bri um and dynam i calproperti esoftheA N N N I(axi alnext-nearest-nei ghborIsi ng)chai n atthem ul ti phase poi nt. A n i nteresti ng property ofthe system i sthe m acroscopi c degeneracy ofthe ground state l eadi ng to ni te zero-tem perature entropy. In our equi l i bri um study w e consi der the e ect of softeni ng the spi ns. W e show that the degeneracy of the ground state i s l i fted and there i s a qual i tati ve change i n the l ow tem perature behavi our of the system w i th a w el l de ned l ow tem perature peak of the speci c heat that carri es the therm odynam i c \w ei ght" of the ground state entropy. In our study of the dynam i cal properti es, the stochasti c K aw asaki dynam i cs i s consi dered. T he Fokker-Pl anck operator for the process corresponds to a quantum spi n H am i l toni an si m i l ar to the H ei senberg ferrom agnet butw i th constrai ntson al l ow ed states. T hi s l eads to a num ber ofdi erences i n i ts properti es w hi ch are obtai ned through exact num eri caldi agonal i zati on,si m ul ati ons and by obtai ni ng vari ous anal yti c bounds. PA C S num bers: 05. 50. + q,02. 50. Ey,05. 40. -a I. IN T R O D U C T IO N Pl anck operator and quantum chai ns has often been ex- T he A N N N I chai n i s one ofthe si m pl est system s w i th com peti ng i nteracti ons. It i s de ned by the fol l ow i ng Isi ng spi n H am i l toni an: H = L X i= 1 (J 1 s i s i+ 1 + J 2 s i s i+ 2 ); s i = 1:(1) For J 2 > 0,the i nteracti ons are com peti ng and one can have di erent ground states dependi ng on the rel ati ve strengthsofthe i nteracti ons.A speci al l y i nteresti ng case i s the poi nt J 1 = 2J 2 , the so-cal l ed m ul ti phase poi nt, w here the ground state i s no l onger uni que. It can be show n that any spi n con gurati on,w hi ch does not have threeconsecuti vespi nsofthesam esi gn,i sa ground state. For a chai n of l ength L, the num ber of ground states L ,w here = ( p 5 + 1)=2 i s the gol den m ean. T hus there are an exponenti al l y l arge num ber of degenerate ground statesand the system has ni te zero-tem perature entropy perspi n.T hem odelhasbeen extensi vel y studi ed both i n one and hi gherdi m ensi ons and i s know n to have a ri ch and i nteresti ng phasedi agram [ 1] .In thi spaperwe consi der som e aspects ofthe equi l i bri um and dynam i cal behavi our ofthe A N N N I chai n at the m ul ti phase poi nt. In ourequi l i bri um study we consi derthe e ectofsofteni ng the spi ns,thati sal l ow i ng them to take conti nuous i nstead of di screte val ues. It i s usual i n the study of spi n m odel sto consi dersoft-spi n versi onsofdi screte spi n m odel s. A wel l -know n exam pl e i s the G i nzberg-Landau H am i l toni an w hi ch i sa conti nuum versi on ofthe di screte Isi ng m odel . O ther exam pl es occur i n the study ofspi n gl ass m odel s. For i nstance,the soft spi n versi on ofthe Sherri ngton-K i rkpatri ck (SK ) [ 2]m odelwas studi ed [ 3] i n the contextofdynam i cs.T he reason forgoi ng to softspi n versi ons i s that they are often m ore am enabl e to theoreti calapproaches.Iti susual l y expected thatqual itati vel y the softand hard spi n versi onsshoul d show si mi l ar behavi our. For m odel s w i th m ul ti pl e ground states,ari si ng outof frustrati on,softnessm ay howeverchange the degeneracy com pl etel y,as m ay be seen i n a three-spi n exam pl e,or as i n the present case as we shal lshow here. T he e ect ofspi n-softeni ng i n system s w i th com peti ng i nteracti ons has been studi ed earl i er by several authors. Seno and Yeom ans [ 4]have l ooked at the e ect ofsofteni ng spi ns at the m ul ti phase poi nt ofa cl ock-m odel . T hey nd,usi ng a perturbati ve m ethod,that as a softness param eter i s vari ed the system goes through a seri es of di erent ground states.In thi swork we use a si m i l arperturbati ve m ethod to prove that the m acroscopi c degeneracy ofthe ground state i n the A N N N Im odeli s l i fted by the sm al lestam ountofsoftness. W e then show expl i ci tl y how the rel ease of the zero-tem perature entropy resul ts i n quali tati ve di erences i n the l ow tem perature properti es of the system . T hi s i s si m i l arto the recentl y observed phenom ena of entropy rel ease i n spi n-i ce system s [ 5] . W e al so construct an e ecti ve hard-spi n H am i l toni an to descri be the l ow -tem perature properti es of the soft-spi n m odel .W e have al so perform ed M onte C arl o si m ul ati ons on the soft-spi n m odeland veri ed the l ow -tem perature predi cti ons ofthe e ecti ve H am i l toni an. In the second part of the paper we l ook at the dynam i calproperti es of the system . A s noted before,the A N N N Im odelat the m ul ti phase poi nt has a l arge number of degenerate ground states. It i s, therefore,of i nterest to l ook at dynam i calproperti es ofthe system at l ow tem peratures. H ere we use K awasakidynam i cs to evol ve the system and consi der zero tem perature properti es onl y. T hus two nearest nei ghbor spi ns i p w i th a rate , provi ded both m agneti zati on and energy are conserved. T hi s dynam i cs has been studi ed earl i er by D as and B arm a [ 6] . In thi s paper we extend thei r studi es by usi ng the correspondence between W -m atri ces for stochasti c processes and quantum spi n chai ns. T he correspondence between the stochasti c Fokker-pl oi ted to deri ve dynam i calproperti es. Fori nstance,the scal i ng,w i th system si ze,ofthe rst exci ted state ofthe quantum H am i l toni an gi ves the dynam i calexponent of the stochasti c process.A wel l -know n exam pl e w here thi s correspondence has been used i s i n excl usi on processes [ 7] , w hi ch are stochasti c m odel s of hard-core di usi ng parti cl es. For such processes, i t has been possi bl e to exactl y cal cul ate the dynam i c exponent by sol vi ng the correspondi ng quantum m odel , nam el y the H ei senberg m odel [ 8] . T he dynam i cs consi dered by us i s very si mi l ar to the sym m etri c excl usi on process (SEP) but w i th added restri cti onson al l owed m oves.To see thi s we rst note that w i th the present dynam i cs, nearest nei ghbor spi nsw i th opposi te si gns i p,provi ded thatthe resul ti ng con gurati on sati s es the ground state constrai nt ofno three successi ve spi ns havi ng sam e si gns. Identi fyi ng up spi nsw i th parti cl esand dow n spi nsw i th hol esweseethat the dynam i cs i s equi val ent to hard core parti cl es di usi ng on a l atti ce w i th the constrai ntthat there cannotbe three successi ve parti cl es or hol es. A n i nteresti ng questi on i s w hether these rather strong constrai nts m ake the system di erent from the SEP. Earl i er num eri cal work [ 6]seem s to suggest that the dynam i cs sti l lbehaves l i ke the SEP.W e note that there have been som e other recent studi es on excl usi on processes w i th constrai nts on al l owed con gurati ons [ 9] . T hese cases are sol vabl e by the B ethe ansatz and show the sam e behavi our as the unconstrai ned m odel . H ere we address thi s questi on of the e ect of the constrai nts by studyi ng the quantum H am i l toni an. B y m eansofexactnum eri caldi agonal i zati on for ni techai ns and through anal yti c bounds, we have tri ed to understand the di erences and si m i l ari ti es of the present H am i l toni an w i th the H ei senberg H am i l toni an for the SEP.W e al so di scuss the di erent sym m etry properti es ofthe two quantum m odel s. T he H ei senberg m odelhas ful lrotati onalsym m etry and thi s has severali m portant i m pl i cati ons som e ofw hi ch are ofdi rectrel evance i n understandi ng the ori gi nalstochasti c process. For exam pl e i t i m pl i es that two-poi nt ti m e correl ati ons i n the SEP do not depend on the num ber ofparti cl es. T he present m odel , on the other hand i s onl y i nvari ant under rotati ons i n the X Y pl ane. T he rest ofthe paper i s di vi ded i nto two secti ons. In secti on (II), we consi der equi l i bri um properti es of the soft-spi n m odelw hi l e i n secti on (III)we consi derthe dynam i cs ofthe hard-spi n m odel . Secti on (IV ) contai ns a sum m ary of our m ai n resul ts and a few concl udi ng rem arks. H s = X i J(2s i s i+ 1 + s i s i+ 2 )+ a g(s 4 i =4 s 2 i =2); s i 2 ( 1 ;1 ) (2) w here a i s a di m ensi onl ess param eter w hi ch control s the am ountofsoftness. In the l i m i ta ! 1 we getthe hardspi n m odel . W e w i l lset g = 1 si nce there i s no l oss of general i ty i n doi ng so. Let us rst l ook at the ground states ofthe soft-spi n H am i l toni an gi ven by Eqn. (2). To do so we l ook at the extrem a ofH s w hi ch areobtai ned by setti ng @H =@s i = 0 for al li. T hi s gi ves: 2J(s i+ 1 + s i 1 )+ J(s i+ 2 + s i 2 )+ a(s 3 i s i )= 0 (3) Sol vi ng thi ssetofcoupl ed nonl i nearequati onsi n general i s very di cul t. H owever for sm al lval ues ofthe parameter 1=a we can obtai n the sol uti ons perturbati vel y. For a ! 1 al l con gurati ons, fs i g, w i th s i = 0; 1 are sol uti ons. T hose w i th fs i = 1g correspond to the m i ni m a. For ni te butl arge a we try to obtai n the sol uti ons perturbati vel y w i th 1=a acti ng as the perturbati on param eter. W e denote the unperturbed m i ni m a by the set ft i = 1g. Let us try the fol l ow i ng perturbati ve expansi on: s i = 1 X n = 0 t (n ) i ( 1 a ) n ;(4) w herethe coe ci entst J[ 2(t i+ 1 + t i 1 )+ (t i+ 2 + t i 2 )]+ J a [ 2(t (1) i+ 1 + t (1) i 1 )+ (t (1) i+ 2 + t (1) i 2 )]+ 2t (1) i + 1 a (3t i (t (1) i ) 2 + 2t (2) i )+ O ( 1 a 2 )= 0: Equati ng di erent powers of1=a to zero we then get: t (1) i = J 2 [ 2(t i+ 1 + t i 1 )+ (t i+ 2 + t i 2 )]+ O ( 1 a ) t (2) = J 2 [ 2(t (1) i+ 1 + t (1) i 1 )+ (t (1) i+ 2 + t (1) i 2 )] 3 2 t i (t (1) i ) 2 and so on. T hus we get 2 L perturbed m i ni m a gi ven by the above perturbati on seri es. T he energi es correspondi ng to these m i ni m a can now be found by putti ng these sol uti ons i nto the expressi on for energy i n Eqn. (2). W e thus get E = E 0 + E 1 + E 2 + O (1=a 2 ); w here E 0 = La 4 E 1 = X i J(2t i t i+ 1 + t i t i+ 2 ) E 2 = 1 a X i [ (t (1) i ) 2 + 2J(t i t(1)i+ 1 + t i+ 1 t (1) i )+ J(t i t(1)i+ 2 + t i+ 2 t (1) i )] = J 2 2a X i (5 + 4t i t i+ 1 + 4t i t i+ 2 + 4t i t i+ 3 + t i t i+ 4 ):(5) In the above expansi on, E 0 corresponds to the unperturbed energy,w hi l e E 1 and E 2 representthe correcti ons resul ti ng from the perturbati on. In the a ! 1 l i m i t the term E 1 causes the energy l evel s of the 2 L m i ni m a to spl i t, w i th separati on between them O (J). W e recogni ze E 1 as the H am i l toni an for the hard-spi n A N N N I m odel . T hus the l owest energy l eveli s sti l l L -fol d degenerate.T he term E 2 then causes a further spl i tti ng of the ground statesi nto l evel sw i th separati on O (J 2 =a). To see w hether or not the m acroscopi c degeneracy of the ground state survi ves,we need to consi der the i nteracti on H am i l toni an correspondi ng to theenergy term E 2 . Si nce we are i nterested i n the spl i tti ng ofthe l owest energy l evelofE 1 ,we onl y consi derthe restri cted subspace ofspi n con gurati ons w hi ch are ground states ofE 1 . In thi s subspace the H am i l toni an correspondi ng to E 2 can be rew ri tten as H 2 = 3LJ 2 2a J 2 2a X i (2t i t i+ 2 + 4t i t i+ 3 + t i t i+ 4 ): (6) T husal lthe i nteracti onsare ferrom agneti c.H oweverthe ground stateofH 2 i snotthestatew i th al lspi nsup,si nce thi s does not bel ong to the subspace ofground states of E 1 . To nd the ground state,we w ri te the second term i n H 2 ,w hi ch we denote by h 2 ,i n the fol l ow i ng form (the constant factor J 2 =(2a) i s suppressed): h 2 = X i (2t i t i+ 2 + 4t i t i+ 3 + t i t i+ 4 ) = X i= (4n + 1) (t i ;t i+ 1 ;t i+ 2 ;t i+ 3 jt i+ 4 ;t i+ 5 ;t i+ 6 ;t i+ 7 ) w here (t 1 ;t 2 ;t 3 ;t 4 jt 5 ;t 6 ;t 7 ;t 8 )= t 1 t 3 + t 2 t 4 + 2t 3 t 5 + 2t 4 t 6 + t 5 t 7 + t 6 t 8 + 2t 1 t 4 + 4t 2 t 5 + 4t 3 t 6 + 4t 4 t 7 + 2t 5 t 8 + t 1 t 5 + t 2 t 6 + t 3 t 7 + t 4 t 8(7) and the i ndex n runs from 0 to (L=4 1) (we take L to be an i ntegralm ul ti pl e of4). B y enum erati ng the m atri x el em ents (t 1 ;t 2 ;t 3 ;t 4 j t 5 ;t 6 ;t 7 ;t 8 ) for al lal l owed spi n con gurati ons we nd that the l owest energy con gurati on i sobtai ned fortheperi odi csequence(""#""# : : : )and the ve othercon gurati onsobtai ned by transl ati ng and i ppi ng thi s.T huswe nd thatthe i n ni te degeneracy of the ground state i srem oved and i nstead we geta si x-fol d degenerateground state.W enotethattheprocedurejust outl i ned provi des a strai ght forward m ethod of ndi ng the ground state of any spi n H am i l toni an. B y num erical l y sol vi ng Eqn.(3)forsm al ll atti ce si zes(L = 12)and ndi ng the m i ni m um energy con gurati ons for a l arge enough (a = 50),we have veri ed that the perturbati ve sol uti ons are qui te accurate. T he fact that softeni ng ofthe spi ns resul ts i n rem oval ofthe exponenti aldegeneracy ofthe ground state m eans that the ni te zero tem perature entropy i s rel eased and we expect i t to show up i n the behavi our of the l ow tem perature speci c heat. T hi s l eads to the soft-spi n m odelhavi ng l ow -tem perature properti es very di erent from the hard-spi n versi on as we shal lnow see. In order to com pare the properti es of the soft-spi n m odel w i th those of the hard-spi n one, i t i s necessary to subtract from the soft-spi n free energy a part correspondi ng to the conti nuum degrees offreedom . W e thus l ook at the fol l ow i ng free energy: F = ( 1= )[ l n T re H s + L l n(2) l n T re H g ] ; (8) w here H s i sasi n Eqn.(2),H g = P i a(s 4 i =4 s 2 i =2) ,and T r i ndi cates i ntegrati on over al lspi n vari abl es. W e note thattheaboveexpressi on forthefreeenergy i sequi val ent to w ri ti ng the parti ti on functi on i n the form Z = T re H P ( s) w i th P ( s)= Y i 2e a(s 4 i =4 s 2 i =2) R ds i e a(s 4 i =4 s 2 i =2) :(9) H bei ng the ori gi nalhard-spi n H am i l toni an and P ( s) a probabi l i ty di stri buti on over the spi n vari abl es. In the l i m i t a ! 1 thi s exactl y reduces to the hard-spi n partiti on functi on w hi l e at T ! 1 one gets Z = 2 L . >From our si m ul ati ons we get properti es correspondi ng to the rst part of the free energy i n Eqn. (8). T he second part si m pl y corresponds to a noni nteracti ng system and i ts properti es can be easi l y com puted num eri cal l y. In Fi g. 1 we pl ot the speci c heat data C (T ) for both the soft-spi n and hard-spi n m odel s.T he hard-spi n resul t i sexactand correspondsto i n ni te system si ze w hi l e the soft-spi n data i s from si m ul ati ons on a chai n of l ength L = 24. T he val ues of vari ous param eters used i n the si m ul ati on werea = 50 and J = 1.T hehi gh tem perature (T > 1) data was obtai ned by averagi ng over 10 6 M onte C arl o steps w hi l e the l ow tem perature data i s over 10 7 steps. A s expected we nd a second peak i n the speci c heat at l ow tem peratures. For the hard-spi n case the totalarea underthecurveforC (T )=T i sequalto l n(2= ). T he ground state entropy,l n( ),w hi ch i s rel eased w hen the spi ns are softened, i s m ostl y accounted for by the area under the l ow tem perature peak. T he l ow tem perature properti es are qui te wel lreproduced by the e ecti ve H am i l toni an,H 2 ,w hi ch descri bes theenergy l evel si n thel owestband.T hetherm odynam i c properti es of H 2 can be exactl y cal cul ated by transfer m atri x m ethods,both for ni te system si zes and i n the i n ni te si ze l i m i t. In Fi g. 2 we pl ot the soft-spi n l ow tem perature si m ul ati on data C (T ) for two system si zes and com pare them w i th resul ts obtai ned from the e ecti ve H am i l toni an. W e see good agreem ent between the two. W e al so show the i n ni te system si ze C (T ) curve obtai ned from the H am i l toni an H 2 . It i s i nteresti ng to notethatthepeak val ueofthespeci cheat rsti ncreases w i th system si zeand then startsdecreasi ng beyond a certai n si ze. III. K A W A SA K I D Y N A M IC S O F T H E H A R D -SP IN A N N N I M O D E L A T T H E M U LT IP H A SE P O IN T A sfortheusualexcl usi on process,thequantum H am i ltoni an correspondi ng to ourprocesscan be easi l y w ri tten and i s gi ven by: H = P f L X k= 1 [ ( + k k+ 1 + k + k+ 1 )+ 1 2 ( z k z k+ 1 1)] P k gP(10) w here k are the usualPaul im atri ces,P k are l ocalprojecti on operatorsgi ven by P k = (1 k 2 k 1 )(1 k+ 2 k+ 3 )=4(11) and P = Q L k= 1 P k i s a gl obalprojecti on operator w hi ch projects onto the space ofal l owed states,i.e those that sati sfy the ground state constrai nt. T he spi n-i p rate, ,has been set to uni ty. A l ternati vel y we can w ri te the Fokker-Pl anck operator i n the fol l ow i ng form : H = L X k= 1 ( k + 2 k ) w here k = P ( + k k+ 1 + k + k+ 1 )P :(12) T he term P T he other com ponents of the total angul ar m om entum S x and S y however do not com m ute w i th H . T hus the present H am i l toni an has U (1) sym m etry i nstead of the SU (2) sym m etry of the H ei senberg m odel . A l so even though the ground states are degenerate,w i th one state i n every S z sector,there i s no anal ogue ofthe rai si ng/l oweri ng operatorS .Iftherewere such an operator then the enti re ei genval ue spectrum i n the n-parti cl e sector woul d be a subset ofthe (n 1)-parti cl e sector (for n < L=2). B y l ooki ng at the spectrum for ni te si zed l atti ces we have veri ed that thi s i s not so. To study the presence of l ong-range order i n the ground-state, we have cal cul ated the two-poi nt stati c correl ati on functi ons c z (r) = < z 0 z r > and c (r) = < + 0 r > i n theground stateforthehal f-l l ed sector.T he si m pl e characteri zati on ofthe ground states i n term s of di sal l owed subsequences enabl es cal cul ati on of groundstate expectati on of any operator by m eans of transfer m atri ces. T he transfer m atri x m ethod sum s over al l the di erent parti cl e sectors, but i n the therm odynam i cl i m i tthehal f l l ed sectordom i nates,and so weget correct resul ts (To com pute expectati on val ues i n other sectors one woul d need to i ntroduce a chem i calpotenti al ). T hus we nd that c z (r) = A cos( 2 r=3)e r= w here = 1=l og((3 + p (5))=2) = 1: 03904: : :and A and are constants that have di erent val ues on odd and even si tes. Fouri er transform i ng c(r) gi ves the structure factor < z ( q) z (q) > w hi ch has the form show n i n Fi g. 3. W e note that i t i s non-vani shi ng at al lq. T he o -di agonalcorrel ati on can si m i l arl y be obtai ned usi ng transfer m atri ces but the cal cul ati on becom es extrem el y cum bersom e. Instead we have com puted thi s correl ati on num eri cal l y for ni te l atti ces and nd that i t saturates,for l arge r,to a constant val ue,w hi ch i s gi ven by < > 2 = 0: 02917: : :(w hi ch has been obtai ned by usi ng the transfer m atri x m ethod). T hus we nd that ground-state correl ati on functi ons show the sam e behavi our as i n the H ei senberg chai n. A way from the two i sotropi c poi nts ( = 1),thi s has the sam e sym m etry as the presentm odel . Ithasno l ong range order i n the gapl ess phase ( 1 < < 1) and al l correl ati ons < (0) (r) > have power l aw decays. In the ferrom agneti c phase ( > 1), the m odelhas a gap and ful l ferrrom agneti c l ong-range order i n the ground state, w i th ul tra-l ocal l ongi tudi nal correl ati ons nam el y c z (r) = 1=4. T hus we see that as far as ground state correl ati ons are concerned the present m odeli s di erent from the ani sotropi c X X Z chai n even though they have the sam e sym m etry properti es. O ur m odeli s m ore si mi l ar i n properti es to the ferrom agnet ( = 1) but has a nontri vi aldepl eti on ofthe condensate,as wel las a nontri vi al< z 0 z r > correl ati on. Fi nal l y we note that rotati onal i nvari ance of the H ei senberg m odelm eansthattwo-poi ntti m ecorrel ati ons are com pl etel y determ i ned by si ngl e m agnon exci tati ons and so have the sam e behavi our i n any S z sector [ 8,11] . T hi sresul tdoesnothol d i n thecaseofthepresentm odel . A second conserved quanti ty i n the m odeli s the total l i near m om entum . T hi s fol l ow s from the transl ati on i nvari ance ofH . T he m om entum operatorcom m utes both w i th H and S z so that i n each S z sector energy states can be l abel l ed by thei r m om entum . C l earl y the ground state has zero m om entum . W e note here that though i t i s usual l y the rst exci ted state that determ i nes the decay ofcorrel ati ons i n the stochasti c process,i t i s possi bl e to construct correl ati on functi ons w hose decay i s governed by som e other ei genval ue. A s an exam pl e consi der the operator Q = e i L P k k z k . T hi s i s the so-cal l ed tw i st operator, rst studi ed by Li eb,Schul tzand M atti s [ 12] .In thi scase,the decay ofthe correl ati on,< Q (0)Q (t)> i s determ i ned by the l owestei genval ue atm om entum si nce the operator carri es m om entum . In Fi g. 5 we show the decay constantasdeterm i ned from the correl ati on decay fordi erent system si zes and com pare them w i th those obtai ned from exact di agonal i zati on. T he correl ati on functi on i s obtai ned from M onte C arl o si m ul ati ons and can al so be used forl argersystem si zesatw hi ch num eri caldi agonali zati on becom es too di cul t. T Q T y = e i L P k k z k + 1 = e i2 d Q w here d = n=L i s the l l i ng fracti on of parti cl es. If j 0 n > i stheground statei n then-parti cl esector,then the states z (q)j 0 n > and Q j 0 n > have m om enta q and 2 d respecti vel y and for q 6 = 0 are orthogonalto the ground state, w hi ch has zero m om entum . H ence the fol l ow i ng expectati on val uesgi ve ustwo di erentupperboundson the gap: (a) e z = < z ( q)H z (q)> < z ( q) z (q)>(14)(b) e Q = < Q y H Q >(15) w here < : : : > denotes ground state expectati ons. W e now eval uate (a) and (b). W e shal l henceforth restri ct oursel vesto the hal f-l l ed sectoronl y,though extensi ons to other sectors can be done. (a)To eval uatee z we rstnotethatthenum eratorand denom i natori n Eqn. (14)can be w ri tten i n the fol l ow i ng equi val ent form : < z ( q)H z (q)> = 1 2 X l e iql < [ z 1 ;[ H ; z l+ 1 ] ]> < z ( q) z (q)> = X l e iql < z 1 z l+ 1 >(16) T he com m utatoroccurri ng i n the above equati on can be eval uated and gi ves: Inserti ng thi s i n Eqn. (16), and usi ng transl ati onali nvari ance ofthe ground state we nal l y obtai n: < z ( q)H z (q)> = 4[ 1 cos(q)]< P P 1 ( + 1 2 + 1 + 2 )P y > = 2[ 1 cos(q)]< P P 1 (1 z 1 z 2 )P >(18) w here the l aststep hasbeen obtai ned usi ng the factthat < 0j H j 0 > = 0. A s noted before ground-state expectati ons ofany operator can be com puted usi ng transfer m atri ces. T he expectati on val ue on the rhs ofEqn. (18) i s thus found to have the l i m i ti ng val ue (as L ! 1 ) < P P 1 (1 z 1 z 2 )P > = 8 16= p 5.T heFouri ertransform of c z (r),w hi ch gi vesthe structure factor< z ( q) z (q)> , hasal ready been obtai ned and waspl otted i n Fi g. 3.W e note thati ti snon-vani shi ng atal lq. Fi nal l y,from Eqns. (14,16) we get e z w hi ch i s pl otted i n Fi g. 6 al ong w i th the exactresul tsfrom ni te si ze di agonal i zati on.Putti ng q = 2 =L and putti ng i n al lnum eri calfactors,we getthe fol l ow i ng resul t: 19: 78 2 L 2(19) (b) W e now obtai n the other bound usi ng the tw i st operator,Q . W e rst note the fol l ow i ng properti es ofQ : Q y + l l+ 1 Q j f g > = e i 2 L + l l+ 1 j f g > Q y l + l+ 1 Q j f g > = e i 2 L l + l+ 1 j f g > : (20) U si ng these rel ati ons we obtai n < 0jQ y H Q j 0 > = X k < 0j Q y P ( + k k+ 1 + k + k+ 1 )P k P Q j 0 > X k 1 2 < 0j Q y P ( z k z k+ 1 1)P k P Q j 0 > = cos(2 =L) X k < 0j P ( + k k+ 1 + k + k+ 1 )P k P j 0 > X k 1 2 < 0j P ( z k z k+ 1 1)P k P j 0 > = L 2 [ 1 cos(2 =L)]< 0j P (1 z k z k+ 1 )P k P j 0 > ; (21) w here i n the l aststep we have agai n used < O j H j 0 > = 0 and transl ati onali nvari ance ofthe ground state.T he expectati on val ue above has al ready obtai ned so that we get,for l arge L,the fol l ow i ng bound for the gap at m om entum q = . 0: 845 2 L(22) In Fi g. 4 we have pl otted both the boundsand the exact ni te si ze resul ts at q = 2 =L and q = as functi ons of the system si ze. IV . SU M M A R Y In sum m ary we have studi ed a one-di m ensi onalspi n m odelw i th com peti ng i nteracti ons and studi ed i ts l owtem perature equi l i bri um and dynam i cal properti es. In the equi l i bri um case we have show n that l ow tem perature properti es ofthe soft-spi n and hard-spi n versi onsof the m odelcan be very di erent. T he hard-spi n versi on of the m odelhas an i n ni tel y degenerate ground-state. T hrough a perturbati ve cal cul ati on we have show n,that assoon as we i ntroduce the sl i ghtestam ountofsoftness, the degeneracy i s l i fted. T he ground state energy l evel s spl i t to form a band w hi ch i s separated from hi gher l evel s by E = O (J). T he energy l evel s w i thi n thi s l owest band are descri bed by an e ecti ve hard-spi n H am i l toni an, contai ni ng ferrom agneti c i nteracti ons upto fourth nei ghbourterm s. T hi s can be used to approxi m atel y deri ve the l ow tem perature properti es of the m odel . W e nd reasonabl y good agreem entw i th resul tsfrom M onte C arl o si m ul ati ons ofthe soft-spi n m odel . O ur resul ts i ndi cate that the xed-l ength (a ! 1 ) l i m i ti s a si ngul arone i n ourm odelatl ow tem peratures. Si ncetheground stateofthesoft-spi n m odelforl argebut ni tea i sonl y si x-fol d degenerate,i twoul d orderi nto one ofthese si x statesasT ! 0. T hi si m pl i esthe occurrence ofa zero-tem perature phase transi ti on and the exi stence of an appropri atel y de ned correl ati on l ength that diverges as T goes to zero. In the xed-l ength (a ! 1 ) l i m i t,on the other hand,averagi ng over al lthe degenerate ground statesl eadsto a ni te correl ati on l ength even at T = 0. T hese resul ts suggest that i t woul d be i nteresti ng to study the e ects ofsofteni ng the spi ns on the therm odynam i c behavi oroftwo-and hi gher-di m ensi onal hard-spi n m odel sw i th extensi ve ground-stateentropy.A wel l -know n m odel of thi s ki nd i s the nearest-nei ghbor Isi ng anti ferrom agnet on a tri angul ar l atti ce [ 13] . T hi s m odeldoesnotexhi bi tany phasetransi ti on ata non-zero tem perature.T hedegeneracy-l i fti ng e ectofi ntroduci ng m agni tude uctuati ons found i n our study suggests that soft-spi n versi ons of thi s and other si m i l ar m odel s m ay exhi bi t ni te-tem perature phase transi ti ons. Further i nvesti gati on ofthi s questi on woul d be very i nteresti ng. W ebel i evethattherem ovaloftheexponenti algroundstate degeneracy by the i ntroducti on ofspi n-softness i n the m odelstudi ed here i s a speci alcase ofa m ore generalphenom enon i n w hi ch the presence ofaddi ti onaldegrees of freedom al l ow s the system to rel i eve frustrati on and thus reduce the num ber of degenerate ground states. C oupl i ng the hard spi ns to other degrees offreedom ,such as el asti c vari abl es descri bi ng possi bl e deform ati ons of the underl yi ng l atti ce, woul d probabl y have si m i l ar e ects on the degeneracy ofthe ground state. It i si nteresti ng to note i n thi s contextthata \deform abl e" Isi ng anti ferrom agneton a tri angul arl atti ce i n w hi ch the Isi ng spi ns are coupl ed to el asti c degrees offreedom exhi bi ts [ 14]a Pei erl s-type phase transi ti on at a non-zero tem perature.T he orderi ng ofthe spi nsatthi stransi ti on i s accom pani ed by a di storti on ofthe l atti ce. In general , i t i s expected that i n real , physi calsystem s, such coupl i ngs to other degreesoffreedom ,howeverweak,woul d i nduce som e ki nd of orderi ng of the spi ns as the temperature i s reduced toward zero, thereby avoi di ng the unstabl e si tuati on ofhavi ng a non-vani shi ng entropy per spi n at T = 0. M any di sordered spi n system s,such asspi n gl asses [ 16] , exhi bi t a l arge num ber ofnearl y-degenerate m etastabl e states ari si ng out of frustrati on. To take an exam pl e, the SK m odel [ 2] of i n ni te-range Isi ng spi n gl ass i s know n [ 16]to have an exponenti al l y l arge num ber ofl ocal m i ni m a of the free energy (l ocal l y stabl e sol uti ons of the TA P equati ons [ 15] ) at su ci entl y l ow tem peratures. T hese l ocal m i ni m a of the free energy becom e l ocalm i ni m a ofthe energy at T = 0. T he presence ofa l arge num ber (di vergent i n the therm odynam i c l i m i t) of nearl y-degenerate m etastabl e states i s cruci ali n the devel opm ent ofthe presentunderstandi ng [ 16]ofthe equil i bri um and dynam i c properti es of thi s system at l ow tem peratures. O ur resul ts about the l i fti ng of degeneracy by the i ntroducti on of spi n-softness rai se the foll ow i ng i nteresti ng questi on: woul d the l ow -tem perature properti es of a soft-spi n versi on of the SK m odeldi er i n any si gni cant way from those ofthe ori gi nalm odel ? W hi l e soft-spi n versi onsofthe SK m odelhave been used i n studi es [ 3]ofthe dynam i cs,questi ons about how the num ber and properti es of the m etastabl e states of thi s m odelchange as the spi ns are m ade soft have not been addressed i n detai l . Further i nvesti gati on ofthese i ssues woul d be m ost i nteresti ng. Fi nal l y i t i s i nteresti ng to note that a si m i l ar way of l oweri ng frustrati on i s to m ake the coupl i ng constants soft w hi l e keepi ng the spi ns hard. For exam pl e, i n the caseoftheEdwards-A nderson Isi ng spi n gl assm odel ,two versi ons have been studi ed [ 17] . O ne i s the J m odel w here the nearest-nei ghborcoupl i ng constantsrandom l y take the di screte val ues J w i th equal probabi l i ty. In the other case,the Js are chosen from a gaussi an di stributi on. In d = 2,both these cases are bel i eved to have zero-tem peraturephasetransi ti ons,butthe natureofthe transi ti on i s di erent i n the two cases. T hi s di erence agai n ari ses because of the di erent ground-state degenaraci es i n the two cases.In the J m odel ,the groundstatei sexponenti al l y degenerate,w hi l ei ti suni que(m odul o a gl obali nversi on ofal lthespi ns)i n thegaussi an case. H owever i n hi gher di m ensi ons w here the transi ti on temperature i s ni te,cri ti calproperti es near the transi ti on appear to be the sam e i n both cases. In our nonequi l i bri um studi es we consi dered the K awasakidynam i cs and studi ed the quantum H am i l toni an correspondi ng to the Fokker-Pl anck operatorforthe stochasti c process. T he spectrum ofthe H am i l toni an i s obtai ned by num eri cal di agonal i zati on of ni te chai ns. A n i nteresti ng crossover of the rst exci ted state from m om entum to 2 =L i sobserved w i th i ncreasei n system si ze. W e have found anal yti c upper bounds on the gaps at these two m om enta. T hese, al ong w i th our num ericaldi agonal i zati on resul ts suggest that the gap vani shes as 1=L 2 and so the dynam i cs i s di usi ve as i n SEP. W e have al so com pared the sym m etry properti es ofour H am i l toni an w i th the H ei senberg m odel . W e nd that w hi l e the m odelhas the sym m etry ofthe X X Z m odel , i ts ground-state properti es are cl oser to those ofthe ferrom agneti c i sotropi c poi nt. In sum m ary we have show n that our m odeli s a very nontri vi alcousi n ofthe H ei senberg ferrom agnet. T he excl usi on of three adjacent l i ke spi ns essenti al l y changes the m odel dynam i cs, and resul ts i n a nontri vi al l y depl eted condensate i n < x 0 x r > and a nontri vi algapped < z 0 z r > correl ati on functi on. T he exi stence ofa groundstatei n every S z sectori squi te obvi ous from the stochasti c poi nt ofvi ew but a nontri vi alonew i thi n the fram ework ofthe quantum system (e. g. the absence ofa operator),and requi re a deeper understandi ng. consi der the fol l ow i ng soft spi n versi on of the A N N N I m odel : = 1 correspond to the unperturbed sol uti ons i n the l i m i t a ! 1 . Pl uggi ng thi s i nto Eqn. (3),we get and one can exactl y com pute vari ous therm odynam i c properti es. In the soft-spi n case the transfer-m atri x ei genval ue equati on becom es an i ntegral equati on w hi ch we have not been abl e to sol ve.H ence we have studi ed the m odelby M onte C arl o si mul ati ons. W e have used a dynam i cs w hi ch al l ow s three ki nds ofprocesses; (i ) si ngl e spi n-i p m oves, (i i ) m oves i n w hi ch two nearest nei ghbor spi ns are sim ul taneousl y i pped and (i i i ) m oves w hi ch change the l ength ofa spi n. A l lthree ki ndsofprocessesoccurw i th usualM etropol i s rates. T he reason for al l ow i ng both si ngl e and doubl e spi n-i ps i s the fol l ow i ng. W e nd that i n the hard-spi n case,equi l i brati on ti m es,w i th a si ngl e-spi n i p dynami cs,becom e very l argeatl ow tem peratures.O n the other hand,al l ow i ng fortwo-spi n i psresul tsi n very fastequil i brati on.T hi si srel ated to the factthatw hi l e the si ngl e spi n-i p dynam i cs at T = 0 i s non-ergodi c, i ncl udi ng doubl e-i ps m akes i t ergodi c. W e expect a si m i l ar si tuati on even i n the case ofsoft-spi ns and so have i ncl uded both (i )and (i i ). Fi nal l y (i i i )i snecessary si nce the spi ns are now conti nuous vari abl es and we need to be abl e to change thei r l engths. sthe di agonalterm si nce i tcorresponds to i ppi ng an unequalpai rtw i ce.Iti si m portantto w ri te the di agonalpart careful l y. For i nstance i fi n Eqn.(10), the l ocalprojecti on operators,P k ,were not present,the o -di agonalel em ents ofH woul d sti l lbe correctbut the di agonalones woul d be w rong.W e now study the properti es ofthi s quantum H am i ltoni an. O ur i nterests are(a) to com pare the sym m etry properti esand conservati on l aw softhe presentH am i l toni an w i th thatofthe H ei senberg m odeland (b)to obtai n resul tson the energy gap and hence the dynam i calexponent.A . Sym m etry properties and conservation law s of the quantum m odel W e rstobservethatthez-com ponentofthetotalspi n, S z ,com m utes w i th H . T hi s si m pl y i m pl i es conservati on of spi n or num ber of parti cl es i n the stochasti c m odel . T huswecan cl assi fy energy statesi nto sectorsl abel l ed by num berofparti cl esn.T he constrai ntson al l owed con gurati onsm eansthat fora l atti ce ofl ength L the num ber ofparti cl escan vary overthe range[ L=3] n L [ L=3] w here [ L=3]denotes the sm al l est i nteger greater than or equalto L=3. It can be show n that except i n the l owest and hi ghest sectors,i n every other case the dynam i cs i s ergodi c. It then fol l ow s from detai l ed bal ance that the steady state i s one i n w hi ch al l al l owed con gurati ons i n a gi ven sector occur w i th equalprobabi l i ty. For the quantum m odelthi s m eans that the ground state i n any sector i s an equal l y wei ghted sum over al l states ( For the speci al case w here L i s a m ul ti pl e of 3, the l owest and hi ghestsectorshave 3-fol d degenerate ground states ). For the H ei senberg m odelc z (r) i s del ta-correl ated w hi l e c (r) saturates to the val ue 1=4 (w hi ch i s m uch l arger than i ts val ue i n the present m odel ). T he presence of o -di agonall ong range order m eans that U (1) sym m etry i s broken i n the ground state. T hi s i s anal ogous to the breaki ng ofSU (2) sym m etry i n the ground state of the H ei senberg m odel . O n the other hand,consi der the X X Z chai n [ 10]de ned by the H am i l B . R esults on the energy gapA s i s wel l know n the rst exci ted state of H determ i nesthe decay ofcorrel ati onsforthe stochasti cprocess. T hus the energy gap 1=L z and thi s determ i nes the dynam i c exponent z. For the SEP, w hi ch corresponds to the H ei senberg ferrom agnet,i t i s know n that z = 2. T hi ssi m pl y re ectsthe di usi ve m odesi n the dynam i cs.T hedynam i csstudi ed herei svery si m i l arto the SEP but w i th the constrai nts on the al l owed num ber ofsuccesi ve parti cl es and hol es. A n i nteresti ng questi on i s w hether these ratherstrong constrai ntschange the dynam i calexponent. U nl i ke the H ei senberg m odelw here the B ethe ansatz i s appl i cabl e and yi el ds i nform ati on on the ei genval ue spectrum , the H am i l toni an i n Eqn.(10) i s m uch m ore com pl i cated and we have not been abl e to use the B ethe ansatz. W e have l ooked at the ei genval ue spectrum by num eri caldi agonal i zati on ofH forsm al lsystem si zesand al so through M onte C arl o si m ul ati ons.W e al so obtai n vari ous anal yti c bounds on the energy l evel s. (i )Resul ts of num erical diagonal ization of H and M onte C arl o sim ul ations W e have carri ed out exact di agonal i zati on of the H am i l toni an i n Eqn. (10) for chai ns of l ength upto L = 22 athal f l l i ng. T he di agonal i zati on hasbeen done i n the m om entum basi s. T hi s m akes the H am i l toni an bl ock di agonaland enabl es us to go to qui te l arge chi an si zes. W e nd that for sm al lL the rst exci ted state occurs at totall i near m om entum q = and the gap seem s to decreases as 1=L. H owever from L = 22 onwards, the rst exci ted state shi fts to q = 2 =L and the gap at thi s m om entum decreases as 1=L 2 . In Fi g. 4 weshow the num eri cal l y obtai ned gapsatthe two m om enta as a functi on of system si ze. W e al so pl ot correspondi ng upper bounds on the gaps (to be deri ved i n the next secti on). nd upper bounds on the rst exci ted state. C onsi derthe sectorw i th states w hi ch have n overturned spi ns. T he bounds are obtai ned by constructi ng tri al wavefuncti onsorthogonalto theground statei n each sector. T hus consi der the operators z (q) = 1 ikq and the tw i stoperatorQ de ned i n the previ oussecti on. U nder transl ati on these operators transform as p L ( l;L 1 l;L )] P y FIG . 1 . 1Si m ul ati on data C (T )forthe soft-spi n m odelon a l atti ce ofsi ze N = 24. A l ow tem perature peak can be seen. For com pari si on w e have al so pl otted the hard-spi n resul ts. M ost ofthe entropy rel eased ( 85% ) i s contai ned w i thi n the l ow tem perature peak w hi l e the rem ai ni ng occurs i n the hi gh tem perature regi on (shaded porti on). FIG .2. T he pl otofC (T )atl ow tem peraturesasobtai ned from si m ul ati ons and from the e ecti ve H am i l toni an for di fferent system si zes. W e al so show the e ecti ve H am i l toni an resul t for i n ni te system si ze. FIG . 3 . 3T hedi agonalstructurefactorpl otted asa functi on ofthe totalw ave num ber q. FIG . 4 . 4In thi s gure the exact energy gaps at the tw o m om enta and 2 =L are pl otted agai nst i nverse system si ze. A l so pl otted are exact bounds at the tw o m om enta. FIG .5. T he energy gap, as obtai ned from the decay of the correl ati on functi on < Q (0)Q (t) > i s pl otted as a functi on ofi nverse system si ze. A l so pl otted are the resul ts from exact num eri cal di agonal i zati on and the upper bound. 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